t w j«myffN4»;i4iiiimHii i 
 
 FOR S ECO^Mf^ SGH 00 IJ 
 
 THURSTON 
 
 CMILLAN'S 
 OMMERCIAL 
 
GIFT OF 
 
f 
 
BUSINESS ARITHMETIC 
 
 FOE 
 
 SECONDAEY SCHOOLS 
 
^"^^v^- 
 
BUSINESS AEITHMETIC 
 
 FOR 
 
 SECONDARY SCHOOLS 
 
 BT 
 
 ERNEST L. THURSTON 
 
 SUPEBINTBNDENT OF PLBLIC SCHOOLS, DISTRICT OF COLUMBIA 
 
 "^ ^•> . « ' 
 
 
 Neto ¥0rlt 
 THE MACMILLAN COMPANY 
 
 LONDON: MACMILLAN & CO., Ltd. 
 
 1917 
 

 Copyright, 1913, 
 By the MACMILLAN COMPANY. 
 
 Set up and electrotyped. Published March, 1913. Reprinted 
 August, September, igi3; April, August, 1914; February, 1916; 
 August, 1917. • • 
 
 * t • • • • 
 
 Norteooti ^rtjjji: 
 Berwick & Smith Co., Norwood. Mass., U.S.A. 
 
PREFACE. 
 
 The use and applications of number vary with the changes 
 and development of our industrial, commercial and social 
 relations. The arithmetic of to-day is not the arithmetic of 
 ten years ago. The fundamental principles and processes do 
 not alter, but new applications constantly develop and older 
 applications lessen in usefulness. In this sense, arithmetic is 
 a living subject which deals with living phenomena. 
 
 In these days arithmetic is commonly used not only for the 
 necessary routine computations incident to our private, busi- 
 ness, or scientific affairs, but its processes are called upon to 
 study facts through number and to interpret these facts — to 
 put number into things rather than to take number from them. 
 Moreover, the modern use of number often takes the form of 
 a language, as in the construction or reading of numerical 
 illustrations, and statements or statistics which are designed 
 to impress upon others certain facts or conditions. Thus 
 modern arithmetic may be said to have a use as a tool, i. e., 
 for routine computation, an interpretive use, and a language 
 use. 
 
 But accuracy and facility in the use of most working instru- 
 ments presuppose a knowledge of the fundamental parts of 
 the instrument, of the interrelation of these parts, of their 
 working principles, and of the classes of material or work to 
 which they may be applied. Thus, facility in the three " uses '* 
 of the arithmetic instrument is founded, primarily, on a logical 
 working knowledge of the fundamental operations and prin- 
 ciples. This is the more necessary because the arithmetic 
 problems of actual life do not have a text-book form of state- 
 ment. If formulated at all, they may offer no direct clue 
 to a particular subdivision of the subject. Often the problem 
 must first be formulated, at least mentally, the power of 
 
 4f?9940 
 
VI PREFACE. 
 
 selection being exercised to choose from a group of number 
 facts those essential to the computation desired. 
 
 This book is an attempt to construct the arithmetic instru- 
 ment logically, in its simplest form, as a branch of mathematics; 
 then to apply this tool to general computation, and to the 
 study, preparation and interpretation of varied number 
 material. Involved in this is the purpose to develop arith- 
 metic as a language of business, or as a means of interpretation 
 and study of business, and economic conditions; to acquire 
 the capacity properly to present numerical facts by tabulations 
 or graphs; to cultivate clearness of thought and expression, 
 and an appreciation of the value of order and system in applied 
 number work. 
 
 In the preparation of the subject matter of this book, no 
 attempt has been made to draw from other text-books, al- 
 though much may be found here that is in harmony with 
 them. Material has been gathered only from living and 
 reliable sources. The material has been selected with the 
 purpose of emphasizing how arithmetic is applied for a definite 
 object. The problem exercises are not intended, as is so 
 often the case, as a means of conveying to the mind indigestible 
 information on most known subjects. The author believes 
 that the book will be found none the less rich in content because 
 it aims at the digestion of facts, rather than at surfeiting 
 with them. 
 
 Much thought has been given to the problem work and, 
 as one result, considerable variety will be noticed in form of 
 statement. Thus the old time "problem in question form" 
 is used, also the incomplete statement, the statistical table, 
 the business form, the written article, the memorandum, etc. 
 Place is also given to series or related problems, composite 
 problems, and "central" problems, to be viewed from different 
 standpoints. Exercises in the preparation of original problems 
 are included for the purpose of cultivating powers of selection 
 
PREFACE. vii 
 
 and statement, and of distinguishing between essentials and 
 non-essentials. Mental development, and growth in power of 
 analysis, must necessarily follow the doing of the work that 
 these exercises require. 
 
 A method of development has been followed, in condensed 
 and simplified form, and rules have been reduced to an absolute 
 minimum. It has been assumed, however, that the pupil has 
 had considerable experience in grade arithmetic, before under- 
 taking the study of this book. For this reason illustrative 
 examples are frequently given only in outline. To give 
 facility in the fundamental operations and main processes, 
 a considerable quantity of abstract work has been introduced. 
 In this line, special emphasis has been laid on the fundamental 
 processes as applied to whole numbers, decimals and fractions. 
 Short methods have been emphasized for paper work, and 
 methods of checking have been introduced or suggested. It is 
 felt that every important computation should be checked. 
 
 Oral exercises, planned with the utmost care to serve as 
 models, have been given throughout the book, but they should 
 be supplemented by further work along the lines that they 
 suggest. The exercises should vary in form, being, at dif- 
 ferent times, sight exercises, exercises by dictation, and exer- 
 cises based upon some fundamental expressions that are 
 placed upon the board, and around which individual examples 
 are built. Many types of exercises will be found illustrated in 
 the following pages. 
 
 In all oral work, it should be remembered that facility will 
 come (1) from constantly varying the simple number com- 
 binations, or "ringing the changes" on fundamentals, and 
 (2) from fitting the examples to the individual pupil, giving 
 him something that he can do and that will develop him in the 
 doing. Attention to these requirements will bring a rich 
 return in the increased capacity of the pupils. 
 
 The author wishes to acknowledge his indebtedness to Mr. 
 
viu PREFACE. 
 
 Charles Hart, head of department of business practice, Busi- 
 ness High School, Washington, D. C, who read the first proof, 
 and whose wise counsel and careful criticism have been helpful. 
 
 E. L. T. 
 Washington, D. C. 
 
EDITOR'S INTRODUCTION. 
 
 No subject of the curriculum has received recently more 
 discussion and criticism than has arithmetic. In that most 
 suggestive book "School and Society'' John Dewey dwells on 
 the facts that there should be established a natural connection 
 of the everyday life of the child with "the business environ- 
 ment of the world around him," and "it is the affair of the 
 school to clarify and liberalize this connection." Observation 
 confirms the dictum of John Stuart Mill that the child is by 
 nature logical — he seeks the reasons for things. The teaching 
 of mathematics as an abstract science based on phenomena 
 outside of the child's experience serves to dull his natural 
 logical habit of mind. 
 
 The three-fold aim of teaching arithmetic is now generally 
 conceded, and a new text can scarcely fail to take these into 
 account. First, it should give the ability to compute correctly 
 and with some rapidity. Second, it should sustain the interest 
 of the learner, that is, arithmetic should present a body of 
 knowledge that is fresh and attractive. Third, the teaching 
 of arithmetic should develop the power of applying number 
 concepts to the activities of everyday life. 
 
 The statement of these ideals is easy: the difficulties come 
 in their realization. Arithmetic can be made an instrument 
 of education, both for what has commonly been termed culture 
 and for practical affairs. No school study offers better oppor- 
 tunities for teaching pupils to think or for directing their 
 thinking to the sort of phenomena with which they will deal 
 when school is at an end. The author and the editor of this 
 book recognized these ideals when the book was planned, 
 eight years ago. In preparing the book certain fixed prin- 
 
 ix 
 
X EDITOR'S INTRODUCTION. 
 
 ciples have been borne in mind. One is Herbert Spencer's — 
 that "so far as possible the child should make his own in- 
 vestigations and draw his own conclusions." Constantly in 
 this book the pupil is asked to investigate for himself, to for- 
 mulate or arrange his data, and to deduce his own conclusions. 
 
 Secondly, this book is built on the assumption that through 
 the use of concrete problems, it is possible to do three things 
 at once, namely, to give drill, to keep up interest, and to train 
 in the facility of practical application. In the selection of 
 material little or no regard has been given to text-books here- 
 tofore existing. Problems have been drawn almost entirely 
 from home, store, farm, shop, the engineer's office, and other 
 like sources. 
 
 We have progressed too far educationally to spend time in 
 arguing that it is saner to teach pupils to think on the data 
 that they will later use, than to try to teach them from the 
 obsolete data of the books. Not only does the first method 
 arouse more interest; it gives the only assurance that the 
 pupils will be trained for practical affairs. 
 
 The author would seem to have had an almost ideal prep- 
 aration for the writing of this book. In the first place he 
 was broadly trained in mathematics, and for several years 
 had experience as a teacher in an institution of higher learning. 
 In addition to this he had some years experience as a teacher 
 of arithmetic in the Business High School at Washington, 
 and more recently has served as a supervisor in methods of 
 teaching in elementary schools. Best of all he is a trained 
 and careful investigator, who has been willing to forsake the 
 traditions of arithmetic to work out by practical investigation 
 a book along new lines. 
 
 This book gives but limited attention to the fundamental 
 operations, and the attention that is given is only to emphasize 
 the relative value of the fundamental processes. As with the 
 other book of the series, decimals are made to follow im- 
 
EDITOR'S INTRODUCTION. xi 
 
 mediately after whole numbers, thus combining these closely 
 related parts of our notation. 
 
 The difficulty with much arithmetic teaching is that the 
 pupil considers arithmetic entirely a matter of the book, 
 having nothing to do with practical affairs. The pupil who 
 works through this book should have an intelligent notion of 
 the modern organization and operation of business, a grasp 
 of himself and arithmetical processes, and such facility in the 
 application of these processes as will enable him to go out and 
 make himself useful in the work-a-day world. 
 
 C. A. H. 
 
 GiRARD College, 
 January 1, 1913. 
 
TABLE OF CONTENTS. 
 
 Chapter Pagb 
 
 I. Units and Numbers 1 
 
 II. Notation and Numeration 3 
 
 III. Addition . . 7 
 
 IV. Subtraction 18 
 
 V. Multiplication 25 
 
 VI. Division 35 
 
 VII. Arithmetical Averaging 43 
 
 VIII. The Equation 46 
 
 IX. United States Money 49 
 
 X. Making Change 54 
 
 XL Postage 56 
 
 XII. Payment for Services 61 
 
 XIII. Business Terms and Accounts 72 
 
 XIV. Advertising 77 
 
 XV. Factors and Multiples 86 
 
 XVI. Reduction of Fractions 91 
 
 XVII. Fractions — Fundamental Processes — Con- 
 version « 95 
 
 XVIII. Fractional Relations of Numbers 106 
 
 XIX. Aliquot Parts 110 
 
 XX. Problem Analysis and Solution 118 
 
 XXI. Involution and Evolution 121 
 
 XXII. Denominate Numbers 132 
 
 XXIII. Practical Measurements 143 
 
 XXIV. Measurement of Time '. 164 
 
 XXV. Practical Measurements — Temperature, 
 
 Composite Units, Formulae 175 
 
 XXVI. The Metric System 182 
 
 xiii 
 
XIV 
 
 TABLE OF CONTENTS. 
 
 Chapter Page 
 
 XXVII. Ratio and Proportion 186 
 
 XXVIII . Graphic Arithmetic — Scales, Plotting, 
 
 Graphs 193 
 
 XXIX. Percentage 213 
 
 XXX. Profit and Loss 225 
 
 XXXI. Commercial Discount 236 
 
 XXXII. Agency 245 
 
 XXXIII. Insurance — Personal and Property 255 
 
 XXXIV. Taxation and Public Revenue — Local, 
 
 State and National 269 
 
 XXXV. Interest — Simple 285 
 
 XXXVI. Interest — Compound 302 
 
 XXXVII. Loans and Payments 306 
 
 XXXVIII. Savings Accounts 326 
 
 XXXIX. Stocks and Bonds 332 
 
 XL. Domestic and Foreign Exchange 355 
 
 XLI. Depreciation 375 
 
 XLII. Cost-Keeping 379 
 
 XLIII. Bids and Estimates 383 
 
 XLIV. Partitive Proportion and Partnership 387 
 
 XLV. Equation of Payments and of Accounts. . . . 397 
 
 XLVI. Billing 403 
 
 XLVII. Storage. . .^ 414 
 
 Appendix 
 
 I. Signs and Symbols 419 
 
 II. Standard Abbreviations 420 
 
BUSINESS ARITHMETIC 
 
 FOR 
 
 SECONDARY SCHOOLS 
 

 BUSINESS AEITHMETIO 
 
 FOE SECONDAEY SCHOOLS 
 
 CHAPTER I. 
 
 UNITS AND NUMBERS. 
 
 In applied arithmetic, constant use must be made of units 
 of value and of measure. Unity means one-ness, but a unit 
 is not necessarily "one," or a single thing, for it may be any 
 quantity definitely measured or stated. One dollar is a unit, 
 but the dollar is one hundred cents, or a certain number of 
 unit grains of gold and alloy, that is, a collection of other 
 units. In practice, units vary with the things with which 
 they are associated. A retailer sells goods by the can — the 
 can is his selling unit; but he buys by the case, say of 24 cans, 
 and the case may be termed his buying unit. The idea of a 
 unit as a definite quantity is emphasized in many occupations, 
 as, for example, in engineering, where a unit distance is often 
 taken as 100 or 1000 feet. 
 
 It is evident that all units are divisible. If the unit is one 
 or a quantity considered as one, it is termed an integral unit. 
 If an integral unit is divided into tenths, hundredths, etc., 
 one of the parts so obtained is termed a decimal unit. If the 
 integral unit is divided into any number of equal parts, it is 
 termed a fractional unit. 
 
 A number is a unit or a collection of units. It is therefore 
 integral, decimal or fractional according to the character of 
 units of which it is composed. If the number is entirely unaf- 
 fected by association with any object or expression it is an 
 2 1 
 
2 • BtSINESS ARITHMETIC. 
 
 ahstr'act ''riurrvber: ' (]6±ainples : 87, 34-) If the number is 
 associated with some idea it is concrete. (Example, 8 foxes.) 
 If the associated expressions are those of standard measures, 
 the number is denominate. (Example, 12 yards.) 
 
 Numbers are like or unlike, according as they have the 
 same or different unit values. Thus "five months" and 
 "eight months" are like numbers, while "Rve months" and 
 "eight pounds" are unlike. 
 
 Numbers are simple if they contain units of the same kind; 
 they are compound if they contain different forms of units 
 which are reducible to a single form of unit. Thus "4 lb." 
 is simple, but "4 lb. 8 oz." is a compound number which may 
 be reduced to the simple number "72 oz." 
 
 EXERCISE. 
 
 1. Name six units of measure. 
 
 2. Name five units employed in different trades or professions. 
 
 3. Classify these numbers and name the unit in each case: 
 
 (a) 976. (d) $3.85. (g) 53^ gal. 
 
 (6) .08. (e) 5 yd. 2 ft. (h) U lb. 
 
 (c) 45 lb. (/) 276 payments. (i) 4 horses, 2 mules. 
 
 4. Name and classify the numbers and units in these expressions: 
 
 (a) "The cost of this cloth is 55 cents per yard." 
 
 (6) "The goods are packed in cases of 48 boxes each." 
 
 (c) "The temperature reached 96° to-day." 
 
 id) " It will take at least half-an-hour and probably forty minutes." 
 
 (e) 240 is H of 480. 
 
CHAPTER II. 
 NOTATION AND NUMERATION. 
 
 INTRODUCTORY EXERCISE. 
 
 1. How do the expressions "3 yd." and ".03 yd." differ in meaning? 
 
 2. Add ciphers to the numbers "40" and ".04" in such a way as not 
 to alter their values. 
 
 3. Where does the insertion of a cipher cause a change of value? 
 
 4. In expressing numbers orally, what are the values of the syllables 
 "teen" and "ty"? 
 
 Arabic Notation. 
 
 The Arabic or Hindu notation is the system universally 
 used in civilized countries. The significant figures or digits, 
 1 to 9, were in use in India 2,000 years ago, but calculations 
 were performed with extreme difficulty until the invention 
 of the tenth symbol, the cipher or zero, 800 years later. The 
 zero made modern computation possible. In the seventeenth 
 century, the invention of the decimal point made possible 
 the present varied use and application of number. The 
 present tendency is toward a continually broadening use of 
 decimals. 
 
 The "significant" figures of the system are those that 
 have a name or face value, and a position value. The zero 
 serves only to alter the position value of other figures. 
 
 PRINCIPLES OF USE. 
 
 1. The name value of each significant figure 'm constant. 
 
 2. The position value increases in ten-fold ratio from right to left. 
 Note. Hence called the "decimal system," from the I^atin decern 
 
 meaning "ten." 
 
 3. A decimal point, or period, separates units and tenths of imits. 
 Note. The portion of the number to the left of the decimal point is 
 
 termed the whole number or integer; the portion to the right is the decimal. 
 
 3 
 
4 BUSINESS ARITHMETIC. 
 
 4. The zero increases the position of significant figures to its left ten 
 fold; it decreases the value of significant figures in a decimal, to its right, 
 to one-tenth their previous value. 
 
 5. In any number, where a significant intermediate figure is lacking, 
 a zero is written in its position. 
 
 6. Place values are named under the French or English system. 
 Under the French system, used in this country, the figures of large numbers 
 are divided into groups or periods of three figures each, as follows: 
 
 Place 131211109 8 7 654 321 . 123 456 
 
 Name 
 
 g a 
 
 fl 2 
 
 iSl Jl_ 211 ^ ^ II Um 
 
 C^Srs !=^c3li: SS^ J^sa a». o ^,C S^'^i 
 
 H WH« «h:^ teHH wh;i3 q hwh hw^ 
 
 Period BilUons Millions Thousands Hundreds Thousandths Millionths 
 
 Illustration of Numeration. The numbers should be read in the 
 simplest way, the word "and," or "decimal," being used for the decimal 
 point, thus: 
 14,264.216 is read: Fourteen thousand, two hundred, sixty-four and two 
 
 himdred sixteen thousandths. 
 2,132. is read: Twenty-one hundred, thirty-two. 
 3.02 is read: Three and two hundredths. 
 30,000.003 is read: Thirty thousand and three thousandths. 
 
 Note. In English tables, the periods contain six places each, called 
 units, millions, etc. Under this system the number 36,000,000,000 would 
 be read: Thirty-six thousand milHons. 
 
 EXERCISE. 
 
 1. Which is the larger, 3 yd. or .030 yd.? 
 
 2. How does the decimal point determine the size of the part taken? 
 
 3. Read: 4. Read: 6. Read: 
 
 52,608 2.43 30.0009 
 
 1,292,723 17.061 2.0703125 
 
 329,624,006 228.0406 .627724 
 
 50,007,064,329 5,060.38001 2.390 
 
 6. Read by both English and French systems: 
 472,693,684,721,125.001. 
 
NOTATION AND NUMERATION. 5 
 
 7. Insert a cipher in the number 17,246 in five different positions in 
 turn, and read the results. In which position has the cipher the least 
 effect? The greatest? 
 
 8. Insert a cipher, in turn, between each two significant figures in 
 the number 86.043201, and read the resulting numbers. 
 
 9. Read .4 as hundredths; .65 as ten-thousandths; .025 as hundredths; 
 .002 as millionths. 
 
 10. What is the decimal of lowest value, that can be written with the 
 figures 3 and 7 and foiu- ciphers? 
 
 Note. In writing large numbers, separate periods by commas, or by 
 spacing. 
 
 11. Write in colunm: Seven billion, two hundred forty-eight million, 
 seventy thousand, eight hundred twenty-eight; two hundred twenty-six 
 million, four hundred thousand, thirty; eighteen hundred thousand, four; 
 six ten-thousandths; five and four milUonths; sixty-three thousand and 
 six hundred thirty hundred-thousandths; four hundred fifty-nine and eight 
 hundred eleven ten-thousandths; fifty-four tenths; fiftj^-two hundred- 
 thousandths. 
 
 12. Write, substituting numbers for numerical words: (a) The cir- 
 cumference of a circle is three and fourteen thousand, one hundred fifty- 
 nine hundred-thousandths times the diameter. (6) A nautical mile is 
 one and one hundred fifty-two thousandths of a statute mile, (c) The 
 total elongation of a bar on a steel bridge, under a maximum load, was 
 two hundred fifty-six thousandths inches. 
 
 Roman Notation. 
 The Roman system employs as number symbols seven 
 letters of the Roman alphabet. These characters have a 
 name value and a variable position value. 
 
 I V X L C DM 
 
 1 5 10 50 100 500 1000 
 
 PRINCIPLES OP USE. 
 
 1. Combinations of symbols are written from left to right in order of 
 value. The total value is the sum of the constituent values. 
 
 Illustrations. CX = 100 + 10 = 110. MDCC = 1000 + 500 
 -f 100 + 100 = 1700. 
 
6 BUSINESS ARITHMETIC. 
 
 2. To avoid the consecutive use of symbols four times, a letter of 
 less value may be written before one of greater value, the combination 
 being equal to the difference of the name values. 
 
 Illustrations. L = 50, but XL = 50 - 10 = 40. IX = 10 - 1 = 9., 
 
 3. The mark " — " over a symbol multiplies its value by 1000. 
 
 Illustration. XIV = 14, but XIV = 14,000. 
 
 EXERCISE. 
 
 1. Write the Roman numerals from 1 to 100. 
 
 2. Spell the Roman numerals from 1854 to 1913. 
 
 3. The corner stone of a pubUc building is marked MDCCCLXVIII. 
 It was laid in what year ?. 
 
 4. Write in Arabic notation: XIV; XCVI; MDVIII; MCDIX: 
 XXlfl; CX; DCXII. 
 
 5. Name three common uses of the Roman notation. 
 
 6. Name objections to the general use of the system. 
 
CHAPTER III. 
 
 ADDITION. 
 
 The fundamental operations on which the working methods 
 of arithmetic are based are: (1) addition; (2) subtraction; 
 (3) multiplication; and (4) division. These processes are 
 applied to whole numbers, decimal fractions and common 
 fractions. They will be considered, first, in their relation to 
 whole numbers and decimals. 
 
 Addition is the process of combining two or more numbers 
 into a single equivalent number, called the sum, the amount, 
 or the total. It is the most universally employed fundamental 
 process, and should be performed with absolute accuracy 
 and reasonable speed. From a business standpoint, this 
 accuracy and speed depend on: (1) careful formation of 
 figures; (2) alignment of figures; (3) capacity to give and take 
 number dictation; (4) capacity to copy figures accurately; 
 (5) working knowledge of "grouping"; and (6) continued, 
 systematic practice. 
 
 Careful Writing. Figures should be uniform in size, legible, 
 simple and distinct in form, evenly aligned vertically and 
 horizontally, and closely spaced but not crowded. 
 
 EXERCISE. 
 
 1. Give reasons for the requirements just stated. 
 
 2. How do carelessly formed figures reduce speed in addition? 
 
 3. Write an original column of 6-place figures. (Exchange papers 
 and criticise for form.) 
 
 4. The money column on page 331 is "unit ruled." What is unit 
 ruling? Of what value is it in addition? 
 
 Number Dictation is frequently incidental and preparatory 
 to addition. One clerk may dictate values to another, who 
 
 7 
 
8 BUSINESS ARITHMETIC. 
 
 writes them in column, or "checks off" a list he holds. In 
 dictation, numbers must be clearly enunciated in an even 
 tone, without unnecessary words, and at uniform speed. 
 
 SUGGESTION FOR TEACHERS. 
 
 1. Dictate a column of figures. Have pupils check accuracy by 
 "dictating back." 
 
 2. Let pupils dictate original columns to each other. Later, check 
 by addition. 
 
 Copying is more common than dictating. Office clerks 
 are continually transferring figures from one record to another. 
 Thus a bank clerk lists a depositor's checks, or a book-keeper 
 posts from a Cash Book to a Ledger. 
 
 SUGGESTION FOR TEACHERS. 
 
 1. Have the class copy a column of blackboard figures. Check by 
 dictation. 
 
 2. Exchange original columns for copjring. 
 
 General Addition. 
 The mental process of adding consists in grouping digits 
 of the same order. The thought should be placed on the 
 result. In adding 5, 8, 9 and 7, think of each sub-total (13, 
 22, 29). Facility in addition is gained by reading instantly any 
 two or three digits as a sum. Thus, "8 + 9" should be read 
 "17." 
 
 DRILL TABLE. 
 
 The forty-five two-figure combinations: Name sums at sight. 
 74241343314221189856455 
 76537623251213199851434 
 
 7 
 
 1 
 
 5 
 
 6 
 
 6 
 
 8 
 
 9 
 
 8 
 
 7 
 
 7 
 
 4 
 
 9 
 
 ' 7 
 
 6 
 
 7 
 
 5 
 
 3 
 
 2 
 
 4 
 
 5 
 
 7 
 
 6 
 
 2 
 
 8 
 
 6 
 
 6 
 
 9 
 
 6 
 
 1 
 
 2 
 
 3 
 
 5 
 
 8 
 
 3 
 
 8 
 
 7 
 
 9 
 
 9 
 
 8 
 
 9 
 
 9 
 
 8 
 
 4 
 
 2 
 
ADDITION. 
 
 EXERCISE. 
 
 1. Name the totals. 4 + 8, 6 + 9, 3 + 1, 4 + 7, 2 + 8, 1 + 7, 3 + 9, 
 
 4 + 5, 4 + 4, 14-2 + 3, 2+4 + 3, 5 + 2 + 7, 6 + 7+3, 8 + 2+4, 
 1 + 1+3+2, 1+8 + 8. 
 
 2. List all possible groups of two or three digits equaling ten. Leam 
 them. 
 
 3. Develop systematically a list of the 165 possible combinations of 
 three digits. 
 
 Illustration. 1 + 1 + 1, 1 + 1+2, 1 + 1 + 3, etc. 
 
 4. Give, orally, the consecutive totals obtamed by adding: 3s to 58 
 (58, 61, 64, etc.), 4s to 127, 9s to 156, 8s to 69, 7s to 32, alternate 3s and 
 8s to 14, alternate 2s and 4s to 52. (Limit 300) 
 
 Grouping. In adding columns of figures, it is advisable to 
 group digits into sub-totals, and to combine these sub-totals. 
 The grouping may be consecutive, irregular or by tens and 
 twenties. 
 
 Illustrations. 
 
 L 11. III. 
 
 3\ 4\ 5 (I.) Read instantly the joined numbers as single 
 
 6 / 9 J 9 \ numbers. The consecutive totals, downward, are : 
 
 9\ 3/ /4j 9,20,30,38,48. 
 
 2 y 8 \ ^ ^ (I^*) ^^^^ upward, grouping simple numbers, 
 4\ 7\ ^6 even if non-consecutive. Totals: 7, 16, 25, 33, 40,. 
 6/ 2) 8\ 49. 
 
 3 \ 9 / 3 I (III.) Since "tens" are easier to add than irreg- 
 6/ Q\ I 2' ular combinations, group "tens" as shown. 
 2\ 1 / \ 7 Totals upward: 5, 15, 25, 35, 45, 50. 
 
 1 ) 49 _5 
 
 J^J 50 
 48 
 
 It is frequently necessary to combine groups of two-place 
 numbers. Some accountants regularly add double columns 
 in place of single columns. Until totals are possible by in- 
 spection, add the tens and then the units, thus: 46 + 57 = 
 40 + 50 + 6 -t- 7. 
 
10 BUSINESS ARITHMETIC. 
 
 ORAL EXERCISE. 
 
 Name the sums of the following: 
 
 36 21 84 73 27 62 81 94 78 55 43 16 21 62 43 67 
 
 43 69 72 69 41 25 73 18 63 57 54 11 21 29 13 58 
 
 — — — — — — — — — — — 14 10 71 2 11 
 
 Complements. The complement of a number is the dif- 
 ference between it and the lowest number of the next higher 
 order. Thus the complement of 8 is 10 — 8, or 2; the com- 
 plement of 87 is 100 — 87, or 13. Complements are an aid 
 to rapid addition. 
 
 Illustration. 376 + 96 = 376 + 100-4. 
 
 ORAL EXERCISE. 
 
 1. Name the complements of the following: 7, 21, 36, 64, 77, 129, 973, 
 9959. 
 
 2. When is it advisable to use the complement in mental addition? 
 
 3. Add by complements: 64 + 97, 88 + 93, 173 + 98, 77 + 84, 
 568 + 92. 
 
 
 
 EXERCISE. 
 
 
 
 Add these colunms 
 
 without copying 
 
 figures, groupmg 
 
 simple numbe 
 
 1. 
 
 2. 
 
 3. 
 
 
 4. 
 
 5. 
 
 524 
 
 3164 
 
 629 
 
 
 234,567 
 
 1 234 567 
 
 689 
 
 43 
 
 61324 
 
 
 123,456 
 
 2 456 673 
 
 306 
 
 721 
 
 399 
 
 
 912,345 
 
 5 928 431 
 
 1247 
 
 2903 
 
 2401 
 
 
 891,234 
 
 4 241 326 
 
 1201 
 
 8127 
 
 88 
 
 
 789,123 
 
 5 261 252 
 
 3824 
 
 655 
 
 1120 
 
 
 678,912 
 
 4 683 392 
 
 6735 
 
 432 
 
 21205 
 
 
 567,891 
 
 9 221 243 
 
 3289 
 
 288 
 
 4630 
 
 
 456,789 
 
 4 322 881 
 
 6476 
 
 179 
 
 880 
 
 
 345,678 
 
 3 333 333 
 
 6. Why is column 1 easier to add than column 3? 
 
 7. What aids to quick addition are found in columns 4 and 5? 
 
 8. Write two original columns, one of 20 six-place figures, the other 
 of 60 two-place figures. Add. Which example is more difficult? Why? 
 
 9. Name ten simple applications of addition, in everyday life. 
 
ADDITION. 11 
 
 Checking. All addition should be "checked," or tested by 
 one of the following methods. 
 
 (a) Reverse direction. Add down after adding up, etc. 
 Why? 
 
 (6) After adding by consecutive numbers or groups, group 
 by tens. Why? 
 
 (c) Divide the column at one or more points by horizontal 
 lines. Determine the total of each partial column. Set out 
 these totals and add. 
 
 (d) Add the unit column, writing the total at one side; 
 add the "tens" column independently, offsetting the total 
 
 as shown. Continue the process. The 
 total of the sub-totals should equal the 
 column total. This method is also used for 
 direct addition by one who is likely to be 
 interrupted frequently. One may stop add- 
 13168 13168 ing at any point, and continue again when 
 opportunity offers. 
 
 (e) Casting or counting out nines. 
 
 To cast out nines from a number, add its digits and when- 
 ever the sum equals or exceeds 9 discard 9, adding the re- 
 mainder to the succeeding digits, etc. 
 
 To check addition, discard the nines from the numbers 
 added and also from their sum. Discard the nines from the 
 remainders of the numbers added. If the final remainder equals 
 the remainder of the sum, the sum may be presumed to be 
 correct. 
 
 3216 
 
 18 
 
 2479 
 
 15 
 
 3821 
 
 20 
 
 3652 
 
 11 
 
 Illustration. 
 
 Final Remainder. 
 
 426 3 
 
 824 5 
 
 936 
 
 1542 3 
 
 3728 2, check remainder. 
 
12 
 
 BUSINESS ARITHMETIC. 
 
 WRITTEN EXERCISE. 
 Below is shown a portion of the circulation statement of a newspaper, 
 for a recent year. Find and check the monthly totals, using the methods 
 just outlined. 
 
 Date 
 
 January 
 
 February 
 
 March 
 
 April 
 
 May 
 
 June 
 
 1 
 
 45578 
 
 59471 
 
 60105 
 
 60150 
 
 58408 
 
 69636 
 
 2 
 
 505Jt2 
 
 59723 
 
 59886 
 
 48302 
 
 59392 
 
 66880 
 
 3 
 
 56551 
 
 59116 
 
 60036 
 
 59937 
 
 59029 
 
 56746 
 
 4 
 
 57318 
 
 59400 
 
 61096 
 
 59887 
 
 58713 
 
 46006 
 
 5 
 
 57273 
 
 49249 
 
 49512 
 
 61030 
 
 58395 
 
 57192 
 
 7 
 
 57942 
 
 59522 
 
 59469 
 
 60105 
 
 47000 
 
 57745 
 
 8 
 
 46937 
 
 59400 
 
 60107 
 
 6O4OO 
 
 68410 
 
 57679 
 
 9 
 
 57771 
 
 59281 
 
 60484 
 
 47861 
 
 58910 
 
 67163 
 
 10 
 
 57909 
 
 59613 
 
 59960 
 
 6O415 
 
 686I4 
 
 66662 
 
 11 
 
 58330 
 
 59395 
 
 59837 
 
 60634 
 
 58164 
 
 46349 
 
 12 
 
 58533 
 
 59035 
 
 48821 
 
 60485 
 
 68084 
 
 57165 
 
 13 
 
 58106 
 
 59372 
 
 601 49 
 
 60332 
 
 67966 
 
 66571 
 
 U 
 
 58428 
 
 59306 
 
 59575 
 
 59953 
 
 47006 
 
 57876 
 
 15 
 
 48O47 
 
 59536 
 
 59680 
 
 59939 
 
 67966 
 
 57957 
 
 16 
 
 57797 
 
 59469 
 
 59839 
 
 47505 
 
 58102 
 
 57859 
 
 17 
 
 57945 
 
 59470 
 
 60663 
 
 58577 
 
 57664 
 
 66226 
 
 18 
 
 58443 
 
 59477 
 
 59950 
 
 60825 
 
 57335 
 
 46547 
 
 19 
 
 58696 
 
 49410 
 
 48875 
 
 60134 
 
 57349 
 
 57287 
 
 20 
 
 58393 
 
 59526 
 
 6OI43 
 
 61332 
 
 56774 
 
 57684 
 
 21 
 
 58697 
 
 59394 
 
 60329 
 
 60874 
 
 46346 
 
 57107 
 
 22 
 
 48676 
 
 53016 
 
 60260 
 
 60504 
 
 67296 
 
 67208 
 
 23 
 
 58396 
 
 60116 
 
 59825 
 
 48023 
 
 57197 
 
 56767 
 
 H 
 
 59334 
 
 59823 
 
 60366 
 
 59989 
 
 57927 
 
 56350 
 
 25 
 
 59015 
 
 59934 
 
 60060 
 
 69751 
 
 67884 
 
 46069 
 
 26 
 
 59453 
 
 49187 
 
 48110 
 
 59658 
 
 57362 
 
 56293 
 
 27 
 
 59544 
 
 59981 
 
 60250 
 
 59170 
 
 57281 
 
 56989 
 
 28 
 
 59075 
 
 69732 
 
 60023 
 
 69147 
 
 46522 
 
 67203 
 
 29 
 
 49327 
 
 
 60569 
 
 68858 
 
 57068 
 
 66927 
 
 30 
 
 59580 
 
 
 60316 
 
 47412 
 
 49684 
 
 56919 
 
 31 
 
 59381 
 
 
 60064 
 
 
 57070 
 
 
 JUiAfy 
 
 
 
 
 
 
 
 ibia£ 
 
 
 
 
 
 
 
 Horizontal Addition. It is often necessary to find the sum 
 of several numbers written in a horizontal row, without re- 
 writing them in vertical columns. Such addition is common in 
 billing, accounting and statistics. 
 Illustration. 
 
 2472 
 
 3014 
 
 5207 
 
 Total Col. 
 13962 
 
ADDITION. 13 
 
 In adding, follow the usual rule. Add the units (22) writing the unit value 
 of the total and carrying '2'; add the tens, etc. Be careful of positions 
 after passing the second place. 
 
 EXERCISE. 
 
 Sales recapitulation sheets are used in department stores, factories, 
 etc., to sum up, for jBxed periods, the total sales of each department, of 
 each class of output, of each salesman, etc. Their totaling often involves 
 both horizontal and vertical addition, as in the following: 
 
 Recapitulation of Department Sales. 
 
 February, 19 — . 
 
 Day Dept. A Dept. B Dept. C Dept. D Dept. E Dept. F Totals 
 
 1 $460.77 $344.00 $450.55 $305.50 $355.06 $455.56 ? 
 
 2 544.04 550.00 405.60 366.67 460.60 758.24 
 
 3 477.07 637.89 400.33 536.56 367.89 457.65 
 
 5 477.08 460.70 512.45 435.56 404.40 460.70 
 
 6 546.87 315.85 428.67 345.67 546.82 467.19 
 
 7 478.89 211.81 543.78 377.69 512.87 601.03 
 
 8 213.56 110.87 232.16 215.46 127.83 105.68 
 
 9 136.46 98.67 205.38 122.69 200.80 149.03 
 10 568.67 511.84 690.57 477.89 624.35 589.98 
 
 12 477.21 647.29 489.39 436.29 527.69 672.39 
 
 13 567.85 437.27 512.11 532.56 455.21 532.24 
 
 14 547.54 512.37 544.65 522.22 499.05 542.26 
 
 15 600.25 590.01 504.71 488.34 473.56 532.56 
 
 16 580.35 543.26 379.67 476.58 532.23 433.59 
 
 17 555.88 496.36 665.00 299.99 548.84 477.25 
 
 19 788.27 564.88 675.50 590.54 693.47 657.76 
 
 20 433.19 459.23 435.67 467.89 590.12 534.56 
 
 21 512.25 511.11 561.00 500.15 509.90 503.32 
 
 22 485.20 500.44 503.67 483.74 468.49 589.70 
 
 23 522.16 460.69 544.36 467.78 488.29 512.61 
 
 24 501.44 577.10 600.00 503.92 605.58 505.45 
 
 26 473.36 546.46 516.17 563.10 718.14 503.25 
 
 27 494.58 536.25 500.26 468.87 600.21 580.41 
 
 28 456.67 478.75 601.37 503.22 496.59 502.45 
 
 Totals 
 
 (a) Total the vertical columns. What do they show? 
 (6) Total the horizontal columns. WTiat do they show? 
 (c) Suggest a "check" for the vertical and horizontal totals. 
 
14 BUSINESS ARITHMETIC. 
 
 (d) Indicate the possible effects of weather, season, special sales, and 
 other causes, on values appearing in departmental columns. Show what 
 values in the above table might be specially accounted for. 
 
 Note. Most tabular addition in business offices, and considerable 
 computation requiring other fundamental operations, is now done by 
 machinery. As early as A.D. 968, Gilbert, Archibishop of Rheims, per- 
 fected some ingenious computing machines, which, however, sometimes 
 failed to compute correctly. Pascal (1642-45) invented a machine correct 
 in theory, which, while not practically useful, formed the basis for some 
 modern machines of great capacity. 
 
 The modern machines generally perform the fimdamental operations. 
 They are either "listing" or "non-listing" machines. One standard 
 machine has eight or more "banks" of keys, each containing nine digits 
 and having its own place value. The keys to represent a certain number 
 are pressed down and the movement of a lever, or the pressing of an electric 
 button causes the printing of the number on a continuous roll of paper. 
 The next number is printed below the first, the changing total appearing 
 in the machine. A special movement prints the final total on the paper, 
 and "clears the machine" for the next operation. Some machines list 
 abstract numbers; others list money values, fractions, etc. Some are at- 
 tached to typewriters so that bills of merchandise may be written and 
 automatically totalled. 
 
 TABULATIONS AND STATISTICS. 
 EXERCISE. 
 
 These two news items have appeared in print: 
 
 (Tabulated.) (Untabulated.) 
 
 The Department of Agriculture The losses caused by insects 
 
 estimates the minimum annual to the products of the country, 
 
 damage done to crops by our most during an average year, are es- 
 
 destructive insects, as follows : timated by the Department of 
 
 Chinch-bug $60,000,000 Agriculture, as follows: Cereals, 
 
 Grasshopper 50,000,000 $200,000,000; animal products, 
 
 Hessian fly 40,000,000 $175,000,000; forests and forest 
 
 Corn-root worm 20,000,000 products, $111,000,000; products 
 
 Corn-ear worm 20,000,000 in storage, $100,000,000; hay 
 
 Cotton-boll weevil 20,000,000 and forage, $53,000,000; truck 
 
 Codling moth (Apple) . 20,000,000 crops, $53,000,000; cotton, $50,- 
 
 Armyworm 15,000,000 000,000; tobacco, $5,300,000; 
 
 Cotton-boll worm 12,000,000 fruits, $27,000,000; sugars, 
 
 Grain weevil 10,000,000 $5,000,000; miscellaneous crops, 
 
 San Jose scale 10,000,000 $5,800,000. Total, | ? . 
 
 Cotton-leaf worm. .... 8,000,000 
 
 Potato-bug 8,000,000 
 
 Cabbage worm 5,000,000 
 
 Total ? 
 
ADDITION. 15 
 
 (a) Find the missing totals. In which case is it easier? Why? 
 
 (6) Wliich statement is read more easily? Which is more noticeable? 
 
 (c) From which statement is it easier to select, or reclassify values? 
 Why? 
 
 (d) Compare capitahzation in the two cases. Compare use of "$." 
 
 (e) Tabulate the second illustration, using a brief title in place of the 
 introductory clause. 
 
 The illustrations given above show the value to the statis- 
 tician of the "column" and "total" in conveying numerical 
 information. The column tabulation attracts attention, and 
 brings each numerical value in sharp contrast with other 
 values, while allowing comparison with the total. In tabulat- 
 ing, names attached to values should be brief, but definite. 
 Each tabulation should have a brief descriptive title. 
 
 EXERCISE. 
 
 1. Tabulate and find the lowest cost of construction of a building for 
 which the following bids were submitted: 
 
 For masonry, iron work, carpentry, glazing, tinning, plastering, etc. — 
 A. B. Ryan, $49,845; J. D. Norton, $41,900; C. A. Olhaus, $42,344; Robert 
 Adams, $48,275; Brown & Mann, $35,691; Wm. Harris, $37,573; W. B. 
 Norton, $44,678; A. S. Keene, $43,536; Northern Cons. Co., $40,547; Jas. 
 Davis, $35,386. 
 
 For materials and labor necessary to furnish, install and complete a 
 low pressure, steam-heating apparatus: — Robert Davis, $4,646; Morris 
 Heating Cons. Co., $3,674; Evanson & Son, $3,940; E, T. Roberts Co., 
 $3,970; Barton Mfg. Co., $4,180; Geo. A. Sanderson, $4,900; Brown & Mann, 
 $4,950; Wm. Harris, $3,398. 
 
 For materials, labor, etc., for plumbing work: — D. F. Kean, $3,325; 
 Robert Harris Co., $5,132; Wm. Harris, $4,736; A. B. Ryan, $4,728; Harper 
 & Son, $4,198; Brown & Mann, $5,763. 
 
 From the tabulation just made, find : (a) what bidder will complete 
 the work for the lowest sum; (h) the lowest sum for which the work may 
 be done by giving sub-contracts to different bidders. 
 
 Decimal Addition. 
 
 INTRODUCTORY EXERCISE. 
 1. Annex a cipher to .6, and read result. 
 
16 BUSINESS ARITHMETIC. 
 
 2. Reduce 4 to tenths. 
 
 3. Reduce 53 to hundredths, 
 
 4. Add .3 and .6. 
 6. Add A, .6, .9, .8. 
 
 6. Add .04, .83, .07, 001. 
 
 7. Add .047 and .07. (.07 = how many thousandths?) 
 
 8. How does reduction to a common order aid in decimal addition? 
 
 There is practically no difference between integral and 
 
 decimal addition. Integral or decimal units of the same order 
 
 are added. In writing in column, be sure that the decimal 
 
 points are in vertical alignment, as this will ensure proper 
 
 "order" position, and also the position of the decimal point 
 
 in the total. 
 
 Illustration. 
 
 46.205 
 
 392.0872 
 
 1245.01232 
 
 254.67005 
 
 1937.97457 
 
 ORAL EXERCISE. 
 
 1. How does the number of decimal places in the total compare with 
 the nimiber of places in the constituent numbers? 
 
 2. May the total contain a less number of decimal places than any 
 significant number? 
 
 Add: 
 
 3. .4 and .85. 8. 2.33, 1.062 and 2.0382. 
 
 4. .072 and 0096. 9. 3.25, 1.0006 and 8.0129. 
 6. .12, .065 and .363. 10. .46, .05, .8 and .002. 
 
 6. .013, .21, and .220. 11. 6.3205 and .00061. 
 
 7. .01, 2.006 and 15.0126. 12. .001, 2 01 and .0352. 
 

 ADDITION. 
 
 
 
 EXERCISE. 
 
 
 Total these columns: 
 
 
 
 (1) 
 
 (2) 
 
 (3) 
 
 23.075 
 
 123.456729 
 
 4.37 
 
 06.5082 
 
 54.08296 
 
 83.070096 
 
 16.09327 
 
 5.4729825 
 
 5.9276 
 
 5463.382473 
 
 .007369 
 
 329.832 
 
 1.29634 
 
 527.8308896 
 
 4.00892 
 
 50069.8002173 
 
 5.2100009 
 
 .0006732 
 
 542.06345276 
 
 67.31438 
 
 1.01 
 
 12.47234 
 
 5729.61 
 
 15.2729 
 
 5067.30821 
 
 80.8 
 
 .888005 
 
 Note. A large proportion of decimal addition deals with mone- 
 
 (4) 
 
 (5) 
 
 (6) 
 
 $ 326.925 
 
 £ 456.29 
 
 45.B25 fr. 
 
 48.065 
 
 39.625 
 
 129.8 
 
 723.8 
 
 1233.40 
 
 46239.03 
 
 9654.47 
 
 56.1235 
 
 596.008 
 
 830.425 
 
 5.60 
 
 63.9 
 
 65.72 
 
 1283.55 
 
 1321.48 
 
 12756.23 
 
 14756.125 
 
 32635.835 
 
 52.08 
 
 677.0875 
 
 4.006 
 
 926.00 
 
 1239.05 
 
 5282.56 
 
 7259.12 
 
 36.075 
 
 43.45 
 
 5.69 
 
 5296.5125 
 
 6.735 
 
 627.33 
 
 83.0125 
 
 529.62 
 
 46.09 
 
 962.825 
 
 88.9 
 
 17 
 
 7. Find the totals, in units of a thousand feet, of these orders for 
 lumber: 6246 ft., 15,389 ft., 7500 ft., 4382 ft., 965 ft., 45 ft., 12,925 ft., 6750 ft. 
 
 8. Total these engineering measurements, and express total in units 
 of 1000 ft.: 564', 296.5', 4521', 885.2', 750', 1325', 632.8', 983.7', 66.8'o 
 
CHAPTER IV. 
 SUBTRACTION. 
 
 INTRODUCTORY EXERCISE. 
 
 1. Name the difference between 2 and 9, 16 and 21, 
 
 2. From 129 subtract 4s. 
 
 3. From 138 subtract alternate 3s and 5s. 
 
 4. Give several illustrations of the use of subtraction in everyday life. 
 
 Subtraction is one phase of addition. It is the process of 
 finding the number to be added to the lesser of two given 
 numbers in order that the sum may equal the greater. 
 
 lLL'j3rRATiON. Find the difference between 774 and 1969. 
 
 Process. Add to the subtrahend, digit by digit, until 
 iQftQ miniioTiri *^® miuucud is obtained, thus: 4 (sub.) +5 = 9 
 
 774 SSend ("^^^•)- '^"^^ ^' ^« "^^^ difference. 7 (sub.) + 9 = 16 
 n^ suDtranena. ^^^^^^ ^^^^^ 9 ^^ ^^^,^ pj^^^ ^^ difference. 7 (sub.) 
 
 1195 difference. + 1 = 8 (9 - 1, min.). (sub.) + 1 = 1 (min.). 
 
 Note. In formal work, it is often inconvenient to write the lesser 
 number under the greater. Position, however, makes no difference in 
 method. The above example might be written: 
 
 (a) 774 sub. (b) 1969 - 774 = 1195. 
 
 1969 min. 
 1195 diff. 
 
 EXERCISE. 
 
 1. Show that these subtraction "checks" are true. 
 
 (a) The sum of subtrahend and difference equals minuend. 
 (6) The minuend less the difference equals the subtrahend. 
 Solve by inspection: 
 
 2. Find by addition the complements of 36, 29, 64, 83, 91, 896. 
 
 3. Solve by addition: 
 
 54 - 13 76 - 23 148 - 93 168 - 72 
 
 59 - 37 43 - 29 124 - 58 249 - 68 
 
 18 
 
SUBTRACTION. 19 
 
 Note. Oral work is sometimes simplified by subtracting the orders 
 of the subtrahend independently, beginning with the highest. This is 
 termed subtraction from the left. Thus 578 - 364 = 578 - 300 (278) 
 - 60 (218) - 4 = 214. 
 
 4. Solve, by subtraction from the left : 
 
 249 - 67 328 - 243 1527 - 892 968 - 871 
 
 1116 - 452 1793 - 896 529 - 381 4473 - 269 
 
 Note. Irregular numbers may be subtracted mentally by subtracting 
 an even number of the next higher order, correcting the difference of the 
 numbers by adding the difference of the subtrahends. Thus 384 — 47 
 = 384 - 50 (334) + 3 = 337. 
 
 5. Solve, explaining the process: 
 
 362 - 98 473 - 57 734 - 96 263 - 99 
 
 654 - 319 328 - 246 899 - 473 346 - 228 
 
 Note. Often, in practical work, the subtrahend must first be obtained 
 by addition. 
 
 6. 624 - (32 + 16 + 11) = ? 
 
 7. 1754 - (200 + 311 + 40) = ? 
 
 8. Sales of 72, 14 and 11 pounds from a stock of 360 lb. leave how 
 many pounds on hand? 
 
 EXERCISE. 
 
 1. 59,768,003 - 407,019 = ? 
 
 2. $3,876,547 3. $567,839 4. 987,654,321 
 
 108,976 203,678 123,456,789 
 
 ? ? . ? 
 
 In the following, subtract the last number from the sum of the balance 
 of the colunm: 
 
 7239 
 
 6. 1209 
 
 7. 
 
 213 
 
 8. 32139 
 
 2146 
 
 9660 
 
 
 32 
 
 4781 
 
 1299 
 
 5138 
 
 
 921 
 
 61020 
 
 881 
 
 1234 
 
 
 4238 
 
 3121 
 
 4731 
 
 8108 
 
 
 1119 
 
 5288 
 
 9-12. Repeat the last four examples, subtracting the first number 
 from the sum of the balance of the colunm. 
 
 Note. Subtraction is often combined with direct addition, especially 
 in business forms and statistical tables. 
 
 13. This form is one of many used in business to keep account of stock 
 on hand. 
 
20 
 
 BUSINESS ARITHMETIC. 
 
 STOCK RECORD OF 
 
 LIMIT 
 
 MO 
 
 WHERE KEPT 
 
 STOCK RECEIVED 
 
 STOCK SOLD 
 
 ON 
 HAND 
 
 DATE 
 
 FROM WHOM 
 
 QUAN. 
 
 DATE 
 
 TO WHOM 
 
 QUAN. 
 
 QUAN. 
 
 19— 
 
 
 
 19- 
 
 
 
 
 3-3 
 
 ^rtAP&nid^'u, 
 
 172 
 
 3-5 
 
 *^3762 
 
 9 
 
 163 
 
 
 
 3-8 
 
 3964 
 
 27 
 
 
 
 
 
 3-9 
 
 4287 
 
 15 
 
 
 
 
 
 3-12 
 
 4561 
 
 42 
 
 
 3-21 
 
 ja/rtAoifn, Vo. 
 
 50 
 
 3-21 
 
 5265 
 
 18 
 
 
 
 
 
 3-24 
 
 5290 
 
 54 
 
 
 
 
 
 3-26 
 
 5380 
 
 21 
 
 
 
 
 
 3-29 
 
 5620 
 
 20 
 
 
 (a) Extend the form. 
 
 (6) Explain the writing of the date. 
 
 (c) How are entries in the "On hand" column obtained? 
 
 (d) Explain the solution for Mar. 21. 
 
 (e) How many desks are on hand on Mar. 11? 
 
 (/) How may the "Received" and "Sold" columns be checked? 
 
 14. Many private families, living on a "cash" basis, keep simple 
 record, showing in detail household income and expenditures. (Page 21.) 
 
 (a) Explain the first items in the "Total paid out" and "Balance" 
 columns. 
 
 (6) Find the remaining values for these columns. 
 
 (c) Total all the vertical columns except the "Balance" column. 
 
 (d) Why is not the "Balance" column totaled? 
 
 (e) Find a check for the "Total paid out" total. 
 (/) Suggest a check for the final balance. 
 
 (g) Suggest how amounts in the different columns might vary with 
 the season of the year. 
 
 The statistician who is tabulating crops, wealth, railway 
 mileage, bank clearings, exports, etc., frequently gives figures 
 for earlier periods and shows by a difference column the amount 
 of change, the differences often lending special significance to 
 the gross values. The business man, also, gives special atten- 
 
 * Order number identifies purchaser. 
 
SUBTRACTION. 
 
 21 
 
 
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22 BUSINESS ARITHMETIC. 
 
 tion to the increases or decreases, as well as to the total of his 
 sales, profits or expenses. In the exercise that follows, note 
 how the differences add weight to the presentation of facts. 
 
 EXERCISE. 
 
 1. These figures from a financial article show certain facts concerning 
 the business of the country for two consecutive years. Fmd the differences, 
 placing decreases in the "increase" column, but marking them with a 
 minus sign. 
 
 Two Years' Record. 
 
 1910 1911 Increase 
 
 Bank clearings $158,450,000,000 $162,914,100,000 ? 
 
 Railway earnings 2,395,800,000 2,423,400,000 ? 
 
 Exports 1,867,605,000 1,637,256,000 ? 
 
 Imports 1,392,500,000 1,426,200,000 ? 
 
 Grain crops 2,702,458,000 2,489,000,000 ? 
 
 EXERCISE. 
 
 1. Bring to class four illustrations of " difference " columns in statistical 
 tables. 
 
 2. Sum up the different methods of checking addition and subtraction 
 used or suggested in this chapter. 
 
 Subtraction of Decimals. 
 
 INTRODUCTORY EXERCISE. 
 
 1. From 36 hundredths subtract 12 hundredths. 
 
 2. 5.46 - 2.39 = ? 
 
 3. .62 = ? thousandths. .52 - .412 = ? 
 
 The only difference in process between whole nun^iber and 
 decimal subtraction consists in the necessity for a reduction, 
 in the latter, of both subtrahend and minuend to decimals of 
 the same order. 
 
 Illustration. Find the difference between 5.029 and 3.0014. 
 Solution. 5.029 is the same as 5.0290. 5.0290 
 
 3.0014 
 
 2.0276 
 
SUBTRACTION. 23 
 
 EXERCISE. 
 By inspection find the difference between: 
 
 1. .51 and .402. 6. .01024 and .08. 
 
 2. 3.6 and 1.32. 7. .000004 and .0004. 
 
 3. .9 and .09. 8. 10.003 and .230. 
 
 4. 1. and .0321. 9. From 3.75 subtract .05s. 
 
 5. 2.0004 and 1.3. . 10. From .324 subtract .004s. 
 11. From one and one-hundredth, subtract nine ten-thousandths. 
 
 EXERCISE. 
 
 1. 4.0739625 - 2.00100376 = ? 
 
 2. 5506.38 - 296.54325 = ? 
 
 3. 100,109.0001003 - 1202.12345 = ? 
 
 4. From 46 subtract the first of the following decimals; from the 
 remainder subtract the second, and so on: 5.0621, 2.010309, 1.0042, .98396, 
 25. 1013096, 1 .24965. How may the accuracy of the final answer be checked? 
 
 Decimal subtraction is common in higher mathematics and 
 in scientific and estimating problems. Two-place decimal 
 subtraction is common in money computations. Usually sub- 
 traction is incidental to some other process. 
 
 EXERCISE. 
 
 1. A farmer's poultry account for January is as follows: Value of fowls, 
 Jan. 1, $1800; sales of fowls during the month, $247.53; value of fowls 
 at close, $1750; cost of food and care, $112.50; sales of eggs, $105.82; 
 miscellaneous expenses, $54.27. Make a neat statement showing the net 
 profit for the month. 
 
 2. A metal bar 16 inches long, is contracted under pressure .8583765 
 inches. What is its length imder pressure? 
 
 3. A yard measure is equal to 0.914,402 meters. A meter exceeds a 
 yard by what decimal of itself? 
 
 4. A certain cloth shrinks .1525 of a foot per yard in dyeing. A yard 
 of the undyed cloth produces how many yards of dyed material? 
 
 5. The needle on a recording machine travels from 3.256 ft. on the 
 scale to 4.105 ft., a distance of ? ft. 
 
24 
 
 BUSINESS ARITHMETIC. 
 
 6. In one city, having 5c car fares, the cost of carrying each passenger 
 is 3.9246c. In another city, having a 4c fare, the cost is 3.1024c. Find 
 the profit per passenger in each case, and the difference in cost per city. 
 
 7. Complete this memorandum of costs and profits. 
 
 The Novelty Manufacturing Company. 
 Memorandum of Costs and Profits per Article. 
 
 Article. 
 
 Cost to Make, ^. 
 
 Net Sell. Pr. ff. 
 
 Net Profit. 
 
 Butterfly cards 
 
 Dime savings banks 
 
 Remembrance cards ... 
 
 3.0245 
 14.308 
 .0965 
 
 4.85 
 13.0625 
 
 7.5 
 16.5 
 
 2.15 
 
 6.1125 
 15. 
 
 ? 
 ? 
 ? 
 
 Surprise box 
 
 ?" 
 
 Three of a kind 
 
 ? 
 
CHAPTER V. 
 MULTIPLICATION. 
 
 Multiplication, as a number process, is second in importance 
 to addition, and has extensive and varied applications. It is 
 really an addition "short cut," for expressions of multiplica- 
 tion, such as "5 X 381" may be written in the form "381 
 + 381 + 381 + 381 + 381." As an independent process, 
 also, it lends itself to so many short methods as to make it 
 worth a special study. 
 
 Illustration. Multiply 3264 by 529. 
 
 3264 (multiplic^d) 
 529 (multiplier) 
 
 29376 (units) 
 6528 (tens) 
 16320 (hundreds) 
 
 1726656 (product) 
 In the product there are 9 X 3264 units; 2 X 3264 tens and 5 X 3264 
 hundreds. 
 
 Note. The multiplicand and multiplier are called factors of the product. 
 
 While the multiplier is invariably an abstract number, the 
 multiplicand and product may be either abstract or concrete, 
 but they are always like numbers. As a result, the product is 
 not altered by changing the order of the factors considered as 
 abstract numbers. 
 
 Illustration. The weight of 8724 bx. of 65 lb. each is either (8724X65) 
 lb., or (65 X 8724) lb. 
 
 FOR DISCUSSION. 
 
 1. What is the objection to solving the examples illustrated 
 above by addition? 
 
 2. At what point, in regular multiplication, does addition 
 occur? 
 
 25 
 
26 
 
 BUSINESS ARITHMETIC. 
 
 3. In finding the product of 873,926 by 1001, which factor 
 should be used as multiplier? Why? 
 
 4. Why is this statement incorrect: 
 
 "$16 X 15 bbl. = $180"? 
 
 ORAL EXERCISE. 
 
 1. Find the products of: 3 X 14, 7 X 18, 3X3X6, 4 X 2 X 22, 
 8 X 40 bu , 5 X 81c, 2 X 3 X 8 X 4, 2 X 24 ft. 
 
 2. Multiply each integer under 21, by itself and by each smaller 
 integral number. These are the most used factors. 
 
 3. A camping party of three men, planning a sixty day trip, order 
 rations in the proportions given below. Determine the required quantity 
 of each article. 
 
 Supplies per Man per Thirty Days. 
 
 Flour 241b. 
 
 Oatmeal 7 " 
 
 Beans 6 " 
 
 Sugar 12 " 
 
 Cond. milk. ... 3 en. 
 
 Tea, cocoa.. ,1 lb. 
 
 Coffee 2 " 
 
 Rice 4 " 
 
 Salt 1 " 
 
 Prunes 2 " 
 
 Butter.... 21b. 
 Bak. pow.. 1 " 
 Bacon.... 11 " 
 Pepper ... 1 " (6 men) 
 
 4. A certain contractor foimd constant need for the products of the 
 factors 80 to 100 by the factors 20 to 40. He prepared the form outlined 
 below, the product of each two factors being written at the intersection 
 of the proper horizontal and vertical columns. The product of 84 by 22 
 is shown. Compute the other products mentally. Show how to use an 
 addition method for the work. Suggest a method of checking the compu- 
 tations. 
 
 
 80 
 
 81 
 
 82 
 
 83 
 
 84 
 
 85 
 
 Etc. 
 
 20 
 
 
 
 
 
 
 
 
 21 
 
 
 
 
 
 
 
 
 22 
 
 
 
 
 
 1848 
 
 
 
 Etc. 
 
 
 
 
 
 
 
 
 5. Construct a similar multiplication table for the factors 1 to 25 in- 
 clusive. 
 
 EXERCISE. 
 
 Find the products of the following and check accuracy by reversing 
 factors: 
 
MULTIPIJCATION. , 27 
 
 1. 726,943 X 120,301. 4. 373,259 X 129. 
 
 2. 3,629,007 X 20,905. 5. 12,345 X 67,890. 
 
 3. 16,730,021 X 802,001. 6. 20,103 X 1,387,654. 
 
 7. A railway company sends out eight engineering parties of 19 men 
 each, on reconnoisance and track location work. Rations are required 
 for 200 days, based on a U. S. Government ration Ust. Prepare a tabu- 
 lation showing the quantity of suppUes to be provided. 
 
 Ration List for One Man for One Hundred Days. 
 (Take first named article.) 
 100 lb. fresh meat, including fish and poultry. 
 50 " cured meat, canned meat, or cheese 
 15 " lard. 
 
 80 " flour, bread, or crackers. 
 
 15 " corn meal, cereals, macaroni, sage, or com starch. 
 5 " baking powder, or yeast cakes. 
 40 " sugar. 
 
 1 gal. molasses. 
 121b. coffee. 
 
 2 " tea or cocoa. 
 
 10 en. condensed milk, or 50 qt. fresh milk. 
 101b. butter. 
 
 20 " dried fruit, or 100 lb. fresh fruit. 
 20 " rice or beans. 
 
 100 " potatoes or other fresh vegetables. 
 30 " vegetables or fruits. 
 
 4 oz. spices. 
 
 4 " flavoring extracts. 
 
 8 " pepper, or mustard. 
 
 3 qt. pickles. 
 1 " vinegar. 
 41b. salt. 
 
 8. As a first result of the survey, the Company decides to build 19 
 miles of single and 38 miles of double track. Compute the total quantity 
 of the following supplies which are to be used in the construction: Cross- 
 ties, at rate of 2640 per mile of single track; steel rail, at 90 pounds 
 per yard; spikes, at the rate of 11,380 per mile of single track; 68 angle 
 irons for special construction, 34 feet long, weighing 16 poimds per foot; 
 240 I-beams, 28 feet long, weighing 65 pounds per foot. Give quantities 
 of iron and steel by weight. 
 
28 BUSINESS ARITHMETIC. 
 
 Short Methods. 
 Most fundamental processes of number may be abbreviated 
 — often with marked increase in rapidity of solution and 
 accuracy of result. The short methods used, however, should 
 be such as one naturally develops for himself, or those that 
 instinctively appeal to one as being simpler and more direct 
 than the regular methods. Memorized rules for short methods 
 are dangerous aids, as an inexact memory may cause an in- 
 correct application of the rule at some critical moment. Short 
 methods are commonly used to do away with computation 
 on paper, or to simplify such computation, or to check the 
 accuracy of computations performed by other methods. 
 
 Short Methods in Multiplication. 
 
 Some typical cases follow. Suggest other methods. In 
 each case, show wherein the value of the method consists. 
 
 Multiplying by 10, 100, etc. 726 X 10 = ; 1428 X 100 
 
 = ; 345 X 1000 = . Compare the significant figures 
 
 of the product with those of the multiplicand. How may the 
 product of such factors be written at sight? What effect 
 has the annexing of a cipher? Of two ciphers? 
 
 ORAL EXERCISE. 
 
 1. 167X10 = ? 4. 100X26 lb. = ? 7. 1,000X80,201=? 
 
 2. 100X3,802 = ? 5. 10,000X131 ft. = ? 8. 70,073X100,000 = ? 
 
 3. 100X50,201=? 6. 10X7,384 yd. = ? 
 Find the weight of: 
 
 9. 246 bbl, averaging 100 lb. ea. 
 10. 1,246 loads, averaging 1000 lb. 
 
 Multiplying by 11, 21, SI, etc., and by multiples of 11, Outline 
 a short method for these multipliers, suggested by the follow- 
 ing illustrations: 
 
 (1) 11 X 64 = 10 X 64 + 64 = 640 + 64 = 704. 
 
 (2) 31 X 126 = 3 (10 X 126) + 126 = 3780 + 126 = 3906. 
 
 (3) 101 X 48 = 100 X 48 + 48 = 4800 + 48 = 4848. 
 
MULTIPLICATION. 29 
 
 ORAL EXERCISE. 
 Find the product of: 
 
 1. 11 X 16. 3. 11 X 242. 5. 21 X 360. 7. 21 X 19. 
 
 2. 21 X 28. 4. 31 X 18. 6. 41 X 22. 8. 1001 X 287. 
 Find the weight of: Find the cost of: 
 
 9. 240 ft. iron at 111b. per. ft. 11. 12 lb. butter, @ 31c. 
 10. 165 kegs, 31 lb. each. 12. 142 sq. yd. tiling, @ 21c. 
 
 Multiplying by 9, 19, 29, etc. After studying the illustrations, 
 outline the short method and compare with the previous case. 
 
 Illustrations. 
 
 (1) 9 X 148 = 10 X 148 - 148 = 1480 - 148 = 1332. 
 
 (2) 19 X 320 = 2 (10 X 320) - 320 = 6400 - 320 = 6080. 
 Used in paper work, also. 
 
 (3) Example. Multiply 32,689 by 19. 
 
 Long method. Short method. 
 
 32689 Factors not re-written. 
 
 19 653780 (2 X 10 X 32689, written at sight) 
 
 294201 32689(1X32689) 
 
 32689 621091 (by subtraction) 
 
 621091 
 
 Note. The short method substituted the multiplication by 2 for the 
 more difficult multiplication by 9. 
 
 ORAL EXERCISE. 
 Find the product of: 
 
 1. 9 X 87. 3. 29 X 16. 5. 9 X 269. 7. 29 X 17. 
 
 2. 19 X 42. 4. 109 X 120. 6. 9 X 424. 8. 59 X 11. 
 9. Find the weight of 720 pc. castings, weighing 9 lb. each. 
 
 EXERCISE. 
 
 Using short methods, find the product of: 
 
 1. 19 X 32,967. 4. 29 X 6,376,951. 7. 99 X 8,703,129. 
 
 2. 9 X 1,213,027. 5. 19 X 12,587,328. 8. 1,009 X 784,562. 
 
 3. 39 X 47,201. 6. 109 X 630,851. 
 
 Multiplying hy 5, 15, etc. 
 
 Illustrations. 
 
 (1) 5 X 620 = One-half of 10 X 620 = ^^ = 3100. 
 
 (2) 15 X 48 = 10 X 48 + H of 10 X 48 = 480 + 240 = 720. 
 
30 BUSINESS ARITHMETIC. 
 
 ORAL EXERCISE. 
 
 1. In multiplying a number, such as 7,268,316, by 5, why may it b« 
 simpler to use the short method? 
 
 2. Multiply by 5: 16, 21, 32, 54, 123, 1309, 677, 1893. 
 
 3. Multiply by 15: 24, 68, 152, 19, 137, 240, 62, 192, 1012. 
 
 4. Find the costs of: 124 en. at 5c; 16 lb. nails at 5c; 68 yd. dress 
 goods at 15c. 
 
 EXERCISE. 
 
 Using short methods, and omitting the re-writing of factors, find the 
 products of: 
 
 1. 5 X 7236. 3. 15 X 724 yd. 5. 15 X 8039 qt. 
 
 2. 15 X 83,963 lb. 4. 5 X 12,463 qt. 6. 15 X 7204 oz. 
 
 Multiplying by factors of multiplier. 
 
 Illustration : 36 X 120 = 6 X 6 X 120 = 6 (6 X 120) = 6 X720 = 4320. 
 
 ORAL EXERCISE. 
 i. Compare the short and the direct methods. 
 
 2. Is the short method of real advantage in oral or in written work? 
 
 3. How do the factors themselves limit the use of the method? 
 
 4. Reduce to two or mor€ simple factors: 65, 24, 49, 18, 72, 96, 48, 
 45, 54, 84, 110, 39, 92, 120. 
 
 5. Determine these products by factoring multiplier, and check by 
 other short methods: 
 
 25 X 160 122 X 12 99 X 48 16 X 32 
 
 240 X 18 12 X 118 15 X 72 20 X 155 
 
 ' Multiplying by parts (multiplication from the left). 
 
 Illustration. (1) Find the weight of 722 bbl. of 112 lb. each. 
 112 X 722 = 100 X 722 + 10 X 722 + 2 X 722 = 72200+7220+1444 = 80864 
 (2) Written work. Multiply 738,421 by 321. 
 
 Regular method. Method by parts. 
 
 738,421 Factors not re-written. 
 
 321 221,526,300 (3 X 100 X number) 
 
 738421 14,768,420 (2 X 10 X number) 
 
 1476842 738,421 (1 X number) 
 
 2215263 237,033,141 
 
 237033141 
 
MULTIPLICATION. 31 
 
 Note. The advantage of this method for oral work, lies in adding con- 
 tinuously decreasing sub-totals to a growing total, each new product being 
 added as found. 
 
 ORAL EXERCISE. 
 Find the product of: 
 
 1. 115X36. 3. 23X42. 5. 62X312. 7. 103X428. 
 
 2. 111X249. 4. 22X124. 6. 210X122. 8. 51X122. 
 
 9. What is the output in 16 hours of a machine making 240 doz. cans 
 per hour? 
 
 EXERCISE. 
 Find the product of: 
 
 1. 212X324,567. 2. 1325X88,459,876. 3. 1245X87,960,345. 
 
 SPECIAL EXERCISE. 
 
 1. Write a brief illustrative paper on "Short methods in Multipli- 
 cation." 
 
 2. Write a paragraph on the dangers and sources of error in the use 
 of short methods. 
 
 3. ^rite a brief paper on "Methods of Checking Multiplication." 
 
 DECIMAL MULTIPLICATION. 
 INTRODUCTORY EXERCISE. 
 
 1. One-tenth of 6 = ? One-hundredth of 6 = ? 
 
 2. Ten times .6 = ? One hundred times .6 = ? 
 
 3. Compare these numbers as to value: .06, .6, 6, 6000. 
 
 4. How is the value of a significant figure affected by moving the decimal 
 point to the right. 
 
 5. Suggest a short method for multiplication of decimals by 10, 100, 
 1000, etc. Illustrate, using .006 and 4.2 as multiplicands. 
 
 6. One-tenth of 68 = 68 ^?; or 68 X ? 
 
 7. Using 72.6 show the effect on its value of moving the decimal point 
 to the left. 
 
 8. Suggest short methods of multiplication by .1, 01, 001, etc. 
 
 The process of multiplication of decimal fractions differs 
 in no essential from the midtiplication of integers, except in 
 the determination of the decimal point of the product. Notic- 
 ing that .1 X .01 = 1/10 X 1/100 = 1/1000 or .001, it is 
 
32 
 
 BUSINESS ARITHMETIC. 
 
 evident that the number of decimal places in the product 
 equals the sum of those in the factors. In multiplication, 
 therefore, decimals may be treated as integers, the proper 
 number of places being pointed off after the numerical product 
 is obtained. 
 
 Illustrations. (1) Multiply 5268.39 by 1.6. 
 
 Solution and analysis. 5268.39 contains 526,839 hundredths. 
 
 1.6 contains 16 tenths 
 526,839 hundredths X 16 tenths = 8,429,424 thousandths, 
 or 8429.424 
 
 (2) Find the product of 7.26 multiplied by 3600. 
 
 Solution. 3600 X 7.26 = 36 X 100 X 7.26 = 36 X 726. 
 
 Note. Notice how the decimal is cancelled before the multi- 
 plication of significant figures is begun. This is often possible. 
 
 5268.39 
 1.6 
 
 3161034 
 526839 
 8429.424 
 
 726 
 
 36 
 
 4356 
 
 2178^ 
 
 26136 
 
 (e) .00008X4.00007 
 (/) 9.001X1.0000401 
 
 (g) 1.002 X. 021 
 (h) .06X6000 
 (i) 7.25 X. 0002 
 
 ORAL EXERCISE. 
 
 1. How many decimal places in the products of: 
 (a) 3.6 X. 004 (c) 58 X. 020 
 (6) 50.29 X. 00609 (d) 1573.829 X .032 
 
 2. Find the products of: 
 (o) 1.2X.12 (d) 4003X.0002 
 (6) .003 X. 0004 (e) 6.01 X. 005 
 (c) 2.01 X. 00002 (/) .000003 X. 3 
 
 3. Compute the cost of: 
 
 (a) 700 lb. @ 32c. (b) 2000 bu. @ 45c. (c) 2960 lb. @ Jl.lO. 
 
 EXERCISE. 
 (Check all products.) 
 
 1. Multiply 7206.021 by 3.001029. 
 
 2. Multiply 739.65482 by 549.03296. 
 
 3. Multiply 7.2098 by .02604, and the product by 100.09. 
 
 ORAL EXERCISE. 
 * (Use the short methods given for integral numbers.) 
 
 1. 100X26yd. = ? 4. 1000 X. 005 yd. = ? 7. 9X.8 = ? 
 
 2. 380X.2X.01 = ? 5. 21X.4 = ? 8. 19Xl.2 = T 
 
 3. 11X3.6 = ? 6. 101 X. 08 = ? 
 
MULTIPLICATION. 33 
 
 GENERAL EXERCISE. 
 
 Note. In the computation of costs and quantities by multiplication, 
 decimals are usually involved. The following are tj'pical illustrations. 
 
 1. For accounting purposes, Martin & Co. take an inventory, or 
 valuation of their stock of skins, pricing them at the lowest market quo- 
 tation, except for the gray fox which they price at the highest rate. Using 
 the following quotations, make a neat tabulation showing the valuation of 
 the entire stock. 
 
 WOOL AND HIDES.— Quotations for fm-s on No. 1 articles only. 
 Wool, washed, free of burrs, per lb., 38a40; wool unwashed, per lb., 30a33; 
 hides, green, per lb., 10; dry, per lb., 15al7; sheepskins, green, each, 1.00a 
 1.25; dry, each, 25a75; calf -skins, green, each, 1.00al.50; muskrat, each, 
 12al8; skunk, each 25a75; mink, each, 3.00a4.00; rabbit skins, each, 10; 
 opossum, each, 25a27; raccoon, each, 25a90; fox, red, each, 2.00a2.50; 
 gray, each, 75a90. 
 
 The stock of Martin & Co. consists of: 2480 lb. wool, washed; 1600 lb. 
 wool, unwashed; 750 lb. hides, green; 2375 lb. hides, dry; 128 calf skins; 46 
 muskrat skins; 256 rabbit skins; 13 gray fox skins; 129 opossum skins; 382 
 raccoon skins. 
 
 2. Tabulate individual and gross weights of the following order: 
 
 200 16' I-beams, weighing 85.2 lb. per ft. 
 162 19' I-beams, weighing 23.5 lb. per ft. 
 200' trough plates, weighing 16.32 lb. per ft. 
 1150 18' channel irons, weighing 33 lb. per ft. 
 450' deck beams, weighing 27.23 lb. per ft. 
 620 14' angle irons, weighing 17.2 lb. per ft. 
 145 Z-bars, 12', weighing 29.8 lb. per ft. 
 
 3. This is an itemized tabulation of the cost of a masonry wall con- 
 taining 12,642 cu. yd. of masonry. Compute the total cost. 
 
 Per cu. yd. Total cost. 
 
 Quarrying stone $0,415 $ ? 
 
 Loading and hauling stone 912 
 
 Hoisting stone 605 
 
 Excavating 300 
 
 Loading and hauling sand 355 
 
 Cement 345 
 
 Mixing and dejivering mortar 205 
 
 Masons and helpers 915 
 
 Carpentry 885 
 
 Blacksmithing 190 
 
 Tools and general supplies 175 
 
 Superintendence, foremen, etc .725 
 
 (How check?) 
 4 
 
34 BUSINESS ARITHMETIC. 
 
 4. Bids a, h, c, d and e have been submitted by different parties, ttj 
 
 cover the cost of a granite block pavement. Find and tabulate the tota.« 
 cost of each item and of each bid. Determine what combination of dif- 
 ferent bids will give the lowest cost. 
 
 a h c d e 
 
 400 cu. yd. concrete $4.00 S3.00 $3.90 $2.55 $3.98 
 
 135 lin. ft. granite crossings 1.20 1.13 1.25 1.00 1.24 
 
 1178 lin. ft. 5" granite curb 1.275 1.30 1.30 1.29 1.30 
 
 162 lin. ft. circular curb 2.00 2.10 1.95 2.00 2.0G 
 
 65 lin. ft. curb, reset, etc 21 .15 .20 .32 .18 
 
 48250 cu. yd. grading , 43 .425 .41 .45 .50 
 
 2476 sq. yd. granite paving 3.25 3.18 3.25 3.00 3.30 
 
 2476 sq. yd. sub grade 06 .06 .06 .06 .06 
 
 268 lin. ft. 24" dram pipe 2.78 1.75 2.00 2.10 1.95 
 
 73 cu. yd. brick masonry 12.00 11.25 12.50 10.90 11.75 
 
 21 cu. yd. rubble masonry 5.75 6.00 4.25 5.00 4.80 
 
 12 pc. Palmer inlets. 40.00 40.00 40.00 40.00 40.00 
 
 Totals 
 
CHAPTER VI. 
 DIVISION. 
 
 INTRODUCTORY EXERCISE. 
 
 1. Add by 4s from to 48. There are how many 4s in 48. 
 
 2. Subtract by 9s from 189 to 0. There are how many 9s m 189. 
 
 3. Compare division with addition and subtraction. 
 
 4. I agree to pay $40 cash and $6 per week for a $100 tjrpewriter. 
 Name the amount due after each payment. Determine the number of 
 payments (1) by addition, (2) by subtraction, (3) by division. Lowering 
 the weekly payment to $4 extends the time of settlement ? weeks. 
 
 5. 172 -^ 4 = ? ? X 4 = 172. 
 
 6. Compare multiplication and division. 
 
 Division may be termed a short method in addition and 
 subtraction, and the reverse of multiplication. It is the 
 process of determining the unknown factor of a product from 
 the product and a known factor. The known factor is the 
 divisor, the product is the dividend, and the factor to be deter- 
 mined is the quotient. The remainder is the portion of the 
 dividend that is left when division is not exact. 
 
 ORAL EXERCISE. 
 
 If the dividend is 64 and the divisor 8, show that: 
 
 1. The dividend divided by the divisor equals the quotient. 
 
 2. The quotient times the divisor equals the dividend. 
 
 3. The dividend divided by the quotient equals the divisor. 
 
 FACILITY EXERCISE. 
 (Solve by inspection.) 
 
 1. Divide by 2: 16, 38, 57, 96, 248, 1749, 62,482. 
 
 2. Divide by 3: 12, 21, 48, 69, 125, 683, 952, 667. 
 
 3. Divide by 2 and the quotient by 3: 36, 124, 88, 962. 
 
 4. Divide by 4: 76, 320, 888, 96, 154, 723. 
 
 35 
 
36 BUSINESS ARITHMETIC. 
 
 6. Divide by 5: 25, 65, 83, 195, 217, 1259. 
 
 6. Divide by 6: 42, 80, 172, 620, 38, 415. 
 
 7. Divide by 7: 49, 35, 73^, 1420, 203, 633. 
 
 8. Divide by 8: 72, 30, 960, 1240, 884, 1720. 
 
 9. Divide by 9: 30, 75, 810, 640, 1850, 27630. 
 
 10. Multiply 24 and successive products by 2 (6 times). Divide by 
 successive 4s. 
 
 Short Division. In short division the divisor and dividend 
 are written, and the figures of the quotient, as obtained, but 
 intermediate remainders and products are omitted. 
 
 Illustration. Divide 7248 by 8. 
 
 Solution. Analysis. 72 hundreds -^ 8 = 9 hundreds; 4 tens are 
 
 8 )7248 not divisible by 8; 4 tens + 8 units = 48 units. 48 units 
 
 906 -^ 8 = 6 units. 
 
 Note. At times, when the divisdr is reducible to very simple factors, 
 it is possible to employ continued short division by the factors in turn. 
 In such cases, of course, each factor must be an exact divisor until the last. 
 
 EXERCISE. 
 
 Divide: 
 
 
 1. 239,872,452 by 4. 
 
 5. 6888 by 56 (continued division), 
 
 2. 8,703,265 by .5. 
 
 6. 24,496 by 16. 
 
 3. 364,285,968 by 6. 
 
 7. 193,625 by 25. 
 
 4. 123,456,789 by 7. 
 
 8. 4,829,672 by 32. 
 
 ILLUSTRATIVE EXERCISE. 
 
 1. Solve: (a) 36 -^ 4 = ? 
 
 
 (6) 36 ft. -^ 4 = ? 
 
 (c) In 36 bbl. there are ? lots of 4 bbl. each. 
 2. What is "common" to these problems? What changes? 
 
 It is essential, in division, to determine whether the 'quotient 
 and the remainder are abstract or concrete. Division by an 
 abstract divisor is called partition, and by a concrete divisor, 
 measuring. Thus, in 1 (6) in the preceding exercise " 36 ft." is 
 parted into 4 parts, and in 1 (c) "4 bbl." is used as a measure 
 to determine the number of "4 barrel" lots in 36 bbl. It 
 
DIVISION. 37 
 
 should be noted that if divisor and dividend are concrete, the 
 quotient is abstract; while if the dividend is concrete and the 
 divisor abstract, the quotient is a concrete number of the same 
 order as the dividend. 
 
 ILLUSTRATIVE EXERCISE. 
 
 1. Find the remainders in the following: 
 
 (a) 725 ^ 4 = ? 
 
 (6) 420 yd ^ 8 = ? 
 
 (c) For 88c I can buy ? yd. of cotton at 8c per yd. 
 
 2. When is the remainder abstract? When concrete? 
 
 Long Division. In long division, intermediate products and 
 remainders are shown. 
 
 Illustrations. (1) Divide 76,945 by 162. (2) 
 
 Solution. 474 (Quotient) Table of Multiples of 162. 
 
 Byl 
 
 162 
 
 2 
 
 324 
 
 3 
 
 486 
 
 4 
 
 648 
 
 5 
 
 810 
 
 6 
 
 972 
 
 7 
 
 1134 
 
 8 
 
 1296 
 
 9 
 
 1458 
 
 (Divisor) 162)76945 (Dividend) 
 (400 X 162) 648 
 
 1214 
 (70 X 162) 1134 
 805 
 (4 X 162) 648 
 
 157 (Remainder) 
 Check. 474 X 162 + 157 = 76945. 
 
 If, as often happens in business and statistics, the same 
 divisor is constantly employed, a table of its multiples (see 
 (2) from 1 to 9, is prepared, and used for reference. This 
 avoids repeated multiplication, and serves to show, without 
 trial, each successive figure in the quotient. 
 
 EXERCISE. 
 
 Divide, expressing remainder separately: 
 
 1. 146,732 by 1396. 3. 9,820,605 by 89,243. 
 
 2. $372,486 by 2013. 4. 620,548 lb. by 63 lb. 
 5. In 468,246 lb. of grain there are ? bu. of 56 lb. each. 
 
38 BUSINESS ARITHMETIC. 
 
 6. Divide the following by 1237, using a table of multiples: 
 
 (a) 673,892. (6) 960,478. . (c) 1,214,734. 
 
 7. Divide the following by 864, using a table of multiples: 
 
 (a) $4,327,432. (b) 8,921,065. (c) 1,234,056,789. 
 
 8. Find the number of tons of coal (2240 lb. ) in each of these shipments, 
 ignoring fractions of a ton. 
 
 Car No. Wt. of Loaded Car. Empty Car. Net Weight. Tons. 
 32,064 134,200 26,420 ? ? 
 
 38,296 129,100 27,020 
 
 43,129 126,000 24,320 ? ? 
 
 Short Methods. 
 
 Division does not lend itself to so many easily applied short 
 methods as does multiplication. The following are the most 
 common : 
 
 Division by 10 y 100 ^ etc. 
 
 ORAL EXERCISE. 
 
 1. What effect has the decimal point on the comparative values of 
 these numbers: 6800, 680 0, 68.00, 6.800? 
 
 2. Suggest a short method of division by powers of 10. Find the 
 quotient of: 
 
 3. 720,000 ^ 10. 6. 30,000,000 lb. ^ 10,000. 
 
 4. 726,000 -^ 1000. 7. 368,000 ^ 100. 
 
 5. $725,900 -h $100. 8. $850 -^ 10. 
 
 Division by multiples of 10. When a power of ten is a factor 
 of both dividend and divisor, the example may be simplified by 
 the principle of continued division. 
 
 Illustration. 4720 -^ 80 = (4720 -^ 10) -^ 8 = 472 4- 8 = 59. 
 
 EXERCISE. 
 
 Find the quotient by inspection. 
 
 1. 840 ^ 70. 4. 320 lb. -=- 20. 7. $650 ^ 130. 
 
 2. 725,000 -^ 600. 5. $7,600 ^ 40. 8. $987,500 -J- 500. 
 
 3. 870 -^ 30. 6. 800,000 lb. -^ 40. 
 
DIVISION. 39 
 
 DECIMAL DIVISION. 
 INTRODUCTORY EXERCISE. 
 
 1. 4.2 is equivalent to ? tenths 
 
 2. 4.2 -i- 6 may be expressed: ? tenths -^ 6. The quotient is — . 
 
 3. Divide by 6: $480, 48 lb , 48 T., .048 cu. yd., .00048. 
 
 4. Divide these same values by .6. What remains unchanged in the 
 quotient? What changes? 
 
 It is evident that the location of the decimal point in the 
 quotient is the new feature of the division process met with 
 in passing from whole numbers to decimals. 
 
 Illustrations. (1) Divide 49.776 by 16. 
 
 Hon 
 
 
 Condensed solution. 
 
 
 3111 
 
 3,111 
 
 
 16)49.776 
 
 16)49.776 
 
 (3 X 16) 
 
 48 
 
 , - 48 
 
 
 1 776 remaining. 
 
 17 
 
 (.1 X 16) 
 
 16 
 
 16 
 
 
 .176 remaining. 
 
 17 
 
 (.01 X 16) 
 
 .16 
 
 16 
 
 
 .016 remaining. 
 
 16 
 
 (.001 X 16) 
 
 .016 
 
 16 
 
 Note. If the divisor is integral, the position value of each figure in 
 the quotient is the lowest position value in the partial dividend used in 
 obtaining it. Thus "49" is the partial dividend used in obtaining "3" 
 of the quotient, and its lowest position value is ''unity." The advantage 
 of writing quotient over dividend is evident, since the "placing" of the 
 first figure of the quotient practically determines the place value of the rest. 
 
 (2) Divide 7.6845 by .15. 
 
 Solution. Multiply each term by 100, by short method. 
 
 7.6845 H- .15 = 768.45 -^ 15. 
 • 51.23 
 
 15)768.45 
 
 Note. In case the divisor is decimal in form, it may be cleared of 
 decimals, by mo^dng the decimal point to the right in both dividend and 
 divisor. The example then comes under the class shown in the previous 
 illustration. 
 
40 BUSINESS ARITHMETIC. 
 
 EXERCISE. 
 
 Using short methods, if convenient, find by inspection the quotient of; 
 
 1. 3.6^6. 4. .0004^2. 7. 3.824-r-8. 10. 52.08 T.-^100. 
 
 2. .427-^7. 5. 2.644-3. 8. 16.860-J-12. 11. 3.62-^10. 
 
 3. .084-^16. 6. 1.64-4. 9. .00129-^30. 12. 29000.01 -MOOO. 
 In the following, clear the divisor of decimals: 
 
 13. .844-1.2. 14. 63.08-i-.002. 15. .00034- 006. 16. 2.40604-1.2. 
 Divide, by inspection: 
 
 17. 1 by .01, .001, .0001, 10000. 22. 200, 4000, .08 and .0006 by 2. 
 
 18. .01 by 10, 100, .01 and .00001. 23. .00012 by 6, 600, .3, .03, .006. 
 
 19. .0001 by .005, .05, .02, 100. 24. 4.24 by .0006 and by 6. 
 
 20. 4.8 by 1.2, .012, .0006, .3. 25. 63 by .007, .09, .7, .0003. 
 
 21. 60 by .002, .3, .0005, .12. 26. 660.15 by .015. 
 
 Exact and Approximate Results. Decimal ciphers may be 
 added to the' dividend indefinitely, to secure an accurate 
 quotient to any number of decimal places, the final remainder 
 being expressed fractionally. Commonly, however, results 
 are required exact to some stated number of places, the remainder 
 being unstated. Thus a business man may compute costs 
 "to the nearest cent," or an engineer make measurements to 
 "hundredths of an inch." Many working tables, also, are 
 computed to four, five or six decimal places, remainders being 
 ignored because, as a rule, they are too small to affect appre- 
 ciably results computed by them. Often, if the remainder is 
 one-half or more of the preceding decimal figure, that figure 
 is increased by 1. Thus a quotient of 7.3086 may be written 
 "to three decimal places" as 7.309. 
 
 When merely an approximate quotient is desired, the 
 approximate method is sometimes used. This consists usually 
 in direct division until the integral portion of the quotient is 
 obtained. From that point, no further figure of the dividend 
 is brought down but, at each step, the right hand figure of 
 the previous divisor is cancelled. 
 
DIVISION. 
 
 41 
 
 Illustration. Divide 684.394683 by 56.138. 
 Contracted Soluiion. 
 12.1914 
 56.138)684.3946839 
 56138 
 123 014 
 112 276 
 10 738 
 (Div. 56.13) 5 613 
 5 125 
 (Div. 56.1) 5 049 
 
 76 
 (Div. 56) 56 
 
 20 
 (Div. 5) 20 
 
 EXERCISE. 
 
 Divide: 
 
 1. $726 48 by 12. 4. .0297 by .31. (2 places ) 
 
 2. 114,269 lb. by 11. (2 places.) 5. 12.873 by 151.3. (1 place.) 
 
 3. 4629.2 cu. yd. by .24. 6. 842,097 by .0036. (Approx.) 
 
 7. 23.0567385 by .07525. (3 places approximate.) 
 
 8. $7296.47 by 123. (3 places.) 
 
 9. 91.38 by 1.02545. (3 places, approximate.) 
 
 10. A speed of 592 knots in 24 hours is a speed of ? knots per hour. 
 (2 places.) 
 
 EXERCISE. 
 
 Note. The weight of any material, as compared with an equal bulk 
 of pure water, is termed its specific gravity. Thus a brick having a specific 
 gravity of 1.602 weighs 1.602 times as much as the same volume of water. 
 
 Assuming the weight of a cubic foot of water as 62.355 lb., find the 
 specific gravity of the following building materials: 
 
 Material. Pounds per Cu. Ft. Specific Gravity. 
 
 Asphaltum 87 
 
 Hard brick 125 
 
 Clay 120 to 145 to 
 
 Limestone 170 to 210 to 
 
 Plaster of paris 75 to 80 to 
 
 Slate 165 to 180 to 
 
 Tile 106 to 122 to 
 
 Sand stone 140 to 155 to ■ 
 
42 BUSINESS ARITHMETIC. 
 
 Selected Individual Assignments. 
 
 1. Write a brief on "Methods of Checking Arithmetical Computa- 
 tions/' giving illustrative examples. 
 
 2. Write a brief on "Short Methods in Computation." Illustrate 
 short methods in common use. If possible, report on business men's 
 attitude towards the use of short methods. 
 
CHAPTER VII. 
 ARITHMETICAL AVERAGING. 
 
 INTRODUCTORY EXERCISE. 
 
 1. What six equal numbers have a total equal to the sum of 12, 14, 
 30, 42, 18 and 10? 
 
 2. Five men give to a public cause, respectively, $40, $35, $20, $60, $25. 
 The same sum would have been secured by their contributing equally, $ — . 
 
 3. The total profit of a business for three years was $19,654.21. As- 
 siuning that there was no variation from year to year, what was the annual 
 profit ? 
 
 4. The values just determined are called averages, or average numbers. 
 What is an average number? What processes of arithmetic are employed 
 in averaging? 
 
 5. What does a physician mean when he says "My office calls average 
 12 per day"? 
 
 Averages offer a common and effective means of comparing 
 values, and are used constantly in statistical work and in 
 business. Thus the average output, expense and profit of a 
 business, for past years, will guide a possible purchaser to a 
 decision, or aid an owner in remedying defects. The average 
 monthly rainfalls, in a certain farming section, will guide an 
 intending purchaser in his decision as to possible crops. Other 
 illustrations will be found throughout the book. 
 
 Ordinarily, the average, mean, or normal, is obtained by a 
 process of addition, followed by division. At times, the addi- 
 tion may be mechanically accomplished, as in the case where 
 a number of castings are placed on a foundry scale and 
 weighed at one operation, the total weight being then divided 
 by the number of castings to determine the average weight. 
 
 In determining averages care must be used to include all 
 influencing factors. 
 
 . 43 
 
44 BUSINESS ARITHMETIC. 
 
 Illustration. On April 1, Brown deposits, in a bank that pays 
 interest on average monthly balances, the sum of $2000; a^d on April 21, 
 reduces this sum lo $1600. Determine his average balance for April. 
 
 Solution. The average is not $2000 + 1600 divided by 2, for the 
 individual balances existed for different times. $2000 was the balance 
 for 20 days, and $1600 of 10 days. There were 20 daily balances of $2000 
 and 10 of $1600. The average is, therefore, 
 
 20 X $2000 4- 10 X $1600 $56000 ^.^^^^^ 
 30 ="^0~ =^1866.67 -. 
 
 EXERCISE. 
 
 Note. These sentences are taken at random from news reports and 
 general publications. Determine the missing values. 
 
 1. "The Lusitania made 598, 606, 612 and 603 knots in four successive 
 days, averaging — knots per day." 
 
 2. "England records a West Coast train from London to Aberdeen, 
 September, 1895, 540 miles in 512 minutes miles per hour." 
 
 3. "In 1895 the number of depositors in the savings banks of the 
 country was 7,696,000, having total deposits of $3,093,236,119, or an 
 average of each." 
 
 4. "The distance is 192 miles, but I can average 12 miles an hour. If 
 I leave at 7 o'clock to-morrow, that would get me there by -" 
 
 5. "Those cattle ought to average 940 lb. each, or lb. for the 
 
 lot of 37." 
 
 6. "The output with the old machinery averaged 46,000 yd. per 10 
 hour day. With the new equipment, it is 48,000 yd. in 9 hours. The 
 average hourly output has increased yd." 
 
 7. "Mr. Newbold testified that the improvement had increased their 
 
 average daily output from 143.2 tons to 158.5 tons — an increase of 
 
 tons per year." (1 year = 300 days.) 
 
 EXERCISE. 
 
 Note. Each example shows a different application of the average. 
 
 1. In a history test, the individual marks obtained were: 47, 90, 87, 
 82, 100, 100, 96, 90, 40, 35, 82, 87, or a class average of—. (2 dec. places.) 
 
 2. Twenty pupils in a class were present on the twenty-one school 
 days in March, four were absent one-half day, three for one day, two for 
 three days, and one for five and one-half days. Find the average daily 
 attendance. 
 
AVERAGING. 45 
 
 3. In an Eastern city, having a normal maximum temperatm-e for 
 July of 76.5" a hot wave caused maxima, for successive days, of 90°, 93°, 
 95.5°, 89°, 96°, 98°, 101°, 92°, an average of — ° above normal. 
 
 4. During November, 1907, individual rainfalls were .45 in., 2.15 in., 
 .08 in , 1.37 in., and .78 in. Was the rainfall excessive as compared with 
 (4.88 in.) the normal for past years? 
 
 5. The Rankin Mfg. Co. pays 60 men $2.15 per day; 80 men, $3 00; 
 12 men, $4.50; and 8 men, $5.75. How does its average daily wage compare 
 with that of the Morgan Co., which has an average for the same class of 
 work of $3,225? 
 
 6. Find the average daily and monthly circulation of the newspaper 
 whose circulation statement is printed on. page 12. 
 
 7. Find the average speed on these long-distance runs: 
 
 Course. Distance. Hours and Min. Average. 
 
 Jersey City and Chicago 912 mi. 17:41 
 
 Jersey City and Pittsburgh ... 443 8:41 
 
 New York and Buffalo 440 8:15 
 
 New York and Chicago 979.52 18:00 
 
 London and Edinburgh 393 7:45 
 
 London and Glasgow 401.50 8:00 
 
 Paris and Bayonne 486.25 8:59 
 
 8. Compare the profits of the company mentioned below for the last 
 year with the average for the past five years. 
 
 Net Profits of the Cropleigh Wheel Company. 
 
 1906 $17,246 1908 $18,076.50 1910 $20,132.61 
 
 1907 21,032 1909 16,548 1911 23,328.46 
 
 1912 24,736.59 
 
 9. Explain the difference in meaning of these expressions: 
 
 (a) "126 pieces of castings, averaging 84 pounds each." 
 (6) " 126 pieces of casting weighing 92 pounds each." 
 (c) "45 pieces of castings weighing 2746 pounds." 
 
CHAPTER VIII. 
 THE EQUATION. 
 
 INTRODUCTORY EXERCISE. 
 
 1. Use the word "equal" in comparing the cost of two articles. 
 
 2. Use the same word in expressing the cost of one article. 
 
 3. Use it in comparing the time necessary to do two pieces of work. 
 
 4. Use the word in expressing the time it takes to go from Philadelphia 
 to New York, by rail. 
 
 5. Use it in comparing the size of two objects, one greater by one-half 
 than the other. 
 
 An equation is formed by using the equality sign (=) be- 
 tween two equal numbers or expressions, called the members or 
 sides of the equation. 
 
 Illustrations. 
 
 My business for May = that for April. 
 
 The cost of 5 bbl. at $6 = the cost of 6 bbl. at $5. 
 
 The quotient X the divisor = the dividend. 
 
 Certain self-evident truths (axioms) aid us in the practical 
 use of the equation: 
 
 1. Things equal to the same thing are equal to each other. 
 
 2. The equality of two expressions is not destroyed: 
 (a) By adding equals to each term. 
 
 (6) By subtracting equals from each term. 
 
 (c) By multiplying each term by the same factor. 
 
 {d) By dividing each term by the same factor. 
 
 Illustrations. 
 
 1. If one barrel of apples cost $3 and one barrel of sweet potatoes 
 costs $3, then — 
 
 The cost of 1 bbl. apples = the cost of 1 bbl. sweet potatoes. 
 If a = 6 and 6 = 6, then a = 6. 
 
 46 
 
THE EQUATION. ' . 47 
 
 2. If the cost of 1 bbl. apples = cost of 1 bbl. sweet potatoes, then 
 
 {a) Cost of 1 bbl. apples + $4 = cost of 1 bbl. potatoes + S4. 
 (6) Cost of 1 bbl. apples — $1 = cost of 1 bbl. potatoes — $1. 
 (c) Cost of 6 bbl. apples = cost of 6 bbl. potatoes. 
 ,r^ Cost of 6 bbl. apples cost of 6 bbl. potatoes 
 id) 2 = 2 • 
 
 3. If a = 6, then — 
 
 (a) a + X = b -\- X. 
 (]b) a-y =h'-y, 
 
 (c) 6a = 66. 
 
 (d) a/4 = 6/4. 
 
 Transpositicm. A term in an equation may be changed from 
 one side to the other provided its algebraic sign is changed 
 from plus to minus, or from minus to plus. 
 
 Illustration. 
 Ifx+I0 = y 
 
 a; + 10 - 10 = ?/ — 10. (Subtract 10 from each side.) 
 X = y — 10. That is, the + 10 on the left has been changed to a 
 — 10 on the right. 
 
 In a similar way, the equation " a—b=c" may be written "a=c+6." 
 
 If in an equation there occurs an expression whose value 
 is not given, the value may be determined, in most cases, by 
 transposition. This is termed solving the equation. 
 
 Illustration. 
 
 Let a + 6 = 9 + 4, to find the value of a; then 
 
 a = 9 + 4 — 6 (transposing 6), and 
 
 a = 7. 
 
 In handling "signs," it is essential to have their order of 
 rank, or precedence, understood. Of the four signs, +, — , X 
 and -^ , the multiplication and division signs have precedence 
 over those of addition and subtraction. 
 
 Illustration. 
 
 Let a =« 52 — 6 X 8. The multipUcation sign is considered first, and 
 we have a = 52 — 48 = 4. 
 
 The signs X and -i- have equal weight, and terms contain- 
 ing them are reduced to simpler form by performing each 
 
48 . ' BUSINESS ARITHMETIC. 
 
 operation as its sign is met. The signs of addition and sub- 
 traction also have equal weight. 
 
 Illustrations. 
 
 9 X 8 4- 3 = 72 -i- 3 = 24. 
 
 4 + 7-3 = 11 -3 = 8. 
 
 Note. The signs of addition and subtraction refer to all following 
 values until the next plus or minus sign. Thus in"3+8X6-^2+ 8," 
 the *' + " refers to the result of "8 X 6 -r 2." The expression equals 
 3 + 48/2 + 8 = 35. 
 
 • EXERCISE. 
 
 1. Illustrate the five axioms, using the equation a + 8 = 6 — 4. 
 
 2. Find the "cost" from the equation "The cost of a house + $275 
 = $8450 - $90." 
 
 3. Find the value of 1 A. from the equation: Cost of 21 A. = $640.50 
 
 4. Solve for ar, if a; - 70 + 4 = 20 X 6 -^ 5. 
 
 5. Dividend X 8 = 64. What is the dividend? 
 
 6. Solve: 2+4-7 = aX6-^2-5. 
 
 7. A number + 16 = 128. Find the number. 
 
CHAPTER IX. 
 
 UNITED STATES MONEY. 
 
 INTRODUCTORY EXERCISE. 
 
 1. What is your own definition of a coin? 
 
 2. What denominations of coin have you handled? What others 
 can you name? 
 
 3. Read the wording on a one-dollar bill. What is the difference 
 between it and a silver dollar? What makes the paper money of value? 
 
 4. WTiat causes the difference in size between a silver dollar and a gold 
 quarter eagle? 
 
 5. WTiy should not the Government coin iron, as money, in place 
 of gold? 
 
 6. In what ways would a man be inconvenienced, who had much 
 property but absolutely no money? Could he get along comfortably 
 without money? 
 
 7. What is the disadvantage in trading one kind of property for 
 another — instead of using actual money to pay for the desired article? 
 
 One function of a modern government is to provide a system 
 of money or currency to serve as a " medium of exchange.'* A 
 primitive people may use anything as a standard, as corn, 
 wampum, horses. As commerce grows, and one who desires 
 an article cannot meet directly a distant producer who has it, 
 property must pass through several hands; the primitive 
 property standards then become difficult of use, and also 
 subject the users to loss and fraud. Then the government, as 
 a representative of all the people, adopts as standard something 
 easily transferable, and puts its mark upon it with such care 
 and in such a way that to every holder a certain quantity may 
 represent the same value. Thus gold is the standard of this 
 country, and the coming standard of the world, because high 
 values are imputed to small quantities of it, making it easy 
 of transfer, and because, when coined, exact uniformity of 
 5 49 
 
50 BUSINESS ARITHMETIC. 
 
 weight and value may be given artificially to each piece. 
 Small denominations are coined in metals of less value than 
 gold, because otherwise the coin would be too small to handle 
 and it would be easily lost. Such coin, in suflficient sums, may 
 be exchanged for gold. Likewise, principally for convenience 
 in transferring large sums, paper certificates are issued in ex- 
 change for gold and silver and are payable on demand in coin. 
 The currency now in use in the United States consists of metal 
 coins, and paper notes and certificates. 
 
 UNITED STATES CURRENCY. 
 Metal. 
 
 Gold. Silver. Nickel. 
 
 Double eagle . . $20.00 Dollar $1.00 Nickel $.05 
 
 Eagle 10.00 Half dollar 50 Brmize. 
 
 Half eagle 5.00 Quarter dollar. . .25 Penny (one cent 
 
 Quarter eagle. . 2.50 Dime 10 piece) 01 
 
 Paper. 
 
 1. Gold certificates. Denominations: $20, upward to $20,- 
 000 . Issued for gold deposited in the United States Treasury, 
 and payable in gold on demand. 
 
 2. Silver certificates. Denominations: $1.00, $2.00, $5.00, 
 $10.00, $20.00, $50.00, $100.00. Issued for silver deposited 
 in the Treasury, and payable, on demand, in the same metal. 
 
 3. United States notes (Greenbacks). Denominations: 
 $10, $20, $50, $100, $500, $1000. They are a part of the debt 
 of the United States and consist of promises to pay in coin the 
 number of dollars specified on demand. 
 
 4. National bank notes. Denominations $5, and upward to 
 $1000. They are issued by national banks, under government 
 supervision, and are redeemable in lawful money. 
 
 Note. Legal tender is a term applied to money that may be legally 
 ofiFered in payment of debts. Gold and silver coin of standard weight, and 
 gold and silver certificates, are legal tender for all debts. United States 
 
UNITED STATES MONEY. 
 
 51 
 
 and national bank notes are legal tender, except for duties on imports and 
 interest on the national debt. 
 
 The Government has five mints, or coin factories, where gold 
 or silver bullion is received for coinage, and seven assay offices. 
 The assay offices, as well as the mints, receive deposits of gold 
 and silver, and also manufacture refined bars of standard 
 purity. The bars, when returned to the depositors, are used in 
 the arts, or are exported. The metal left in bars far exceeds 
 in value the metal coined. The mints are situated at Phila- 
 delphia, San Francisco, New Orleans, Carson City and 
 Denver. The assay offices are scattered throughout the 
 country at the main points of metal production and importa- 
 tion. Some idea of the volume of business may be obtained 
 from these figures for a recent year: 
 
 Coinage of the United States. 
 
 Denomination. 
 
 Pieces. 
 
 Value. 
 
 Gold 
 
 Double eacles . • 
 
 1 495 035 
 
 1 494 795 
 
 1 559 435 
 
 142 509 
 
 $ ? 
 
 Eagles 
 
 
 Half eagles 
 
 
 Quarter eagles 
 
 
 Total 
 
 ? 
 
 1 621 700 
 
 4 422 700 
 
 20 998 431 
 
 ? 
 
 Silver. 
 
 Half dollars 
 
 ? 
 
 Quarter dollars 
 
 
 Dimes 
 
 
 Total 
 
 ? 
 
 46 047 950 
 
 89 588 480 
 
 ? 
 
 Minor. 
 
 Nickels 
 
 ? 
 
 Pennies 
 
 
 Total 
 
 ? 
 
 ? 
 
 Total Coinage 
 
 ? 
 
 ? 
 
 EXERCISE. 
 
 Find the missing values in this and the following tables. 
 
 Worn or abraded coins are collected through the banks and 
 
52 
 
 BUSINESS ARITHMETIC. 
 
 are recoined. The loss through abrasion is considerable, as 
 the following table shows: 
 
 
 Recoinage of Uncurrent Coin. 
 (Six recent years.) 
 
 
 
 Face Value. 
 
 New Coin. 
 
 Loss. 
 
 1 
 
 2 
 3 
 4 
 5 
 6 
 
 $3 832 280.69 
 3 333 437.06 
 3 008 747.98 
 2 828 384.90 
 1964 476.11 
 1 414 963.90 
 
 $3 613 021.59 
 3 141 548.04 
 2 829 890.71 
 2 656 104.21 
 1 839 219.24 
 1 322 834.27 
 
 ? 
 
 Total, 6 yr. 
 Average 
 
 ? ? 
 ? ? 
 
 
 While the total amount of money in circulation seems very 
 large, the actual amount of money in circulation per individual 
 of our population seems remarkably small, and shows to what 
 extent we carry on transactions by. means of checks and other 
 forms of credit, without actually handling currency. 
 
 Money in Circulation. 
 (Five recent years.) 
 
 Population. 
 
 84,662,000 
 86,074,000 
 87,496,000 
 88,926,000 
 90,363,000 
 
 Money in Circulation. 
 
 $2,736,646,628 
 2,772,956,455 
 3,038,015,488 
 3,106,240,657 
 3,102,355,605 
 
 Circulation 
 per Capita. 
 
 $32 32 
 ? 
 ? 
 ? 
 ? 
 
 IS a 
 
 As has already become evident, the "money value" 
 common factor in a large part of the arithmetic calculations 
 incident to our daily life, yet in the majority of cases the 
 actual calculations are based on dollars and decimals thereof — 
 and no knowledge of denominations is essential to solutions. 
 The following classification of United States money is common, 
 
UNITED STATES MONEY. 53 
 
 although the dime and eagle are usually omitted, being con- 
 sidered simply the names of coins. 
 
 Table of United States Money. 
 10 mills = 1 cent. Expressed c, ct., ^, $.01. 
 
 10 cents = 1 dime.* " d. or $.10. 
 
 10 dimes = 1 dollar. " $1.00. 
 
 10 dollars = 1 eagle. " E. 
 
 When, as a result of a calculation (as, for example, the deter- 
 mination of cost or wage) a payment of money must be made, 
 knowledge of denominations of coins and certificates becomes 
 of distinct value. A cashier, a bank teller, or a pay clerk, 
 must have extended knowledge even of issues of coins and 
 notes and of the minute details that distinguish the genuine 
 from counterfeits. Moreover, such employees must be able 
 to make change quickly and must be able to plan out and 
 arrange for the quantities of coins of different denominations 
 necessary during each day for the easy transaction of business. 
 
 In connection with the cash payments, many notes and 
 checks are handled. These, and similar papers, usually have 
 the money value stated twice, once in words, as briefly as pos- 
 sible, and once in figures. Portions of a dollar are usually 
 expressed in figures, fractionally. See page 323. 
 
CHAPTER X. 
 MAKING CHANGE. 
 
 Speed and accuracy in change making and in making money 
 payments are essential in an increasing number of positions. 
 
 Making change is a process of addition. The cashier adds 
 mentally to the amount of the purchase each coin or note he 
 takes from his tray, until he reaches the total of the sum given 
 in payment. In most cases he does not know the exact 
 amount of the change returned. 
 
 iLTiUSTRATioN. A $10.00 note is given in payment for a purchase 
 of $2.37. The cashier takes from his tray three pennies, a dime, a half- 
 dollar, a two-dollar note and a five-dollar note, saying aloud, if dealing 
 with a customer, "Thirty-eight, thirty-nine, forty, fifty, three dollars, 
 five, ten dollars." 
 
 Question. Why is this better than a subtraction method? 
 
 The careful salesman, on receiving a note, states its value 
 
 to the one who gave it to him, before sending it to the cashier, 
 
 in order to avoid argument when the change is returned. In 
 
 the above case, the salesman might remark, "You gave me 
 
 $10.00." 
 
 Question. State some simple misunderstanding that might arise from 
 neglect of this precaution. 
 
 ORAL EXERCISE. 
 
 Note. In practising change-making, name the sub-totals until the 
 amount is reached. When not reciting, watch the totals given to determine 
 the currency denominations used. Suggest other combmations. 
 
 1. Name the common United States coins, certificates and notes. 
 
 2. Make change for the foUowmg: 
 
 irchase, 
 
 . Pavment. 
 
 Purchase. Payment. 
 
 Purchase. Payment. 
 
 $5.47., 
 
 . . .$io.oo 
 
 $15.43.... $20.00, 
 
 $.01.... $.10 
 
 3 19. 
 
 ... 20.00 
 
 .08 25 
 
 .18 50 
 
 .67., 
 
 . . 2.00 
 
 1.32.... 5.00 
 
 3.21.... 3.50 
 
 .11., 
 
 . . . .50 
 
 .21 50 
 
 .83..., 5.00 
 
 1.14. 
 
 ... 5.00 
 
 .43... 2.00 
 64 
 
 2.17.... 4.00 
 
MAKING CHANGE. 55 
 
 3. In each of the above cases, name the denomination of coin or notes 
 given in payment. 
 
 4. Why should not a customer give a salesman $4.27 for a purchase of 
 $3.09? 
 
 Make change for: 
 
 5. $10.00, given in payment for two chairs @ $1.17. 
 
 6. $5.00, in payment for May gas bill, $2.53. 
 
 7. $10.00, in payment for one pair of shoes, $3.75. 
 
 8. 50c, in payment for 17 Ic stamps. 
 
 Note. Cashiers may have to break notes into smaller denominations, 
 or to make specified payments. 
 
 9. Change a $1 00 bill into nickels and dimes. 
 
 10. " Break up " a $10.00 bill. 
 
 11. Pay a check for $124, in denominations of $5.00 and under. 
 
 12. Count out $15.42 for a pay envelope. 
 
 13. Pay your milk man, $3.41. 
 
 14. Pay John Brewer $700 in Treasury notes. 
 
 15. Pay John Smith $445 in gold coin. 
 
 FOR DISCUSSION. 
 
 1. Do all businesses require, for change-making, the same proportions 
 of coins of each denomination? 
 
 2. Inquire of employees of ten distinctive businesses what coins are 
 most needed for change-making. Report to class. 
 
 3. What businesses receive more change than they need? 
 
 4. Name businesses receiving large quantities of pennies, nickels and 
 dimes. 
 
 5. Find out the proportions of coins required by a commercial bank 
 in meeting the demands of its customers for change. 
 
 6. At what seasons of the year do banks make special preparations for 
 supplying small change and bills? 
 
 7. Investigate and report on money machines, such as coin trays, 
 cash registers, change-making machines, etc. 
 
CHAPTER XI. 
 POSTAGE. 
 
 INTRODUCTORY EXERCISE. 
 
 1. Is postage a tax? 
 
 2. Are stamps used on all mail matter? 
 
 3. Why do the stamp denominations differ in part from our coin 
 denominations? 
 
 4. What government stamps are issued for other objects than the 
 direct payment of postage? 
 
 5. Name some limitations as to acceptable mail matter? 
 
 6. How does the modern postal service help business? 
 
 7. What is "rural free delivery"? 
 
 8. Why is the regular postage rate in a country the same, irrespective of 
 distance the matter is carried? 
 
 9. TMiat is the " parcels post "? 
 
 The Post Office Department is one of the greatest of business 
 organizations under government control. It is not run for 
 profit, but its enormous expenses are met, largely, by postage 
 and money order receipts. Some idea of its cost may be 
 obtained from these figures for five recent years: 
 
 Post Offices and Post Routes. 
 
 Year. 
 
 Post 
 Offices 
 Num- 
 ber. 
 
 Extent 
 of Post 
 Routes 
 Miles. 
 
 Revenue of 
 
 Department 
 
 Dollars. 
 
 Expended for Trans- 
 portation. 
 
 Total Ex- 
 penditure of 
 Department. 
 
 Defi- 
 
 Domestic 
 Mail. 
 
 Foreign 
 Mail. 
 
 cit. 
 
 1 
 2 
 3 
 4 
 5 
 
 65,600 
 62,659 
 61,158 
 60,144 
 59,580 
 
 478,711 
 463,406 
 450,738 
 448,618 
 447,998 
 
 167,932,782 
 183,585,006 
 191,478,663 
 203,562,383 
 224,128,637 
 
 72,944,352 
 
 77,471,917 
 78,174,988 
 80,901,899 
 81,709,433 
 
 2,895,756 
 2,988,849 
 2,982,732 
 2,804,170 
 3,164,254 
 
 178,449,779 
 190,238,288 
 208,351,886 
 221,004,102 
 229,977,224 
 
 
 Total 
 
 
 
 ? 
 
 ? 
 
 ? 
 
 ? 
 
 
 56 
 
POSTAGE. 57 
 
 POSTAL INFORMATION. 
 1. Domestic Mail Matter. 
 
 Postage rates apply also to mail matter for Canada, Canal Zone, Cuba, 
 Guam, Hawaii, Mexico, Philippines, Porto Rico, Republic of Panama, 
 Tutuila, and the U. S. postal agency at Shanghai. The domestic rate, 
 for letters only, applies to Germany, Great Britain and Ireland, and 
 Newfoundland . 
 
 First Class. Letters and sealed matter: 2c for each ounce or fraction. 
 Postal cards and post cards: Ic eacL Limit of weight, 4 pounds. 
 
 Second Class. Newspapers and periodicals: Ic per 4 ounces or fraction. 
 No weight Umit. Publisher's rate, Ic per pound. 
 
 Third Class. Books and miscellaneous printed matter: Ic per 2 ounces 
 or fraction. Limit of weight, 4 pounds, except for a single book. 
 
 Fourth Class. (Parcels-Post.) All matter not in other classes. Seeds and 
 plants, Ic for 2 oz. For other matter the coimtry is divided into unit 
 areas and ei^t zones. The first zone includes territory within a radhis of 
 50 miles of any unit area. The radius of the second zone is 150 miles, of 
 the third zone, 300 miles, etc. Postage rates: Under 4 ounces, Ic per 
 ounce; over 4 ounces, for delivery in first zone, 5c for first pound or 
 fraction, and 3c for each additional pound. 
 
 For each additional mile of delivery there is an increase in the rate. 
 Special stamps must be used. Limit of weight, 11 pounds. Consult your 
 postmaster as to rates and size limitations. 
 
 Special Rates. For registering for safe delivery, 10c in addition to 
 regular postage. Indemnity limit, $50. For "special delivery" or im- 
 mediate dehvery within the carrier-delivery limit of city free delivery and 
 within one mile of any other United States post oflice, 10c in addition to 
 postage. 
 
 Notes. 
 
 Matter that is harmful, libellous or threatening, which has to do with 
 lotteries or fraudulent schemes, is unmailable. 
 
 At least 2c postage must be prepaid on first class matter. All other 
 matter must be prepaid in full. Stamps are not affixed to publications 
 mailed by the publishers, which are weighed in bulk. Stamps need not be 
 affixed to third and fourth class matter consisting of at least 2000 identical 
 packages. In these cases, a special "permit" is printed on the wrapper 
 to show that postage has been paid. 
 
 Postage Stamps and Postal Cards. 
 The denominations of stamps are 1, 2, 3, 4, 5, 6, 8, 10, 15 and 50 cents; 
 and 1 dollar. Stamped envelopes are issued of different sizes, for de» 
 nominations of 1, 2, 4 and 5 cents. Postal cards are issued for Ic, single, 
 
68 BUSINESS ARITHMETIC. 
 
 and for 2c, reply. Private post cards must be within fixed limits as to size 
 and must have a Ic stamp attached. 
 
 2. Foreign Mail Matter. 
 Rates and Regulations. 
 
 Letters and Sealed Matter. 5c for first ounce; 3c for each additional 
 ounce or fraction. See Domestic Rates, for certain coimtries. 
 
 Postal Cards. 2c, single; 4c, double- 
 
 Commerical Paper. Ic for each 2 ounces or fraction; not less than 5c 
 per package. 
 
 Printed Matter. Ic for each 2 ounces or fraction 
 
 Samples of Merchandise. Ic for each 2 ounces or fraction, but at 
 least 2c per packet. 
 
 Parcels-Post. 12c per pound. There are so many conditions as to 
 such mail that it is advisable to consult your postmaster before sending 
 package. Conditions are not uniform for all countries. 
 
 (For other information see United States Official Postal Guide, or a 
 booklet on Postal Information, issued for public distribution.) 
 
 EXERCISE. 
 
 Classify this mail matter. Determine rate and cost of postage. Des- 
 tination, the United States unless otherwise stated. 
 
 1. A 2 oz. letter to London. 9. 9 oz. of seeds. 
 
 2. 40H oz. sealed ckculars. 10. 12 oz. seeds, registered. 
 
 3. 18 oz. merchandise samples H* A. 6 lb. package to Bremen. 
 
 toBerlm. 12. 3 lb. 7 oz. of merchandise. 
 
 4. Issue of 60,000 4-oz. maga- (^h-st zone.) 
 
 zines. 13. A Christmas card, IH oz. 
 
 5. 8 souvenir postals to London. 14. 4750 lb. of newspapers. 
 
 6. IH lb. of gunpowder. 15. 9 oz. of proof sheets. 
 
 7. Mdse., 3 lb. 9 oz., to Zurich. 16. A book weighing 50 oz. 
 
 8. 83^ oz. commercial paper to Paris. 
 
 17. Estimate the total postage on: 21 letters; 150 circulars, single page; 
 4 pkg. merchandise, respectively 9 oz., 21 oz., 16 oz., 13 oz.; 1 pkg. seeds, 14 
 oz.; 2 registered letters; 5 special delivery letters; 25 postals. (Pkg.— first 
 zone.) 
 
 18. Estimate the publisher's postage bill for his December issue of 
 147,456 copies, weighing 11 oz. each. 
 
 19. Compute postage on the following, and state the simplest stamp 
 denominations to use: (a) On a 4-oz. letter, by special delivery, (h) On 
 a 10-oz. re^stered package of merchandise. (Second zone.) 
 
POSTAGE. 
 
 59 
 
 (c) On a 53^ oz. sample of merchandise. (Fifth zone.) 
 
 (d) On a book weighing 47 oz. 
 
 (e) On a 3^ oz. registered letter to London. 
 
 Most business organizations, sending out hundreds or 
 thousands of pieces of mail matter per day, keep accurate 
 postal records, such as the following : 
 
 Postage Report. 
 For the month ending January 31, 190-. 
 
 Dr7 
 
 Total 
 Bought. 
 
 
 
 
 
 Departments 
 
 Total 
 
 of 
 Mo. 
 
 Balance. 
 
 Hand. 
 
 Pur- 
 chase. 
 
 Adrer- 
 tising. 
 
 Sales. 
 
 Acct'ng. 
 
 Miscel- 
 laneous. 
 
 for the 
 Day. 
 
 1 
 2 
 3 
 4 
 5 
 
 145 
 115 
 90 
 200 
 120 
 
 20 
 90 
 
 25 
 ? 
 ? 
 ? 
 ? 
 
 
 ? 
 ? 
 ? 
 ? 
 ? 
 
 
 9 
 
 4 
 
 8 
 
 6 
 
 16 
 
 16 
 90 
 52 
 10 
 
 82 
 
 7 
 5 
 
 12 
 9 
 
 13 
 
 25 
 40 
 
 82 
 75 
 40 
 
 Ill 
 
 92 
 
 47 
 
 105 
 
 116 
 
 40 
 
 85 
 32 
 86 
 45 
 
 6 
 
 3 
 
 4 
 
 8 
 
 11 
 
 15 
 27 
 52 
 21 
 06 
 
 3 
 
 1 
 
 10 
 09 
 96 
 72 
 52 
 
 ? 
 ? 
 ? 
 ? 
 
 ? 
 
 Note. At the close of the day the maiUng clerk enters under each 
 department its cost for postage. The ''Total Bought" column shows 
 each day's purchases; "Balance" shows the quantity from the previous 
 day. Together, they equal the amount on hand. The subtraction of 
 the total for the day determines the new balance. 
 
 Two forms follow, which are designed to show even the 
 denomination of stamps used, and a classification of the mail 
 matter (page 60). 
 
 On hand 
 
 Received 
 
 Total 
 
 Used 
 
 Balance 
 
 Value of bal. 
 
 Denominations. 
 
 Ic 
 
 54 
 700 
 
 ? 
 528 
 
 ? 
 
 ? 
 
 2c 
 
 110 
 500 
 
 ? 
 602 
 
 ? 
 
 ? 
 
 3c 
 
 4c 
 
 90 
 400 
 
 ? 
 128 
 
 ? 
 
 ? 
 
 5c 
 
 45 
 
 100 
 
 ? 
 
 46 
 ? 
 ? 
 
 6c 
 
 150 
 
 ? 
 
 51 
 ? 
 ? 
 
 8c 
 
 200 
 
 ? 
 
 40 
 
 ? 
 
 10c 
 
 300 
 
 400 
 
 ? 
 
 161 
 ? 
 ? 
 
 15c 
 152 
 
 50c 
 
 40 
 
 100 
 
 ? 
 
 45 
 
 ? 
 
 ? 
 
 1.00 
 
 The Postage 
 Account 
 
 On hand 
 
 Rec'd 
 
 Total 
 
 Used 
 
 Bal. 
 
 EXERCISE. 
 Complete the extension of the above forms, and of the form on page 60, 
 
60 
 
 BUSINESS ARITHMETIC. 
 
 is" 
 
 g 
 
 o 
 
 QQ 
 
 I 
 
 o 
 
 ^ 
 
CHAPTER XII. 
 
 PAYMENT FOR SERVICES. 
 
 INTRODUCTORY EXERCISE. 
 
 1. Name advantages and disadvantages of being a wage-earner. 
 
 2. What payment for services does a private householder make? 
 
 3. Name typical cases in which compensation is spoken of as a wage 
 As a salary. 
 
 4. What name is given the payment of a lawyer? Of a physician? 
 How do these payments differ from wages or salary? 
 
 The modern organization of business demands a careful 
 accounting of all sources of expense. Of these, the cost of 
 labor is a leading item and requires special attention. Some 
 idea of the sums paid for services may be obtained from the 
 following recent Census figures for a few selected industries: 
 
 
 No. of Estab- 
 lishments. 
 
 Wage-Earners. 
 
 Ayerage 
 Wage. 
 
 Industry. 
 
 Av. Num- 
 ber. 
 
 Wages. 
 
 Agricultural implements 
 
 648 
 1316 
 18227 
 4956 
 4504 
 9423 
 
 605 
 624 
 
 47,394 
 
 149,924 
 
 81,284 
 
 60,722 
 
 137,190 
 
 402,914 
 
 242,640 
 79,601 
 
 $25,002,650 
 69,059,680 
 43,179,822 
 30,878,229 
 57,225,506 
 
 229,869,297 
 
 141,426,506 
 26,767,943 
 
 $? 
 ? 
 
 Bakery products 
 
 Carriages, etc '. 
 
 ? 
 ? 
 ? 
 
 
 ? 
 
 General iron and steel 
 works 
 
 ? 
 
 Silk, gilk goods 
 
 ? 
 
 Methods of payment vary, and the calculations involved are 
 often so numerous and technical as to rank this class of com- 
 putation as an independent subject. With millions of wage- 
 earners, it is evident that the subject affects, directly or in- 
 directly, our entire population. 
 
 61 
 
62 BUSINESS ARITHMETIC. 
 
 In business, the recompense paid, or stipulated to be paid, 
 to persons, at regular intervals, for services rendered, is called 
 salary or wage. In general, a stipend based on long intervals 
 (i. e., year or month) is called a salary. Recompense reckoned 
 by brief intervals (i. e., week, day, or hour) is usually called a 
 wage. There are many exceptions to this classification. In 
 some businesses, payment for distinctively mental labor is 
 termed salary and manual labor wage. Thus a clerk in an 
 office receives a salary, while a high-class mechanic, working at 
 a machine, may receive a far higher wage. 
 
 In manufacturing establishments payment for services is 
 frequently based on the quantity of work done. This work is 
 called " piece work." 
 
 Salaries. 
 
 Salaries are usually paid monthly or semi-monthly. The 
 rate is commonly expressed in dollars per year. A month is 
 considered one-twelfth of a year, but in many cases, pay for 
 a fraction of a month is based on the exact number of days in 
 the month. (How would one day's earnings in February 
 compare with one day's in January?) Payments are usually 
 made by check or in cash. Sometimes employees are credited 
 on the books and may draw funds as needed. Usually a pay 
 roll is prepared and is signed by each person when he receives 
 payment. A pay roll is shown on the opposite page. 
 
PAYMENT FOR SERVICES. 
 
 63 
 
 o 
 
 CO 
 
 6 6 ^ o ^ ^ o 
 '"'*"' o '"' o o '"' 
 
 a> 
 
 Sis fe 2^ 
 2 q3 2 
 
 o3 o g 
 £ > ^ c^ H 
 
 u 
 
 o 
 
 i I a 
 
 a ^ o 
 
 ai rt Eh OJ r! 
 
 ^ 9 fl a E-< 
 
 P^ H? W 4 O 
 
 •c.a 
 
 c- o 
 
 gl 
 
 § 
 
 8 
 
 05 05 05 05 
 
 
 11 
 
 5 
 
 r 
 
 CO i-H 1 
 
 CO T^ 1 
 
 
 i 
 
 i 
 
 CO i-iO 
 
 <N- 
 
 l1 
 
 5^ 
 
 1 
 
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 s 
 
 1—1 
 
 s 
 
 1— ( 
 
 a et 
 
 5s 
 
 II 1 1 III 
 
 1 
 
 1 
 
 11 ill 
 
 I- 
 
 
 0) 
 
 
 (N CO rt< lO 
 
64 BUSINESS ARITHMETIC. 
 
 EXERCISE. 
 Reckon salaries for June, in the following cases: 
 
 1. Annual salaries of $1200, $1600, $900, $2500, $450, $840. 
 
 2. Promoted from $1200 to $1300 on June 20. 
 
 3. Resigned from a $900 position to take effect on June 4. 
 
 4. Appointed at $960 to take effect June 4. 
 
 5. Rule up and extend the March pay roll for the general oflfice force 
 of a factory. Assume salaries and positions. Use names of pupils in 
 the class. Have the pupils sign the pay roll. See that signatures are 
 correct. Illustrate promotions, dismissals, appointments and "salaries 
 advanced." 
 
 6. Prepare a similar pay roll for the teaching force of a school. 
 
 7. Special topic. (For individual pupils.) Secure blank forms of pay 
 rolls that are in use in typical establishments in your neighborhood. 
 Submit a report showing how the forms are adapted to special needs, and 
 illustrating any unusual entries or calculations. 
 
 Wages. 
 
 Wages are usually paid by the week and are quoted at a 
 week, day or hour rate. Some workers are paid by the piece 
 system, the amount of wage depending on the amount of work 
 done. In other cases, a boniis of some form is added for extra 
 efficiency. Many establishments pay one and one-half, or 
 double rates, for overtime and necessary Sunday work. 
 
 In many small organizations, the pay rolls in use contain a 
 receipt clause and a column for signature. In large organiza- 
 tions, the pay roll is simply a record sheet. Since relatively 
 few workmen keep business bank accounts, employees are paid 
 in cash, the pay envelopes being prepared beforehand; or in 
 orders on the treasurer of the organization, payable on pre- 
 sentation. A more complicated pay roll is shown opposite. 
 
PAYMENT FOR SERVICES. 
 
 65 
 
 ^ 
 
 j; 
 
 
 -d 
 
 o3 
 
 a 
 
 03 
 
 <v 
 
 rC3 
 
 OP 
 
 
 f$ 
 
 
 
 n 
 
 (U 
 
 M 
 
 •5 
 
 M 
 
 ^ 
 
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 •c 
 
 ^ 
 
 73 
 
 (^ 
 
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 rg 
 
 V 
 
 s 
 
 nS 
 
 5 
 
 p3 
 
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 Q) 
 
 cq 
 
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 ^ 
 
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 S 
 
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 fi 
 
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 fs 
 
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 hf 
 
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 i 
 
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 c .S . 
 
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 ^ .^ -►^ 
 
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 ^Q 
 
 1=1 
 
 « So 
 
 1' 
 
 bS 
 
 HS 
 
 
 
 Si 
 
 00 
 
 
 s p^ 
 
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 4r?i^«4 
 
 s 
 
 c-e^c- c^ 
 
 IS 
 
 ":»<N 
 
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 OO lO -*(NQ000(NOO 
 
 iOTt< "^ oo-^eoeoeococo 
 
 00 CO GO C^» O 00 00 •«*' TjH c^. c^. 
 
 TjH "^ 1-1 TjH •^ (N C^5 
 
 OOOO 00 
 
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 v^ w ^v^^ 
 
 
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 CO l>00050 r-(C^ cc^ cof;- 
 
66 
 
 BUSINESS ARITHMETIC. 
 
 From the "time in hours" and "remarks" columns explain how each 
 person worked, and how his wage was determined? {Note. In the 
 above case, Sunday time and overtime is paid at double rates.) 
 
 Do you notice anything unusual in the signatures? 
 
 ORAL EXERCISE. 
 
 Reckon wages for the week ending March 9, as follows: 
 
 1. 36 hr. at 15c; 42 hr. at 18c; 48 hr. at 20c; 47 hr. at 27c; 39 hr. at 19c; 
 42 hr. at 30c; 54 hr. at 33ic; 46 hr. at 25c. 
 
 2. Reckon wages in the following cases, basing calculation on a 54 hour 
 week, with double pay for overtime: 42 hr. at 15c; 54 hr. at 30c; 54 hr. 
 at 32c; 48 hr. at 17c; 60 hr. at 25c; 56 hr. at 30c; 45 hr. at 35c. 
 
 EXERCISE. 
 
 Rule up and extend a pay roll for the current week for some shop, 
 factory or contracting firm. Add to the pay roll a column to show work- 
 man's position or department. Follow directions in example 5, page ^. 
 
 Schedule of Change Desired. 
 
 By 
 
 Address 
 
 In 1-cent pieces ^ 
 
 " 5-cent " ■ 
 
 " 10-cent " • 
 
 " 25-cent " • 
 
 " 50-cent " • 
 
 " Silver Dollars 
 
 1 
 
 1 
 
 2 
 
 14 
 
 14 
 40 
 60 
 75 
 50 
 
 In Notes or Certificates- 
 Si 
 
 41 
 
 36 
 
 20 
 
 100 
 
 
 2 
 
 
 ' 5 
 
 
 io 
 
 
 20 
 
 
 50 
 
 
 
 100 
 
 
 
 
 
 
 Total S 
 
 217 
 
 39 
 
 When the pay envelope method is used the pay clerk com- 
 putes, or if he is an expert he approximates, the quantity of 
 each coin denomination required to enable him to *'make- 
 
PAYMENT FOR SERVICES. 
 
 67 
 
 up" each employee's exact "amount due." Sometimes this 
 coin classification is incorporated in the pay roll. Often 
 a currency memorandum similar to that shown opposite is 
 prepared, to be presented at the bank when the check for 
 the total of the pay roll is cashed. 
 
 EXERCISE. 
 
 1. Prepare a denomination slip for this series of payments: $4.78, $9.62, 
 $11.21, $11.45, $12.00, $12.65, $13.11, $14 29, $14.43, $15.80. 
 
 2. Prepare a currency memorandum for the pay roll shown on page 65. 
 
 3. Prepare a wage pay roll and currency slip for a contractor's force 
 for the current week. The force consists of 1 foreman, 4 drivers, and 17 
 loaders. One man is discharged and two are hired during the week; one 
 is out sick for one day. 
 
 The pay clerk who is obliged to handle a large pay roll 
 based on fixed rates, often buys or computes wage tables as 
 aids to rapid work. These give the wages due for any number 
 of hours or half hours, at given rates per week. The original 
 computations are made with absolute accuracy, and are then 
 written in the table to the nearest cent. 
 
 EXERCISE. 
 The Rankin M.^j^ufacturing Company. 
 
 $12.00 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 48 hours. 
 
 Hour. Wage. Hour 
 
 Wage. Hour. 
 
 Wage. Hour. Wage. Hour. Wage. Hour. Wage. 
 
 k 
 
 13 
 
 8^ 
 
 2 
 
 13 
 
 16i 
 
 4 
 
 13 
 
 24i 
 
 6 
 
 13 
 
 32i 
 
 8 
 
 13 
 
 m 
 
 10 
 
 13 
 
 1 
 
 25 
 
 9 
 
 2 
 
 25 
 
 17 
 
 4 
 
 25 
 
 25 
 
 6 
 
 25 
 
 33 
 
 8 
 
 25 
 
 41 
 
 10 
 
 25 
 
 1* 
 
 i38 
 
 9^ 
 
 2 
 
 38 
 
 17^ 
 
 4 
 
 38 
 
 25^ 
 
 6 
 
 38 
 
 33^ 
 
 8 
 
 38 
 
 4U 
 
 10 
 
 38 
 
 2 
 
 
 50 
 
 10 
 
 2 
 
 50 
 
 18 
 
 A 
 
 50 
 
 26 
 
 6 
 
 50 
 
 34 
 
 8 
 
 50 
 
 42 
 
 10 
 
 50 
 
 2^ 
 
 
 63 
 
 10^ 
 
 2 
 
 63 
 
 18^ 
 
 4 
 
 63 
 
 26^ 
 
 6 
 
 63 
 
 34-^ 
 
 8 
 
 63 
 
 42-^ 
 
 10 
 
 63 
 
 3 
 
 
 75 
 
 11 
 
 275 
 
 19 
 
 
 75 
 
 27 
 
 6 
 
 75 
 
 35 
 
 8 
 
 75 
 
 43 
 
 10 
 
 75 
 
 3^ 
 
 
 88 
 
 IH 
 
 2i88 
 
 19^ 
 
 
 88 
 
 27-^ 
 
 6 
 
 88 
 
 35i 
 
 8 
 
 88 
 
 43^ 
 
 10 
 
 88 
 
 4 
 
 100 
 
 12 
 
 300 
 
 20 
 
 5 
 
 00 
 
 28 
 
 7 
 
 00 
 
 36 
 
 9 
 
 00 
 
 44 
 
 11 
 
 00 
 
 4^ 
 
 1 
 
 
 1 
 
 20^ 
 
 
 
 
 
 
 
 
 
 
 
 
 1. Complete the above table. (The upper half only is given.) 
 
 2. What error would result from constructing the table by com- 
 putations based on the approximate value for the first half hour? 
 
68 
 
 BUSINESS ARITHMETIC. 
 
 3. Read off a man's wage, at the given rate, for 37 hours, 21 hr,, 13 hr., 
 12 >^ hr., 19 hr., 363^ hr., 50 hr. (double), 52 hr. 
 
 4. How might the table be used for a wage rate of $6.00 per week? 
 Of $18.00? 
 
 5. Construct a table for a wage of $15.00 and a week of 54 hr. 
 
 6. Construct a 48 hour wage table showing overtime at double rate up 
 to 60 hours. 
 
 Teacher's Note. Assign each pupil a different rate. Have tables 
 prepared and exchanged for testing. 
 
 In some factories and large stores time clocks are used to 
 record the arrival and departure of employees. In some 
 systems, the employee inserts his key in the clock, thus auto- 
 matically recording his number and the time. In other 
 systems, the employee inserts a card in the clock, the time 
 being stamped automatically. Generally, the form is used 
 to show whether or not the employee is on time — small frac- 
 tions of hours being ignored. In other 
 organizations, a few moments* tardiness 
 may cause the docking of the employee 
 to the extent of one-half, or one hour. 
 
 EXERCISE. 
 
 1. Compute the cost of labor on the piece of 
 work shown by ticket A. Reckon to the nearest 
 one-fourth hour. The rate given is the rate per 
 week. 
 
 2. Compute the time of eight men for Mon- 
 day, and determine the wages due at 31 ^c per 
 hour: 
 
 FM>r 
 
 kCT 
 
 OYE 
 
 CRY EXPENSE 
 .0. ^7S 
 
 
 WOR 
 
 f €2e-^ 
 
 
 DATE 
 
 V27 
 
 
 
 1 
 
 DAY 
 
 IN 
 
 OUT 
 
 IN 
 
 OUT 
 
 rOTAl 
 
 s 
 
 A.M 
 
 
 
 
 
 
 P.M 
 
 
 
 
 
 
 s 
 
 A.M 
 
 
 
 
 
 
 P.M 
 
 
 
 
 
 
 M 
 
 A.M. 
 
 6A« 
 
 11^1 
 
 
 
 
 P.M. 
 
 2253 
 
 5«2 
 
 
 
 
 T 
 
 A.M. 
 
 6l» 
 
 &*2. 
 
 11^1 
 
 V>J1 
 
 
 P.M. 
 
 
 -. 
 
 
 
 
 W 
 
 A.M. 
 
 :^c^ 
 
 wiA 
 
 ed 
 
 
 P.M. 
 
 
 
 
 
 
 T 
 
 A.M. 
 
 
 
 
 
 
 P.M. 
 
 
 
 
 
 
 F 
 
 A.M. 
 
 
 
 
 
 
 P.M. 
 
 
 
 
 
 
 p 
 
 //S^ff 
 
 fiOUT ' 
 
 
 Ticket A. 
 
 No. 
 1 
 2 
 3 
 
 4 
 5 
 6 
 
 7 
 8 
 
 In. 
 6:59 
 7:01 
 6:58 
 6:59 
 
 6:58 
 6.56 
 6:50 
 
 Out. 
 12:01 
 12 .02 
 11.59 
 10:01 
 
 12:00 
 12:01 
 12:02 
 
 In. 
 
 1:00 
 
 1:01 
 12:58 
 
 2:00 
 12:59 
 12:58 
 12:57 
 12:55 
 
 Out. 
 4:59 
 5:03 
 5.01 
 5:02 
 5:03 
 3:02 
 4:59 
 5:04 
 
PAYMENT FOR SERVICES. 
 
 69 
 
 Premiums or Bonus Work. The premium plan of labor 
 payment is intended to reduce the time of production in 
 factory work, by the payment of a bonus for extra efficiency. 
 Usually, as a result of hundreds of experiments, a fixed average 
 time for a certain operation is determined. The employee 
 is then told that his employer will share with him any saving 
 in working time as compared with this average. For example, 
 the offer might be "one-half of the wage saved." If the 
 average time to make an article is 10 hours, and the wage 
 $4.00, the employee earns 20 cents extra for eVery hour saved. 
 The table below shows the effect of the plan. 
 
 Time Con- 
 sumed in 
 Hours. 
 
 Wage per 
 Piece. 
 
 Premium. 
 
 Total Cost of 
 Work. 
 
 Workman's 
 Hr. Rate. 
 
 Workman's 
 Day Rate. 
 
 10 
 9 
 
 8 
 7 
 6 
 
 $4.00 
 
 3.60 
 
 3.20 
 
 2 80 
 
 ? 
 
 .20 
 
 .40 
 
 ? 
 
 9 
 
 $4.00 
 3.80 
 3.60 
 ? 
 
 $.40 
 
 .422 
 
 .45 
 
 ? 
 
 ? 
 
 $4.00 
 
 4.22 
 
 4.50 
 
 ? 
 
 ? 
 
 Note. Often a set number of pieces or operations is required per hour, 
 and any increase in quantity is paid for at a special rate. Often, if the 
 wage is stated as a price per piece, the wage rate is increased for a quantity 
 over a certain fixed number. 
 
 EXERCISE. 
 
 1. Complete the above form. 
 
 2. Prepare a similar table showing one-half hour time reductions from 
 6 hours to 4 hours, the 6-hour standard rate being $3.60. 
 
 3. A workman, whose wage is 38c per hour, reduces his working time 
 on a stove from 4^ to 3 hours, earning a premium of 16c per hour. He 
 raises his wage for a 9-hour day to $ — . 
 
 4. The standard number of pieces required of a polisher per hour is 12. 
 The rate is 5c each, or 6c for extra efficiency. He averages 13^ per hour, 
 for an 8-hour day, earning $ — . 
 
 Bills for Professional Service. 
 Professional men, as doctors, and lawyers, and many inde- 
 pendent artisans, charge fees for their work, and render hills 
 for services. As a rule the computations are very simple. 
 
70 BUSINESS ARITHMETIC 
 
 B. C. Student & Co., 
 No. 848 Payne Ave., 
 
 To Charles Rawlingson, Dr., 
 Attorney at Law. 
 
 Lawyer's Bill. 
 
 Cleveland, Ohio, Dec. 21, 19—. 
 
 For professional services rendered in suit against Salton & 
 
 Bro. 
 Court Costs, 
 Stenographer, 26 hr. at 30c 
 
 Note. Bills for the service of an agent are considered in the chapter 
 on Agency. 
 
 EXERCISE. 
 
 1. Draw up a lawyer's bill for services. Fee, $275; court costs, $30; 
 stenographer, 17 hr. at 32c; registration fee, $3.50. 
 
 2. Draw up a cabinet maker's bill for remodelUng an antique side- 
 board. Material, $1.75; delivery, 50c; labor $ — . 
 
 3. Draw up an original bill for services. 
 
 4. From the doctor's memorandum card illustrated on the following 
 page, showing house treatments (numerals), and office treatments (O), 
 draw up a bill. The rates are $2 and $1 respectively, for house treat- 
 ments and office visits. Supply names. 
 
PAYMENT FOR SERVICES. 
 
 71 
 
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CHAPTER XIII. 
 
 BUSINESS TERMS AND ACCOUNTS. 
 
 All business is an exchange of one kind of value for anothei 
 kind. The values exchanged may be general property, funds, 
 or services. In this connection, the business man employs 
 two terms: (1) Debit, as an expression of value received or 
 due; and (2) credit, an expression of value delivered or owed. 
 
 Illustrations. (1) A dealer gives $200 (a credit) for a horse (debit). 
 (2) The dealer pays his clerk $20 salary. The debit, or value received, 
 is services; the credit is $20 cash. (3) He buys 40 bbl. of flour from Norton, 
 giving his note for 30 days. The debit is merchandise;^ the credit a note 
 payable. Notice that in every transaction the debits equal the credits. 
 
 Accounts are collections of debits or credits relating to a 
 particular person or thing. Thus the Cash Account is a record 
 of all cash received or paid out. Other accounts with which 
 the average person meets are: Merchandise, Expense and 
 Personal. 
 
 If, in the course of business, a transaction, or a series of 
 transactions, produce more than they cost, a gain results; 
 if the sum spent exceeds the returns, a loss results. The 
 excess of gains over losses, for an entire business, or for a 
 certain series of transactions, is termed the net gain; the 
 excess of losses over gains is a net loss. Any property owned 
 by, or owed to, a person is termed his resource; any amount he 
 owes is termed his liability. If resources exceed liabilities, 
 a person has a net capital, or present worth, equal to the differ- 
 ence. If liabilities are greater, the excess represents his net 
 insolvency. Properly kept accounts show the resources, lia- 
 bilities, losses and gains of a business. 
 
 The Cash Account shows the debits and credits of cash, the 
 
 72 
 
BUSINESS TERMS AND ACCOUNTS. 
 
 difference, or balance, representing the amount of cash 
 hand, as follows: 
 
 Cash, 
 
 73 
 
 on 
 
 19— 
 
 RECEIVED 
 
 
 
 
 19- 
 
 
 2000 
 
 — 
 
 €ct. 
 
 18 
 
 Vfi 
 
 — 
 
 
 20 
 
 485 
 
 60 
 
 
 24 
 31 
 
 2625 
 
 60 
 
 PAID 
 
 * nee 
 
 30 
 
 m 
 
 2625 60 
 
 Note. The halnnce is entered on the smaller side, so that the sides 
 "total" the same. The source of receipt or payment is sometimes entered 
 in the explanatory space. 
 
 The Merchandise Account shows the cost of merchandise 
 bought (debits) and the receipts from sales (credits). In 
 general, explanations are not written. Here, however, they 
 show, among other things, the people who owe, or are owed, 
 for merchandise. The balance shows a loss, if the debit side 
 exceeds the credit, or a gain, if the credit side is in excess. 
 See below. 
 
 Merchandise, 
 
 19- 
 
 MDSE. REC D. 
 ^oodi. on, Aa/nc6 
 
 5.K>. So. ^ane 
 
 i6\ifr^. ^ou^ 
 
 18,^i^uA, 
 3lf^aMt 
 
 
 
 19— 
 
 
 845 
 
 20 
 
 J^' 
 
 5 
 
 278 
 
 40 
 
 
 8 
 
 840 
 
 60 
 
 
 15 
 
 500 
 
 — 
 
 
 29 
 
 ??? 
 
 — 
 
 
 31 
 
 2683 
 
 SO 
 
 MDSE. DELIVERED. 
 
 
 40 
 120 
 
 H 
 
 2146 
 
 50 
 
 60 
 20 
 
 30 
 
 Note. If unsold goods are on hand their cost value (inventory) is 
 entered on the credit side. This is equivalent to subtracting the inventory 
 from the debit side, in order to find the cost of the goods sold, and thus 
 to determine the gain. 
 
 The Expense Account, as illustrated on the following page, 
 shows the cost of running a business: 
 
74 
 
 BUSINESS ARITHMETIC. 
 Expense, 
 
 19— 
 
 
 
 
 
 19— 
 
 
 
 
 ~ 
 
 J5^. 
 
 2 
 
 ^o^la^ 
 
 ^^ 
 
 20 
 
 ^l 
 
 28 
 
 ^'n/tten/c^tt^ 
 
 2 
 
 m 
 
 
 6 
 
 "^oa/ 
 
 SO 
 
 — 
 
 
 
 {^43S^Cl^) 
 
 
 
 
 15 
 
 ^^^ 
 
 60 
 
 — 
 
 
 28 
 
 ,^'n/v^9tto'?^ 
 
 8 
 
 50 
 
 
 21 
 
 C^^'yleA. 
 
 38 
 
 50 
 
 
 
 {cc^) 
 
 
 
 
 26 
 
 J^^^ 
 
 12 
 
 47 
 
 
 28 
 
 %^S 
 
 m 
 
 
 
 156 
 
 17 
 
 156 
 
 17 
 
 A Personal account shows the amounts owed to, or by, the 
 person whose name appears in the title. 
 
 Henry Brown, 1456 K St., N, W. 
 
 19— 
 
 ^Md^. 
 
 ^Mc{&e. 
 
 240 
 165 
 
 — 
 
 19— 
 
 10 
 10 
 
 25 
 
 405 
 
 — 
 
 176 
 
 50 
 
 176 
 
 50 
 
 
 5 
 JfiO 
 
 105 
 
 48 
 
 'i'i 
 
 Note. Whenever the two sides of a personal account are equal, it is 
 ruled up to show no balance or debt. If debits exceed credits the person 
 named owes the firm the balance; if the credits are in excess the firm owes 
 the person named. 
 
 The Proprietor's or Partner's Account shows his relation to 
 the business. The credits usually represent investments 
 and gains; the debits show the losses and withdrawals. A 
 credit excess represents the present worth, or net capital; a 
 debit excess, the net insolvency. A proprietor's account is 
 shown on the next page. 
 
BUSINESS TERMS AND ACCOUNTS. 
 S. p. Mason, Proprietor. 
 
 75 
 
 19— 
 
 11 
 
 
 85 
 
 f 
 
 60 
 
 19— 
 >-- 
 
 1 
 1 
 
 31 
 
 
 12865 SI 
 
 
 JfiOO 
 80G0 
 
 865 
 
 12865 
 
 ORAL EXERCISE. 
 
 1. What are debits? Credits? WTiat is an account? 
 
 2. When is an account debited? When credited? 
 
 3. What sum of cash was received, according to the cash account? 
 What sum paid out? What sum was on hand on October 21? For what 
 was cash paid out during October? 
 
 4. From the Merchandise account it is evident that, $ ? worth of 
 merchandise was bought during January, and that, $ ? was sold. $ ? 
 was sold for cash; $ ? for a note; and $ ? to be paid for later by ?. 
 
 5. Henry Brown's account shows that he bought merchandise on what 
 dates? For which purchases has he paid in full? For which has he 
 evidently paid in part? 
 
 6. 
 
 8. 
 $2500? 
 
 Name the debit and the credit in each of the following transactions: 
 (a) The fii-m buys merchandise for cash, $200. 
 (6) The firm sells A. B. Bolton, merchandise, $400, for which he 
 
 agrees to pay in ten days, 
 (c) The firm pays cash for office furniture, $120. 
 {d) The firm receives cash for some real estate. (Real Est. Ac.) 
 (e) The firm pays cash in settlement of a gas bill. 
 
 Classify the followdng as resources, liabilities, losses and gains: 
 (a) The balance of the Cash Account. 
 
 (6) A personal account with debits of $500 and credits of $400. 
 (c) Merchandise account with debits, .$600, credits, $700. No 
 
 goods on hand. 
 id) Expense account with debits of $46.50 and credits of $13.00. 
 (e) Merchandise on hand, $125. 
 
 What is the present worth, if resources are $12,800 and* liabilities 
 
 9. What is the net loss if gains are $560 and losses $764? 
 
76 
 
 BUSINESS ARITHMETIC. 
 
 EXERCISE. 
 
 1. Copy the Cash Account on page 73 through Oct. 24. Add these 
 entries: Oct. 25, received for merchandise, $250; Oct. 26, paid for horse 
 and wagon, $185; Oct. 27, lost $2; Oct. 29, received from C. B. Drew, 
 $89.60. Balance the account. 
 
 2. Alter the Merchandise Account as follows: Goods unsold on Jan. 31, 
 $2003.15; amount paid by A. B. Brown, Jan. 15, $36.40. Balance the 
 account. 
 
 3. Find the balance again, assuming that all goods had been sold. 
 What does the account show? 
 
 4. Write an account for D. Poynter, Proprietor, showing invest- 
 ments on Jan. 1, of cash, $2500; merchandise, $3867; and real estate, $4500. 
 Withdrawals, Jan. 10, $265 cash; Jan. 18, $1645 in merchandise. Net 
 gain, Jan. 31, $211.46. 
 
 NAME 
 
 ADDRESS 
 
 t^Hewi^Ae^ ^e^t9t. 
 
 FILE NO. 
 
 19— 
 
 16 
 21 
 
 ^«fe^ ^ 3620 
 4125 
 
 360 
 
 — 
 
 91- 
 
 17 
 31 
 
 30 d. rto£i^ 
 
 360 
 
 214 
 
 — 
 
 
 214 
 
 85 
 
 85 
 
 
 30 
 
 4356 
 
 561 
 
 40 
 
 J^.' 9 
 
 " 
 
 245 
 
 — 
 
 J^. 
 
 3 
 
 ■ 4581 
 
 311 
 
 27 
 
 20 
 
 60 d. rvole 
 
 900 
 
 — 
 
 
 19 
 
 4936 
 
 1216 
 
 03 
 
 
 20 
 
 SS^ ccoA, 
 
 316 
 
 03 
 
 
 21 
 
 26 
 
 6011 
 5203 
 
 1008 
 65 
 
 60 
 
 40 
 
 
 25 
 28 
 
 
 17 
 f 
 f 
 
 25 
 
 
 f 
 
 
 
 
 (a) Brewer still owes $ ? 
 
 (6) Between what dates did Brewer owe nothing? 
 
 (c) Between what dates did he owe the least amount? 
 
 {d) How do his cash payments compare with his payments by note? 
 
 (c) Explain the credit item for Feb. 25. 
 
CHAPTER XIV. 
 
 ADVERTISING. 
 
 Modern advertising, or solicitation through public notice, 
 really dates from the insertion of such notices into the maga- 
 zines in 1864. In that year, the most prominent monthly con- 
 tained, in its October number, three and one quarter pages. 
 During this period advertising rates rose enormously, as did 
 the number and circulation of periodicals. Advertising ex- 
 tended in heavy volume to newspapers, and then to bill boards, 
 complicated catalogues, street car signs, show window dis- 
 plays and demonstrations. In a recent year it was estimated 
 that $600,000,000 was spent in these main sources of adver- 
 tising. 
 
 Practically everyone reads advertisements; and many people, 
 regularly or on some special occasions, advertise. The cost 
 of advertising is one of the heaviest expenses of many a modern 
 business. The methods of computing costs and returns, how- 
 ever, are based on the simplest arithmetical processes. 
 
 INTRODUCTORY EXERCISE. 
 
 1. What is advertising? 
 
 2. Name at least five ways of advertising merchandise in print. 
 
 3. Name other ways of advertising. 
 
 4 What is a trade mark? Is it a form of advertising? 
 
 5. What effects have climate, season and prosperity on advertising 
 matter? 
 
 6. Is the paper of largest circulation necessarily the best advertising 
 medium? 
 
 7. What is the value of a "catch phrase" or title? 
 
 8. Why do not local dealers advertise in magazines? 
 
 77 
 
78 BUSINESS ARITHMETIC. 
 
 9. Does the character of newspaper advertising differ from that oi 
 magazines? 
 
 10. Can you name any article brought into general use by advertising? 
 
 11. Some advertisements give no dealer's name. Why? 
 
 12. What need has the salaried man or private householder to adver- 
 tise? 
 
 13. Special, Bring to class specimen advertisements, securing cost of 
 insertion, where possible. 
 
 Wall Signs, for advertising purposes, are painted on blank 
 walls of buildings, fences, bulletins, etc. Contracts are usually 
 for the year, including painting, and exclusive of the vary- 
 ing commissions or fees for rental. For bulletins, the price is 
 quoted per running foot. 
 
 EXERCISE. 
 
 1. A manufacturer of breakfast foods rents for $200, a wall space of 
 40 ft. by 24 ft., and pays 6c. per sq. ft. for the sign. Find the entire cost. 
 
 2. Lawton & Co. rent spaces as follows: 26' by 20' for $250; 20' by 18' 
 for $4 per sq. ft. ; 32' by 21' $2.50 per sq. ft. The charge for placing signs 
 is $4.25 per sq. ft. What is the entire cost? 
 
 3. I contract for six months, for a 20' sign on a standard bulletin board 
 (10' high) at 45 c per running foot per month. What is the cost of ad- 
 vertising? 
 
 4. What is the cost of a one-year contract for 60 feet of bulletin space 
 at 55 c per foot per month? 
 
 Posters for advertising purposes are shown on bill boards. 
 These boards are designed to accommodate posters four sheets 
 high and one to six sheets wide. Complete posters are spoken 
 of as 4s, 8s, 16s, etc., according to the number of sheets. 
 The sheets measure 28 in. high by 42 in. wide. A bill-poster's 
 space is rented by the month of four weeks, or the year of 
 forty-eight weeks. Prices are usually quoted for four weeks' 
 display. 
 
 EXERCISE. 
 
 1. Find the dimensions in feet and inches of 4, 8, 12 and 16 sheet 
 posters. 
 
ADVERTISING. 79 
 
 2. Allowing for 1/10 rene\\:als of damaged sheets, how many sheets are 
 required for 450 24-sheet posters? 
 
 3. To post 100 16-sheet posters for ten weeks cost a cereal company 
 16 c per sheet per month, or $ ?, 
 
 4. Extend the following: 
 
 Estimate of the Cost of Two Thousand 16-sheet Posters. 
 
 Number required 2000 
 
 For renewal, 1/5 ? 
 
 For 16 s. at $269 per M $ 
 
 Freight and drayage 26.45 
 
 To be posted 12 weeks 
 
 560 in small towns at 14 c $ 
 
 1200 in specified cities at 16 c 
 
 240 in San Francisco at 12 c 
 
 $ = 
 
 Street Car Posters are printed on cards, usually 1 1 in. by 
 21 in., though some of other size are used. The rates 
 average about 40 c per card per month, on yearly contracts, 
 45 c on contracts for six months, and 50 c on three months 
 contracts. 
 
 EXERCISE. 
 
 1. The manufacturer of a toilet soap contracts for 3500 cards at $47.50 
 per M., paying S18.50 for the design, and $16.00 for the necessary plates, 
 a gross cost of $ . 
 
 2. He distributes 3/5 of his supply (on six months contracts) throughout 
 Middle West cities at prices given above. This cost him $ ? 
 
 3. The balance he distributes in Pennsylvania and New York on a one 
 year contract, at the above rate, except for 80 for which the price is 35 c. 
 Find the cost of the year contract? 
 
 4. Prepare a statement showing the entire cost to the manufacturer. 
 
 Newspaper and periodical advertising absorbs the larger 
 portion of the money spent on publicity. Periodical adver- 
 tising is usually the more costly. The common measure of 
 space is the agate line, of which fourteen equal one inch of 
 column space. One periodical charges $6.00 per insertion 
 per agate line, or $4000 for a full page of 9f '' by 14j". 
 
80 BUSINESS ARITHMETIC. 
 
 For a standard magazine page, 5^" by 8", rates may run up 
 to $500 per insertion. 
 
 Extensive general advertisers usually engage agencies to 
 prepare copy, select publications, and arrange for insertion. 
 The agency is paid either a lump sum or a commission. In 
 some cases the commission is paid by the publisher. 
 
 EXERCISE. 
 
 1. yxom the following table find to two decimal places the average 
 number of copies issued for each member of our population. 
 
 Newspaper and Periodical Circulation. * 
 
 (Round numbers). 
 
 Year Population Gross Circulation Copies per capita. 
 
 1850 23,000,000 426,000,000 ? 
 
 1860 31,000,000 928,000,000 ? 
 
 1870 39,000,000 1,509,000,000 ? 
 
 1880 50,000,000 2,068,000,000 ? 
 
 1890 63,000,000 4,681,000,000 ? 
 
 1900 76,000,000 8,168,000,000 ? 
 
 1910 92,000,000 13,105,000,000 ? 
 
 Find necessary values for the following : 
 
 2. The recent holiday numbers of ten standard magazines contained 
 respectively, 192, 181, 175, 171, 150, 150, 148, 193, 138, and 127 pages of 
 advertising, a total of pages, and an average of pages. 
 
 3. In the same issue, the agate lines of advertising were 43,776, 41,496, 
 39,900, 39,047, 34,980, 34,392, 34,088, 33,306, 32,172 and 27,132 Imes. 
 Total lines . Average lines . 
 
 4. Five great railway systems recently spent for advertising in one year, 
 respectively; $348,457; $150,647; $84,335; $147,564; $25"l,532— an 
 average of $ . 
 
 5. A half-page advertisement in a monthly, $448 per page less one-fifth 
 for time, costs a dealer $ for twelve insertions. 
 
 6. The magazine referred to above contained 168 pages of advertising 
 in a given number. Estimate the minimum income. Why might the 
 income be larger? 
 
 7. A novelty desk pad is advertised by its maker in a 20-line space for 
 3 months, in five monthlies, having line r^ies as follows: $3.50, $2.75, $2.50, 
 $2.00 and $1.20. Compute the cost. 
 
ADVERTISING. 
 
 81 
 
 Note. Agencies receive net or gross rates from publishers. They 
 charge advertisers the net rate, plus a commission ; or, if under the gross 
 rate agreement, charge the advertiser this rate, receiving a commission or 
 rebate from the publisher. 
 
 8. Complete this statement. 
 
 New York, N. Y. Nov. 30, 19 . 
 Mr. Brown C. Norton 
 
 To J. F. Jamison, Dr. 
 Advertising Agent. 
 
 
 
 
 Net 
 
 Gross 
 
 The Johnstone Weekly 
 
 ip. 
 
 12 times 
 
 
 
 5400 
 
 
 The World Magazine 
 
 1 col. 
 
 12 times 
 
 
 
 2700 
 
 
 The Hour 
 
 ip. 
 
 12 times 
 
 3880 
 
 
 
 
 Every Moment 
 
 1^ col. 
 
 12 times 
 
 7650 
 
 
 
 
 The Monthly 
 
 Ip. 
 
 12 times 
 
 3648 
 
 
 
 
 Literary Magazine 
 
 Ip. 
 
 12 times 
 
 
 
 960 
 
 
 Livelihood 
 
 Icol. 
 /lO 
 
 12 times 
 
 1260 
 
 
 
 
 Commission 1 
 
 ? 
 ? 
 
 
 ? 
 ? 
 
 
 In newspaper advertising, which is largely local, many dealers make 
 contracts for a fixed amount of space per annum, to be used as required. 
 This space is usually expressed in line measure. For "want" advertising 
 and minor notices, rates vary with the advertising matter. The rate cards 
 of newspapers are often very complicated. Opposite is a small section of 
 one. 
 
 Extract from a Newspaper Rate Card. 
 
 General Display Agate Line. 
 
 Run of paper 50c. 
 
 Special portion, last page, opp. editorials, etc. 55c. 
 Space Discounts. One Year Limit. 
 
 5000 Unes 1/20 off. 20000 lines 3/20 ofif. 
 
 10000 " 1/10" 40000 " 1/5 " 
 
 Classification Rates. 
 Agate Line. 
 
 Amusements 50c Une. Public notices 20c line. 
 
 Art sales, etc 20c " Railway notices 15c " 
 
 Auctions 20c " 
 
 Birth, marriage, death 
 
 notices $1.00 ea. 
 
 Financial notices 45c line. 
 
 Reading notices, financial page, $2.00. 
 
 Railway time-tables, daily by the 
 month 20c line. 
 
 Railway time-tables, daily by the 
 year 15c line. 
 
82 BUSINESS ARITHMETIC 
 
 Want Advertisements. 
 
 Per Line 1 time 3 7 Per line 1 time 3 7 
 
 For sale 18c 25c 40c Lost and Found. . .8c 20c 35c 
 
 Help wanted 8 20 35 Positions wanted. .5 10 15 
 
 Note. Eight words to the line: five words if set in caps. 
 
 ORAL EXERCISE. 
 
 Use above rates where none are stated in problem. Find the cost of: 
 
 1. 60-line advertisement, run of paper. 
 
 2. 28-line advertisement, general display. 
 
 3. 5-inch advertisement, special position on last page. 
 
 4. 100-line advertisement, special position surrounded by reading 
 matter — the charge being 5c. extra per line. 
 
 5. 5000 lines, contracted for 55c. 
 
 6. 20,000 lines, contracted for 35c. 
 
 7. 10,000 lines, quoted rates. 
 
 8. 4-inch public notice. 
 
 9. Three insertions of a four-line liOst and Found notice. 
 
 10. One month's insertion of a 56-line railway time table. 
 
 11. A six-inch reading notice on the financial page. 
 
 12. Seven insertions of a six-line " position wanted " advertisement. 
 
 13. A ten-inch advertisement of a circus. 
 
 EXERCISE. 
 
 1. A 12-in. advertisement in a Chicago newspaper @ $4.92 per 
 in. costs $ ? per insertion, or for 26 insertions @ $3.92 per inch, per in- 
 sertion, $ ? 
 
 2. A Seattle dealer contracted for 12,000 lines @ 49c. During January 
 he inserted advertisements as follows : 120 lines, 65 lines, 140 lines, 210 lines, 
 560 lines (twice running), 28 lines. What is his bill for the month? 
 
 3. A 70-line double column advertisement in a Kansas City paper, @ 
 $1.40 per inch, costs how much for two insertions? 
 
 4. Compute the cost of this advertisement at quoted rates. 
 
 WANTED— EXPERIENCED SALESWOMEN IN 
 books, stationery, handkerchiefs, notions, neckwear, 
 toys, dolls, umbrellas, &c. Also bright young women 
 from 8th grade and high school. 
 Box 462, Boston, Mass. 
 
ADVERTISING. 83 
 
 5. An agency agrees to insert the above "want ad" in 103 city papers 
 for $35.00 for 21 words, and $11.65 for each additional 7 words. This 
 will cost $ ? 
 
 Advertisers watch the returns from advertising and dis- 
 card pubHcations that fail to bring customers. Returns are 
 traced in many ways. For example, an advertiser may alter 
 his street number, writing it "234 State Street" in one period- 
 ical, and "236 State Street" in another; or he may offer a 
 different bargain in each magazine; or he may insert in the 
 advertisement a reply-slip, properly keyed with a number 
 or letter for identification. The replies are then classified, 
 and the papers yielding the lowest cost per inquiry, the greatest 
 number of orders, or amount of sales, are rated as the best 
 advertising media. 
 
 EXERCISE. 
 
 1. The form on p. 84 illustrates a cost per inquiry card. It is arranged 
 as a calendar showing the number of inquiries received on each date. 
 Why are no replies received on Jan. 8, 15, etc.? Each horizontal line total 
 represents what? The vertical total? Extend the form and find all 
 values. 
 
 2. To trace advertisements month by month, and to see which of a 
 series of advertisements draws best, the key in a magazine may be changed 
 in each issue. For example, the magazines may be numbered from 10 to 99. 
 To them is affixed a number from 1 to 12 to show the issue. Magazine 
 No. 10, for a Februarj^ issue would be represented by 102, because February 
 is the second month. The form on p. 85 is made out for June, and shows 
 on the left the classification of calendar replies. The blocks on the right 
 (1 to 12) are a classification of replies by the advertisements that called 
 them forth. Find the totals and explain the form in full. 
 
84 
 
 BUSINESS ARITHMETIC. 
 
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CHAPTER XV. 
 FACTORS AND MULTIPLES. 
 
 Thus far, attention has been given to numbers, their expres- 
 sion, the units of which they are composed and the funda- 
 mental operations involving them. Certain properties com- 
 mon to numbers should now be noted. 
 
 1. Any number may he considered as the product of two or more 
 factors. The process of determining these factors is termed 
 factoring. 
 
 ORAL EXERCISE. 
 
 1. Define a factor. 
 
 2. Name two factors of 35, 64, 120, 42, 18, 168. 
 
 3. 2 X 3 X ?= 48. 
 
 4. Name three factors of 36, 160, 54, 846. 
 
 5. 2 is a factor of which of these numbers: 68, 45, 32, 19, 127, 388? 
 
 6. Name some factor that is common to 48 and 36; 42 and 280. 
 
 7. Name two factors of 17, 29, 41, 53, 13. ^Vhat is conmion about the 
 factors of these numbers? 
 
 Classified according to factors, integral numbers are even 
 if two is a factor, and odd if two is not a factor. 
 
 Integral numbers are prime numbers if the number itself 
 and one are the only factors. A series of numbers are mutually 
 prime if one is the only common factor. All numbers not 
 prime are composite and contain two or more factors. 
 
 ORAL EXERCISE. 
 
 1. Name and classify the numbers from 1 to 150, as even, odd, prime 
 or composite. 
 
 2. Name four groups of numbers that are mutually prime. 
 
 3. May the factors of a number be composite? 
 
 Note. There are certain tests of divisibility that are useful in factor- 
 ing and general computation. Thus a number is divisible by: 
 
 86 
 
FACTORS AND MULTIPLES. 
 
 87 
 
 2, if it is even. 
 
 3, if the sum of its digits Ls divisible by 3. 
 
 4, if the number represented by its two right hand figures is divisible 
 by 4. 
 
 5, if the right hand figure is 5 or 0. 
 
 6, if even, and the sum of its digits is divisible by 3. 
 
 7, (No simple test). 
 
 8, if the number expressed by the three figures to the right is divisible 
 by 8. 
 
 9, if the sum of the digits is divisible by 9. 
 10, if the last figure is 0. 
 
 EXERCISE. 
 
 1. Suggest tests of divisibility by 100, 1000, 10000, 20, 90, 18, 12. 
 
 2. If a number is divisible by 8, it is necessarily divisible by ?, ?, or ? 
 
 3. If a number is divisible by 12, it is necessarily divisible by ?, ?, ?, or ? 
 
 4. 9,684,324 is divisible by what factors under 13? 
 Using divisibility tests, determine at least two factors of: 
 
 5. 144. 7. 1728. 9. 2240. 11. 5280. 13. 4589. 
 
 6. 320. 8. 19,200. 10. 63,360. 12. 12,345. 14. 12,726. 
 
 EXERCISE. 
 
 Reduce to prime factors: 
 
 1. 7668. 
 
 Being even, 2 is a factor. The resulting factor, 3834, is 
 even, and is therefore again divisible by 2; the sum of the digits 
 of 1917 (18) is divisible by 3, a prime factor. The remaining 
 factors are found by inspection. 
 
 Check 2X2X3X3X3X71= 7668. 
 
 2. 246. 4. 4225. 6. 1728. 8. 2496. 
 
 3. 389. 5. 4128. 7. 63,360. 9. 7884. 
 Find the prime factors common to: 
 
 10. 378 and 864. 
 
 Since 7 and 16 have no common factor, division 
 need not be carried further. It is evident that the 
 factors common to 378 and 864, are 2, 3, 3 and 3. 
 
 11. 984 and 676. 12. 1425 and 5000. 13. 
 
 Solution. 
 2 7668 
 
 3834 
 
 1917 
 
 639 
 
 213 
 
 71 
 
 Solution, 
 2 378 
 
 864 
 
 3 189 
 
 432 
 
 3 63 
 
 144 
 
 3 21 
 
 48 
 
 7 
 672 and 396. 
 
 16 
 
88 BUSINESS ARITHMETIC. 
 
 The product of the common factors (in Ex. 10, 2 X 3 X 3 
 X 3) of two or more numbers is termed their highest common 
 factor (h.c.f.). The highest common factor was once of 
 extreme importance in compHcated fractional work, but it is 
 now little used, owing to the substitution of decimals for all 
 but simple fractions. 
 
 EXERCISE. 
 
 Find the highest common factor of: 
 
 1. 678 and 420. 2. 5280 and 320. 3. 546 and 384. 
 
 EXERCISE. 
 
 Factoring is often employed to simplify and shorten the processes of 
 multiplication and division. This is done by rejecting common factors 
 from numbers that stand in the relation of dividend and divisor. 
 
 1. Why is the quotient of 5 X 8 divided by 5 X 4 the same as 8 -i- 4? 
 
 2. Divide 4X7X8 by 3X7X9. Reject the common factor and 
 divide. 
 
 What is the effect of dropping the common factor? 
 
 3. Divide 12 X 42 X 18 X 6 by 6 X 8 X 9 X 11. 
 
 Solution. 
 
 3 ^ 
 
 *gX 42 X IS X^ ^ 3 X42 ^ 126 
 
 ^X-&XS^X11 11 11 
 
 ■2- 
 
 11 A 
 
 Indicate the products and division as shown. Cancel the common factor, 6. 
 Cancel the factor 9 of the divisor with the 9 in 18 of the dividend, leaving 2. 
 Cancel the 2 from dividend and from 8 in the divisor, leaving 4. Cancel 4 
 from the divisor and from 12 in the dividend, leaving 3. Multiply the 
 remaining factors. 
 
 4. 14 X 8 X 64 divided by 11 X 16 X 4 X 7 = ? 
 
 6. 3X21X84X9-^69X8X6X14 = ? 
 6 52 X 16 X 5 X 11 ^ ^ 
 
 • 13 X 28 X 12 X 15 • 
 
 7. The output of a cracker factory is 840 doz. packages per hour, 
 for a nine-hour day. This is equivalent to how many cases of 8 cartons, 
 containing six packages each? 
 
FACTORS AND MULTIPLES. 89 
 
 8. 420 cases of 4 doz. cans each, are equivalent to how many packages 
 of each containing 6 cans? 
 
 9. At the rate of $9.60 per doz., what is the cost of five units? 
 
 10. In a given year the coal consumption for heating an office building 
 was 1825 tons. The price paid was $3.90 per ton. What was the average 
 daily cost for fuel? 
 
 The multiple of a number is any integral number of times 
 the number. A common multiple of two or more numbers is 
 one that is an integral number of times each of the numbers. 
 The least number that is such a multiple, is termed the least 
 common multiple (l.c.m.). 
 
 EXERCISE. 
 
 1. Name multiples of 8, 5, 21, 4. 
 
 2. 12 is a common multiple of what smaller numbers? 
 
 3. Name a common multiple of 15 and 25. 
 
 4. Name two common multiples of 4 and 6. What is the least common 
 multiple? 
 
 5. Is the product of two numbers a multiple of them? 
 
 6. Is it ever the least common multiple? 
 
 7. Find the 1 c. m. of 36, 84 and 96. 
 
 To be a common mutiple the number must contain the 
 prime-factors of each of the given numbers. 
 
 Illustration. 
 
 Resolve 36, 84 and 96 into their prime Solution. 
 
 factors. The multiple must have the fac- 36 = 2X2X3X3 
 tors "2" five times because 96 contains 84 = 2X2X3X7 
 that number; the factor "7" once, because 96 = 2X2X2X2X2X3 
 84 contains it ; and two " 3's " because both 
 36 and 96 have them. 
 
 The least common multiple is therefore 
 2X2X2X2X2X3X3X7, or 2016. 
 
 If the factors are not determined by in- 
 spection, arrange the numbers in a hori- 2 
 zontal line and divide by factors common 2 
 to any two numbers. 3 
 
 Bring down undivided quotients that 3 7 8 
 
 are not exactly divisible. Continue until 
 
 Solution. 
 36 84 
 
 96 
 
 18 42 
 
 48 
 
 9 21 
 
 24 
 
90 BUSINESS ARITHMETIC. 
 
 all quotients are mutually prime. The 1. c. m. is equal to the product 
 of the divisors and the final quotients. 
 
 1. c. m. = 2 X 2 X 8 (2 X 2X 2) X 3 X 7 = 2016. 
 Find the 1. c. m. of: 
 
 8. 4, 9, 6, 2. 10. 4, 20, 30, 18. 12. 5, 35, 60, 75. 
 
 9. 9, 12, 15. 11. 4, 2, 8, 32. 13. 3, 6, 5, 92, 18. 
 
 14. A steamship company has two steamers leaving New York, the 
 fii'st maldng a round trip in 10 days and the second in 16 days. How often 
 do they reach New York at the same time? 
 
 15. A manufacturing company has on the road, four drummers, each 
 following circuits that take respectively 7, 6, 4 and 12 days. How often 
 can the manager confer with the four men together? 
 
 16. Two wheels of 20' and 16 ft. circumference are geared together. 
 How often do the wheels complete their revolutions at the same instant? 
 
 17. A novelty manufacturer uses cloth in strips of 3", 4", 9", 6" and 
 12" in width. What least width of cloth should he buy in order that he 
 may cut it into any of the given widths without waste? 
 
 FOR DISCUSSION. 
 
 1. Find the h. c. f. and 1. c. m. of any two numbers. Compare the 
 product of these values with the product of the numbers. Explain. 
 
 2. The dollar of our currency is a multiple of what other denomina- 
 tion of coin? 
 
 3. Name some of our systems of measures that are based on a multiple 
 system. 
 
 4. What are the advantages of multiple over mutually prime measures? 
 
 5. Show that in determining the 1. c. m., all numbers that are factors 
 of other given numbers may be disregarded. 
 
CHAPTER XVI. 
 REDUCTION OF FRACTIONS. 
 
 A fraction is one or more of the equal parts into which a 
 whole may be divided. One of the equal parts is a fractional 
 unit. 
 
 All fractions may be expressed as common fractions by two 
 numbers or terms separated by a horizontal line. The 
 denominator (lower term) names the parts or size of the frac- 
 tional unit; the numerator (upper term) shows the number of 
 fractional units in the given fraction. In reading, the numera- 
 tor is invariably read first. 
 
 ^ five (numerator) 5 „ • i ^t 
 
 Illustration. . , ., ,j . . \ = s = nve-eightns. 
 eighths (denominator) 8 
 
 INTRODUCTORY EXERCISE. 
 
 1. Name the unit in the expressions: 15 lb., 7 da., 3 wk., 1/2, 5/6, 
 four-ninths, 7/8 yd. 
 
 2. Which term of the common fraction is missing in the decimal 
 fraction? How is it expressed? 
 
 3. Which is greater, 1/4 or 1/8? How is the value of a fraction afifected 
 by increasing its denominator? 
 
 4. Arrange in order of value: 1/5, 1/98, 1/72, 1/5000. 
 6. Can 1/3 lb. and 1/4 A. be compared as to value? 
 
 6. How many thirds is 1? WTiich is greater, 1/3 or 2/3? How is 
 the value of a fraction affected by increasing the numerator? 
 
 7. Arrange m order of value: 3/9, 7/9, 11/9, 5/9. 
 
 It is evident that any increase in the numerator or decrease 
 in the denominator increases the value of the fraction, and 
 that the value is decreased by decreasing the numerator or 
 increasing the denominator. Moreover, if two fractions have 
 like numerators the larger fraction has the smaller denomina- 
 
 91 
 
92 BUSINESS ARITHMETIC. 
 
 tor; if they have like denominators, the larger fraction has the 
 larger numerator. 
 
 Common fractions are proper fractions if the numerator is 
 less than the denominator; otherwise they are improper 
 fractions. 
 
 Fractions are compound, if each term is fractional I e. g., 
 
 3; 1. A mixed number is an expression containing an integer 
 
 and a fraction (e. g., 6f ). 
 
 The tendency in all business and general computation is 
 toward the use of decimals in place of complicated or irregular 
 fractions, so that only the simple fractions are in common use. 
 The business series covers halves, thirds, fourths, fifths, sixths, 
 eighths, twelfths, sixteenths and twentieths. To these our 
 main attention will be given. 
 
 EXERCISE. 
 
 1. This is one inch of a ruler divided into eight parts. Name the 
 unit and the smallest fractional unit. 
 
 2. In the whole there are how many halves, 
 fourths, eighths? 
 
 3. Multiply numerator and denominator of 1/2 
 by 4. How is the value affected? 
 
 4. Divide numerator and denominator of 6/8 
 by 2. How is the value affected? 
 
 5. Why is value unaffected? {Suggestion. Note independent effect 
 of first dividing numerator and then denominator.) 
 
 It is evident that a change of expression by multiplying or 
 dividing both terms of a fraction by the same number, causes 
 no change in value. A fraction is reduced to lower terms when 
 smaller numbers are found to express the same value; it is 
 reduced to higher terms when larger numbers are used. 
 
 ORAL EXERCISE. 
 
 1. Reduce to thirds: 8, 17, 21, 14, 8, 64. 
 
 "!,f ; 
 
 ■/. '/, % 
 
 ■'A \ '/, 
 
 
 1 1 
 
REDUCTION OF FRACTIONS. 93 
 
 2. Reduce to 16ths: 3, 4, 12, 9, 21, 5. 
 
 3. Reduce to 12ths: 7, 13, 5, 3, 12, 15. 
 
 4. Reduce to 20ths: 3/4, 5/4, 7/5, 3/10, 3/2. 
 Illustration. 3/4 = ?/20. 20 -^ 4 = 5; 1/4 = 5/20; 
 
 3/4 = ^-^ = 15/20. 
 
 5. Reduce to 16ths: 3/4, 5/8, 5/2, 7/8, 9/4. 
 
 6. 7/6 = ?/36; 5/8 = ?/40; 7/3 = ?/18; 3/7 = ?/49. 
 
 7. Is there any limit to the increase of size of denominator in reduction? 
 
 8. Reduce to improper fractions: 3i, 5^, 16f, 27f. 
 Illustration. In 3 there are 3 X 4 fourths. In 31 there are 12/4 
 
 + 1/4 or 13/4. 
 
 9. Reduce to lOths: If, 2f, 12 i 
 
 (Suggestion. Reduce first to improper fractions.) 
 10. 3i = ?/12; If = ?/24; 7i = ?/20; 9| = ?/6. 
 
 EXERCISE. 
 
 1. Reduce 76 to a fraction with a denominator of 842. 
 
 2. Reduce 117 to 43ds. 
 
 3. 3/7 = ?/427; 118/142 = ?/710. 
 
 4. Reduce: 16 j^ to 24ths; 192^2^ to 45ths. 
 
 It is often advisable, in computations, to simplify a fraction 
 by reducing it to lower terms. In final results, proper fractions 
 are usually expressed in lowest terms, and improper fractions 
 are expressed as whole or mixed numbers. It has been noticed 
 that the value of a fraction is not affected by dividing numera- 
 tor and denominator by the same number. Hence reduction is 
 possible by continued reduction by common factors. When 
 the reduced terms are mutually prime, the fraction is reduced 
 to its lowest terms. 
 
 Illustration. Reduce 585/675 to its lowest terms. 
 
 Analysis and Solution. By inspection, 5 is a common 585 4- 5 _ 117 
 
 factor of both terms. 675 4- 5 ~ 135 
 
 By inspection, 9 is seen to be a second common factor. 
 
 This really amounts to dividing the terms by their 117 -^ 9 _ 13 
 
 greatest common di\isor. 135 -i- 9 ~ 15 
 
94 BUSINESS ARITHMETIC. 
 
 ORAL EXERCISE. 
 
 1. Reduce to 8ths: 3/4, 2, 10/16, 15/24. 
 
 2. Reduce to 12ths: 3/4, 5/6, 18/24, 12/36, 40/60. 
 
 3. Reduce to lowest terms: 15/18 cu. yd., 45/69, 720/2400 T. 
 
 4. R-duce to lowest terms: 160/4, 325/5, 56/8, 120/3. 
 
 5. Reduce to lowest terms, or to mixed numbers: 9/2, 15/4, 73/3 
 45/12 lb . 52/8 bu., 162/16. 
 
 EXERCISE. 
 
 1. Reduce to lowest terms: 56/124, 128/264, 985/1200, 172/465. 
 
 2. Reduce to whole or mixed numbers: 725/80, 392/56, 1982/1250. 
 
 3. Simplify: 285/16 T., 3924/5280 mi., 453/144 sq. ft. 
 
CHAPTER XVII. 
 
 FRACTIONS— FUNDAMENTAL PROCESSES. 
 
 Addition of Fractions. 
 
 Since only like numbers can be added, it follows that only 
 fractions or parts of like units can be added. Moreover, since 
 the denominator of a fraction is but a name showing the 
 relative size of the fractional unit, only similar fractions, those 
 having the same denominator, can be added. If fractions 
 differ only in denominator, addition is possible after reduction 
 of the fractions to a common denominator. In the cases 
 commonly arising in everyday usage, the least common 
 denominator is usually determined by inspection. 
 
 Illustration. Add 3/5 and 5/6. 
 
 Solution. By inspection, the common denominator is 30. 
 
 3/5 = 18/30. 
 
 5/6 = 25/30. 
 
 3/5 + 5/6 = 18/30 + 25/30 = 43/30, or 1^. 
 
 ORAL EXERCISE. 
 
 Which of these groups can be added 
 
 1 without intermediate steps? 
 
 Which cannot be added at all? 
 
 
 
 1. 4/5 + 8/5 + 5/5. 
 
 3. 
 
 5/4 ft. + 2 lb. 
 
 2. 1/4 + 1/2 bu. 
 
 4. 
 
 1/4 + 1/5 + 1/6. 
 
 5. 1/4 yd. + 31 yd. 
 
 
 
 Find the sum of: 
 
 
 
 6. 1/4, 3/4, 2/4, 9 fourths. 
 
 14. 
 
 3/8, 4/5, 7/10. 
 
 7. 1/3 yd., 2/3 yd., 5/3 yd., 3/3 yd. 
 
 15. 
 
 7/12, 2/3, 1/4. 
 
 8. 3/16, 4/16, 7/16, 11 sixteenths. 
 
 16. 
 
 1/2, 2/5, 7/10. 
 
 9. 1/5, 3/5, 7/5, 4/5, 11/5, 9/5. 
 
 17. 
 
 3/20, 4/5, 1/2. 
 
 10. 2/3 +4/5+ 9/15 (= ?/45). 
 
 18. 
 
 1/3, 3/2, 5/6, 1/12. 
 
 11. 1/3, 1/4. 
 
 19. 
 
 1/3, 2/9, 5/6. 
 
 12. 3/4 ft., 5/8 ft 
 
 20. 
 
 3/4 ft,, 2/3 ft., 5/2 ft. 
 
 13. 1/2, 3/4, 5/8, 1/16. 
 
 
 
 95 
 

 2. 
 
 3i 
 
 51, 6f. 
 
 
 4. 
 
 5/12 
 
 , 1^1, 1.5. 
 
 
 6. 
 
 4H, 
 
 85/r. 
 
 32961 
 
 
 
 9. 1292U 
 
 562i 
 
 
 
 1828^ 
 
 14729t5i 
 
 
 
 13461 
 
 862^ 
 
 
 
 4621 
 
 928xV 
 
 
 
 18f 
 
 6621 
 
 
 
 
 96 BUSINESS ARITHMETIC. 
 
 EXERCISE. 
 
 Note. If the numbers to be added are mixed numbers, the integers 
 and fractions rnay be added separately and the two results combined. 
 This, however, is not necessary with very simple numbers. 
 
 1. U, U, 5i 
 
 3. 32i, 7f, 8f. 
 
 5. 3i, 41, 12tV. 
 
 7. 2141 
 
 319xV 
 
 4291 
 
 296i 
 
 Note. In totalling lengths of cloth, the exponents, 1, 2, 3, are used to 
 represent one-fourth, two-fourths and three-fourths respectively. Thus 
 7621 yd. are 7621 yd. 
 
 10. 27 pieces of percale are sold a customer on an order. Find the 
 total quantity if the respective piece lengths ai"e; 44, 40, 43, 42^, 41^, 40*, 
 42S 423, 432^ 431^ 433^ 442^ 42, 43S 422, 431^ 422^ 421, 412^ 403, 44S 43S 42', 
 43^ 42S 43S 403. 
 
 11. Find the cost of thirty-one pieces of Scotch cheviot, @ 37c per yd., 
 the pieces measuring respectively: 38S 37S 37^, 39, 39^, 40S 42^, 40S 38^, 
 391, 40, 382, 391^ 40, 41, 392, 383, 392^ 40, 401, 402, 39^, 38^, 39S 39^, 40S 
 39', 392, 391, 40S 412. 
 
 EXERCISE. 
 
 1. Find the sum of any two fractions that have the same numerator. 
 Study the solution and suggest a short method for adding any two such 
 fractions. 
 
 Add: 
 
 2. 1/8, 1/3. 4. 1/11, 1/16. 6. 5/8, 5/11. 
 
 3. 1/9, 1/4. 6. 4/9, 4/21. 7. 2/3, 2/9. 
 
 Subtraction of Fractions. 
 
 Since subtraction and addition are reverse processes, the 
 
 principles of units and common denominators apply as in 
 
 addition. 
 
 Illustration. Find the difference between 3/7 and 1/6. 
 Solution. 3/7 = 18/42. 1/6 ^ 7/42. 
 
 3/7 - 1/6 = 18/42 - 7/42 = 11/42. 
 
SUBTRACTION OF FRACTIONS. 97 
 
 EXERCISE. 
 
 (Solve mentally if possible.) 
 Find the difference between: 
 
 1. 1/4 and 1/5. 5. 7/3 and 4/5. 9. 7/8 yd. and 2/3 yd. 
 
 2. 6/7 and 1/4. 6. 4/3 ft. and 3/4 ft. 10. 7/40 and 3/20. 
 
 3. 3/3 and 2/9. 7. 7/4 and 4/5. 11. 1/3 and 1/4. 
 
 4. (1/4 + 1/8) and 3/4. 8. 7/8 and 7/20. 12. 6/9 and 6/11. 
 
 Note. If mixed numbers are given, whole numbers and fractions may 
 be subtracted separately, except that in case the fraction of the subtrahend 
 is greater than the fraction of the minuend, it may be necessary to reduce 
 an integral unit of the subtrahend to fractional units. 
 
 Find the value of: 
 
 13. 5^ - 3i 18. 5| - 2.1. 
 
 14. 6| - 1t\. 19. 388f - 172r\. 
 
 15. 2f yd. - 1| yd. 20. 32,^^ + 87| - 23^. 
 
 16. If - 5/6 21. 582i - 96^. 
 
 17. 17/^ - .8. 
 
 Multiplication of Fractions, 
 introductory. 
 
 '6/8' may be reaxi in the form '5 eighths.' 
 
 16 times 5 eighths are ? eighths. 
 
 One-eighth of 16 is ? ; five-eighths of 16 are ? . 
 
 It is evident that the product of a fraction by a whole 
 number is equal to the product of the numerator and the 
 whole number, divided by the denominator. Thus, either 
 fraction or whole number may be taken as multiplier. 
 
 Illustration. Multiply 5/6 by 124. 
 Solutim. 124 X 5/6 = i?^^ = ^ = 103i 
 
 EXERCISE. 
 
 (Solve mentally if possible.) 
 
 1. Multiply 320 by each of these fractions: 1/4, 1/2, 3/4, 5/8, 4/5, 
 3/2, 3/16. 
 
 2. Find the following fractional parts of 48: 1/3, 3/4, 5/6, 7/8, 2/3, 7/6. 
 8 
 
3. Find 2/3 of 7, 3, 1 
 
 I 18, 
 
 42, 6, 27, 49, 56, 129. 
 
 Multiply: 
 
 
 
 4. 425 by 2/5. 
 
 8. 
 
 2/7 by 147. 12. 
 
 5. 25 by 3/4. 
 
 9. 
 
 2/9 by 117. 13. 
 
 6. 37 by 5/8. 
 
 10. 
 
 5/8 bu. by 726. 14. 
 
 7. 3/8 by 5280. 
 
 11. 
 
 5/6 by 8 X 42. 15. 
 
 98 BUSINES ARITHMETIC. 
 
 320 rd. by 9/16. 
 142 by 2/3. 
 7/12 by 30 X 9. 
 364 T. by 4/3. 
 
 If one factor is a mixed number, it may be reduced to an 
 improper fraction, although direct multipHcation is common. 
 Illustration. Multiply 36 by 2^. 
 
 Solution (1). 36 X 2i = 36 X 11/5 = ?i^JJ: = 395/5 = 791. 
 Solution (2). 36 
 
 7i (1/5 of 36) 
 Z2_ 
 79i 
 
 EXERCISE. 
 
 (Solve mentally if possible.) 
 Find the product of: 
 
 1. 16 X U. 5. 165 X If 
 
 2. 424 X 7f . 6. 3i X 16 T. 
 
 3. 21 X ^. 7. 2f X $120. 
 
 4. 4i X 160 ft. 8. 168 X 68t^^. 
 Find the cost of: 
 
 9. 7| yd. dress goods @ $3; 8^ yd. lining @ 17c; 3/4 yd. velvet 
 @ $3.50. 
 
 10. 420 bu. corn @ 42Jc; 652 bu. wheat @ 81|c; 1900 bu. oats @ 
 32ic. 
 
 Fractions Multiplied by Fractions. 
 
 INTRODUCTORY. 
 
 1/2 in. = ? fourths of an inch =^ ? eighths. 
 1/4 of 1/2 in. = 1/4 of ? eighths, or ? eighth. 
 
 The product of two fractions is evidently a fraction whose 
 numerator is the product of the given numerators and whose 
 denominator is the product of the given denominators. The 
 computation may be simpHfied, in some cases, by cancellation. 
 
MULTIPLICATION OF FRACTIONS. 
 
 99 
 
 Illustrations. Multiply 3/85 by 5/6. 
 3X5 
 
 15 
 
 Solution 1. 
 
 3/85 X 5/6 =gg-g=^^^ = 1/34. 
 
 Solution 2. 
 
 By cancelling common factors. 
 
 
 3/85 X 5/6 = ^^-^ = n X 2 = i 
 17 2 
 
 ORAL FACILITY EXERCISE. 
 
 Find the product of . 
 
 1. 1/4 X 1/3. 4. 4/5 X 7/12. 7. 3/8 X 8/3. 
 
 2. 1/2 X 3/4. 5. 5/6 X 6/7. 8. 3/8 X 5/6 X 4/5. 
 
 3. 3/8 X 5/2. 6. 3/2 X 8/9. 
 
 EXERCISE. 
 
 Simplify 
 
 1. 3/4 of 9/24 of 7/15 of 852 bu. 
 
 194. 
 
 2. 3/84 X 5/72 X ^ X $8290. 
 
 15 
 
 3. i^ X 84/36 X 21/22. 
 
 Products of Mixed Numbers. 
 With large integral numbers and small fractions, multiplica- 
 tion by parts is common. In simple cases, the mixed numbers 
 may be reduced to improper fractions and the products 
 obtained as in the previous case. 
 
 Illustrations. 
 Solviion, 
 
 (1) Multiply 429f by 127|. 
 
 127J 
 
 i (1/3X3/5) 
 
 143 (1/3 X 429) 
 
 76i (127 X 3/5) 
 
 3003 ' 
 
 858 \ (127 X 429) 
 
 429 J 
 
 54702f Ans. 
 
100 BUSINESS ARITHMETIC. 
 
 (2) Multiply 4i by Gf. 
 
 
 Solution. 4i : 
 
 = 13/3. 6f 
 
 = 51/8. 
 
 17 
 
 
 13/3 X 51/8 
 
 13 X-BT 221 
 
 -a-xs ~ 8 ~^^^' 
 1 
 
 EXERCISE. 
 
 
 (Solve 
 
 mentally if possible.) 
 
 Multiply: 
 
 
 
 1. U by 2f. 
 
 
 7. 4f by 15i 
 
 2. 2f by 3/8. 
 
 
 8. 7f by 14^. 
 
 3. If by 3f 
 
 
 9. 2341 by 65i 
 
 4. If by li 
 
 
 10. 3451 by 168i 
 
 5. 4| by If. 
 
 
 11. 1420f by 17281. 
 
 6. li by 2f. 
 
 
 12. 182f by 14^. 
 
 Short Methods. 
 Few fractional short methods, aside from cancellation, have 
 a broad use. The most valuable method is that applying to 
 mixed number factors having the fraction "J." 
 Case I. Integers alike. Example. Multiply 4| by 4^. 
 Analysis and Solution. 
 
 By parts, 4^ X 4| = 4 X 4 + 4 X 1/2 + 1/2 X 4 + 1/2 X 1/2. 
 = 4 fours + 2 halves of 4 + 1/4. 
 = 5 fours +1/4 
 Condensed solution. 
 
 4^X4^=5X4 + 1/4=201. 
 
 In general, multiply the integer by the next higher integer and add 1/4. 
 Case II. Integers unlike. Example. Multiply 85 by 11^. 
 Analysis and Solution. 
 By parts, 8i by lU = 11 X 8 + 11 X 1/2 + 1/2X8 + 1/2 X 1/2- 
 
 = 88 + 1/2 (11 + 8) +1/4. 
 
 = 88 + 9^ + 1/4 = 97f . 
 In general, add to the product of the integers one-half their sum, plus II4. 
 
 EXERCISE. 
 
 Multiply: 
 
 
 
 
 
 1. \2h by 12i. 
 
 5. 
 
 6^ by 6i. 
 
 9. 
 
 m by 30i. 
 
 2. 8§ by 8^. 
 
 6. 
 
 61 by 8i. 
 
 10. 
 
 15^ by 5.5. 
 
 3. 15i by 15i 
 
 7. 
 
 10.5 by 12i. 
 
 11. 
 
 461 by 46i. 
 
 4. 9.5 by 9.5. 
 
 8. 
 
 Uh by 16i. 
 
 12. 
 
 124^ by 6U. 
 
DIVISION OF FRACTIONS. 
 
 101 
 
 DIVISION OF FRACTIONS. 
 • 1. Fractions by Integers, 
 introductory. 
 
 12 ft. 4- 3 = how many feet? 
 
 12 thirteenths -r- 3 = how many thirteenths? = ?/13. 
 - 15/16 -^ 5 = 1/5 of 15/16 = 1/5 X 15/16 = ? 
 
 It is possible, evidently, to divide the fraction by dividing 
 the numerator, or multiplying the denominator by the integral 
 divisor. 
 
 The reciprocal of a number is 1 divided by that number. 
 Multiplying the denominator by the integral divisor is equiva- 
 lent to multiplying the fraction by the reciprocal of the 
 divisor. 
 
 Illustration. 
 
 Divide 18/23 by 
 
 6. 
 
 
 Solutions. 
 
 
 
 3 
 
 48- 3 
 
 23 X-^ 23 
 
 1 
 
 (i)i-« 
 
 18 -J- 6 3 
 23 23 • 
 
 (2)1-6 = 
 
 
 (3)l|^6 = 
 
 3 
 .^y 1_3 
 "23^^ 23* 
 
 
 Divide: 
 
 1. 28/37 by 4. 
 
 2. 5/8 by 6. 
 
 3. 2/3 by 12. 
 
 4. 24/15 by 6. 
 
 EXERCISE. 
 
 (Solve mentally if possible.) 
 
 5. 3/8 by 5. 
 
 6. 21/25 by 7. 
 
 7. 2/3 by 7. 
 
 8. 5/9 by 8. 
 
 9. 5/8 by 15. 
 
 10. 142/56 by 14. 
 
 11. 36/89 by 18. 
 
 12. 22/45 by 33. 
 
 2. Division of Mixed Numbers by Integers. 
 
 Since the mixed number may be reduced to an improper 
 fraction, the method does not differ from that just shown. It 
 is common, however, to divide directly. 
 
 Illustration. Divide 54 1 by 6. 
 
102 BUSINESS ARITHMETIC. 
 
 'Sobkima. :(1] 
 
 '-t = f. 
 
 491 491 1 491 
 9 • *" 9 ^6" 54 "^^^• 
 
 (2) 6)541 
 
 V 
 
 
 54 
 
 
 
 f (5/9 - 
 
 r 6 = 5/9 X 1/6 = 5/54) 
 
 
 
 EXERCISE. 
 
 
 (Solve 
 
 mentally if possible.) 
 
 Divide: 
 
 
 
 1. If by 3. 
 
 6. 
 
 16i by 5. 9. 382f by 26. 
 
 2. 12f by 4. 
 
 6. 
 
 4f by 3. 10. 193A by 15. 
 
 3. 27f by 9. 
 
 7. 
 
 12t\ by 4. 11. 6721 by 12. 
 
 4. 8f by 3. 
 
 8. 
 
 2321 1 by 8. 12. 458^^ by 15. 
 
 Fractional Divisors. 
 
 INTRODUCTORY. 
 In 4 there are how many thirds? How many groups of 2 thirds? 
 
 4^2=iX?=? 
 * 3 2 
 
 Evidently the quotient of an integer by a unit fraction is 
 the product of integer and denominator. If the fractional 
 divisor is not a unit fraction, its terms may be inverted (the 
 reciprocal taken) and the quotient is then the product of the 
 dividend (fractional or integral) and the reciprocal fraction. 
 
 Illustrations. (1) Divide 425 by 5/6. 
 Solution. 425 -^ 5/6 is equivalent to 425 X 6/5. 
 
 85 6 
 JiQS'X^= 510. 
 
 1 
 
 (2) Divide ^gby|. 
 
 9 3 9 4 
 
 Solution, rs -^ r is equivalent to ,-5 X o • 
 lb 4 lb o 
 
 3 
 ^^-3- 4- 
 
DIVISION OF FRACTIONS. 103 
 
 EXERCISE. 
 
 (Solve mentally if possible.) 
 Divide: 
 
 1. 1/2 by 2/3. 4. 1/6 by 1/7. 7. 5/8 by 7/12. 
 
 2. 2/3 by 1/5. 5. 2/5 by 5/6. 8. 3/8 by 9/16. 
 
 3. 3/4 by 4/5. 6. 3/7 by 4/11. 
 
 Note. Simplify the process in the following by reducing to a common 
 denominator. 
 
 9. 3/4 by 7/8. 12. 426 by 3/4. 15. 3/15 by 5/36. 
 
 10. 2/3 by 5/6. 13. 129 by 2/3. 16. 5/8 by 17/140. 
 
 11. 5/12 by 1/3. 14. 3545 by 5/18. 17. 256 by 19/24. 
 
 Mixed Number Divisors. 
 
 Since mixed numbers may be reduced to improper fractions, 
 this case reduces to that last given. If the fractions are simple 
 it is often advisable to reduce to a common divisor. 
 
 EXERCISE. 
 (Solve mentally if possible.) 
 Find the quotient of: 
 
 1. U -^ U. 3. U -^ U. 5. 5/2 -^ 1|. 
 
 2. 2 ^ If 4. 2| ^ 31. 6. 51 4- 4|. 
 
 Divide : 
 
 7. 5461 by li 11. 416f by 12^ 
 
 8. 672H by 14^ 12. 6546^ by 2f. 
 
 9. 1291 by 66|. 13. 599^ by 7f. 
 10. 5821 by 13f . 14. 59^^ by If. 
 
 Conversion of Fractions. 
 
 Owing to the growing use of the decimal fraction, it is 
 necessary to have facility in expressing fractions in either 
 common or decimal form. It is evident that the denominator 
 of a decimal fraction, though not expressed, is always deter- 
 mined by the decimal point. 
 
104 BUSINESS ARITHMETIC. 
 
 Illustration. Express .0825 as a common fraction. 
 
 Solution. .0825 = 825 ten-thousandths = -|^ = ^. 
 
 lUOUU 400 
 
 EXERCISE. 
 Express as common fractions reduced to lowest terms: 
 
 1. .125. 5. 18. 9. .096. 13. .19275. 
 
 2. .25. 6. 1.54. 10. .392. 14. .088. 
 
 3. .331. 7. 7.45. 11. .875. 15. 62.045. 
 
 4. .0084. 8. 375. 12. .0625. 16. 1.0458. 
 
 The reverse process of converting common fractions to 
 decimal fractions is far more common than that just given. 
 It is performed by fractional reduction, or, more commonly, 
 by executing the represented division. 
 
 Illustration. Reduce 28/85 to the nearest decimal of three places. 
 Sohdion. Carry out the represented division. 
 
 .329 Ans. 
 
 85)28.00000 
 
 25 5 
 '2 50 
 
 170 
 
 800 
 
 765 
 35 
 
 EXERCISE. 
 
 (Solve mentally if possible.) 
 
 1. Convert into decimal form: 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, (.14|), 
 1/8, 1/9, 1/10, 1/11, 1/12, 1/15, 1/16. 
 
 2. Reduce to decimals: 3/4, 2/3, 5/8, 3/8, 1/30, 5/6, 2/9, 5/16. 
 Note. Common measures are frequently expressed as decimals of 
 
 other measures. 
 
 3. Express 1 ft. aa a decimal of 1 yd.; 1 qt. as a decimal of 1 gal. 
 
 EXERCISE. 
 Reduce to equivalent decimals: 
 
 1. To two decimal places with fractional remainder: 3/12, 5/11, 5/24, 
 11/25, 13, 16, Ui 17/48, 5^, 8,^^. 
 
DECIMALS AND FRACTIONS. 105 
 
 2. To the nearest four-place decimal: 47/52, 5/14, 6/28, 9/32, 21/56, 
 9/125, 12/250, 13/80, 11/60. 
 
 3. Express 100' as a decimal of a mile (5280'). 
 
 4. Mechanics often use conversion tables, giving fractions of inches 
 as decimals of a foot. Complete the following table for each 1/8" up to 
 one foot, computing to four decimal places. 
 
 Inch. Fraction of Foot. Decimal. 
 
 1/8 1/96 .0104 
 
 1/4 1/48 .0208 
 
 QUESTIONS FOR DISCUSSION. 
 
 1. Why are not 7ths and 9ths common business fractions? 
 
 2. What has led to the general use of the common business fractions? 
 
 3. Are our common measures adapted to the business fractions? 
 
 4. What fractions are used in measuring cloth? Give similar illus- 
 trations of the use of particular fractions in specific trades or professions. 
 
 5. Why are decimals used in place of fractions, in statistical tables? 
 
 6. Name some fractions that are simpler, for business use, than their 
 equivalent decimals? 
 
 7. What is the shortest method of comparing the relative value of 
 several irregular fractions? 
 
 8. Why is the value of a fraction altered by adding the same amount 
 to both numerator and denominator? 
 
 9. What is true of the product of two reciprocal fractions? Why? 
 
CHAPTER XXII. 
 
 FRACTIONAL RELATIONS OF NUMBERS. 
 
 It is often necessary to determine the fractional relations 
 of numbers. There are two common cases: 
 
 I. To Find What Part One Number Is of Another. 
 
 INTRODUCTORY. 
 
 8 is what part of 12? Reduce to lowest terms. Which number repre- 
 sents the part? Which number represents the 
 whole to which the part is referred? 
 
 2 sevenths is what part of 4-sevenths? 
 
 3/4 in. = ?/8 in. 3/4 m. = what part of 7/8 
 m.? 
 
 INCH 
 
 Vs V4 % h Vs % v& 
 
 I I I I I 
 
 -v^ 
 
 7/8=- SMALLEST UNITS 
 
 3/4= -SMALLEST UNITS H [^ cvidcnt thatthc part that one 
 
 number is of another is expressed as a 
 
 fraction having for a numerator the 
 
 number expressing the part, and for the 
 
 denominator that whole to which it is referred. This is equally 
 
 true of integral, fractional, decimal, abstract or concrete 
 
 numbers. 
 
 Illustration. Example. I buy 3/8 interest in a business from one 
 who owns 4/5 interest in it. What part of his holdings do I buy? 
 
 SoMion. ^^^^^ II = relation. 
 
 1^ = 3/8 -^ 4/5 = 3/8 X 5/4 = 15/32 the part bought. 
 
 4/5 
 
 EXERCISE. 
 
 (Solve mentally if possible.) 
 What part of: 
 
 1. 14 is 22? 3. 1§ yd. is 7^ yd.? 5. 14 lb. is 6^ lb.? 
 
 2. 11 is 15? 4. 1/2 is 1/3? 6. 7 bu. is 3i bu.? 
 
 106 
 
FRACTIONAL RELATIONS OF NUMBERS. 107 
 
 7. $2.50 is 70 c? 9. 4/11 is 13/16? 11. 5.01 is 1.29? 
 
 8. 3/4 is 5/8? 10. $326.50 is $420? 12. 1728 is 1264? 
 
 13. The stands at a ball park have a seating capacity of 13,500, of 
 which 3200, or ?/?, are grand stand seats; 1/4 are bleachers and the balance, 
 or ? seats, are in the paviUon. 
 
 14. In a certain coal mine an investigating commission discovered 
 that for every ton of coal secured for market, 6.45 tons are wasted. What 
 part is wasted? 
 
 15. Compare the speed of two trains, one of which covers four milea 
 in 3 5 minutes, and the other 7 miles in 6 minutes. 
 
 16. 726 tickets were taken up at the entrance to a hall seating 1200. 
 The hall was approximately what part filled? 
 
 Note. WTien an approximate relation is called for, find the nearest 
 simple fractional relation. 
 
 II. To Find the Whole When a Part Is Known 
 
 INTRODUCTORY. 
 
 If 16 ft. equal 1/5 of a quantity, find 5/5 of it. 
 
 If 16 ft. equals 4/5 of a distance, find 1/5 of the distance. Fmd the entire 
 distance. 
 
 Thus the whole may be determined by finding the value of 
 a fractional unit of the given part, and multiplying this value 
 by the number of fractional units in the whole. 
 
 Illustration. 3/8 is 2/3 of what number? 
 Analysis and Solution. 
 
 3/8 = 2 thirds of a number. 
 
 1/2 of 3/8, or 3/16 = 1 third of the number. 
 
 3 X 3/16, or 9/16 = 3 thirds of the number, or the number itself. 
 In brief: If 3/8 = 2/3 of a number, the number = 3/2 of 3/8, or 9/16. 
 
 Find missing values: 
 
 1. 16 = 1/2 of — . 
 
 2. 35 = 5/6 of — . 
 
 3. 42 bu. = 3/7 of — . 
 
 4. $5.60 = 4/5 of — . 
 
 5. 2/3 = 1/2 of — . 
 
 6. 1/2 - 2/3 of — . 
 
 7. 750 lb. = 2/3 of — . 
 
 EXERCISE. 
 
 
 8. 
 
 4280 = 
 
 = 1/4 of — . 
 
 9. 
 
 192 lb 
 
 . = 4/3 of — . 
 
 10. 
 
 127 = 
 
 3/4 of — . 
 
 11. 
 
 2420 lb. = 3/15 of - 
 
 12. 
 
 $752 = 
 
 = 3/8 of — . 
 
 13. 
 
 592 = 
 
 4/13 of — . 
 
108 BUSINESS ARITHMETIC. 
 
 14. A railway speed of 14 miles in IS minutes is equivalent to a speed 
 of how many miles per hour? 
 
 15. A steam shovel excavated 47^ cu. yd. of earth in 4| minutes. 
 This was at the rate of how many cubic yards per hoiu"? 
 
 16. A yield of 26 bushels on a trial plot of 2/5 of an acre is equivalent 
 to a yield of how many bushels per acre? 
 
 17. A certain cloth shrinks 1/32 in dyeing. To have 930 yd. of 
 finished product, one must dye how many yards? 
 
 18. A man who offered me $750 for a 3/8 interest in a certain patent, 
 evidently valued the patent at % — . 
 
 19. A contractor has completed 2/5 of a grading contract in 48 working 
 days. At the same rate, how many days work remain? 
 
 20. A manufacturer replaces his old machines with new ones of one- 
 fifth greater productive power. He runs on a ten hour day basis. His 
 new machines should give him his former weekly output with a saving of 
 how many hours. 
 
 ORAL FACILITY EXERCISE. 
 
 Find the single term equivalent to each group of two or more tei*ms. 
 Distinguish amounts less than zero by the term minus, thus 3 — 4 = — 1. 
 
 1. Halves and thirds. Two is what part of three? Three of two? 
 Fmd 1/2 + 1/3, 1/3 + 1/2, 1/3 of 1/2, 1/3 - 1/2, 1/2 - 1/3, 1/2 - 2/3, 
 1/3 -^ 1/2, 1/2 -^ 1/3, 1/3 ^ 2/3, 1/2 •- 2/3, 1/3 X 1/2, 1/2 X 2/3, 2/3 
 of 1/2, 2/3 divided by 1/3, 1/2 of .4, 2/3 of 15, 6 = 1/2 of ? , 12 = 1/3 of ? , 
 1/2 = 1/3 of ? , 1/3 = 1/2 of ? , 1/3 is what part of 1/2, 1/2 is what part 
 of 2/3? 
 
 2. Thirds, sixths and twelfths. Find 1/3 + 1/6, 2/3 - 5/6, 1/6 - 2/3, 
 1/6 ^ 1/3, 5/6 -^ 5/3, 2/3 -i- 1/6, 2/3 ^ 4/6, 1/3 of 1/6, 2/3 X 5/6, 1/3 
 + 1/16 + 1/12, 1/6 -^ 1/12, 1/3 ^ 1/12, 7/12 ^ 1/3, 5/12 ^ 5/6, 5/6 
 - 5/12, 1/3 of 2.4, 5/6 of 72, 7/12 of 18, 4 == 1/6 of ?, 1.5 = 5/12 of ?, 6 
 = 1/12 of 1/2 of ?, .1/6 is what part of 1/12? Of 1/3? Of 5/6? 5/6 is what 
 part of 2/3? Of 7/12? 1/6 is 1/12 of ? 5/12 is 1/3 of ? 
 
 3. Halves, fourths, eighths and sixteenths. Find 1/2 + 3/4, 3/4 + 5/8, 
 7/8 - U, 9/16 - 5/8, 1/4 -^ 3/16; 3/4 of 3/16, 5/8 X 1/2, 5/8 of 3/4, 
 1/16 - 3/8, 5/16 ^ 3/8, 3/8 of 5/16, 1/4 + 1/8 -4- 1/16, 1/2 times 7/16, 
 1/2 -^ 9/16, 1/2 X 15/16. 1/2 is what part of 7/8? Of 15/16? Of 5/4? 
 3/4 is what part of 3/8? Of 15/16? Of If? What is 1/2 of 3.6? Find 
 3/4 of 64. What is 5/8 of 12? Find 3/16 of 80. 11/16 of 3.2 = ? 14 
 = 1/2 of ? 10 = 5/16 of ? 12 - 3/8 of ? 12 = 9/16 of ? 1/2 = 3/4 of ? 
 1/8 = 1/2 of ? 5/8 = 3/4 of ? 9/16 =- 1/8 of ? 9/16 = 3/4 of ? 
 
FRACTIONAL RELATIONS OF NUMBERS. 109 
 
 4. Use the answer, at each step, as the basic value for the next com- 
 putation. 
 
 4 + 12 = ? Add 8. Divide by 2. Multiply by 1/2. Divide by 1/3. 
 Multiply by 2/3. Divide by 4. Increase by 2/3 of itself. Multiply by 
 4/5. Divide by 2/3. Multiply by 3/4. Change to eighths. What ia 
 the final result? 
 
 Teacher's Note. Similar original exercises should be prepared by 
 the teacher and given to the class, in order to cultivate a high degree of 
 accuracy in handling the simple business fractions. 
 
CHAPTER XXIII. 
 
 ALIQUOT PARTS. 
 
 Numbers that exactly divide a given number are termed 
 its aliquot parts. Thus 1, 1^, 1/5, 2, 3, etc., are aliquot parts 
 of 6. 
 
 ORAL EXERCISE. 
 
 1. Name 6 aliquot parts of 12. What fractional part is each aliquot 
 part? 
 
 2. Find 1/4 of 200; of 10. Find 1/8 of 10, of 100, of 1000, etc. 
 
 3. What parts of $1.00 are 50c, 10c, 33 ic, 20c, 2c? 
 
 4. . Which of these numbers are ahquot parts of 10: 1, I5, 2, 2^, 3, 3i, 
 31, 4? 
 
 5. Complete the following table: 
 
 Aliquot Part. 
 
 Number. 
 
 1/2- 
 
 1/3 
 
 1/4 
 
 1/5 
 
 1/6 
 
 1/8 
 
 1/9 
 
 1/10 
 
 1/12 
 
 1/15 
 
 1/16 
 
 1/20 
 
 1. 
 10. 
 100. 
 1000. 
 
 50 
 
 33i 
 
 25 
 
 20 
 
 16! 
 
 12^ 
 
 IH 
 
 10 
 
 8i 
 
 6f 
 
 6i 
 
 5 
 
 1. Multiplication. 
 
 Aliquot parts are frequently used to simplify multiplication 
 and division. The aliquot parts of 10c and $1.00 are espe- 
 cially useful in business. 
 
 INTRODUCTORY. 
 
 1. Express as aliquot parts of $L00, 25c, 12ic, 2^0, 4c. 
 
 2. How does the cost of several articles @ 25c compare with their 
 cost @ $1.00? 
 
 3. The cost of 48 yd. @ 25c = the cost of 48 yd. at what price? 
 
 4. 622 X 33ic = 522 X $1/3 = 1/3 of $ ? = $ ? 
 
 110 
 
ALIQUOT PARTS. Ill 
 
 5. The cost of 522 articles @ $1.00 is $ ? The cost at 33 fc is 
 what part of the cost at Sl.OCV 
 
 It is evident that the cost of an article at any aliquot price 
 of one dollar is its simple fractional part of the cost at one 
 dollar. 
 
 Illustration. Find the cost of 568 yd. cloth @ 12^c per yd. 
 Analysis and Solution. 12^c = $1/8. 
 
 Cost of 568 yd. @ 12^c = Cost of 568 yd. @ $1/8 
 = 1/8 of $568 = $71. 
 Note. Notice that in some of the following examples the aliquot 
 lactor is the figure representing the quantity rather than the price. 
 
 EXERCISE. 
 
 Find cost of: 
 
 1. 840 yds. wash silk @ 25c; at 33ic. 
 
 2. 1620 yds. plaids @ $1.25; at 33i; at 50c. 
 
 3. 580 yds. cotton at 6fc, at 6^c. 
 
 4. 864 yds. prints at 12§c; 16ic, 20c. 
 
 5. 164 yds. Iming at 12|c, 10c, 8ic 
 
 6. 422 yds percale at 25c, 20c, 16|c, 12^0. 
 
 7. 360 doz eggs at 12^c, at 25c. 
 
 8. 560 lb. lard at 10c; 8ic, 12^c. 
 
 9. 48 lb. chocolate @, 25c, 16fc, 33ic, 
 
 10. 450 lb. tapioca @ Q\c, S^c. 
 
 11. 62 hats @ $2.50 (aliquot part of $10). 
 
 12. 156 T. old iron @ $16f ton. 
 
 13. 1520 A. @ $12.50 per A. 
 
 14. 1500 blank books @ 25c. 
 
 15. 196 doz. buttons @ 16fc. 
 
 16. 124 pr. shoes @ $1.25 ($l+25c). 
 
 17. 2500 pineapples @ 12ic. 
 
 18. 1425 posts @ 33ic. 
 
 19. 625 yds. wire netting @ 25c yd. 
 
 20. 364 lbs. coffee @ 33^0. 
 
 21. 48 lbs. tea @ 50c; @ 33^0. 
 
 Note. The principle of interchanging multiplier and multiplicand is 
 frequently employed. Thus the cost of 25 yd. @ 88c is equivalent to 
 the cost of 88 yd. @ 25c. 
 
112 BUSINESS ARITHMETIG. 
 
 Compute the cost of: 
 
 22. 33i yd. @ 78c. 24. 25 bu. beans @ $2.60 (260c) 
 
 23. 16f yd. lace @ 96c. 25. 50 bu. clover seed @ $4.20. 
 
 ILLUSTRATIVE EXERCISE. 
 100 X 720 = ? 
 
 100 
 25 X 720 = what part of 100 X 720? 25 X 720 = -y- X 28 = ? 
 
 Suggest the short method. 
 
 It is evident that the aliquot method greatly simplifies computation, 
 often substituting oral for written work. 
 Illustration. Multiply 672 X 33i 
 
 Solution-aliquot. Solution-regular. 
 
 672 X 33i = 1/3 of 672 X 100.. 672 
 
 100 X 672 = 67200. 33i 
 
 67200 -^ 3 = 22400 224 
 
 2016 
 2016 
 22400 
 ORAL EXERCISE. 
 
 Suggest short methods for multipHcation by 12|, 6i, 6f , 25, 50, 
 To multiply by 12^, multiply by 100 and divide by 8. 
 
 1. 
 
 Suggest she 
 
 ►rt me 
 
 161. 
 
 
 
 Illustration. 
 
 Tom 
 
 WTiy? 
 
 
 
 Find the products of: 
 
 2. 
 
 25 X 36. 
 
 
 3. 
 
 33i X 42. 
 
 
 4. 
 
 12i X 184. 
 
 
 5. 
 
 61 X 960. 
 
 
 6. 
 
 16! X 640. 
 
 
 7. 
 
 20 X 85. 
 
 
 8. 
 
 33i X 456. 
 
 
 9. 
 
 6i X 64. 
 
 
 10. 
 
 6! X 90. 
 
 11. 
 
 50 X 588. 
 
 12. 
 
 25 X 692. 
 
 13. 
 
 25 X 12.4. 
 
 14. 
 
 8i X 72. 
 
 15. 
 
 3i X 420. 
 
 16. 
 
 12^ X 8.48. 
 
 17. 
 
 2i X 880. 
 
 WRITTEN EXERCISE. 
 
 Find the products of: 
 
 1. 426 X 25 4. 6! X 968. 
 
 2. 33i X 2672. 5. 6\ X 884. 
 
 3. 4892 X m. 6. 25 X 672. 
 
ALIQUOT PARTS 113 
 
 7. 2.5 X 876. 12. 6i X $6.40. 
 
 8. 250 X 968. 13. 25 X 8.004. 
 
 9. 25 X 48/49. 14. 2^ X 8.4. 
 
 10. 33i X 66f. 15. 3i X .0696. 
 
 11. 6f X $1.50. J6. 333i X 6.0096. 
 
 MULTIPLES OF ALIQUOT PART. 
 
 The fraction representing an aliquot part has a numerator 
 
 of ? 
 
 Find 1/2 of 100; 1/8 of 100; 3/8 of 100; 1/3 of 100; 2/3 of 100; 
 1/4 of 100; 3/4 of 100. 
 
 12i X 248 = . 37| = X 12^. 
 
 S7i = what part of 100. 37^ X 248 = what part of 
 100 X 248? 
 
 Suggest short methods for multiplying by multiples of 
 aliquot parts. 
 
 ORAL EXERCISE. 
 
 1. Multiply 963 by 66i 
 
 Solution. 66f = 2/3 of 100. 66| X 963 = 2/3 of 963 X 100 = 32100. 
 
 Note that the fractional part of 2/3 may be taken before or after mul- 
 tiplication by 100. Also one may multiply by 100 and subtract 1/3 of 
 product. 
 
 Multiply: 
 
 
 2. 124 X 125. 
 
 10. 7i X 640. 
 
 3. 68 X 75. 
 
 11. 66f X 12. 
 
 4. 42 X 66|. 
 
 12. 125 X 86. 
 
 5. 16 X 37i. 
 
 13. 168 X 37i 
 
 6. 24 X 66f . 
 
 14. 96 X 112|. 
 
 7. 24 X 62i 
 
 15. 1220 X 150. 
 
 8. 48 X 87i. 
 
 16, 84 X 75. 
 
 9. 54 X 83i 
 
 17. 328 X 37i. 
 
 
 WRITTEN EXERCISE. 
 
 1. 75 X 874. 
 
 5. 871 X 920. 
 
 2. 661 X 960. 
 
 6. 83i X 97.3. 
 
 3. 37^ X 8480. 
 
 7. 71 X 4520. 
 
 4. 62^ X 7244. 
 9 
 
 S. 3i X 872. 
 
114 BUSINESS ARITHMETIC. 
 
 9. 125 X 1846. 13. 333iX 942 qt. 
 
 10. 133i X 276. 14. 137^ X 840 bu. 
 
 11. 1121 X 488. 15. 183i X 6960. 
 
 12. 233i X 606. 16. 166f X 825. 
 
 ILLUSTRATIVE EXERCISE. 
 
 1. Find 1/4 of a dollar, 3/4 of it, 2/3, 3/8, 5/8, 4/3, 7/8, 3/40. 
 
 2. What part of a dollar are: 25c, 75c, 66|c, 7^c, 15c, 70c, $1.25, 
 $2.50? 
 
 3. The cost of 240 yd. @ 75c is what part less than the cost at $1.00? 
 How many times the cost at 25c? 
 
 4. Suggest two short methods for finding the cost at 75c. 
 
 It is evident that the cost at some multiple of an aliquot 
 price is obtained by taking the aliquot fractional part of the 
 total cost at $1.00. 
 
 Illustration. Find the cost of 726 yd. @ 66fc. 
 Solution. 726 yd. @ $1.00 cost $726. 66fc = 2/3 of $1.00. 
 2/3 of $726 = $484, cost. 
 
 ORAL EXERCISE. 
 
 1. Suggest short methods for finding the cost @ 80c, 37ic, Sl.lOfc. 
 83ic, $1.25. 
 
 Find the cost of: 
 
 2. 290 lbs. @ 40c, 10c, $1.50. 
 
 3. 42 yds. @ 66|c, 50c, 25c. 
 
 4. 85 yds. @ 80c, 10c, 50c. 
 6. 96 lb. @ 62ic, 75c, 83ic. 
 
 6. 24 T. @ $7.50, $2.50, $8.00. 
 
 7. 136 qt. @ 25c, 37ic, 75c. 
 
 8. 48 oz. @ Sic, 6fc, 66fc, $1.33i 
 
 9. 120 sq. yds. @ 7ic, 40c, 75c, 37^c, 66fc. 
 
 10. 90 cu. yds. @ $1.33^ $2.66f, $15,661- 
 
 11. 16 T. @ $2.50, $25, $3.75, $6.25. 
 
 12. 45 ft. @ 20c, 40c, $1.20, $1.60. 
 
 13. 75 yd. @ $1.24, $3.36, $2.40, $.88. 
 
 14. 12i T. @ $4.80, $5.40, $6.00, $12.00. 
 
 15. 37i yds. @ 64c, 72c, 90c, 24c. 
 
ALIQUOT PARTS. 115 
 
 DIVISION. 
 INTRODUCTORY EXERCISE. 
 
 1. $1.00 will buy how many pounds @ 25c; @ 33ic; @ 50c? 
 
 2. $1.00 will buy how many yd. of prints @ 25c? $240 will buy 240 
 times yd. or yd. 
 
 3. Suggest a method of finding quantity when cost and aliquot price 
 are given. 
 
 4. 37^c = what part of a dollar? $840 -J- 37ic = $840 ^ $? = 840 
 X? = — 
 
 5. Suggest a method of division when the price is a multiple aliquot 
 price. 
 
 It is evident that by substituting the aliquot fraction, the 
 operation is redi^ced to that of division by fractions, and thus 
 to multipUcation. 
 
 Illustration. $360 will buy articles @ 33ic; or articles 
 
 @ 37|c. 
 
 Solution. (1) 33ic = $1/3. $360 ^ $1/3 = 360 X 3/1 = 1080, the no. 
 of articles that can be purchased. 
 
 (2) 37^ = $3/8. 360 -i- 3/8 = 360 X 8/3 = 960, the number 
 of articles that can be purchased. 
 
 ORAL EXERCISE. 
 
 1. 
 
 State 
 
 rules for division by 375C, 45c, 75c, 66fc, 
 
 $1.16f. 
 
 
 2. 
 
 Suggest methods of checking such divisions. 
 
 
 
 
 Cost. 
 
 Price. Quantity. Cost. 
 
 Price. 
 
 Quantity. 
 
 3. 
 
 $72 
 
 25c ? 9. $5.20 
 
 $1.25 
 
 
 4. 
 
 40 
 
 12^c ? 10. 48 
 
 .37^ 
 
 
 5. 
 
 46 
 
 16!c ? 11. 60 
 
 1.33i 
 
 
 6. 
 
 45 
 
 6fc ? 12. 54 
 
 1.50 
 
 
 7. 
 
 21 
 
 8ic ? 13. 918 
 
 2.25 
 
 
 8. 
 
 126 
 
 50c ? 14. 468 
 WRITTEN EXERCISE. 
 
 .76 
 
 
 1. 
 
 $758 will buy — lb. @ 80c. 
 
 
 
 2. 
 
 $296 will buy — yd. @ 37ic. 
 
 
 
 3. 
 
 $540 will buy — bu. @ 62^ 
 
 
 
 4. 
 
 $925 will buy — A @ $12.50. 
 
 
 
116 BUSINESS ARITHMETIC. 
 
 5. $5296 will buy — bu. @> 75c. 
 
 6. — cu. yd. © 62^c cost $1600. 
 
 7. — oz. @ 7ic cost $52. 
 
 8. — lb. @ 16f c cost $615.50. 
 
 9. — lb. @ 12ic cost $6040. 
 10. — lb. @ $2.75 cost $810. 
 
 ILLUSTRATIVE EXERCISE. 
 
 1. 1.25 is what part of 100. The quotient obtained by dividing by 
 25 will be times the quotient by dividing by 100. 
 
 2. 7620 4- 100 = . 12^ = of 100. 7620 -^ 12^ = 
 
 times 7620 -h 100. 
 
 3. Suggest short methods for division by aliquot parts of 100, of 1000. 
 
 4. 37i is contained times in 100. 
 
 5. Explain: 9640 ^ 37| = 8/3 of 1/100 of 9640 = . 
 
 6. Suggest short methods for division by 62^, 66f , 7^, 3i, 166|, 125, 
 133|. 
 
 ORAL EXERCISE. 
 Find the quotient of: 
 
 1. 2500 ^ 20. 7. 1600 -J- 80. 
 
 2. 4200 -r- 25. 8. 920 ^ 33i. 
 
 3. 3850 -r- 50. 9. 600 -^ 66f. 
 
 4. 7200 -^ 33i 10. 1200 -J- 133i. 
 
 5. 240 ^ 12i by 16!. H. 800 ^ 125. 
 
 6. 1200 -5- 75. 12. 2760 ^ 150. 
 
 WRITTEN EXERCISE. 
 
 1. 17800 -^ 37i 4. 82984 -^ 1.33i. 7. 84020 -^ 6i. 
 
 2. 52900 ^ 62i. 5. 67500 -^ 125. 8. 94368 -5- 116f. 
 
 3. 4890 ^ 66i 6. 1284 4- 87^ 9. 9218 -^ 16i. 
 
 Price by the Hundred or Thousand. 
 
 In the many cases where articles are sold by the hundred or 
 thousand, computations are simplified by '^ pointing off." 
 
 Illustration. (1) Compute the cost of 920 lb. lead @ $4.00 per c. 
 920 lb. = 9.2 cwt. $4.00 X 9.2 = $36.80 cost. 
 
ALIQUOT PARTS. 117 
 
 EXERCISE. 
 
 (Solve mentally if possible.) 
 
 Compute the cost of: 
 
 1. 425 lb. @ $5.00 per c. 6. 4580 lb. @ S5.50 per c. 
 
 2. 675 lb. @ $12.50 per c. 7. 546 lb. fertilizer @ 85c per c. 
 
 3. 1250 yd. @ $45.00 per c. 8. 2450 ft. lumber @ $33,331 per M. 
 
 4. 786 ft. @ $1.25 per c. 9. 5260 ft. pine @ $40 per M. 
 
 5. 1900 ft. @ $1.33i per c. 10. 4690 ft. oak @ $50 per M. 
 
CHAPTER XX. 
 PROBLEM ANALYSIS AND SOLUTION. 
 
 In general, five steps should mark the analysis and solution 
 of a problem. 
 
 1. A careful reading of the problem to determine what is 
 given and what required. 
 
 2. Analysis, to determine the relations between what is 
 given and what required. 
 
 3. Selection of process and of factors to be used. 
 
 4. The computation. 
 
 5. Checking — of reasoning and of computation. 
 
 A problem should be thoroughly understood before its 
 solution is attempted. The process selected should be the 
 simplest and most direct. The computation should be 
 performed and set down in the briefest and most direct 
 manner — factoring, cancellation and short methods being 
 used to save time and energy. In the written statement, care 
 should be taken to use symbols accurately, and to exercise 
 judgment in the naming of important values. 
 
 There is no uniformity in the method of solution of problems. 
 It is essential, however, that the selected solution should 
 involve no unnecessary labor, and that the results obtained 
 by it should be easily checked. All solutions of applied 
 arithmetic problems require the application of common sense 
 analysis, but this analysis is a simple matter, as the following 
 illustrations will show. 
 
 Illustrations. (1) Andrews offers a 3/8 acre plot of land for $72. 
 This price is equivalent to a price of how many dollars per acre? 
 Analysis and Solution. 
 
 1. Given the price of a part, to determine the price of the whole. 
 
 118 
 
PROBLEM ANALYSIS AND SOLUTIONS. 119 
 
 2. If 3/8 of an acre cost $72, 8/8 acres mil cost 8 times one-third of $72. 
 
 3. All given factors required for solution. 
 
 24 
 
 4. 8 X^^ = $192. The value of one acre is $192. 
 
 5. Check. 3/8 of $192 = $72. 
 
 (2) Working at the rate of 100 strokes per minute, a power pump dehvers 
 iOO gal. of water per minute. How much must the speed be increased 
 to pump 30,000 gal. per hour? 
 
 1. Given present speed and output per minute, and required output 
 per hour, to find speed increase. 
 
 2. If the output is 400 gal. per minute, it is 60 times 400 gal. per hour. 
 The speed must be increased by the same fractional part that the required 
 output is greater than the present output. 
 
 3. By multiplication, find the hourly output. Compare the two out- 
 puts to find increase. By fractional multiplication determine increase in 
 speed. 
 
 4. Soluiion. 
 
 60 X 400 gal. = 24,000 gal., capacity per hour. 
 
 94 000 ~ 4 ' B,equired capacity is 5/4 of present capacity, or 1/4 greater. 
 
 1/4 of 100 strokes = 25 strokes. Therefore, the speed must be in- 
 creased 25 strokes per minute. 
 
 5. Check. An increase of 1/4 in speed wiU increase the delivery per 
 minute 1/4, or from 400 to 500 gal. A deUvery of 500 gal. per minute 
 = a delivery of 60 X 500 gal. per hour, or 30,000 gal. 
 
 Note. Numbers 1,2 and 3 are usually distinct mental processes. They 
 are not represented in paper solutions. 
 
 ORAL EXERCISE. 
 
 1. 32 is 4/7 of what number? 
 
 2. A pump that raises 6400 gal. per day of 8 hours will raise gal. 
 
 if run 10 hours. 
 
 3. At 84c per lb., what is the cost of 3/4 lb. of pepper? 
 
 4. A factory is insured in one company for $6000, and for $4000 in a 
 second. If damaged by fire to the extent of $3600, what loss should each 
 company pay? 
 
 5. An improvement to a certain steam engine is warranted to reduce 
 the coal consumption 2/5. The present consumption being 15 T per day, 
 
120 BUSINESS ARITHMETIC. 
 
 this means a saving of T., or a saving in money, at $3.40 per T, 
 
 of $ . 
 
 6. A train running 1 mile in 4/5 of a minute is running at the rate of 
 miles per hour? 
 
 7. The report of a cannon is heard 20 sec. after firing. How far 
 distant is the cannon if sound carries 65,000 feet per minute? 
 
 8. Divide 10 into two parts, one being three times the other. 
 
 EXERCISE. 
 
 1. A 5 inch line is used on paper to represent a distance of 60 feet. 
 The paper length is what part of the true length? 
 
 2. By improvements in machinery, the output of a factory is increased 
 3/4. If, at the same time, the working time is reduced from 10 to 9 hours, 
 what is the net increase in output? 
 
 3. A blend of tea consists of two grades, the first forming 3/8 of the 
 mixture. By increasing the proportion of second grade 1/4, what part 
 remains first grade? 
 
 4. A pipe discharges 32| gal. per sec. How many minutes will be 
 necessary to discharge 10,000 gal.? What is its discharge rate per hour? 
 
 5. A dredge averages If cu. yard of material per minute. At 
 that rate, it will remove how many cu. yd. in a working day of 8 hours. 
 
 6. How much water must be added to a solution containing 1/10 
 ammonia, to make the solution 1/30? 
 
CHAPTER XXI. 
 INVOLUTION AND EVOLUTION. 
 
 I. Involution. 
 
 The products obtained by taking any number two or more 
 times as a factor are termed powers of that number. The 
 process of finding the powers of numbers is termed involution. 
 
 If the number is taken twice, the product is termed the 
 square of the number; if taken three times, the cube of the 
 number; if taken more times, the corresponding power of the 
 number, as the fourth power, fifth power, etc. 
 
 Powers are expressed by writing the number with a small 
 raised figure or exponent, showing the times it is taken as a 
 factor. 
 
 Illustrations. 
 
 2 X 2 is written 2^. 
 
 2X2X2X2is written 2*. 
 
 Since the exponents represent the number of times the given 
 number is taken as a factor, the product of two powers of the 
 same number may be represented by writing the number with 
 the sum of the given exponents as the exponent of the product. 
 This principle is of value, also, in reckoning high powers of 
 numbers. 
 
 Illustrations. 
 
 (1) Represent the product of 2' and 2*. 
 Solution. 2' = 2 X 2 X 2. Short method. 
 
 2* = 2 X 2 X 2 X 2. 23 X 2< = 23+< = 2\ 
 
 23 x2< =2X2X2X2X2X2X2= 2' 
 (2) Find the value of 3«. 
 Solution. 38 = 3*+* = 3* X 3*. 
 
 3* = 3 X 3 X 3 X 3 = 81. 
 
 38 = 3* X 3* = 81 X 81 = 6561. 
 121 
 
122 BUSINESS ARITHMETIC. 
 
 ORAL EXERCISE. 
 
 Determine the value of: 
 
 1. 3'. 3. 53. 5. 8». 7. 12*. 
 
 2. 2«. 4. 4». 6. 4*. 8. l(fi. 
 
 9. Find the squares of 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 20, 
 25, 100, 120, 400, .2, .5, .12, .006, 1/4, 3/8, 2/3. 
 
 10. Show that a square must end in 0, 1, 4, 5, 6, or 9. 
 
 11. Show that a cube may end in any of the digits. 
 
 EXERCISE. 
 
 Compute, by the simphfied method, the value of: 
 
 1. 168. 3. 6.05^ 5. (f)«. 
 
 2. 121*. 4. .0036«! 6. (B*. 
 
 11. Evolution. 
 
 The process of determining a number from one of its powers 
 is termed evolution. If the number is obtained from its square 
 it is termed the square root; if obtained from the cube it is 
 termed the cube root. Roots of higher powers are named horn. 
 the powers, as fourth root, fifth root, etc. 
 
 Illustrations. The square of 6 is 6 X 6, or 36. 6 is the square root 
 of 36. 
 
 The cube of 4 is 4 X 4 X 4, or 64. 4 is the cube root of 64. 
 
 Roots are expressed with the symbol l/ written over the 
 number of which the root is to be found. The power to be 
 taken is shown by a numeral, except in the case of the square 
 root. 
 
 Illustrations. The square root of 225 is expressed V225, but the 
 square root equals 15. 
 
 The cube root of 729 is expressed ^729, but it equals 9. 
 
 INTRODUCTORY EXERCISE. 
 
 1. Name at sight the equal factors or square roots of 4 (4 = 2 X 2; 
 square root is 2), 9, 16, 25, 36, 64, 81, 100, 10,000, 121, 144. 
 
 2. Find the squares of each digit. How many places in each case? 
 
INVOLUTION AND EVOLUTION. 123 
 
 3. Find the squares of 1, 10, 100, 10,000, .1, .01, .001. tiow do the 
 numbers of places in the squares compare with those in the roots? 
 
 It is evident, from these examples, that the number of 
 places in any square is twice, or one less than twice, the 
 number of places in the root. This fact is important, as it is 
 possible, by pointing off the square from the decimal point 
 into periods of two places, to determine the number of places 
 in the root. 
 
 The method of determining the square root may be under- 
 stood, now, by studying the reverse process, using the method 
 of multiplication by parts. 
 
 Example. Find the square of 86. 
 
 Solution. 86 = 80 + 6. 
 
 80 + 6 86 
 
 80 + 6 86 
 
 (80 X 6) + 62 516 
 
 802 ■!■ (go X 6) 6880 
 
 802 _|_ 2(80 X 6) + 62 = 7396 
 
 It is evident that the square of a number, composed of tens 
 
 and units, is equal to the square of the tens plus twice the 
 
 product of the tens multiplied by the units plus the square of 
 
 the units. It is evident, also, that any number may be 
 
 divided into two parts and its square determined in exactly 
 
 the same way. This composition of the square is made use 
 
 of in determining the root. 
 
 Illustrations. (1) Compute the square root of 1296. 
 Solution. 
 
 12'96(30 + 6 = 36, root. 
 302 =9 00 
 
 3 96(6 
 Trial div. 2 X 30 = 60 
 
 2 X 30 X 6 =360 
 
 62 = 36 3 96 
 
 Analysis. 
 
 Since there are 4 places in the square there are 1/2 of 4, or 2 places in 
 the root. Point off by 2's. 12 represents the ten's place in the root, and 
 
124 
 
 BUSINESS ARITHMETIC. 
 
 96 the unit's place. By trial, the largest square in 12 is the square of 3. 
 .'. the largest square of the tens is 30^, or 900. Subtracting the square of 
 the tens, there are left 396 units. Since the square of the tens has been 
 subtracted, there is left in the perfect square twice the product of tens and 
 units plus the square of the units. As a trial divisor, use twice the tens, 
 or 60. 60 is contained in 396 six times. Try 6 as the required unit. 
 2(30X6) =360; 6X6 = 36. The sum is 396. Evidently 6 is the required 
 unit, and 36 the required root. 
 
 Check. 36 X 36 = 1296. 
 
 (2) Find the square root of 1.6384. 
 
 Solution 
 
 1.63'84(1.28 root. 
 12 j_ 
 
 2X1.0 = 2.0, trial, 63(0.3 
 2X1.0X .2 =40 
 
 .22-= .04 44 
 2X1.20 = 2 40, tFial, .19 84(0.8 
 2 X 1.20 X. 08 =.1920 
 
 .082 = .0064 19 8 4 
 Check. 1.28 X 08"= 1.6384 
 
 or 
 
 Analysis. 
 
 Point off by 2*8, each way from 
 the decimal point. If a perfect 
 square, the root will contain one 
 whole number and two decimals. 
 1 is the nearest square of the unit 
 period and the first figure of the 
 root. Subtract the unit and bnng 
 down the first decimal group. 
 Twice 10 tenths, or 20 tenths is the 
 trial divisor. The resulting quo- 
 tient, 3, is found to be too large, so 
 the next lower number, 2, is taken 
 as the next figure of the root. 
 Complete the square, considering 1 
 as the ten, and .2 as the units. 
 Subtract the value of the com- 
 pleted square from .63 leaving .19 
 and bring down final group. As a 
 trial divisor, use twice the portion 
 of the root already found, or 2.40, 
 obtaining the final figure, .08. 
 Complete the square. 
 
 Thus, to determine the square root: 
 
 1. — Beginning at the decimal point separate the number, 
 each way, into groups of two figures. (The outermost period 
 on the left may have only one place.) 
 
 2. — Find the greatest square of the left hand period. Its 
 root is the first figure in the required root 
 
 Treat as whole number. 
 
 
 1.63'84(1.28 root, 
 
 1« = 1 
 
 1 
 
 Trial, 2X10 
 
 63 
 
 2X10X2=40 
 
 
 22= 4 
 
 44 
 
 Trial 2X120 
 
 19 84 
 
 2X120X8 = 1920 
 
 
 82= 64 
 
 19 84 
 
 
 
 
INVOLUTION AND EVOLUTION. 125 
 
 3. — Subtract the square from its period and bring down the 
 next period. As a trial divisor use twice the root value already 
 found, annexing a cypher. Considering the quotient as the 
 new unit, complete the square by taking twice the product of 
 the tens multiplied by the units plus the square of the units. 
 
 4. — Subtract this value from the last remainder; bring down 
 the next group, and proceed as before. 
 
 Noi'E. (1) In the case of fractions, either take the square root of 
 numerator and denominator separately, or reduce the fraction to the 
 decimal form. 
 
 (2) When the remainder, at any stage, will not contain the trial divisor, 
 write a cypher in the root and bring down the next group. If there are 
 no further periods, periods of two cyphers may be added to the original 
 number and the root determined to any desired number of decimal places. 
 
 Since the product of a second power by itself is a fourth 
 power, it is evident that the square root of a fourth power is 
 the square of the number. If the square root of this square 
 is then taken, the result will be the fourth root of the number. 
 
 EXERCISE. 
 
 Find the square root of: 
 
 1. 256 5. 1025.9209 9. 29,506,624- 
 
 2. 7225. 6. 52.9984. 10. 329,968,821 (2 dec. pi.). 
 
 3. 7744. 7. 134,944.81. 11. .008,907,386,72 (5dec.pl.). 
 
 4. 16,641. 8. .000,000,494,209. 12. 56,847.02938176 (4 dec. pi.). 
 
 m 2209 17,161 85,264 
 
 7569 3969 23,104 1,580,049 
 
 17. Reduce the following to decimals and take the approximate square 
 root to the thu-d decunal place: (a) 5/14; (6) 7/88; (c) 1/21. 
 
 By continued square root, find: 
 
 18. The fourth root of 10,098, 039, 121. 
 
 19. The fourth root of 3.8416. 
 
 CUBE ROOT. 
 INTRODUCTORY EXERCISE. 
 
 1. WTiat is the cube of any number? What is the cube root of the cube? 
 
 2. Find the cubes of the digits. What is the largest number of places 
 in the results? 
 
126 BUSINESS ARITHMETIC. 
 
 3. Find the cubes of 1, 10, 100, 1000, .1, .01. How many places in 
 each cube? 
 
 It is evident that the cube root of a number is one of its 
 three equal factors. It is evident, also, that each period of 
 three places in the cube stands for one place in the root. 
 
 As in the case of square root, the method of determining 
 the cube root may be understood from a study of the reverse 
 process. 
 
 Illustration. Find the cube of 46. 
 Solution. Let 46 be written 40 + 6. 
 
 40 + 6 46 
 
 40+6 46 
 
 (40 X 6) + 62 276 
 
 40^ + (40 X 6) 1840 
 
 402 _|. 2(40 X 6) + 62 2116 
 
 40+6 46 
 
 (402 X 6) + 2(40 X 62) + 6' 12696 
 
 408 ^ 2(402 X 6) + (40 X 62) 84640 
 40^ + 3(40* X -6) + 3(40 X 62) + 6» = 97336 
 
 From a study of the product it is evident that the cube of 
 a two-place number equals the cube of the tens plus three 
 times the product of the square of the tens multiplied by the 
 units, plus three times the product of the tens multiplied by the 
 square of the units, plus the cube of the units. It is now a 
 simple matter to reverse the process and determine the cube 
 root. 
 
 Illustration. Find the cube root of 97,336. 
 Solution. 
 
 97'336(46 Separate into periods of three places. 
 
 43 64 The nearest cube root of the tens group 
 
 Trial 4800 33 336 is 4. Subtract the cube of 4 from 97 
 
 3X402X6=28800 and bring down the next group. Asa 
 
 3X40 X 62 = 4320 trial divisor use three times the square 
 
 6' = 216 33336 of the tens, obtaining 6 as a trial unit. 
 
 Check. 46X46X46 = 97,336. The 33,336 units must be made up of 
 
 three times the product of the square of 
 the tens multiplied by the units, plus three times the tens multiplied by 
 the square of the units, plus the cube of the units. These values are 
 
INVOLUTION AND EVOLUTION. 127 
 
 determmed separately and totalled as shown. Since they total 33,336, 
 the 6 is the correct unit value of the root, and 46 is the cube root of the 
 number. 
 
 As in the case of square root, by always considering the 
 figures of the root, which have been determined at any point 
 in the solution, as tens, it is possible to determine the next 
 figure of the root. 
 
 Illustration. Find to three decimal places, the cube root of 
 32.890033664. 
 
 Solution. 32.890'033'664(3.203 + root. 
 
 3' 27 
 
 Trial, 3X30^ 5 890(2 
 
 3X30^X2 = 5400 
 
 3 X 30 X22 = 360 
 
 2» = _8 5 768 
 
 Trial, 3X3200^ 122 033 664(3 
 
 3X32002X 3 = 92160000 
 
 3X3200 X 32 = 86400 
 
 3» = 27 92 246 427 
 
 29 787 237 remainder. 
 
 Note. In trying for the third figure of the root, the trial divisor proves 
 larger than the new dividend. Therefore a cypher is written in the root, 
 another period brought down, and a new trial divisor used. 
 
 It is evident that the steps to be followed in obtaining a 
 cube root are: 
 
 1. — Beginning at the decimal point, separate the number, 
 each way, into periods of three places each. 
 
 2. — For the first root figure, take the cube root of the great- 
 est cube contained in the left hand period. Subtract the 
 cube, bringing down the remainder and the next period of 
 three places. 
 
 3. — For succeeding root figures, use as a trial divisor, three 
 times the square of the root values already obtained, con- 
 sidered as tens. Use the quotient so obtained as a trial root 
 value arid complete the cube based upon it. If it proves 
 too large reduce the unit value by 1. Complete the cube, 
 subtract from the last dividend and bring down the next 
 
128 BUSINESS ARITHMETIC. 
 
 period. Continue this process until the periods are exhausted, 
 or the desired nlimber of decimal places in the root have been 
 obtained. 
 
 EXERCISE. 
 
 Find the cube root of: 
 
 1. 941,192. 6. 4/5, by reduction 729_ 
 
 2. 9.300746727. to decimal. * 9261 
 
 3. 147,197,152. 7. 8.0378843 (4 dec. 12,167 
 
 4. 25 (4 dec. pi.). pi.). 10,095,813 
 
 5. 1,083,201,548.8. 1007.65 (5dec.pl.). 12. .0000000678 (3 dec. 
 
 9. .001205 (to 3dec.pl.). pi.). 
 
 GRAPHIC ILLUSTRATIONS OF SQUARES AND CUBES. 
 
 Note. If deemed advisable, this section may be taken after the 
 chapter on practical measurements. 
 
 1. Squaees and Square Root. 
 
 It has been shown numerically (page 123) that the square of 
 
 any number composed of tens and units is equal to the square 
 
 of the tens plus twice the product of the tens multiplied by the 
 
 c X D units plus the square of the units. 
 
 This may be shown graphically, as 
 
 ^ follows: 
 
 Illustration. Compute the square of 
 45. 
 
 Solviwm. 
 
 (1) (2) 
 
 40+5 45 
 
 40+5 45 
 
 5X40+5« 225 
 
 4 0^+ 5X40+ 180 
 
 40»+2 X5 X40+52 =2025 2025 
 (3) To any scale, lay off 40, and then 5, in one straight line. The square 
 on this line, ABCD, is evidently the square of 45. Construct the square 
 of 40, AB'C'D', extending the sides meeting at D', to their intersection 
 with the square of 45. This divides the square, as shown, into a square of 
 40, two rectangles measuring 40 X 5, and a small square of 5. The square 
 of 45 plainly contains the square of the tens (40), the square of the units 
 
 *-5(m) 
 
 40 (0 
 
 5U)-* 
 
 5 
 
 
 
 6(u)- 
 
 / 
 
 i 
 
 
 
 
 
 40(0 
 
 6(«)- 
 
 
i-^ — 
 
 
 f.\ 
 
 
 
 
 20 
 
 1 
 
 
 I 
 
 5x 
 
 
 100 
 
 
 20 
 
 INVOLUTION AND EVOLUTION. ' 129 
 
 (5) and twice the product of units and tens (2X40X5). But 45 might 
 have been any number of t tens and u units. Substituting t and u for 40 
 and 5 respectively, it is evident, from the figure, that the square of t+u 
 = t^-\-2tu-\-u^. 
 
 In a number containing more than two places, as 125, the principle 
 still holds. 125 = 100+20+5. As shown by the diagram, the square of 
 100+20 = 1002+2X100X20+202. Now as- 
 suming 120 as twelve tens, the square of 125 
 is evidently, 1202+2X5X120+52. It is evi- 
 dent that each square, from the inmost, be- 
 comes in turn the square of the "tens" for the 
 determining of the square of the "tens plus 
 units." 
 
 In the reverse process of obtaining the 
 square root of a number (see page 124) it 
 should be evident, now, that in taking as the 
 highest figure of the root the square root, of ^"^ 
 
 the greatest square contained in the left hand 
 
 group of two figures, one obtains the edge of the innermost square. Con- 
 sidering this as the "tens" value, the next place is computed as the "unit," 
 thus obtaining the next larger square. Then this square is taken in turn 
 as the square of the "tens" and a new "tens plus units" square is obtained, 
 and so on. 
 
 In the process of computing the square root, at each step, after the 
 square of the known "tens" is subtracted from the total square, the 
 remainder equals 2 X tens X units plus the square of the units. The new 
 unit is unknown, but, by dividing the remaining area by the known factor 
 of the large term, "2 tens," as a trial divisor, there is obtained an approach 
 to the exact imit. This imit is then tested to see if it is exact* 
 
 2. Cubes and Cube Roots. 
 
 It has been shown numerically (page 126) that the cube of 
 any two-place number equals the cube of the tens plus three 
 times the product of the square of the tens multiplied by the 
 units, plus three times the product of the tens multiplied by 
 the square of the units, plus the cube of the units. As in the 
 case of squares, this may be shown graphically as well as 
 numerically. 
 
 Illustration. Compute the cube of 15. 
 10 
 
130 
 
 BUSINESS ARITHMETIC. 
 
 ions. 
 
 (1) 
 
 
 (2) 
 
 15 
 
 
 10+5 
 
 15 
 
 
 10+5 
 
 75 
 
 
 5X10+52 
 
 15 
 
 102+ 5X10 
 
 225 
 
 102+2X5X10+52 
 
 15 
 
 
 10+5 
 
 1125 
 
 5X102+2X52X10+5' 
 
 225 
 
 103+2X5X102+ 
 
 52x10 
 
 3375 = 103+3X5X102+3X52X10+5' 
 (3) To any scale, construct a cube of one "ten" edge. Extending the 
 edges meeting at one point, construct over the cube, a second cube of an 
 edge equal to one " ten " and five " units." Assume the faces of the smaller 
 
 cube extended to cut the faces 
 of the larger. A study of the 
 subdivided figure will show 
 that the entire cube is made 
 up of a cube of 10 edge; three 
 rectangular solids. A, B and 
 C, each measuring 10X10X5; 
 three rectangular solids, D, E 
 and F, measuring 10X5X5; 
 and a cube, G, of 5 edge. Thus, 
 as in the numerical statement, 
 the cube contains the cube of 
 the tens, plus three solids whose 
 contents equal the square of 
 the tens multiplied by the units 
 (10X10X5) plus three solids 
 whose contents equal the pro- 
 Original cube 1, 2, 3, 4, 5, 6, 7, 8 shown duct of the tens multipHed by 
 hy th.Q light Wne^. the square of the units (10X5 
 
 X5) plus the cube of the units 
 (53) , Since this is general for two place numbers, or for numbers treated 
 as tens and units, the cube, in general, will equal t^-\r^t'hj,-\rZtu^-\-v?. 
 
 In reconstructing this cube from the known cube of the tens, it is evident 
 that there must be added to the cube of 10, 3375 - 1000, or 2375, cubic units. 
 These additions must be in the form of three solids having an outer surface 
 area of 10X10, or 100 square units; three solids having an outer surface 
 length of 10 and for width the unit value still unknown; and a cube having 
 the unknown unit for an edge. All these additions have a uniform thick- 
 
INVOLUTION AND EVOLUTION. 131 
 
 ness equal to the desired unit of the root. Since the numerical value 
 of the cubic contents, when divided by the numerical value of surface area, 
 gives a Unear measure as a quotient, the trial process in finding the second 
 and succeeding figures in a cube root consists in dividing the cubic contents, 
 remaining after the cube of the known root is subtracted by the surface 
 area of the new additions, so far as known. The known part is 3t^-\-St. 
 Since this does not represent the complete outer surface area of the addi- 
 tion, it is used as a trial divisor, and the resulting quotient is tested to see 
 if it is exact, or too large or small. It is tested of course in the formula 
 
CHAPTER XXII. 
 
 DENOMINATE NUMBERS. 
 
 Standard Measures. 
 
 The modern systematizing of business, and the simplifying 
 of methods of handling merchandise, have had a marked effect 
 on our systems of measures. It is rarely, now, that commercial 
 quantities are expressed in several denominations. Instead, 
 some measure of a table is selected as a unit or standard and 
 quantities are expressed as multiples, decimals, or simple 
 fractions of that unit. The unit varies for different trades, 
 professions, or different classes of merchandise. Thus the 
 civil engineer measures by units of 1000 ft., 100 ft., 10 ft., 
 tenths of a foot, etc.; the mechanic, in inches and decimals. 
 The contractor often estimates in cubic yards and decimals. 
 Scales are now made to weigh in tons and decimals, and clocks 
 to mark time in hours and hundredths. The grocer uses frac- 
 tional pounds rather than ounces, as the dry goods dealer uses 
 the yard and its fractions. Many milk dealers use the pint 
 as a unit. Grain, and some vegetables, may be measured by 
 the pound unit instead of by dry measure. Practically all 
 forms of statistics use a decimal system based on a single unit. 
 
 The tendency toward unit measures is aided and reflected 
 by the increasing sale of merchandise in unit packages, and 
 by the increasing use of automatic weighing and computing 
 machines, practically all of which work on a decimal basis. 
 
 The simplicity in actual use is reflected, naturally, in com- 
 putations. Except in computations of time and English 
 money, it is seldom necessary to handle three denominations 
 of any table. 
 
 132 - 
 
DENOMINATE NUMBERS. 
 
 133 
 
 Tables of Measures. 
 
 This section includes the standard tables in general use, 
 and a few units and special values common to certain type 
 businesses. 
 
 MEASURES OF CAPACITY. 
 
 Dry Measure. 
 2 pints = 1 quart. 
 8 quarts = 1 peck (pk.). 
 4 pecks = 1 bushel (bu.). 
 1 dry qt. =67.2 cu. in. 
 1 bushel = 2150.42 cu. in. 
 
 A heaped bushel, used for corn 
 in the ear, apples, etc., contains 
 2747.71 cu. in. 
 
 Liquid Measure. 
 4 gills (gi.) = 1 pint (pt). 
 2 pints = 1 quart (qt.). 
 4 quarts = 1 gallon (gal.). 
 
 1 gallon = 231 cu. in. 
 
 Barrels and hogsheads vary in size. 
 For general estimates. 
 3U gal. = 1 barrel (bbl.). 
 63 gal. = 1 hogshead. 
 
 MEASURES 
 
 Avoirdupois Weight. 
 16 ounces (oz.) = 1 poimd Qh ). 
 100 pounds = 1 hundred- 
 
 weight (cwt.). 
 2000 pounds or 20 
 
 hundredweight = 1 ton (T,). 
 In measuring mining products 
 and in custom house business, the 
 long ion of 2240 lb. is used. 
 
 Comparative Table. 
 Troy and Avoirdupois Weight. 
 1 oz. Avoir. = 437^ gr. 
 1 lb. " = 7000 gr. 
 1 oz. Troy = 480 gr. 
 1 lb. " = 5760 gr. 
 
 OF WEIGHT. 
 
 Troy Weight. 
 24 grains (gr.) = 1 penny- 
 weight (pwt.). 
 20 pennyweights = 1 ounce. 
 12 ounces =1 pound. 
 
 Troy weight is used in the 
 measurement of the precious 
 metals. Jewels, such as diamonds 
 and pearls, are weighed by the 
 carat which = 3.2 Troy grains. 
 The term "carat" is also used to 
 express the proportion of pure gold 
 in composition metals. If pure 
 gold, the metal is "24 carats fine." 
 If one-half alloy, it is "12 carats 
 fine," etc. 
 
 Apothecaries' Weight. 
 20 grains = 1 scruple (sc. or 9) . 
 3 scruples = 1 dram (dr. or 5)- 
 8 drams = 1 oimce (oz. or 5)- 
 12 ounces or 
 5760 gr. = 1 pound (lb.). 
 
134 BUSINESS ARITHMETIC. 
 
 Special Unit Weights — Avoirdupois. 
 1 bbl. flour = 196 lb. In most states: 
 
 1 bbl. salt = 280 lb. 1 bu. potatoes = 60 lb. 
 
 1 bbl. pork = 200 lb. 1 bu. wheat = 60 lb. 
 
 1 cu. ft. fresh water = 62^1b. 1 bu. com = 56 lb. 
 
 1 bu. oats = 32 lb. 
 
 COUNTING. 
 
 12 things = 1 dozen (doz.). 24 sheets (paper) = 1 quire (qr.). 
 
 12 dozen = 1 gross (gro.). 20 quires, or 
 
 12 gross = 1 great gross (g. gr.). 480 sheets = 1 ream (rm.). 
 
 20 things = 1 score. (Commercially, 500 sheets are 
 
 often used for a ream.) 
 
 MEASURES OF EXTENSION. 
 
 Linear or Long Measure. Surveyor's Long Measure. 
 
 12 inches (in.) = 1 foot (ft.). 7.92 inches = 1 link (Ik.). 
 
 3 feet = 1 yard (yd.). 25 links = 1 rod. 
 
 5^ yards, or 4 rods, or 
 
 16^ feet = 1 rod (rd.). 100 links = 1 chain (ch.). 
 
 320 rods, or 80 chains = 1 mile. 
 
 5280 feet = 1 mile (mi.). City property is measured in 
 
 feet and decimals thereof; large 
 
 Sea Measure. tracts of farm, or unimproved land 
 
 6 feet = 1 fathom. by surveyor's measure. 
 120 fathoms = 1 cable length. 
 6086.7 feet, or about 
 
 1.15 miles = 1 knot, nautical or 
 geographical mile. 
 3 knots = 1 league. 
 
 Circular or Angular Measure. 
 60 seconds (") = 1 minute ('). 90 degrees = 1 right angle. 
 
 60 minutes = 1 degree (°). 360 degrees = 1 circumference. 
 
 1 minute = 1 geographic mile. 
 
 This measure is used in geography, navigation and higher surveying, 
 for the computation of differences of time, locations on the earth's surface, 
 longitude and latitude; and for the measurement of angles. 
 
 Square Measure. Surveyor's Square Measure. 
 
 144 square inches = 1 square foot. 625 square links = 1 square rod. 
 
 (sq. in.) (sq. ft.) 10 square rod« = 1 square chain. 
 
DENOMINATE NUMBERS. 
 
 135 
 
 Square Measure. 
 
 9 square feet 
 30j square yards 
 160 square rods 
 640 acres 
 100 square feet 
 
 = 1 square yard. 
 = 1 square rod. 
 = 1 acre (A ). 
 = 1 square mile. 
 = 1 square. 
 
 Surveyor's Square Meaeure. 
 10 square chains = 1 acre. 
 640 acres = 1 square mile. 
 
 36 square miles = 1 township. 
 
 Cubic Measure. 
 
 1728 cubic inches = 1 cubic foot. 128 cubic feet or 
 
 (cu. in.) (cu. ft.) 8 cord feet = 1 cord (wood). 
 
 27 cubic feet = 1 cubic yard. 1 cubic yard = 1 load (earth). 
 
 16 cubic feet = 1 cord foot (cd. ft.) . 
 
 MEASURE OF TIME. 
 
 60 seconds (sec.) = 1 minute (min.). 
 
 360 days 
 
 = 1 commercial year. 
 
 60 minutes = 1 hour (hr.). 
 
 365 days 
 
 = 1 common year. 
 
 24 hours = 1 day. 
 
 366 days 
 
 = 1 leap year. 
 
 7 days = 1 week (wk.). 
 
 10 years 
 
 = 1 decade. 
 
 30 days = 1 commercial 
 
 100 years 
 
 = 1 century. 
 
 month. 
 12 months (mo.) = 1 year (yr.). 
 
 365 days, 5 hours, 48 minutes, 49.7 seconds = 1 solar year. 
 Centennial years divisible by 400 and other years divisible by 4 are 
 leap years. 
 
 MEASURES OF VALUE. 
 U. S. Money. Canadian Money. 
 
 See pages 50 and 53. The same as for the United States. 
 
 100 cents = 1 dollar = $1. 
 English Money. 
 
 4 farthings (far.) = 1 penny (d.). 
 
 12 pence = 1 shilling (s.) = $0,243 +. 
 
 20 shilUngs = 1 pound sterling (£) = $4.8665. 
 
 French Money. 
 100 centimes = 1 franc 
 
 $0,193. 
 
 German Money. 
 100 Pfennigs = 1 Mark = $0,238. 
 
 Mexican Money. 
 100 centavos = 1 peso = $0,498. 
 
136 BUSINESS ARITHMETIC. 
 
 DENOMINATE NUMBERS. 
 INTRODUCTORY EXERCISE. 
 
 1. Give illustration of abstract numbers, concrete numbers, like and 
 unlike numbers, denominate numbers, simple and compound numbers. 
 (See pp. 1 and 2.) 
 
 2. Name simple denominate numbers expressing: time, distance, area, 
 value, weight. 
 
 3. Name compound denominate numbers expressing: cubic contents, 
 liquid capacity, area. 
 
 4. Name a very common denominate measure used in each of these 
 trades, businesses or professions : grocery, provision, oil, produce, dry goods, 
 plumbing, mechanical engineering, contracting, banking, jewelry, coal 
 and wood. 
 
 5. Name common denominate measures used in speaking of such 
 things as — 
 
 (1) The distance from New York to Chicago. 
 
 (2) The distance across the street. 
 
 (3) The quantity of a meat order. 
 
 (4) The time required to do an errand. 
 
 (5) The amount of imports of the United States for last year. 
 
 6. Name the common measures used in your household and its trading. 
 
 7. What is the advantage of buying by some measure or fraction of it, 
 over buying by two or more denominations of a thing? 
 
 8. What are the advantages to the retailer in selling goods in de- 
 nominate packages, rather than in bulk? 
 
 It is evident from the tables that a number expressed in one 
 denomination may be expressed in another denomination of 
 the same table, and sometimes of other tables, without altering 
 its value. The process of changing the form of expression of 
 a quantity is termed reduction ascending, if the denomination 
 chosen is larger than the original, and reduction descending, if it 
 is smaller. 
 
 REDUCTION DESCENDING. 
 
 Illustrations. (1) Reduce 10 rds. 3 yd. to feet. 
 Analysis and Solution, (a) 1 rd. = 5.5 yd. 10 rd. = 55 yd. 55 yd. 
 + 3 yd. = 58 yd. 1 yd. = 3 ft. 58 yd. = 3 X 58 ft. or 174 ft. 
 
DENOMINATE NUMBERS. 
 
 137 
 
 (6) 1 rd. = 16.5 ft. 10 rd. = 10 X 16.5 ft. or 165 ft. 
 3 yd. = 3 X 3 or 9 ft. 165 ft. + 9 f t. = 174 ft. 
 (2) Reduce .45 gal. to lower denominations. 
 Analysis and Solution. 
 
 1 gal. = 4 qt. .45 of 4 qt. 
 
 1 qt. =2 pt. 
 1 pt. =4 ^. 
 
 .-. .45 gal. = Iqt. 1 pt. 2.4 gi. 
 
 1.8 qt. 
 .8 of 2 pt. = 1.6 pt. 
 .6 of 4 gi. = 2.4 gi. 
 
 1 yd = 3 ft. 
 
 Find missing values: 
 
 1. 3 gal. = qt. 
 
 2. 2 cd. = cu. ft. 
 
 3. 2 sh. 6 d. = d. 
 
 4. 2\ gal. = pt. 
 
 5. 3/4 bu. = pt. 
 
 6. 2 m. = rd. 
 
 7. 2 bu. = pt. 
 
 8. 1 gr. 3 doz. = units. 
 
 ORAL EXERCISE. 
 
 9. 
 
 .4 pk. = qt. 
 
 10. 
 
 3.5 sq. yd. = sq. f 
 
 11. 
 
 U' 6" = ". 
 
 12. 
 
 65 score = units. 
 
 13. 
 
 1 mi. 40 rd. = rd. 
 
 14. 
 
 1.3 T. = lb. 
 
 15. 
 
 £2.25 = sh. 
 
 EXERCISE. 
 
 3. 5.6 gal. to gills. 
 
 4. 1.375 sq. mi. to A. 
 
 Reduce: 
 
 1. 7.3 cu. yd. to cu. in. 
 
 2. 31 T. to oz. 
 6. 7.475 mi. to ft. 
 
 6. Find the cost of 5f A. @ 5c per sq. ft. 
 
 7. What will it cost to excavate 942.5 cu. yd. @ 15c per cu. ft.? 
 
 8. Express 2 mi. in equivalent engineer's unit of 1000 ft. 
 
 9. Reduce 11 gt. gross to a fraction of 1000 units. 
 
 10. What part of 1 rd. are 6 in.? 11. What part of 1 mi. are 2 y4.? 
 
 REDUCTION ASCENDING. 
 
 Illustrations. (1) Express 729 ft. in higher denominations. 
 Solution. Since 3' = 1 yd. 3)729 
 
 1 rd. = 5i ft. 5.5)243 = no. yd. 
 
 (mult, by .2) or 11)486 
 
 729 ft. = 44 rd. 2 yd. 44 = no. rd. 
 
 2 yd. rem. 
 
138 BUSINESS ARITHMETIC. 
 
 (2) Express 2 ft. 9 in. as a decimal of a yard. 
 Solution. ■: 2 ft. 9 in. = 33 in. and 1 yd. = 36 in. 
 
 .-. 2 ft. 9 in. = 33/36 of a yd. 33 -^ 36 = .916 +. 
 
 .-. 2 ft. 9 in. = .916+ yd. 
 
 ORAL EXERCISE. 
 
 Reduce to the next higher denomination: 
 
 1. 52 ft. ' 3. 
 
 12 yd. 
 
 5. 65 in. 7. 8^ sh. 
 
 2. 27i pt. (dry). 4. 
 
 27,000 lb. 
 
 6. 148 oz. 8. 50 cu. ft. 
 
 Find the cost of: 
 
 
 
 9. 2 lb. @ 6c per oz. 
 
 13. 
 
 3i gal. @ 5c per pt. 
 
 10. 5 qt. @ 8c per pt. 
 
 14. 
 
 3 pk. @ 60c per bu. 
 
 11. 4 yd. @ 5c per in. 
 
 15. 
 
 40 ft. @ 30c per yd. 
 
 12. 2 bu. @ 30c per pk. 
 
 16. 
 
 900 lb. @ $36 per T. 
 
 EXERCISE. 
 
 (Decimals to 4 places.) 
 
 1. Express as decimals of a mile: 1 rd.j 1 ft.; 1 yd.; 7.35 in.; 8.4 yd. 
 
 2. Express as decimals of a bu.: 1 pk.; 3 qts.; 1 pt.; 3^ pt. 
 
 3. Express as decimals of a cu. yd.: 1 cu. ft.; 20 cu. in. 
 
 4. Express as decimals of a £: 1 sh.; 1 d.; 7 sh. 6 d.; £ 3 8 sh. 5 d. 
 
 5. Express as decimals of a T. : 11 lb.; 1 oz.; 745 lb.; 212 lb. 8 oz. 
 
 6. Express as decimals of an acre : 1 sq. rd. ; sq. yd. ; 96 sq. ft. ; 45 sq. yd. 
 
 7. Reduce to higher denominations: 326,425 in.; 8429 ft. (dry); 
 11,927 cu. in 
 
 8. What quantity (gallons) of ohve oil must be bought to fill 90 gross 
 of pint bottles? 
 
 The Fundamental Processes. The fundamental pro- 
 cesses, as applied to denominate or compound numbers, differ 
 relatively little from the same processes in simple numbers. 
 The main point of difference is in the substitution of an irreg- 
 ular reduction scale for the decimal scale. 
 
 ADDITION. 
 
 Illustration. (1) Add 567 and 329. (2) Add 5 bu. 3 pk. 5 qt. and 
 3 bu. 2 pk 4 qt. 
 
DENOMINATE NUMBERS. 139 
 
 Sohdions. (1) (2) (2) Condensed. 
 
 567 5 bu. 3 pk. 5 qt. 5 bu. 3 pk. 5 qt. 
 
 329 3 2 4 3 2 4 
 
 896 8 bu. 5 pk. 9 qt. 9 bu. 2 pk. 1 qt. 
 
 or 9 bu. 2 pk. 1 qt. (Reduction ascending.) 
 Note. (2) In the condensed form the reduction is performed mentally, 
 as the total of each measure is obtained. Thus the total of the quart 
 column is 9 qt. Since 9 qt. = 1 pk. 1 qt., the 1 quart is written in the sum, 
 and the 1 peck is carried forward to the next column. 
 
 ORAL EXERCISE. 
 
 Find the sum 
 
 of: 
 
 
 
 
 
 
 1. 
 
 
 
 2. 
 
 
 
 3. 
 
 5 gal. 2 qt. 1 pt. 
 
 
 
 6 bu. 1 pk. 
 
 
 
 1 T. 12 cwt. 
 
 2 1 1 
 
 13- 
 
 
 3 
 
 5qt. 
 
 
 3 9 
 
 4. 
 
 5. 
 
 
 6. 
 
 15 cwt. 84 lb. 
 
 
 7] 
 
 mi. 46 rd. 
 
 
 6sq. 
 
 yd. 5 sq. ft. 
 
 7 19 
 
 
 2^ 
 
 300 
 
 
 1 
 2 
 
 1 
 8 
 
 7. To 3 gal. add consecutive pints. % 
 
 EXERCISE. 
 
 1. Find the total if the items of a purchase from a London merchant are : 
 £5 3 s. 6 d.; £ 18 11 s. 4 d.; £12 5 s. 8 d. 
 
 2. In St. Clair County the following stretches of country road have 
 been macadamized in the past year. 1 mi. 160 rd., 2 mi. 145 rd., 8 nai. 
 176 rd., 1 mi. 46| rd., a total of ?. 
 
 3. A lumberman, who already owns stumpage rights on 3 sq. mi. 
 260 A., buys rights to neighboring tracts of 84 A., 1 sq. mi. 52 A., 526 A. 
 40 sq. rd., and 64 A. 120 sq. rd. He then has rights to ? sq. mi. ? A. or 
 to ? A. For the rights to lumber these new tracts, he paid $85 per A. or 
 $ ? m aU. 
 
 SUBTRACTION. 
 Subtraction involves reduction descending. 
 
 Illustrations. (1) (2) 
 
 5 bu. 3 pk. 6 qt. 5 T. 7 cwt. 16 lb. 
 
 2 1 3 2 5 46 
 
 Differences 3 bu. 2 pk. 3 qt. 3 T. 1 cwt. 70 lb. 
 
 Reduce 1 cwt. of the minuend to pounds. 116 lb. — 46 lb. = 
 70 lb.; 6 cwt. — 5 cwt. = 1 cwt., etc. 
 
140 BUSINESS ARITHMETIC. 
 
 ORAL EXERCISE. 
 Find the difference between : 
 
 1. 2. 3. 4. 
 
 5 bu. 3 pk. 5 gal. 2 qt. 16 cwt.18 lb. 5 lb. 11 oz. 
 
 12 23 4 20 32 
 
 Find the difference between the larger and smaller quantity in ex. 1-5 
 of the last oral exercise. 
 
 EXERCISE. 
 
 1. What check can you suggest for compound addition and subtraction? 
 Illustrate with examples. 
 
 2. On a contract for the laying of 752 cu. yd. of masonry, 15 cu. yd. 
 
 7 cu. ft. are laid one day, 31 cu. yd. 14 cu. ft. the second day, 68 cu. yd. 
 
 8 cu. ft. the third and 88 cu. yd. 11 cu. ft. the fourth. How much work 
 remains to be done? 
 
 3. From 15.0739 mi. subtract 2 mi, 126 rd. 2 yd. 
 
 4. Make the extensions in this form: 
 
 Gross Cost. Discount. Net cost. 
 
 £ 16—5— 8 £ 2— 6—4 £ 
 
 4—9— 3 8—7 
 
 84—7—11 3—15—8 ————■ 
 
 Totals ? ? ? 
 
 MULTIPLICATION. 
 That no new principle is involved is seen from this 
 Illustration. Find the product of 3 bu. 2 pk. 1 qt. by 8. 
 Solution. Analysis. 
 
 3 bu. 2 pk. 1 qt. 8 X 1 qt, = 8 qt. = 1 pk. 
 
 S_ 8 X 2 pk. + 1 pk. = 17 pk. = 4 bu. 1 pk. 
 
 28 bu. 1 pk. qt. 8 X 3 bu. + 4 bu. = 28 bu. 
 
 EXERCISE. 
 
 (Solve mentally if possible.) 
 Multiply: 
 
 1. 3 ft. 2 in. by 6. 
 
 2. 1 yd. 2 ft. by 8. 
 
 3. 3 lb. 2 oz. by 30. 
 
 4. 2 pk. 3 qt. by 12. 
 9. How much silver is required for five ornaments each containing 
 
 1 oz. 7 pwt.? 
 
 5. 
 
 2 qt. 1 pt. by 8. 
 
 6. 
 
 3 rd. 27 Unks by 9. 
 
 7. 
 
 3 cwt. 60 lb. by 10. 
 
 8. 
 
 1 T. 4 cwt. by 20. 
 
DENOMINATE NUMBERS. 141 
 
 10. A factory manager reduces the length of a metal bar in a machine 
 he manufactures 3f in. This means a saving of how much metal in 
 12,000 bars? 
 
 11. Determine the cost of 75 yd. extra broadcloth, purchased at £1 
 2 8. 3 d. per yd. 
 
 12. A factory expert discovers that the length of a bolt used in the 
 company's product may be reduced 1/4 in. This means a saving of how 
 many feet if 250,000 bolts are used per year? 
 
 13. Compute the cost of a car load containing 60 bbl. of apples, each 
 containing 2 bu. 2 pk., at 75c per pk. 
 
 DIVISION. 
 
 In division of denominate numbers the divisor may be either 
 abstract or denominate. 
 
 Illustrations. (1) Compute the cost, per yd., of 80 yd. cloth invoiced 
 at £180. 
 
 Solution (a). Solution and Analysis (5). 
 
 80 ) £180 The cost per yd. is evidently 1/80 of 
 
 £ 2.25 = £2 5 8. (cost) £180. 1/80 of £180 = £2 and a 
 
 remamder of £20. £20 = 400 s. 
 1/80 of 400 8. = 5 8. .*. 1/80 of 
 £180 = £2 5 8. 
 (2) At a price of £1 2 s. per yd., how many yards may be purchased for 
 £89 2 8.? 
 
 Solution. 
 £ 1 2 s. = 22 8. The numbers being concrete, re- 
 
 £89 2 s. = 1782 s. duction to a common denominator is 
 
 81 = no. of yd . necessary. 
 
 ~22)1782 
 
 ORAL EXERCISE. 
 Divide : 
 
 1. 6 bu. 3 pk. by 3. 
 
 2. 5 gal. 2 qt. by 2. 
 
 3. 3 T. 6 cwt. by 200. 
 
 4. 41b. by 8. 
 
 5. 3 pk. 6 qt. by 10. 11. 5 bu. 1 pk. by 1 bu. 3 
 
 6. 5 yd. 1ft. by 4. 
 
 7. 
 
 3 sq. yd. 3 sq. ft. by 6. 
 
 8. 
 
 3 gal. by 2 qt. 
 
 9. 
 
 4 lb. by 8 oz. 
 
 10. 
 
 2 ft. 6 in. by 3 m. 
 
142 BUSINESS ARITHMETIC. 
 
 EXERCISE. 
 
 1. Divide 960 cu. yd. 5.4 cu. ft. by 11.5, checking by multiplication. 
 
 2. A carriage wheel revolves two times in 7 yards, or times in one 
 
 mile. 
 
 3. Divide 5 bu. 7 pk. 3 qt. by 1 pk. 2 qt. 1 pt. Check the result by 
 multiplication. 
 
 4. A dealer has 2 bbl., 31| gal. each, of olive oil, to be bottled in 
 half-pint bottles. He requires — bottles. 
 
 5. A newspaper press printed 57,384 copies in 2 hr. 5 min., at the 
 average rate of — copies per minute. 
 
 Individual Original Work. 
 
 Report on "Common Measures Used in Business." Disregard text- 
 book classifications, and find out what parts of tables are in actual common 
 use. Classify under: (1) Common to business in general; (2) limited to 
 particular trades; (3) non-text-book measures. Give illustrative examples. 
 
CHAPTER XXIII. 
 
 PRACTICAL MEASUREMENTS. 
 
 Many simple measurements in business and science are 
 based on elementary geometric principles. A slight knowledge 
 of some of these principles, and of the geometric figures to 
 which they apply, is of value to everyone. 
 
 Geometric Conceptions — Plane Figures. 
 
 A geometric line is considered to have extension but neither 
 length nor breadth. A straight line is the shortest distance 
 between two points. Lines are parallel if they are the same 
 distance apart throughout their entire length.ZZ 
 
 An angle is the divergence of two lines having a common 
 point. AXB is the angle of the lines AX and XB, If the 
 
 two lines meet so that the two angles on the same side of one 
 line are equal, the angles are right angles, and the lines are 
 perpendicular. Thus AYC and BYC are right angles and CY 
 is perpendicular to AB. An acute angle is less than a right 
 angle. An obtuse angle is greater than a right angle. 
 
 A geometrical surface has length and breadth but no thick- 
 ness. A plane, or plane surface, is a level surface, such as that 
 of still water. A figure in a plane is a plane figure. 
 
 143 
 
144 
 
 BUSINESS ARITHMETIC. 
 
 A triangle is a plane figure bounded by three straight lines. 
 It is called equilateral, isosceles, or scalene, according as it has 
 
 three sides equal (A), two sides equal {B),ot no sides equal (C). 
 A right angled triangle has one right angle (D). 
 
 A quadrilateral is a plane figure bounded by four straight 
 lines. If the opposite sides of a quadrilateral are parallel, the 
 figure is called a parallelogram (E-H). The parallelogram is 
 a rectangle if it has four right angles (F). It is a square if the 
 
 sides are equal and the angles right angles (G). It is a 
 rhombus, if the four sides are equal but the angles are not 
 right angles (H), 
 
PRAlCTICAL MEASUREMENTS. 
 
 145 
 
 A circle is a plane figure bounded by a curved line, called the 
 circumference, every point of which is equidistant from a point 
 within called the center. The diameter 
 is any straight line passing through 
 the center and terminating in the cir- 
 cumference. The radius is one-half 
 of the diameter. An arc is any part 
 of the circumference, and is measured 
 by degrees of angular measure. 
 
 The circumference is approximately 
 3.1416 (approximately 3y) times the 
 diameter. This factor is called by 
 
 the Greek letter ''tt," (pronounced, "pi.") If the diameter 
 is known, the circumference may be determined by multiply- 
 ing it by 3.1416. If the circumferenpe is known, the diameter 
 is determined by finding the quotient of the circumference 
 divided by 3.1416. 
 
 The perimeter of a plane figure is the distance around it. 
 The ha^e is the side on which it is assumed to rest. The 
 altitude is the perpendicular distance from the base to the 
 most distant point of the figure. 
 
 EXERCISE. 
 
 1. Draw four circles of 4", 5", 6", and 8" radius, respectively. In 
 each case, measure carefully the circumference and divide by the re- 
 spective diameters. . Take the average of the resulting quotients. How 
 close to the value of "x" do you approach? By computation, find your 
 error in measurement of each circumference. To what is this error due? 
 11 
 
146 
 
 BUSINESS ARITHMETIC. 
 
 2. Compare the diameters and circumferences of two circles, one of 
 which has a diameter of 10' and the other a circumference of 42'. 
 
 3. What is the diameter of a circular path a mile in circumference? 
 
 4. Compute the cost of the fencing around a circular park of 90 ft. 
 radius, at $1.50 per foot. 
 
 5. In racing on a circular track, what gain may result from securing 
 an inside position? Illustrate. 
 
 6. A man has 1000 ft. of movable fencing for a chicken nm. Give 
 dimensions of a square field, a rectangular field and a circular field that 
 he may enclose. 
 
 Areas of Plane Figures. 
 
 The accompanying rectangle is divided by parallel lines into 
 
 unit squares. How many unit squares in each row? How 
 
 many rows? Which dimension shows 
 
 the number of rows? Which the units 
 
 per row? What is the total number of 
 
 units? How obtained? Show that the 
 
 number of square units would be the 
 
 same if "5'' were considered the base 
 
 and "6" the altitude. It is evident that 
 
 the area of the rectangle is equal to the 
 
 product of its length and width — its two dimensions. 
 
 In the case of the parallelo- 
 gram,by cutting off the section 
 X and placing it in the posi- 
 tion X', it is evident that the 
 figure is equivalent in area to 
 a rectangle having the same 
 
 base and an equal altitude. Try this experimentally with a 
 piece of paper. 
 
 In the case of the triangle, as shown by 
 the figure, the area is equal to one-half 
 the area of the rectangle erected on the 
 same base, and with equal altitude. The 
 area is equal, therefore, to one-half the pro- 
 duct of its base and altitude. 
 
PRACTICAL MEASUREMENTS. 147 
 
 If a circle is divided as shown, and the parts rearranged as a 
 series of approximate triangles, it is evident that the area will 
 equal the sum of the areas of the triangles. If the triangles 
 are made small enough they have altitudes equal to the radius 
 of the circle. The sum of their bases equals the circumference 
 
 of the circle. The area of the circle, therefore, equals one- 
 half the product of the radius by the circumference. Since 
 the circumference equals tt X the diameter (2 X radius) the 
 area equals ^RX 2RX 3.1416, or 3.1416 X the square of the 
 radiiLS. 
 
 The area of irregular right line figures may be obtained by 
 dividing them into triangles and parallelograms and deter- 
 mine the areas of these parts. 
 
 ORAL EXERCISE. 
 Compute the areas of the following: 
 
 Figure. Altitude. Base. 
 
 1. Triangle 14 ft. 16 ft. 
 
 2. Rectangle 24 ft. 2^ ft. 
 
 3. Square 15 ft. 
 
 4. Parallelogram 16 in. 2 ft. 
 
 5. Rectangle 2\ii. 4Ht. 
 
 6. Triangle 20 in. 1 ft. 6 in. 
 
 7. Parallelogram 6i ft. 2 ft. 3 in. 
 
 8. Find the difference between 3 sq. yd. and an area 3 yd. square. 
 
 9. How does a 5 in. square compare with an area of 5 sq. in.? 
 
 10. Name two factors of 360. Find two dimensions for a rectangle of 
 360 sq. ft. area. Find other dimensions for a second rectangle of the same 
 
148 
 
 BUSINESS ARITHMETIC. 
 
 area. How many rectangles have the same area? If one dimension is 
 4' what is the other? If one dimension is 12 ft. what is the other? 
 
 11. A series of rectangles have a common area of 480 sq. ft. Find the 
 bases if the altitudes are respectively 4, 16, 12, 6, 8, 15, 20, 40, 1/2 feet. 
 Compare the perimeters. Do rectangles of the same area have the same 
 perimeter? 
 
 12. Give the dimensions of several rectangles that have a common 
 area of 1200 sq. in. 
 
 13. To what is the area of a triangle equal? If the area and one 
 dimension are known, how is the other determined? 
 
 14. Find the altitudes of a series of triangles having a common area 
 of 180 sq. ft. and bases of 4, 3, 20, 9, 18 and 30 ft., respectively. 
 
 15. Name the bases and altitudes of ten triangles that have a common 
 area of 720 sq. in. 
 
 EXERCISE. 
 
 1. Determine the area of a floor measuring 25 ft. 4 in. by 16 ft. 8 in. 
 
 2. Determine the area of the end section of a metal bar measuring 
 3j in. by 1| in. 
 
 3. How many 3-in. squares may be cut from a piece of card board 
 measuring 25 in. by 36 in.? 
 
 4. What is the acreage of a field measuring 720 rd. by 96 rd.? 
 
 5. A field measuring 88 rd. by 
 126 rd. produces 280 bu. of grain, 
 an average of bu. per acre. 
 
 6. Subdivide the field repre- 
 sented by the accompanying dia- 
 gram into rectangles, as shown by 
 the dotted lines. Name the dimen- 
 sions of each rectangle. Determine 
 the total area. Check the com- 
 
 10/ putation by subdividing into an- 
 
 other group of geometric figures 
 and determining area. 
 
 7. How many oblongs, 2 in. by 3^ in., can be cut from sheets of 
 composition metal 24 in. by 36 in.? Does it make any difference which 
 way the oblongs are laid off on the sheet? 
 
 12' 
 
 12' 
 
 6'- 
 
 
 
 to 
 
 
 
 
 •o 
 
PRACTICAL MEASUREMENTS. 
 
 149 
 
 80' 
 
 8. The diagram here shown rep- 
 resents a park with a circular flower 
 bed. The walk is 5 ft. wide. Find 
 the area of the flower bed. Find 
 the number of square yards of sod- 
 ding required for the parking. 
 How is this area determined from 
 the area of the rectangle and the 
 area of the circle? 
 
 Compute the cost of constructing 
 the walk, at $1.25 per sq. yd. Com- 
 pute the cost of curbing for the inner and outer edges of the walk, at 38c 
 per linear foot. 
 
 9. Find the acreage of the plot of land here represented . Express dimen- 
 
 sions in linear measure. Check 
 computation of acreage by using 
 dimensions just found. 
 
 Applications of Square 
 Root. 
 
 Square root frequently is 
 employed, in connection 
 with geometric figures, in de- 
 termining missing dimen- 
 sions. 
 
 1. Squares. Since the area of a square is equal to the 
 product of its two equal sides, it is evident that the length 
 of one side is equal to the square root 
 
 of the area. 
 
 2. Right Triangles. It has been proved, 
 geometrically, that the square of the 
 hypotenuse, or side opposite the right 
 angle, is equal to the sum of the squares 
 of the other two sides. This is illus- 
 trated graphically in the drawing. It 
 is evident, then, that the length of the 
 hypotenuse is equal to the square root 
 
150 
 
 BUSINESS ARITHMETIC. 
 
 of the sum of the squares of the other two sides; and that the 
 length of either other side is equal to the square root of the 
 difference of the squares of the hypotenuse and the known side. 
 3. Since the area of a circle equals tt X the square of the 
 radius, it is evident that the radius equals the square root 
 of the area divided by 3.1416. 
 
 EXERCISE. 
 
 1. Determine the side of a square having the same area as a rectangle 
 measuring 54 rd. by 17.3 rd. 
 
 2. To have an area of 60,000 sq. ft., a square must have an edge of 
 how many feet? 
 
 3. Find the edge of a square that has the same area as a circle of 
 50 ft. radius. 
 
 4. Determine the radius of a circle that has the same area as a 
 square of 80 ft. side. 
 
 5. How many linear feet of fencing are required for a rectangular 
 chicken run, 80 ft. by 120 ft.? What is the difference in amount of fencing 
 required for a square run of the same area? 
 
 6. Compare the perimeters of a square, a rectangle of 20 ft. altitude, 
 and a circle, if all have equal areas of 600 sq. ft. 
 
 7. If the two sides of a right triangle are 8 ft. and 16 ft., what is the 
 length of the hypotenuse? 
 
 Determine the missing values in the figures shown below: 
 
 8. 10. 
 
PRACTICAL MEASUREMENTS. 
 
 151 
 
 Gables. 
 
 The pitch of a roof is the amount of rise of a rafter for each 
 foot in the base of the gable. 
 
 Illustkation. In the cut, the rafter rises 10 ft. for a base of 20 ft. 
 This is a pitch of 10/20, or 1/2. The familiar Gothic pitch is 5/8. 
 
 
 
 EXERCISE. 
 
 
 
 Fmd the missing values: 
 
 Base Width. Height. 
 1. 30 ft. 18 ft. 
 
 Pitch. 
 ? 
 
 Length 
 ? 
 
 of Rafter. 
 
 (Allow 1 ft. Overhang.) 
 
 2. 40 ft. 
 
 ? 
 
 Gothic 
 
 
 ? 
 
 (< H 11 
 
 3. 45 ft. 
 
 20 ft. 
 
 ? 
 
 
 ? 
 
 " li ft. 
 
 4. ? 
 
 16 ft. 
 
 1/4 
 
 
 ? 
 
 (t tl tt 
 
 5. 36 ft. 
 
 ? 
 
 1/2 
 
 
 ? 
 
 tt U tl 
 
 PRACTICAL APPLICATIONS OF SQUARE MEASURE. 
 The following applications are selected to illustrate the use 
 of square measure as modified by business custom. 
 
 1. Flooking. 
 
 Considerable structural work, such as flooring, roofing, 
 tiling, etc., is measured frequently by the square, or 100 sq. ft. 
 More rarely the measurement is expressed directly in sq. ft., 
 or in units of 1000 sq. ft. 
 
 In properly matching and fitting tongue and groove flooring, 
 there is considerable waste of material. Thus, for every foot 
 
152 
 
 BUSINESS ARITHMETIC. 
 
 of flooring in place, 1^ to 1^ ft. of flooring material may be 
 required. 
 
 EXERCISE. 
 
 1. Compute the cost of flooring a hall measuring 80' by 35', at $2.95 
 per square. 
 
 2. Compute the cost of flooring with hard wood, the two front rooms 
 shown in this floor plan of a bungalow, at $40 per M, allowing 1/3 waste, 
 and $10.50 for extras. 
 
 DINING ROOM 
 
 / ^ .o'«'/ 
 
 10' X 12'6 
 
 BED ROOM 
 
 JO' X 10 6' 
 
 -* -r-i b^ 
 
 3. How many tile, 6" square, are required for the right rear room, 
 allowing 1/20 extra for breakage? 
 
 4. A gable roof, 42' long, and having rafters 21' long, is to be covered 
 with roofing tile. At the rate of 300 per square, how many tile are re- 
 quired? How many shingles are required for the same purpose, at the 
 rate of 800 per square? How many bundles of shingles, 250 each? 
 
 Note. Carpenters and builders have tables showing the numbers 
 of tile, shingles, etc., of different sizes, req^uired per square. These are 
 often used as a basis of computation in practical work. 
 
PRACTICAL MEASUREMENTS. 153 
 
 2. Plastering. 
 
 Plastering is commonly measured by the square yard. 
 Allowances for openings are made in various ways, such as 
 (1) by actual measurement, (2) by arbitrary allowance, etc. 
 If working plans for a building are not given, dimensions 
 of rooms are usually stated in this order: Length — width — 
 height. 
 
 Example: 21' X 16' X 9' 6''. 
 
 EXERCISE. 
 
 1. Compute the cost of hard finishing the walls and ceiUng of the two 
 front rooms of the building, page 152, at 43c per sq. yd. Height of room 
 9'. Allow 20 sq. ft. for door openings and 18 sq. ft. for windows. 
 
 2. Compute the cost of rough finishing the bedroom and dining room 
 at 26c. per sq. yd., allowing by actual measurement for the doors, which 
 are 1' by 4', and for windows, which are 6' by 4^'. 
 
 3. Painting. 
 
 Painting is usually measured by the square yard, no allow- 
 ance being made for windows or openings of a similar nature. 
 
 EXERCISE. 
 
 1. Compute the cost of painting a fence, 6' high, around the lot shown 
 on page 148, both sides, at a cost of 24c per sq. yd. 
 
 2. Compute the cost of painting with two 
 
 coats the walls of a building 48' long, and having ^-^''^'^N^ 
 
 this end section. Price for double coat, 45c per 
 sq. yd. 
 
 4. Papering. 
 
 Although dimensions vary, the width 
 of wall paper is usually assumed to be 18 
 inches. The length is 8 yards for single 
 rolls and 16 yards for double rolls. Complete rolls must be 
 bought. 
 
 Allowance is made for openings in various ways, such as 
 
CEILING PLAN 
 
 154 BUSINESS ARITHMETIC. 
 
 by (1) exact measurement; (2) uniform allowance for each 
 opening; (3) allowance for width of opening, and full height of 
 room, the assumption being that remnants of rolls, after 
 cutting full strips, will be sufficient to paper the wall over 
 and under openings; and (4) an arbitrary allowance according 
 to area, as " one roll to 24 sq. ft. regardless of openings.'* 
 
 ORAL EXERCISE. 
 
 1. If paper is laid on the ceiling represented in the accompanying il- 
 lustration parallel to the shorter dimension, what is the length of each strip 
 in yards? How many strips can be cut from each 
 24 FT. double roll? How many strips are needed? How 
 
 many rolls? What is the cost, at 45c per roll? 
 
 2. The walls of the room, the ceiling of which 
 is shown, are also to be papered. What does the 
 border cost, at 15c per yd? The strips of papering 
 on the walls must be 8 ft. long. How many strips 
 are required? How many strips can be cut from 
 each double roll? How many rolls are needed? 
 What is the cost, at 50c per roll? 
 
 It is evident that the simplest method 
 of approximation is to divide the perime- 
 ter of the room, less the width of all openings, by the width of 
 a strip, and to divide the number of strips by the number 
 of complete strips in one roll, in order to determine the 
 number of rolls. 
 
 EXERCISE. 
 
 1. A room 30' by 28' has five openings, each 2\' wide, and one 
 opening 4' wide. It is 9' from baseboard to ceiling. Find the cost of 
 papering sides and ceiling, without border, at 55c per roll, allowing 6" 
 per strip for matching design. 
 
 2. Compute the quantity of ceiling, border and wall paper required 
 for your class room. 
 
 3. Find the quantity of wall, ceiling and border paper required for 
 each of the rooms shown in the diagram on page 152, assuming the wall 
 height, allowing for border, as 9', and allowing 6" per strip for matching 
 design. Assume width of doors 3'; of windows, 2\'. 
 
PRACTICAL MEASUREMENTS. 155 
 
 5. Carpeting. 
 
 Carpeting is measured by the yard 'per length of strip, 
 regardless of width. Oil cloth, linoleum and carpet linings 
 are sometimes measured by the square yard. 
 
 In order to compute the quantity of carpet for a room, it is 
 necessary to determine the number of strips and their length, 
 somewhat after the manner of the computations for papering. 
 The number of strips depends on 
 the direction in which the strips 
 are laid. Allowance must be 
 made, also, for matching design. 
 
 Illustration. Compute the quan- 
 tity of carpet required for the room 
 shown, if laid lengthwise, the width of 
 the carpet being three feet. 
 
 Solution. 19-T-3 = 6^, no. of strips. 
 Therefore, 7 strips are required. 24' = 
 length of one strip = 8 yd. 7X8 =72, 
 no. of yd. required. 
 
 Note. Since each strip is 3' wide, the total width of 19' divided by 3 
 will give the number of strips, a full strip being purchased for each fraction 
 of width. 
 
 EXERCISE. 
 
 1. Find the quantity of carpet required for the room shown above if 
 laid across the shorter dimension? 
 
 2. Compute the cost of Brussels carpet, 27" wide, for the above room, 
 laid as shown, at $1.50 per sq. yd., allowing 6" per strip, after the first 
 strip, for matching. Why need no allowance be made for matching the 
 first strip? 
 
 3. The above robm is to be carpeted with Ingrain carpet, 1 yd. wide, 
 laid lengthwise, and with a border around the room 2' wide. Determine 
 the quantity of border and carpet, allowing 1/20 for matching. Compute 
 the number of square yards of lining required. 
 
 4. Find the cost of carpeting the dining room (page 152) with linoleum, 
 1^ yd. wide, at $1.45 per sq. yd., allowing 2 sq. yd. for matching design. 
 
 5. Compute the cost of carpeting each of the rooms with Wilton 
 carpet, 3/4 yd. wide, at $1.25 per yd. Allow 9" for matching design, and 
 allow for an 18" border, at $1 35 per yd. Lay the carpet in the direction 
 that will be more economical. 
 
156 
 
 BUSINESS ARITHMETIC. 
 
 Geometric Conceptions — Solids. 
 A solid is that which has three dimensions, length, width 
 and thickness. It is bounded by surfaces. 
 
 A polyhedron is a solid bounded by 
 plane surfaces called faces. A poly- 
 hedron having two parallel and equal 
 bases, and three or more sides which 
 are parallelograms, is a prism. 
 
 A prism is a rectangular solid (A) if 
 it is bounded by six rectangular sur- 
 faces. It is a cube if these surfaces 
 are all squares. 
 A pyramid (B) is a polyhedron whose faces, with one 
 exception, meet in a common point called the vertex. The 
 faces of a pyramid are therefore triangles. 
 
 A cylinder (C) is a solid bounded by two eqUal parallel 
 circles and a uniformly curved surface. 
 
 A conical surface is a surface generated by 
 the motion of a straight line that continu- 
 ally intersects a fixed curve, frequently a cir- 
 cle, and passed through a fixed point. A cone 
 (D) is a solid bounded by a closed conical sur- 
 face and a plane surface. 
 
 A spherical surface is a curved surface every 
 
PRACTICAL MEASUREMENTS. 
 
 157 
 
 point of which is equidistant from 
 a point called the center. A sphere 
 (E) is a solid bounded by a spher- 
 ical surface. A hemisphere is one- 
 half of a sphere cut by a plane pass- 
 ing through the center. 
 
 INTRODUCTORY EXERCISE. 
 
 A cubic foot is a solid 1 foot long, 1 
 foot wide and 1 foot thick. How many cubic feet in figure (6) if each 
 edge is 1 foot long? How many cubic feet in each horizontal row of (c)? 
 Hovv' many rows? How many cubic feet in all? How many rows and 
 units per row in one vertical slice of (d)? How many sections? How 
 many units in all ? What is the area of the end face ? Of the top 2 
 What is the total area of all the faces? 
 
 It is evident that the volume of a rectangular solid is equal 
 to the product of its three dimensions. These dimensions 
 must ordinarily be expressed in units of the same denomi- 
 nation. The total surface area is equal, evidently, to the 
 sum of the areas of the individual faces. 
 
 ORAL EXERCISE. 
 
 Compute the cubic contents, or volume, of: 
 
 1. A cube of 6" edge. 
 
 2. A rectangular solid measuring 6' by 5' by 8'. 
 
 3. A rectangular solid measuring 8' by 4^' by 10'. 
 
 4. An excavation, 10' by 20' by 30'. 
 
 5. A piece of metal 6" wide, 12" thick, and 10' long. 
 
158 
 
 BUSINESS ARITHMETIC. 
 
 EXERCISE. 
 1. Compute the number of cubic yards of earth that must be ex- 
 cavated to sink this cellar to a depth of 8'. 
 
 Note. Find area and multiply by 
 32 FT. depth. Check, by dividing into separate 
 
 prisms as shown by dotted lines. Why 
 do these processes bring the same result? 
 
 2. Determine the number of cubic 
 feet in a stone column 4^' by 2' by 18'. 
 
 3. Compute the cost of a masonry 
 wall, 40 long, 6' high and 2' wide, at $4.36 
 per cu. yd. 
 
 4. Determine the cubic contents of a 
 room 60' by 25' by 10'. Determine the 
 surface area of walls, ceiling and floor. 
 
 5. A school room, 36' by 30' by 10', 
 seats 42 people. How many cubic feet of 
 air space is allowed each person? 
 
 6. At the rate of 22 brick per cu. ft., 
 compute the cost at $2 per M. of the 
 
 bricks necessary for a wall 36' long, 20' high and 24" wide for half the 
 height and 18" wide for the remaining height. 
 
 
 
 20 FT. 
 
 u. 
 vo 
 
 
 12 FT. 
 
 ORAL EXERCISE. 
 
 1. Find the cubic contents of this rectangular soUd? 
 effect on the volume of doubling any one dimension? 
 
 2. If the volume of a rectangular 
 solid is 180 cu. in., and the product of 
 two dimensions is 30 sq. in., what is 
 the third dimension? 
 
 3. If two dimensions are 8' and 
 12', what is the third dimension of a 
 solid having a volume of 960 cu. ft.? 
 
 Note. It is evident that the third 
 dimension may be determined by divi- 
 ding the volume by the product of the 
 two known dimensions. 
 
 What is the 
 
 8 FT. 
 
 EXERCISE. 
 
 1. 8' X 4.5' X ? = 720 cu. ft. 
 
 2. Select two dimensions to be used with a height of 9', to determine 
 
PRACTICAL MEASUREMENTS. 159 
 
 a rectangular solid of 120 cu. ft. of volume. How many solutions are 
 possible? 
 
 3. Name three dimensions of a solid of a cubic content of 5600 cu. in. 
 
 4. Determine the height of a room 30' long and 20' 6" wide, if it is to 
 have a cubic content of 5535 cu. ft. 
 
 The Cylinder and Sphere. 
 
 If the surface of a paper cylinder is cut along its length, 
 and the paper rolled back, it will be seen to have the form of a 
 rectangle whose height is the altitude of the cylinder and 
 whose base is the length of the circumference of the base 
 circle. Evidently, the lateral area is the product of the altitude 
 by the circumference. To this must be added, to determine 
 the entire area, the areas of the two base circles. 
 
 ORAL EXERCISE. 
 
 1. If the area of the base of a rectangular solid is 14 sq. ft., what is 
 the volume per foot of height? 
 
 2. If the area of the circular base of a cylinder is 30 sq. in., what is the 
 volume per foot of height? What is the volume if the height is 12 ft.? 
 
 As seems evident from the above, the volume of a cylin- 
 der equals the product of the altitude and the area of the base. 
 
 EXERCISE. 
 
 1. Allowing for 1^" overlap, how much sheet metal is required 
 for the lateral surface of a hollow cylinder of 16' altitude and 3' radius of 
 base? 
 
 2. Determine the cubic contents of this cylinder. 
 
 3. What must be the altitude of a cylinder of 4' diameter to have a 
 cubic content of 600 cu. ft.? 
 
 4. How many cu. ft. of water will a circular cistern of 12' depth and 5' 
 radius contain? 
 
 5. Determine reasonable dimensions for a cistern to contain 5000 cu. ft. 
 
 6. Metal pieces for the lateral surfaces of cans of 2" radius and 6" 
 height are to be cut from sheet metal 24" by 32", no allowance being made 
 for overlap? How many can be cut from each sheet? 
 
160 BUSINESS ARITHMETIC. 
 
 The surface of a sphere is equal to ^ X tt X ^^^ square of the 
 radius; and the volume of the sphere is equal to 4/3 X ir X the 
 cube of the radius, 
 
 EXERCISE. 
 
 1. Find the surfaces of spheres of radii (a) 2"; (6) 6"; (c) 21''. 
 
 2. Find the volume of spheres of radii (a) 5"; (6) 15"; (c) 26' 6". 
 
 3. Find the volume of a hemisphere of .4' 6" radius. 
 
 4. Find the radius of a sphere of 750 sq. in. of surface. 
 
 5. How many square inches of gilding must be done to cover a sphere 
 of 16" radius? 
 
 SPECIAL APPLICATIONS. 
 
 Cord Measure. 
 
 Fuel wood, tan bark, sometimes stone, etc., are measured 
 by the cord, a unit pile 8' long, 4' wide and 4' high, containing 
 128 cu. ft. In many localities, a cord of fuel is a pile 8' long 
 and 4' high, the price varying with the width — as half sawed, 
 quartered sawed, etc. It is evident that if a pile is 4' high 
 and 4' wide, the number of cords may be determined from the 
 length, as measured by the unit length of 8'. Thus a 36' pile 
 contains 4^ cords. In the case of irregular dimensions, the 
 measure is the number of cubic feet in a cord. 
 
 EXERCISE. 
 
 1. A wood lot produces piles of wood of regulation 4' length (width), 
 of dimensions as follows: 10' long, 3' high; 12' long, 5' high; 16' long, 4' 
 high; 20' long, 4' high. How many cords are produced? 
 
 2. A pile of hemlock bark measures 80' by 4' by 21' and contains 
 how many cords? 
 
 3. Sixteen acres of woodland produce a pile of cord wood 12' high 
 and 47' long. What is the product in cords per acre? 
 
 4. Five cords of wood, regulation size, are to be piled in a shed along 
 a wall measuring 15'. How high should the pile be? 
 
 Lumber. 
 
 In the measurement of boards of less than 1" in thickness, 
 square measure of surface is used. In the measurement of 
 
PRACTICAL MEASUREMENTS. 
 
 161 
 
 boards of 1" thickness, or over, such as heavy boarding, 
 planks, joists, scantUng, and heavy dressed timber, the unit 
 measure is the hoard foot. The hoard foot is a hoard 1 ' long, 
 12" wide and 1" thick , or its equivalent. Since the end area 
 is 12 sq. in., the board foot may be said to have a length of 1' 
 and an end area of 12 sq. in. 
 
 Board Feet. In measuring width, except in the more valu- 
 able woods, the next lower quarter inch of width is sometimes 
 taken. Thus a 6|" width is reckoned 6f and a ^\" width, 
 9'', etc. 
 
 In stating quantity and dimensions of an order, or esti- 
 mate, the arrangement is: Number of pieces — thickness in 
 inches — width in inches — length in feet, thus 30 pc, 6" X 8" 
 — 16'. The price is generally stated per M feet, board measure. 
 
 In computation, both on paper and orally, the method of 
 cancellation is valuable. 
 
 Illustration. Compute the number of feet, board measure, in 
 36 pc, 2\" X 10" —16'. 
 
 Solution. 3 2| X 10 ^ no. bd. ft. per ft. of 
 
 = 1200, no. bd. ft. 12 l^"^^^' 
 
 This no. X length X no. pieces 
 = no. bd. ft. 
 
 21X10X16X ^6^ 
 1 
 
 Evidently the number of board feet is the product of the 
 width in inches, hy the thickness in inches, hy the length in feet, 
 by the number of pieces, divided hy 12. The quantity to cancel 
 is always 12. Notice that this is one of the rare cases where 
 dimensions are not expressed in the same unit. 
 12 
 
162 BUSINESS ARITHMETIC. 
 
 ORAL EXERCISE. 
 
 Determine the number of board feet per foot of length in timbers of the 
 following dimensions : 
 
 1. 13"X4". 4. 2"X9". 7. 3"X6". 10. 3"X9". 
 
 2. 8"X16". 5. 2"X8^ 8. 2h"X&'. 11. 2"X3". 
 
 3. 4"X12". 6. 4"X6". 9. 6"X9". 12. 2"X2". 
 Determine the number of board feet in these timbers: 
 
 13. 3"X4"— 16'. 16. 4"X8"— 12'. 
 
 14. 4"X6"— 12'. 17. 2"X9"— 16'. 
 
 15. 8"X10"— 6'. 18. 3"X6"— 12'. 
 Determine the board feet in these items: 
 
 19. lOpc. 2"X6"— 10'. 20. 20pc., 3"X4"— 12'. 
 
 EXERCISE. 
 Determine the number of feet, board measure, in: 
 
 1. 162 pc. hemlock, S"X10"— 32'. 
 
 2. 40 pc. white pine, 2" X 10"— 20'. 
 
 3. 360 pc. maple, 1|"X3"— 18'. 
 
 4. 456pc. oak, 3"X6"— 18'. 
 
 At $26 per M., compute the cost of: 
 
 5. 150 joists, 2"X8"— 16'. 
 
 6. 42 beams, 6"X9"— 24'. 
 
 7. 78 scantling, 2"X4"— 16'. 
 
 8. How much timber is required for a plain board fence around a lot 
 180' by 246', if it consists of two strips of 2"X4" scantling on which is 
 nailed 1" boarding 6' high. This does not include posts. Allow 1/8 
 for waste in cutting. 
 
 Capacity of Bins, Tanks, Etc. 
 
 In measuring the capacity of bins and tanks, it is often 
 
 necessary to express the common dry and liquid measures in 
 
 terms of cubic feet. 
 
 Illustrations. (1) Determine the capacity in gallons of a rectangular 
 iron tank measuring 4' by 8' by 3'. 
 Solution. (Abstract.) 
 4'X8'X3' =96 cu. ft., capacity. 
 96 X 1728 = capacity in cu. in. 
 
PRACTICAL MEASUREMENTS 163 
 
 ^^oqV^^ =^o. of gal. (since 1 gal. =231 cu. in.) =718+ gal., capacity. 
 
 (2) What is the weight of the water the tank will hold? 
 Solution, 
 
 96 cu. ft. = contents. 
 
 1 cu. ft. of water weighs 62^ lb. 
 
 96X621 lb. =6,000 lb., the weight of water. 
 
 EXERCISE. 
 
 1. Determine the capacity in bushels of a rectangular bin measuring 
 8' by 4' by 4'. Determine the capacity, also, by approximate measure 
 (1 bu. = li cu. ft.). 
 
 2. A bin in a certain bam can be allowed a floor space of only 4' by 
 9' 6". What must be its height if it is to have a capacity of 250 bu.? 
 (Approximate measure.) 
 
 3. Determine reasonable dimensions for a bin to contain 400 bu. of 
 potatoes. Use 1 c\i. ft. as equivalent to .63 of a heaped measure bushel. 
 
 4. In order to have a capacity of 1,000,000 gal., a reservoir must have 
 a capacity of how many cu. ft.? 
 
 5. Compute the capacity, in gallons, of: 
 
 (a) A rectangular tank, 4' X 12' X3' 6". 
 
 (b) A cylindrical tank of 6' diameter and 8' high. 
 
 (c) A cylindrical cistern of 2' 6" radius and 16' deep. 
 
 (d) A vat 6' square and 4' 6" high. 
 
 6. A metal tank, known to have a capacity of 60,000 gal., will hold for 
 temporary storage how many bushels? 
 
 7. Construct a table to give the number of standard gallons in volumes 
 from 1 to 1000 cubic feet. 
 
 SPECIAL INDIVIDUAL PROBLEMS. 
 
 1. Estimate the quantity and cost of materials for a cheap board 
 fence to enclose a vacant lot in your neighborhood. Prepare a plot of the 
 land and a diagram showing character of fence. 
 
 2. Prepare plans and estimates for a modem earth croquet court (or 
 tennis court) to be constructed on a vacant lot in your neighborhood. 
 
 3. Report on the total number of square feet of lighting surface in the 
 school building. Tabulate results (1) by rooms, (2) by floors, (3) by com- 
 
 4. Report on the number of cubic feet of air space in the building, by 
 rooms, corridors and floors. 
 
CHAPTER XXIV. 
 
 MEASUREMENT OF TIME. 
 
 1. Longitude and Time. 
 
 Longitude and time are matters of common interest. They 
 are also fundamental to navigation, astronomy and geography. 
 
 Nortli Pole 
 
 Cape Town 
 
 South Pole 
 
 A meridian is an imaginary line on the surface of the earth, 
 extending due north and south and terminating in the poles. 
 It is therefore a semi-circumference of the earth. Any me- 
 ridian used as a reference Hne, from which to locate places on 
 
 164 
 
MEASUREMENT OF TIME. 165 
 
 the earth's surface by determining their distances east or west 
 of it, is a 'prime meridian. It is now the common custom to 
 use the meridian through the Royal Observatory at Green- 
 wich, England, as a prime meridian. 
 
 Distances east or west of the prime meridian are termed 
 respectively east longitude and west longitude. They are 
 measured by degrees, minutes and seconds of circular measure, 
 as shown on page 164, where the meridians intersect the equator. 
 Each place has its own meridian, but two places may have the 
 same meridian even when thousands of miles apart. Longi- 
 tude west is designated by the symbol ' +,' and longitude 
 east by the symbol ^ — .' Thus the longitude of New York, 
 which is west of Greenwich is +73° 58' 25.5'', while the longi- 
 tude of. Brussels, east of Greenwich, is — 4° 22' 9". 
 
 From a study of the diagram, it is evident that the 
 difference in longitude of two places may be obtained by 
 subtraction, if the signs of the individual longitudes are the 
 same; and by addition if they are different. 
 
 Since the earth revolves on its axis once in twenty-four 
 hours, every point on the surface revolves in a circle, or passes 
 through 360° of circular measure, in that time. Hence there 
 is a close relation between time and degrees of circular measure. 
 
 Since 360° correspond to 24 hours. 
 1° is equivalent to 1/360 of 24 hr., or 4 min. 
 1' is equivalent to 1/60 of 4 min., or 4 sec. 
 1" is equivalent to 1/60 of 4 sec, or 1/15 sec. 
 
 Since 24 hours correspond to 360°, 
 1 hr. is equivalent to 1/24 of 360°, or 15°. 
 1 min. is equivalent to 1/60 of 15°, or 15'. 
 1 sec. is equivalent to 1/60 of 15', or 15". 
 
 Regarding the sun as stationary with reference to the earth, 
 the earth, revolving from west to east, carries the meridians 
 under the sun in succession, the latter seeming to move toward 
 the west. Since all meridians pass under the sun in suc- 
 cession each twenty-four hours, they really mark, by differ- 
 
166 BUSINESS ARITHMETIC. 
 
 ence in longitude, the difference in time of places on the earth's 
 surface. In science, difference of longitude is often expressed 
 in time. Local time is determined at each place by calling 
 the instant the sun crosses its meridian 12:00 noon. 
 
 Illustration. If the difiference in longitude of two places is 84° 30', 
 what is the difiference in time? 
 Solution. 
 84° = 84 X 1/15 hr. Since 1° corresponds to 1/15 hr. 
 
 = 5fhr. =5hr. 36min. 
 
 30' = 30X4 sec. = 2 Since 1' corresponds to 4 sec. 
 
 84° 30' =5 hr. 38 sec, difiference in time. 
 
 EXERCISE. 
 
 1. Do points in the east or west of a country receive sunlight first? 
 Which have earlier local time? 
 
 2. If two places have different west longitudes, which has earlier time? 
 Which of two places having east longitude has the earlier local time? 
 
 Evidently the more western points have the later time; 
 the more eastern the earlier. Since time bears an intimate 
 relation to degrees of longitude, it is a simple matter to com- 
 pute one from the other. 
 
 Illustrations. (1) The longitude of Albany is 73° 44' 48", and of 
 Ann Arbor, Mich., 83° 43' 48". Determine their difiference in time. 
 Solution. 
 
 83° 43' 48", long, of Ann Arbor, 
 73° 44' 48", long, of Albany, 
 9° 59' = difiference in longitude. 
 
 9X1/15 hr. =3/5 hr., or 36 min. 
 5dX4sec. =236 sec, o r 3 min. 56 sec. 
 9° 59' = 39 min. 56 sec, difif. in time. 
 
 (2) The difiference in the time of two places is 6 hr. 18 min. 12 sec 
 Determine the difiference in longitude. 
 Solution. 
 
 6 hr. 18 min. 12 sec = difiference in time. 
 6hr. = 6X15° =90° 
 18 min. = 18X15' = 4° 30' 
 
 12 sec =12X15" = 3' 
 
 6 hr. 18 min. 12 sec. =94° 33', difiference in longitude. 
 
Athens, 
 
 -23° 43' 55.5". 
 
 Berlin, 
 
 -13° 23' 43.5". 
 
 Chicago, 
 
 +87° 36' 42.0". 
 
 New York, 
 
 , +74° 0' 3.0". 
 
 Paris, 
 
 - 2° 20' 15.0". 
 
 MEASUREMENT OF TIME. 167 
 
 Note. Both these solutions are based directly on the tables of com- 
 parative values given on page 165. 
 
 Longitude of Well-Known Places. 
 
 Philadelphia, +75° 9' 45.0". 
 Rome, -12° 27' 14.0". 
 
 San Francisco, +122° 25' 40.8". 
 Tokyo, -139° 42' 30". 
 
 EXERCISE. 
 
 1. If the difference in longitude of two places is 71° 18' 0", what is 
 the difference in time? 
 
 2. Compute the difference in local time between (a) Berlin and Phila- 
 delphia; (6) Chicago and San Francisco; (c) Rome and Paris, and (d) 
 Chicago and Tokyo. 
 
 3. A man coming to Chicago found that his watch, set by local time 
 at his starting point, was 48 minutes slow by local Chicago time. Did 
 he come from east or west? From what longitude? 
 
 4. When it is 5:00 P. M. at Chicago, it is what time at (a) San Fran- 
 cisco, (6) Berlin, (c) Rome and (d) New York ? 
 
 5. A ship sailing in longitude 183° 40' 9" is how far in difference of 
 longitude from the port of New York. Near what great body of land is 
 the ship? 
 
 The International Date Line has been selected as the 180th 
 meridian, with slight changes at certain points. This meridian 
 in the Pacific Ocean passes through no important land areas. 
 Going westward ships set their calendars forward one day at 
 this point; going east they set them back one day. 
 
 Standard Time. If every place used its strict local time, it 
 is evident that endless confusion would result in travel, 
 especially from difficulties in constructing time tables. On 
 this account the railways of the United States and Canada 
 agreed on four time belts for the two countries. The belts 
 are approximately 15° wide, and each belt takes the time of 
 its center meridian. Thus the guide meridians are 15°, or 
 one hour apart. The belts are the Eastern, Central, Moun- 
 tain and Pacific, the meridians being +75°, +90°, +105° and 
 
168 BUSINESS ARITHMETIC. 
 
 + 120°. Thus, when it is 10:00 A.M. in any town in the 
 Eastern time belt, it is 9:00 A.M. in the Central belt, 8:00 
 A.M. in the Mountain belt and 7:00 A.M. in the Pacific 
 belt. 
 
 Foreign countries are adopting similar belts, based largely 
 on meridians at 15° intervals from Greenwich. 
 
 2. Commercial Time. 
 
 The accurate computation of time between dates, and of 
 dates of maturity, is of vital importance in business. All 
 interest charges depend on time. Many business and com- 
 mercial papers run for fixed times to dates of maturity. 
 Considerable merchandise is sold on limited time. Failure 
 to correctly compute a payment date may cause a business 
 man to lose standing or credit, by being caught unprepared to 
 meet just demands. 
 
 Many seemingly minor details affect time and date com- 
 putation. Time may be computed either approximately 
 or by exact days, and dates of legal maturity are affected by 
 varying state laws affecting holidays and Sundays (see page 
 312). 
 
 I. Time Between Dates. 
 
 Since the common business time is six months, or less, it is 
 possible to make most time computations without use of paper. 
 Where paper is used, the method is often different. 
 
 (a) Approximate Time. (Paper.) 
 
 Illustration. Find the time between June 21, 1908, and Jan. 17, 
 1910. 
 
 Solution. Follow the method of denomi- 
 nate numbers. In the subtrahend, 1 mo. is 
 reduced to days, making 30+17, or 47 da.; 
 1 yr. 6 mo. 26 da. and 1 yr. is reduced to 12 mo. 
 
 Note. In reduction, one month is considered 30 days, but if a given 
 date contains the thirty-first day it is retained. 
 
 Yr. 
 
 Mo. 
 
 Da. 
 
 1910 
 
 1 
 
 17 
 
 1908 
 
 6 
 
 21 
 
MEASUREMENT OF TIME. 169 
 
 EXERCISE. 
 
 Find the time between: 
 
 1. Jan. 27, 1906, and Aug. 29, 1909. 4. 3/ 3/ '07 and 6/ I/'IO. 
 
 2. Oct. 21, 1907, and Dec. 15, 1912. 5. 7/ 5/ '06 and 10/ 17/ '06. 
 
 3. Jan. 31, 1908, and Sept. 12, 1909. 6. 3/ 21/ '07 and 5/ 18/ '09. 
 
 7. Money borrowed Apr. 16, 1907, and due to-day, has drawn interest 
 for how long? 
 
 8. I am given until Oct. 1 to pay a debt due Feb. 11. For how long is 
 time of payment extended? 
 
 (b) Approximate Time. (Mental.) 
 
 Illustrations. (1) Find the time between Mar. 12, 1906, and May 
 17, 1908. 
 
 Solution. Reason as follows: From Mar. 12, 1906, to Mar. 12, 1908, 
 2 years; from Mar. 12, 1908. to May 12, 2 months; from May 12 to May 
 17, 5 days. Total time, 2 yr. 2 mo. 5 da. 
 
 Note. The process involves adding to the earlier date, on the principle 
 used in making change. 
 
 (2) Find the time between Aug. 28, 1913, and July 12, 1914. 
 
 Solution. Aug. 28 (8th mo.) to June 28 (6th mo.) = 10 mo. June 28 
 to July 12 = 2 + 12 da. = 14 da. Total time, 10 mo. 14 da. 
 
 EXERCISE. 
 
 Compute the time between : 
 
 1. Feb. 11 and Apr. 29. 5. Mar. 16, 1912, and Dec. 15, 1913. 
 
 2. June 16 and Dec. 31. 6. Feb. 28, 1912, and Nov. 29, 1912. 
 
 3. May 31 and Nov. 24. 7. Aug. 13 and to-day. 
 
 4. Mar. 17 and Oct. 9. 8. May 19 and to-day. 
 
 Exact Time. In computing by exact time, the true number 
 of days in each month is taken into account. Usually only 
 one of the extreme days is included, although some bankers 
 include both. Exact time is computed for periods less than 
 a year. Longer terms are reckoned in calendar years and 
 exact days. 
 
 (c) Exact Time. (By Table.) 
 
 The following table shows the calendar year and the day of the year of 
 each calendar day. Thus Feb. 17 is the 48th day of the year. Other 
 forms of tables are very common. 
 
170 
 
 BUSINESS ARITHMETIC. 
 
 
 
 Table 
 
 FOR THE 
 
 Calculation of Time 
 
 
 
 
 Jan. 1 
 
 1 
 
 Feb. 1 
 
 32 
 
 Mar. 1 60 
 
 Apr. 1 
 
 91 
 
 May 1 
 
 121 
 
 June 1 
 
 152 
 
 2 
 
 2 
 
 2 
 
 33 
 
 2 61 
 
 2 
 
 92 
 
 2 
 
 122 
 
 2 
 
 153 
 
 3 
 
 3 
 
 3 
 
 34 
 
 3 62 
 
 3 
 
 93 
 
 3 
 
 123 
 
 3 
 
 154 
 
 4 
 
 4 
 
 4 
 
 35 
 
 4 63 
 
 4 
 
 94 
 
 4 
 
 124 
 
 4 
 
 155 
 
 5 
 
 5 
 
 5 
 
 36 
 
 5 
 
 64 
 
 5 
 
 95 
 
 5 
 
 125 
 
 5 
 
 156 
 
 6 
 
 6 
 
 6 
 
 37 
 
 6 
 
 65 
 
 6 
 
 96 
 
 6 
 
 126 
 
 6 
 
 157 
 
 7 
 
 7 
 
 7 
 
 38 
 
 7 
 
 66 
 
 7 
 
 97 
 
 7 
 
 127 
 
 7 
 
 158 
 
 8 
 
 8 
 
 8 
 
 39 
 
 8 
 
 67 
 
 8 
 
 98 
 
 8 
 
 128 
 
 8 
 
 159 
 
 9 
 
 9 
 
 9 
 
 40 
 
 9 
 
 68 
 
 9 
 
 99 
 
 9 
 
 129 
 
 9 
 
 160 
 
 10 
 
 10 
 
 10 
 
 41 
 
 10 
 
 69 
 
 10 
 
 100 
 
 10 
 
 130 
 
 10 
 
 161 
 
 11 
 
 11 
 
 11 
 
 42 
 
 11 
 
 70 
 
 11 
 
 101 
 
 11 
 
 131 
 
 11 
 
 162 
 
 12 
 
 12 
 
 12 
 
 43 
 
 12 
 
 71 
 
 12 
 
 102 
 
 12 
 
 132 
 
 12 
 
 163 
 
 13 
 
 13 
 
 13 
 
 44 
 
 13 
 
 72 
 
 13 
 
 103 
 
 13 
 
 133 
 
 13 
 
 164 
 
 14 
 
 14 
 
 14 
 
 45 
 
 14 
 
 73 
 
 14 
 
 104 
 
 14 
 
 134 
 
 14 
 
 165 
 
 15 
 
 15 
 
 15 
 
 46 
 
 15 
 
 74 
 
 15 
 
 105 
 
 15 
 
 135 
 
 15 
 
 166 
 
 16 
 
 16 
 
 16 
 
 47 
 
 16 
 
 75 
 
 16 
 
 106 
 
 16 
 
 136 
 
 16 
 
 167 
 
 17 
 
 17 
 
 17 
 
 48 
 
 17 
 
 76 
 
 17 
 
 107 
 
 17 
 
 137 
 
 17 
 
 168 
 
 18 
 
 18 
 
 18 
 
 49 
 
 18 
 
 77 
 
 * 18 
 
 108 
 
 18 
 
 138 
 
 18 
 
 169 
 
 19 
 
 19 
 
 19 
 
 50 
 
 19 
 
 78 
 
 19 
 
 109 
 
 19 
 
 139 
 
 19 
 
 170 
 
 20 
 
 20 
 
 20 
 
 51 
 
 20 
 
 79 
 
 20 
 
 110 
 
 20 
 
 140 
 
 20 
 
 171 
 
 21 
 
 21 
 
 21 
 
 52 
 
 21 
 
 80 
 
 21 
 
 111 
 
 21 
 
 141 
 
 21 
 
 172 
 
 22 
 
 22 
 
 22 
 
 53 
 
 22 
 
 81 
 
 22 
 
 112 
 
 22 
 
 142 
 
 22 
 
 173 
 
 23 
 
 23 
 
 23 
 
 54 
 
 23 
 
 82 
 
 23 
 
 113 
 
 23 
 
 143 
 
 23 
 
 174 
 
 24 
 
 24 
 
 24 
 
 55 
 
 24 
 
 83 
 
 24 
 
 114 
 
 24 
 
 144 
 
 24 
 
 175 
 
 25 
 
 25 
 
 25 
 
 56 
 
 25 
 
 84 
 
 25 
 
 115 
 
 25 
 
 145 
 
 25 
 
 176 
 
 26 
 
 26 
 
 26 
 
 57 
 
 26 
 
 85 
 
 26 
 
 116 
 
 26 
 
 146 
 
 26 
 
 177 
 
 27 
 
 27 
 
 27 
 
 58 
 
 27 
 
 86 
 
 27 
 
 117 
 
 27 
 
 147 
 
 ^ 27 
 
 178 
 
 28 
 
 28 
 
 28 
 
 59 
 
 28 
 
 87 
 
 28 
 
 118 
 
 28 
 
 148 
 
 28 
 
 179 
 
 29 
 
 29 
 
 — 
 
 — 
 
 29 
 
 88 
 
 29 
 
 119 
 
 29 
 
 149 
 
 29 
 
 180 
 
 30 
 
 30 
 
 — 
 
 — 
 
 30 
 
 89 
 
 30 
 
 120 
 
 30 
 
 150 
 
 30 
 
 181 
 
 31 
 
 31 
 
 — 
 
 — 
 
 31 
 
 90 
 
 — 
 
 — 
 
 31 
 
 151 
 
 
 
 Illustration. (1) The time from Aug. 17 to Dec. 21 is ? days. 
 Solution. From table. Dec. 21= 355th da. 
 Aug. 17 = 229th da. 
 The difference equals elapsed time =126 da. 
 (2) Nov. 11 to Jan. 26= ? da. 
 
 Solution (a). Nov. 15 is 315th day. 365 da. -315 da. = 50 da. 
 Jan. 26 is 26th day of next year 26 
 
 Total time 76 da. 
 
 Solution (6). The complementary time between Nov. 11 and Jan. 26 
 is the time between Jan. 26 and Nov. 11. 
 
 Nov. 11= 315th da. 
 Jan. 26= 26th da . 
 Complementary elapsed time =289 da. 
 
 True elapsed time =365 da. -289 da. = 76 da. 
 
MEASUREMENT OF TIME. 
 
 171 
 
 Table for the Calculation of Time. 
 
 July 
 
 182 
 
 Aug. 1 
 
 213 
 
 183 
 
 2 
 
 214 
 
 184 
 
 3 
 
 215 
 
 185 
 
 4 
 
 216 
 
 186 
 
 5 
 
 217 
 
 187 
 
 6 
 
 218 
 
 188 
 
 7 
 
 219 
 
 189 
 
 8 
 
 220 
 
 190 
 
 9 
 
 221 
 
 191 
 
 10 
 
 222 
 
 192 
 
 11 
 
 223 
 
 193 
 
 12 
 
 224 
 
 194 
 
 13 
 
 225 
 
 195 
 
 14 
 
 226 
 
 196 
 
 15 
 
 227 
 
 197 
 
 16 
 
 228 
 
 198 
 
 17 
 
 229 
 
 199 
 
 18 
 
 230 
 
 200 
 
 19 
 
 231 
 
 201 
 
 20 
 
 232 
 
 202 
 
 21 
 
 233 
 
 203 
 
 22 
 
 234 
 
 204 
 
 23 
 
 235 
 
 205 
 
 24 
 
 236 
 
 206 
 
 25 
 
 237 
 
 207 
 
 26 
 
 238 
 
 208 
 
 27 
 
 239 
 
 209 
 
 28 
 
 240 
 
 210 
 
 29 
 
 241 
 
 211 
 
 30 
 
 242 
 
 212 
 
 31 
 
 243 
 
 Sept. 
 
 1 
 2 
 3 
 4 
 
 6 
 7 
 8 
 9 
 10 
 11 
 12 
 13 
 14 
 15 
 16 
 17 
 18 
 19 
 20 
 21 
 22 
 23 
 24 
 25 
 26 
 27 
 28 
 29 
 30 
 
 244 
 
 Oct. 1 
 
 274 
 
 Nov. 1 
 
 305 
 
 Dec. 1 
 
 245 
 
 2 
 
 275 
 
 2 
 
 306 
 
 2 
 
 246 
 
 3 
 
 276 
 
 3 
 
 307 
 
 3 
 
 247 
 
 4 
 
 277 
 
 4 
 
 308 
 
 4 
 
 248 
 
 5 
 
 278 
 
 5 
 
 309 
 
 5 
 
 249 
 
 6 
 
 279 
 
 6 
 
 310 
 
 6 
 
 250 
 
 7 
 
 280 
 
 7 
 
 311 
 
 7 
 
 251 
 
 8 
 
 281 
 
 8 
 
 312 
 
 8 
 
 252 
 
 9 
 
 282 
 
 9 
 
 313 
 
 9 
 
 253 
 
 10 
 
 283 
 
 10 
 
 314 
 
 10 
 
 254 
 
 11 
 
 284 
 
 11 
 
 315 
 
 11 
 
 255 
 
 12 
 
 285 
 
 12 
 
 316 
 
 12 
 
 256 
 
 13 
 
 286 
 
 13 
 
 317 
 
 13 
 
 257 
 
 14 
 
 287 
 
 14 
 
 318 
 
 14 
 
 258 
 
 15 
 
 288 
 
 15 
 
 319 
 
 15 
 
 259 
 
 16 
 
 289 
 
 16 
 
 320 
 
 16 
 
 260 
 
 17 
 
 290 
 
 17 
 
 321 
 
 17 
 
 261 
 
 18 
 
 291 
 
 18 
 
 322 
 
 18 
 
 262 
 
 19 
 
 292 
 
 19 
 
 323 
 
 19 
 
 263 
 
 20 
 
 293 
 
 20 
 
 324 
 
 20 
 
 264 
 
 21 
 
 294 
 
 21 
 
 325 
 
 21 
 
 265 
 
 22 
 
 295 
 
 22 
 
 326 
 
 22 
 
 266 
 
 23 
 
 296 
 
 23 
 
 327 
 
 23 
 
 267 
 
 24 
 
 297 
 
 24 
 
 328 
 
 24 
 
 268 
 
 25 
 
 298 
 
 25 
 
 329 
 
 25 
 
 269 
 
 26 
 
 299 
 
 26 
 
 330 
 
 26 
 
 270 
 
 27 
 
 300 
 
 27 
 
 331 
 
 27 
 
 271 
 
 28 
 
 301 
 
 28 
 
 332 
 
 28 
 
 272 
 
 29 
 
 302 
 
 29 
 
 333 
 
 29 
 
 273 
 
 30 
 
 303 
 
 30 
 
 334 
 
 30 
 
 — 
 
 31 
 
 304 
 
 — 
 
 — 
 
 31 
 
 335 
 336 
 337 
 338 
 339 
 340 
 341 
 342 
 343 
 344 
 345 
 346 
 347 
 348 
 349 
 350 
 351 
 352 
 353 
 354 
 355 
 356 
 357 
 358 
 359 
 360 
 361 
 362 
 363 
 364 
 365 
 
 EXERCISE. 
 
 Compute the time between ; 
 
 1. Feb. 11 and Oct. 16. 
 
 2. Mar. 19 and Nov. 10. 
 
 3. Jan. 15 and Oct. 29. 
 
 4. Apr. 24 and Dec. 27. 
 
 5. Feb. 28 and Aug. 7. 
 
 6. July 15 and Oct. 1. 
 
 7. Mar. 27, 1910, and Oct. 16, 1911. 
 
 8. Apr. 14, 1909, and Nov. 22, 1911. 
 
 9. May 29, 1910, and Dec. 19, 1911. 
 10. Nov. 16, 1912, and Apr. 26, 1914. 
 
 11. Show how to use the table in leap years. 
 
 (d) Exact Time. (Without Tables.) 
 Illustrations. Find the time between Aug. 18 and Oct. 27. 
 
172 BUSINESS ARITHMETIC. 
 
 Solution (a). Addition method. Aug. 18 to 31 = 13 da. 
 
 Sept. 1 to 30 = 30 
 
 Oct. 1 to 7 = 27 
 
 Total time =70 da. 
 Note. Mentally, each group is added to the previous total, as found. 
 One would say 13, 43, 70. 
 
 Solution (&). From approximate time. 
 
 Reason as follows: Aug. 18 to Oct. 27, 2 mo. 9 da., or 69 da. But this 
 period includes a 31st day in August. Therefore the exact time is 69 da. 
 + 1 da., or 70 da. 
 
 Note. Since February has 28 days, this method involves subtraction 
 when the period includes the last of this month. 
 
 EXERCISE. 
 
 Find the exact number of days from : 
 
 1. Aug. 28 to Dec. 29. 6. Jan. 5 to Mar. 8. 
 
 2. Sept. 19, 1907, to Jan. 25, 1908. 7. Nov. 11 to date. 
 
 3. Oct. 1 to Jan. 13. 8. Oct. 16 to date. 
 
 4. Aug. 12 to Dec. 3. 9. Feb. 21 to June 9. 
 
 5. Mar. 28 to Apr. 14. 10. Mar. 3 to Aug. 2. 
 
 II. Dates of Matukity. 
 
 In computing final date? and dates of maturity, it is neces- 
 sary to know the initial date and the "term" or period of time. 
 If the term is stated in months and days, approximate methods 
 are used; if stated in days, as "90 days," exact methods are 
 used. "Day" terms are seldom over 90 days. 
 
 (a) Approximate Method. 
 
 Illustration. A payment due 1 yr. 7 mo. 19 da. from Aug. 11, 1912, 
 is due on what date? 
 
 Solution. (Paper.) Write in denominate form. 1912 8 11 
 (Add and reduce upward.) 1 7 19 
 
 1914 3 30 
 Date due, Mar. 30, 1914. 
 Solution. (Mental.) 
 
 Reason as follows: Aug. 11, 1912, + 1 yr.=Aug. 11, 1913; Aug. 11, 
 1913,+7 mo. = Mar. 11, 1914; Mar. 11 + 19 da. = Mar. 30, 1914. 
 
MEASUREMENT OF TIME. 173 
 
 EXERCISE. 
 
 Find the final date. 
 
 Initial Date. Term. 
 
 1. Jan. 27, 1907 1 yr. 3 mo. 18 da. 
 
 2. Feb. 16, 1913 2 yr. 5 mo. 11 da. 
 
 3. Mar. 9, 1908 1 yr. 11 mo. 3 da. 
 
 4. Apr. 14, 1908 2 yr. 5 mo. 29 da. 
 
 5. May 30, 1908 1 yr. 2 mo. 11 da. 
 
 6. Jmie 14, 1912 9 mo. 19 da. 
 
 7. Aug. 21, 1908 1 yr. 5 mo. 25 da. 
 
 8. Sept. 4, 1913 2 yr. 5 mo. 6 da. 
 
 ORAL EXERCISE. 
 
 1. What date is 1 mo. from Jan. 28? From Jan. 29? Jan. 30? 
 Feb. 1? Mar. 30? Apr. 6? June 20? Aug. 31? 
 
 Note. If there is no corresponding date, the last date in the month 
 is taken. Thus, 1 mo. from Oct. 31 is Nov. 30. 
 
 2. Mar. 6+lmo. = ? • 6. Nov. 30+3 mo. =? 
 
 3. Apr. 20+3 mo. =? 7. Dec. 31+2 mo. = ? 
 
 4. June 30+4 mo.=? 8. Dec. 31+6 mo. = ? 
 
 5. July 31+4 mo. =? 9. July 14+8 mo. = ? 
 
 (b) Exact Method. {By Table, pp. 170, 171.) 
 
 ORAL EXERCISE. 
 
 1. What day of the year is June 14? What day of the year is 60 days 
 later? What date? 
 
 2. Find a date 90 days in advance of June 29. 
 {Solution. June 29 = 180th day. 180 da. +90 da. =270 da. 
 
 By table, the 270th day is Sept. 27.) 
 Find the final dates. 
 
 Initial Date. 
 
 Term. 
 
 
 Initial Date. 
 
 Term. 
 
 3. June 26 
 
 • 90 da. 
 
 7. 
 
 Oct. 14 
 
 125 da. 
 
 4. Aug. 31 
 
 90 da. 
 
 8. 
 
 June 11 
 
 45 da. 
 
 5. Jan. 30 
 
 30 da. 
 
 9. 
 
 July 29 
 
 40 da. 
 
 6. Feb. 26 
 
 60 da. 
 
 10. 
 
 Aug. 27 
 
 165 da. 
 
174 BUSINESS ARITHMETIC. 
 
 (c) Exact (Periodic) Method. 
 
 ORAL EXERCISE. 
 
 1. What date comes 30 days after Jan. 31? After Mar. 31? After 
 April 20? After July 16? 
 
 2. When, for periods of 30 days, is the final date the corresponding date 
 of the proper month? When is it earlier? When later? 
 
 3. Find the date of maturity of a paper running 90 days and dated 
 Jan. 26. 
 
 Solution. Jan. 26+30 da. = Feb. 25, since Jan. has 31 da.; Feb. 25 
 +30 da. = Mar. 27. (Why?) Mar. 27+30 da. =Apr. 26. Therefore, the 
 paper is due Apr. 26. 
 
 Find the dates of maturity. 
 Initial Date. Term. 
 
 Initial Date. 
 
 Term. 
 
 4. Jan. 29 
 
 60 da. 
 
 8. June 27 
 
 90 da. 
 
 5. Feb. 17 
 
 90 da. 
 
 9. Aug. 14 
 
 75 da. 
 
 6. Apr. 30 
 
 60 da. 
 
 10. Sept. 19 
 
 60 da. 
 
 7. May 19 
 
 40 da. 
 
 11. Nov. 30 
 
 60 da. 
 
 INDIVIDUAL ORIGINAL WORK. 
 
 1. Report on "The Arithmetic of Time Table Construction." Give 
 illustrations. Give some idea of the character of information obtainable 
 from time tables. 
 
 2. Prepare a time table and car schedule for a double track street 
 railway. Estimate the number of motors, trailers, and the working force 
 necessary for efficient service. Length of trip 40 minutes from 6 A.M. 
 to 9 P.M.; 35 minutes from 9 P.M. to 6 A.M. 3 minute schedule, 7 to 
 10 A.M.; 3 to 7 P.M.; 5 minute schedule, 6 to 7 A.M.; 10 A.M. to 3 P.M.; 
 7 to 10 P.M. 8 minute schedule,' 10 to 12 P.M. 30 minute schedule, 12 
 P.M. to 6 A.M. 
 
 3. You are to leave Washington on April 30 for a tour of the following 
 places to look after business interests for me: Boston, Portland, Me., 
 Albany, Buffalo, New York, Cincinnati, Memphis, St. Louis, New Orleans? 
 Charleston, S. C, Charlestown, W. Va., and Minneapolis. Examine 
 railway time tables and submit detailed report showng routes selected ^ 
 provisional time table, etc. If possible make an estimate of expense. 
 Plan for at least ten business hours in Boston, New York, Cincinnati, 
 and St. Louis, eight hours in Minneapolis, and at least four hours in each 
 of the other places. 
 
BOILING POINT 
 
 OF WATER +80° 
 
 CHAPTER XXV. 
 PRACTICAL MEASUREMENTS— Coniinwed 
 
 1. Measure of Temperature. 
 
 Ordinary temperatures are measured by thermometers, 
 graduated on one of three scales: (1) the Fahrenheit, commonly 
 used in business and private life; (2) the Centigrade , used in 
 scientific work, and in countries employing the metric system 
 
 and (3) the Reaumur, used on the 
 
 continent of Europe, especially in 
 
 Germany. 
 
 The common thermometer, con- 
 sisting of its column of mercury in 
 a glass tube, cannot be used for 
 extreme high temperatures. For 
 such temperatures, especially those 
 — . over 500°, instruments of many 
 varied type5 are constructed. Sev- 
 eral, for example, measure tem- 
 perature by the relative expansion 
 of two metals, such as copper and iron. 
 
 The level at which the mercury stands, when the thermo- 
 meter is in contact with melting ice is marked 0° on the Cen- 
 tigrade and Reaumur scales, and '+32° on the Fahrenheit 
 scale. When in contact with water commencing to boil, the 
 mercury level is marked +80°, Reaumur; +100°, Centigrade; 
 and +212°, Fahrenheit. These limits being determined, the 
 degrees of the scale are uniformly marked off. Degrees below 
 0° are marked " -," those above " +." 
 
 t 
 
 ORAL EXERCISE. 
 1. Name the number of degrees between freezing and boiling point 
 on each scale. 
 
 175 
 
176 
 
 BUSINESS ARITHMETIC. 
 
 2. Rank the individual degrees in order of size. 
 
 3. 1° R. is what part of 1° F.? Of V C? 
 
 4. rF.iswhatpartofrC? OfTR.? 
 6. 1° C. is what part of 1° R.? Of 1° F.? 
 
 Scale Reduction. 
 
 Owing to the lack of uniformity on the use of scales, the 
 problem of reduction from one to another is very common. 
 
 Illustrations. (1) 40''F. = ?° 
 
 c. 
 
 (2) 12° F. = ?°C. 
 
 Solution. 
 
 
 Solution. 
 
 '.' 180° F. = 100° C, between limits. 
 
 32° -12° = 20° F., distance below 
 
 rF.-^^^g^Jc.,or5/9°C. 
 
 
 0°C. 
 20° F. = -20X5/9° C. 
 
 40° F. = 40° -32° above zero, 
 
 or 8° 
 
 = -11.11° C. 
 
 above zero. 
 
 
 (4) -5°C. = ?°F. 
 
 8 X 5/9° = 4.44 + ° C. 
 
 
 Solution. 
 
 Solution. 
 
 
 5°C. = 5X9/5°F.= 9°F. 
 
 '.- 100° C. = 180° F. 
 
 
 But 0°C.=32°F. 
 
 1° C. = 9/5° F. 
 
 
 .-. -5°C.=32°-9°F.=23°F. 
 
 25° C. =25x9/5° F. ; 
 
 = 45° F., above freezing. 
 But 0° C. = 32° F. 
 .'. 25° C. = 45° F. + 32° F. = 77° F. 
 
 EXERCISE. 
 Express each of the following temperature reading on two other scales: 
 
 1. +196° F. 5. +18.6° F. 9. +182° C. 13. +67.2° R. 
 
 2. +108° F. 6. -18° C. 10. +46.5° C. 14. -50° R. 
 
 3. +64° F. 7. +64° C. 11. +18° R. 15. +112° R. 
 
 4. -13° F. 8. -30° F. 12. -66° C. 16. +100**R. 
 17. Express the following table in Centigrade scale: 
 
 Approved Cold Storage Temperatures. 
 Degrees Fahrenheit. 
 
 Beef 36 to 40. 
 
 Pork 29 to 32. 
 
 Lamb, mutton 32 to 36. 
 
 Fish 25 to 28. 
 
 Oysters 33 to 45. 
 
 Fruits 32 to 36. 
 
 Poultry, frozen 25 to 30. 
 
 Poultry, to freeze 5 to 10. 
 
 Eggs 32 to 34. 
 
 Butter 32 to 38. 
 
 Vegetables 34 to 40. 
 
 Canned goods 38 to 40. 
 
PRACTICAL MEASUREMENTS. 177 
 
 18. Find the missing temperatures in these tables: 
 
 (a) Boiling Points. (6) Melting Points. 
 
 Substance. C. F. Substance. C. F. 
 
 Benzine 176" Copper 2012° 
 
 Chloroform 140" Platinum 1775" 
 
 Mercury 676" Sulphur 115" 
 
 Sulphur 570" Zinc 779° 
 
 19. Construct a conversion table, Centigrade-Fahrenheit, for 0" to 50° 
 Centigrade. 
 
 2. Composite Units of Measure. 
 It sometimes happens that an accurate measure of values 
 cannot be made by a simple standard unit, because of some 
 special influencing condition. Thus the freight traffic of a 
 railway cannot be measured accurately by tons, because some 
 tons are carried only short distances, while others are carried 
 great distances; while if measured by freight earnings, the 
 factor of quantity is lost sight of. To meet the difficulty, a 
 composite unit, the ton-mile, was adopted. This unit is 
 one ton carried one mile. If the ton is carried ten miles the 
 traffic amounts to 10 ton-miles; if 5 tons are carried 8 miles, 
 the traffic amounts to 5X8, or 40 ton-miles. Measured 
 by such a unit the full effect of both quantity and distance 
 is taken into account. Moreover, it makes possible a com- 
 parison of the traffic of different branches, or of independent 
 roads. Composite measures of a similar order are increasingly 
 used in science and in statistics. 
 
 A Few Other Composite Unit Measures. 
 
 Unit of work. A foot-pound — the work done in raising one 
 ■pound one foot. 
 
 Unit of power. The horse-power. It is equal to 33,000 
 foot-pounds per minute, or 550 foot-pounds per second. 
 
 The irrigation unit. The acre-foot. The quantity of water 
 required to cover one acre to a depth of one foot. 
 13 
 
178 BUSINESS ARITHMETIC. 
 
 The, unit of freight traffic. The ton-mile. The equivalent of 
 one ton of freight carried one mile. 
 
 The unit of passenger traffic. The passenger-mile. One 
 passenger carried one mile. 
 
 The unit of train traffic. The train-mile. One train run one 
 mile. 
 
 Illustration. What horse-power in an engine will raise 4,950,000 lb. 
 10 ft. in 5 min.? 
 
 Solution. 
 
 4,950,000X10 = 49,500,000, no. of ft. p. 
 49,500,000 4- 5 = 9,900,000, ft. p. per min. 
 9,900,000^33,000=300, no. of horse power. 
 
 ORAL EXERCISE. 
 
 1. How many foot-pounds of work are performed in raising 48 lb. 
 15 ft.? 
 
 2. How many acre-feet of water are required to supply a 2^ ft. 
 depth to an area of 4^ A.? 
 
 3. What is the ton-mileage of a car load of 60 T., carried a distance 
 of 56 mi.? 
 
 4. Compute the passenger mileage of 32 people, carried an average 
 distance of 15 mi. 
 
 EXERCISE. 
 
 1. Compute the horse-power employed in raising 30,000 lb. an average 
 distance of 48 ft. per min. 
 
 2. Compute the horse-power performed in raising 5000 gal. of water 
 per sec, a distance of 10 ft. 
 
 Note. Express the water in terms of weight. 
 
 3. The owner of the tract of land shown on page 149 contracts for a 
 supply of water of 2^ ft. per annum, for irrigation. What will it cost 
 him, at $46 per acre-foot? 
 
 4. A storage reservoir is to be built with a capacity of 50,000 acre-feet. 
 What must be its capacity in gallons? 
 
 5. A metal casting weighing 48 T. 150 lb. is carried a distance of 169.3 
 mi. Compute the ton-mileage. 
 
 6. In a recent year, the railways of the country carried 1,309,899,165 
 T. of freight an average distance of 133.23 mi. Compute the ton-mileage. 
 
PRACTICAL MEASUREMENTS. 179 
 
 7. In the year above mentioned, 715,419,682 passengers were carried a 
 passenger mileage of 21,923,213,536. What was the average distance 
 each passenger was carried? 
 
 8. In the year above referrred to the passenger train mileage was 
 440,464,866 train miles. What was the passenger mileage per train-mile? 
 
 3. The Formula. 
 
 In estimating of every kind, it frequently occurs that a 
 general rule for the solution of problems of a certain type is 
 reduced in statement to a simple form of an equation, em- 
 ploying words, or more frequently, symbols. 
 
 Illustration. The fact that the area of a triangle is equal to one-half 
 the product of base and altitude may be expressed: 
 
 . r X • 1 Altitude X base 
 Area of a triangle = ^ . 
 
 We may need, however, to compute areas of many different figures. 
 In such a case, we may let "A" stand for the area of any figure, "a" for 
 altitude, "6" for base, "c" for circumference and "i2" for radius, the 
 Greek letter "tt" for the ratio of diameter to circumference. Using 
 symbols, certain common rules for area reduce to the following: 
 
 1. For area of a triangle, A = —^ , or simply -^. 
 
 2. For area of a rectangle, A=ab. 
 
 3. For area of a circle, A = irE^. 
 
 Frequently the determination of the formula is a matter of 
 summing up many experiments (in arbitrary rules — called 
 empirical rules), or of complicated computations by higher 
 mathematics. But once formulated, computations may often 
 be made by simple arithmetic if in any problem numerical 
 values are known for all except one symbol. 
 
 Illustrations: (1) Find the area of a circle of 6" radius. 
 Solution. 
 
 Formula, A = ttB^. 
 
 But TT =3.1416, R = 6, fi» = 36. 
 
 Substituting in the formula, 
 A =3.1416X36. 
 = 113.0976 sq. in. 
 
180 ' BUSINESS ARITHMETIC. 
 
 (2) Find the base of a rectangle whose area is 135 sq. in. and whose 
 altitude is 9". 
 
 Solution. 
 
 Formula, A=ab. 
 But A = 135and6=9. 
 
 Substituting in the formula, 
 135 = 9a. 
 15 = a, the altitude. (Dividing by 9.) 
 
 Note. Care should be taken that denominations are correct. Thus, 
 in above case, if area had been stated in square feet while other factors 
 were in inches, both factors should have been reduced to a common de- 
 nomination of feet or inches. 
 
 EXERCISE. 
 
 1. Express in symbols the value of the circumference of a circle, in 
 terms of the radius and constant ratio. 
 
 2. Express in symbolic form (a) the area of a parallelogram, (6) the 
 surface of a cube, (c) the volume of a rectangular solid, (d) the surface area 
 of such a solid, (e) the perimeter of a parallelogram. 
 
 3. The safe weight of a floor for a certain type of bridge is 80 lb. 
 per sq. ft, Express this fact in formula for any sized bridge of the type. 
 
 4. Find by the bridge formula, the safe load of a bridge 90 ft. long 
 and 17 ft. 6 in. wide. 
 
 xb 
 
 5. If a: = 12, 6 = 6 and c = 5, find A, in the formula A- — . 
 
 c 
 
 6. Find S, in the formula S = iTrR^, if R = 17. 
 
 7. The number of board feet in a rectangular timber = length in feet 
 X breadth in feet X thickness in inches. 
 
 a. Express this formula in symbols. 
 
 h. Use the formula to determine the number of board feet in a 
 timber 12' long, 9" wide and 2" thick. 
 
 8. The pressure of wind blowing directly against an object is sometimes 
 measured by the formula P = .0023 7^ in which P= pressure in pounds 
 per sq. ft., and 7 = velocity in feet per second. Find the pressure per 
 square foot under a wind blowing 40 miles per hour. 
 
 9. A roof is so inclined that the wind pressure is .45 of what it would be 
 if the wind struck it perpendicularly. What is the total wind pressure on 
 the roof, which measures 18' by 60', when the wind is blowing 32 miles 
 per hour? 
 
PRACTICAL MEASUREMENTS. 181 
 
 10. The volume of boards that may be sawed from a round log may be 
 computed by the formula 
 
 L{d-^)dl2 
 
 V = 
 
 8 
 
 in which d = diameter in inches, L = length in feet, and y= volume in feet, 
 board measure. Compute the board measure in a 60 ft. log of 15 in. 
 diameter. 
 
CHAPTER XXVI. 
 
 THE METRIC SYSTEM, 
 i 
 
 The metric system is a decimal system of measures adopted 
 by the French government about 1800. It is used, scien- 
 tifically, the world over, and commercially in many civilized 
 countries. It is authorized in the United States and Great 
 Britain, but is used only in a part of their foreign trade. 
 
 The system is based on the meters a unit of length of approxi- 
 mately 39.37 in., which was assumed to be exactly 1/10,000,000 
 of the distance from equator to pole. Later measurements, 
 however, have proven this computation incorrect. The unit 
 of capacity, the liter, is a cube of -^^ m. edge. The unit of 
 weight, the gram, is the weight of a cube of water of 1/100 m. 
 edge. The tables contain multiples and decimals of units. 
 Multiples are known by the Greek prefixes: myria (10,000), 
 kilo (1000), hekto (100) and deka (10); decimals by Latin 
 prefixes: deci (1/10), centi (1/100), milli (1/1000). Owing to 
 the decimal system, few measures are commonly used. Those 
 used are shown in black-faced type. 
 
 Note. Literature on the Metric System is published by The Bureau of 
 Standards, Washington, D. C. 
 
 
 Linear Measure 
 
 
 1 myriameter 
 
 = 10 kilometers 
 
 = 10,000 meters. 
 
 1 kilometer (Km.) 
 
 = 10 hectometers 
 
 = 1,000 
 
 1 hektometer 
 
 = 10 decameters 
 
 = 100 
 
 1 dekameter 
 
 = 10 meters 
 
 10 
 
 1 meter (m.) 
 
 = 10 decimeters 
 
 = 1 " 
 
 1 decimeter (dm.) 
 
 = 10 centimeters 
 
 .1 " 
 
 1 centimeter (cm.) 
 
 = 10 millimeters 
 
 .01 
 
 1 millimeter (mm.) 
 
 = 
 
 .001 " 
 
 182 
 
THE METRIC SYSTEM. 
 
 183 
 
 sq. 
 
 Square Measure. 
 myriameter = 100 sq. kilometers = 100,000,000 
 
 sq. meters. 
 
 1 sq. kilometer (Km.') = 100 sq. hektometers 
 
 = 1,000,000 
 
 1 sq. hektometer 
 
 = 100 sq. dekameters 
 
 10,000 
 
 1 sq. dekameter 
 
 = 100 sq. meters 
 
 100 
 
 1 sq. meter (m.^) 
 
 = 100 sq. decimeters 
 
 1 
 
 1 sq. decimeter (dm.') = 100 sq. centimeters 
 
 .01 
 
 1 sq. centimeter 
 
 = 100 sq. millimeters 
 
 .0001 " 
 
 1 sq. millimeter 
 
 Land Measure. 
 
 .000001 " 
 
 1 hektare (ha) 
 
 = 100 ares = 10,000 sq. meters. 
 
 1 are 
 
 = 100 centares = 
 
 100 " 
 
 1 centare 
 
 Cubic Measure. 
 
 1 " 
 
 1 cu. myriameter 
 
 = 1000 cu. kilameters =1,000,300,000,000 cu. meters 
 
 1 cu. kilometer 
 
 = 1000 cu. hektometers = 
 
 1,000,000,000 " " 
 
 1 cu. hektometer 
 
 = 1000 cu. dekameters = 
 
 1,000,000 " " 
 
 1 cu. dekameter 
 
 = 1000 cu. meters 
 
 1,000 " " 
 
 1 cu. meter 
 
 = 1000 cu. decimeters = 
 
 1 " " 
 
 1 cu. decimeter 
 
 = 1000 cu. centimeters = 
 
 .001 " " 
 
 1 cu. centimeter 
 
 = 1000 cu. millimeters = 
 
 .0000001 " " 
 
 1 cu. millimeter 
 
 Table of Weight 
 
 .000000001 " " 
 
 1 metric ton (t.) (t 
 
 onneau) =10 quintals 
 
 = 1,000,000 grams. 
 
 1 quintal (q.) 
 
 = 10 myriagrams 
 
 = 100,000 
 
 1 myriagram 
 
 = 10 kilograms 
 
 = 10,000 
 
 1 kilogram (Kg.) 
 
 = 10 hektograms 
 
 1,000 
 
 1 hektogram 
 
 = 10 dekagrams 
 
 100 
 
 1 dekagram 
 
 = 10 grams 
 
 10 
 
 1 gram (g.) 
 
 = 10 decigrams 
 
 = 1 
 
 1 decigram 
 
 = 10 centigrams 
 
 .1 
 
 1 centigram (eg.) 
 
 = 10 miUigrams 
 
 .01 
 
 1 milligram (mg.) 
 
 = 
 
 .001 " 
 
 The metric ton 
 
 or tonneau= weight of 1 n 
 
 I. of water; the kilogram = 
 
 weight of 1 hter of water. 
 
 
 
 Table of Capaciti; 
 
 
 1 hektoliter (HI.) 
 
 = 10 dekaliters 
 
 = 100 liters. 
 
 1 dekaliter 
 
 = 10 liters 
 
 = 10 
 
 1 Hter (1.) 
 
 = 10 deciliters 
 
 = 1 
 
 1 deciUter 
 
 = 10 centiliters 
 
 = .1 " 
 
 1 centiliter (cl.) 
 
 = 10 milliliters 
 
 = .01 " 
 
 1 milliliter (ml.) 
 
 = 
 
 .001 " 
 
184 
 
 BUSINESS ARITHMETIC. 
 
 Table of Equivalents. 
 
 Note. Importers and customs officials have to perform a relatively 
 slight amount of reduction to and from the metric system. Common 
 equivalents follow. 
 
 Measure. 
 Meter 
 
 Liter 
 
 Gram 
 Kilometer 
 Hectare 
 Sq. meter 
 Cu. meter 
 Metric ton 
 Kilogram 
 Mile 
 Yard 
 A. 
 
 Sq. yd. 
 Cu. yd. 
 
 For fairly accurate work. For approximation. 
 
 39.37 in 3J ft. or 1.1 yd. 
 
 f 1.0567 1. qt. ■) 
 
 1 .9081d. qt.J ^'^*' 
 
 15.432 gr 15^ gr. 
 
 .6214 mi 6 mi. 
 
 2.47 A 2^ A. 
 
 1.196 sq. yd T. 1^ sq. yd. 
 
 1.308 cu. yd. . . ?f 1^ cu. yd. 
 
 1.1023 T 1.1 T. 
 
 2.2046 1b 2.2 1b. 
 
 1.6093 km. 
 
 Quart (liquid) 
 
 .9436 1. 
 
 .9144 m. 
 
 Quart (dry) 
 
 1.101 1. 
 
 40.47 a. 
 
 Pound 
 
 .4536 kg 
 
 .8361 m2 
 
 Ton 
 
 .9072 t. 
 
 .7645 m3 
 
 
 
 ORAL EXERCISE. 
 
 1. Read as meters: 
 
 a. 7 km., 5 dekameters and 3 cm. 
 
 b. 9 km., 4 cm., 8 mm. 
 
 2. Read as square meters: 
 a. 9 km2, 2 m^, 3 dm^. 
 h. 5 hm2, 8 m^, 9 cm^. 
 
 3. Read as grams: 4 kg., 8 dkg., 7 eg. 
 
 4. Read as liters: 9 hi., 7 1., 4 cl., 8 inl. 
 
 5. Use decimal values: 
 
 100 m. = ? in. 1000 m. = 
 
 1000 1. = ? 1. qt. 200 t. = 
 
 2 km. = ? mi. 10,000 m2 = 
 
 6. Use approximate equivalents: 
 20 m. = ? ft. 20 1. 
 68.5 1. = ? qt. 200 ha. = 
 80 km. = ? mi. 600 m^ = 
 40 kg. = ? lb. 200 g. = 
 
 7. Express in metric system: 100 1. qt.; 1000 lb. 
 10 T.; 200 gr.; 100 cu. yd.; 10 A.; 200 sq. yd. 
 
 ? cu. yd. 
 ? T. 
 ? sq. yd. 
 
 = ? T. 
 = ? A. 
 
 = ? sq. 
 = ? gr. 
 100 mi, 
 
 yd. 
 
 ; 1000 d. qt.; 
 
THE METRIC SYSTEM. 185 
 
 8. A "hundred meter" dash is equivalent to a race of ? yd. 
 
 9. A man who orders 1000 m. of dress goods should receive ? yd. 
 
 EXERCISE. 
 
 1. If a merchant desires 1200 yd. of French silk, he must order ? 
 meters. 
 
 2. A consgt. of 50 1. olive oil will fill how many pint bottles ? 
 
 3. A measurement on a foreign map is found by scale to be equivalent 
 to 87.56 km. The equivalent distance is ? miles. 
 
 4. An importer is required to pay a tax of 15c per gal. on 120 1. of oil. 
 The tax amounts to S?. 
 
 5. How does a speed of 71 km. per hour for a German express train 
 compare with a record of 71 miles in 69 min. made by a train in this country? 
 
 6. Granite is 2.7 times as heavy as water. Estimate the weight of a 
 piece measuring 3.1 m. by 6 m. by 4.55 m. 
 
 7. The United States requires foreign letters weighing 15 g. or less 
 to pay 5c postage. What is the Umit of weight in our standards? 
 
 8. What is the mileage rate in cents for first class passage from Rouen 
 to Paris, 136 m., if the fare is 15.20 fr. (1 fr. =$.193). 
 
 9. A man who weighs 172 lb. weighs ? kg. 
 
 10. Determine the weight of an aluminum bar 1 m. long and 30 cm. 
 cross-section (specific gravity, 2.71)? 
 
 11. A tank measuring 5'X4'X8' has a capacity of ? m*. 
 
 12. Determine reasonable dimensions for a rectangular tank to have 
 a capacity of 500 1. 
 
 13. How many tons of coal will fill an order for 5000 t.? 
 
 14. How many pieces 8.1 cm. long, may be cut from 70 m. of wire? 
 
 15. A 400 meter running track is ? yd. long. 
 
 Individual Original Work. 
 
 1. Prepare a brief on the metric system contrasting it with our own 
 system. Illustrate by parallel computations. 
 
 2. Construct a six-place table for converting Linear Measure to Metric 
 Measure. 
 
 3. Show how an increasing error is caused in the construction of a 
 working table, by a minor error at the start. 
 
 4. Compare computation by table and by process, giving illustrations 
 to show the value of tables. 
 
CHAPTER XXVII. 
 RATIO AND PROPORTION. 
 
 INTRODUCTORY EXERCISE 
 
 1. Find the difference between 150 and 30. 
 
 2. 30 is what part of 150? 
 
 3. 150 is how many times 30? 
 
 4. $3000 worth of a novelty were sold in one year, and $12,000 worth 
 the next year. Find the increase in sales. The second year's sales were 
 how many times those of the first year? 
 
 It is evident that division oflFers one effective means of 
 comparing numbers. By it one may determine how many times 
 as greaty not how much greater one number is than another. 
 The numbers so compared must be Hke numbers, and conse- 
 quently the quotient is always abstract. This quotient, 
 whether represented or computed, is termed a ratio. 
 
 The ratio may be expressed by any of the signs of division, 
 or by the colon. The numbers involved are called terms, the 
 first term being the antecedent and the second the consequent. 
 
 Illustration. The ratio of a payment of $500 to one of $2500 may be 
 expressed (a) $500/$2500, (6) $500 -^ $2500, or (c) $500 : $2500. 
 
 The value of the ratio is determined by performing the repre- 
 sented division. In the above illustration, the value is 1/5. 
 If the second payment had been referred to the first, the ratio 
 would have been expressed $2500 : $500, and its value would 
 have been 5. The ratio of the first quantity to the second is 
 termed the direct ratio; that of the second to the first the 
 inverse ratio. 
 
 ORAL EXERCISE. 
 
 Express and determine the ratios of: 
 1. 6 to 42. 2. 2 yd. to 2 ft. ' 
 
 186 
 
RATIO AND PROPORTION. 187 
 
 3. 5 qt. to 6 gal. 8. 1' to 1/4'. 
 
 4. 2 sq. ft. to 1 sq. yd. 9. 2 lb. to 4 oz. 
 
 5. $720 to $900. 10. 900 to 1500. 
 
 6. 75c to $2. 11. Measure a rod by a yard. 
 
 7. 1" to 1'. 12. Measure a bushel by three pinta 
 
 13. Name groups of two numbers having the ratios 12, 4, 1/2, 1^. 
 
 14. Add 1 to the antecedent of the ratio 2:5. What is the effect on 
 the value of the ratio? 
 
 15. By experiment, find the effect on the value of the ratio, from: 
 
 (1) Increasing its antecedent. 
 
 (2) Decreasing its antecedent. 
 
 (3) Increasing its consequent. 
 
 (4) Decreasing its consequent. 
 
 (5) Multiplying or dividing the antecedent. 
 
 (6) Multiplying or dividing the consequent. 
 
 (7) Multiplying or dividing the consequent and antecedent by 
 
 the same value. 
 
 16. Compare the above cases with the corresponding cases of simple 
 fractions. 
 
 ^. Antecedent (dividend) -r^ . , .... 
 
 bince -i^ ^ . ,. . — ^ = Ratio (quotient), it is evi- 
 
 Consequent (divisor) 
 
 dent that the antecedent equals the product of ratio value 
 
 and consequent, and that the consequent equals the quotient 
 
 of the antecedent divided by the ratio. It is therefore a simple 
 
 matter to find a single missing term. 
 
 Illustration. A manufacturer issues one design of oblong calendars 
 in assorted sizes, but with the dimensions of length and breadth in a constant 
 ratio of 2|. (a) If the length of one is 15" what is the width? (6) If 
 the width of one is 8" what is the length? 
 
 Solution (a). „„?^, = ratio = 2|. 
 Widtn 
 
 Width =Length-^2^ = 15"X2/5 = 6'^ 
 
 Solution (6). Length = 2^ X the width = 2^X8" =20". 
 
 ORAL EXERCISE. 
 
 Determine the missing values in each of these ratios : 
 
 1. 24 :4= ? 2. ?/18 = 3. 3. 36/? = 2. 
 
188 BUSINESS ARITHMETIC. 
 
 4. ? : 4 = 8. 6. 3/4 : 2/3 = ? 8. 1/4 yd. : ? = 1/8. 
 
 5. 1/2 : 1/4 = ? 7. ? : 3 mi. = 1/4. 9. 2h ft. : U ft. = ? 
 
 Note. If two variable (changing) quantities or dimensions have an 
 unchanging or constant ratio, one is said to vary as the other. Thus the 
 ratio of the circumference of a circle to its diameter is always 3.1416, no 
 matter what the size of the diameter. Quantities are said to vary inversely 
 if one increases as the other decreases, in a constant ratio. In common 
 speech, we refer to approximate changes in the same way. 
 
 10. What do we mean by saying that summer travel from the big cities 
 varies with the heat? 
 
 11. What do we mean when we say that prices vary with demand? 
 
 12. What do we mean when we say that the cost of production of a 
 certain article varies inversely as the quantity produced? 
 
 The ratio method is often used in geometry and its appli- 
 cations, in physics and in the solution of many business 
 principles. It is really a method of fractional analysis and is 
 used, without the name, throughout this book. 
 
 EXERCISE. 
 
 1. The specific gravity of copper is 8.9. What is the weight of three 
 cubic feet? See page 134. 
 
 2. Water is composed of hydrogen and oxygen in the ratio of 2 : 1. 
 How many parts of each gas in 15,000 parts of water? 
 
 3. If a 3" line in a drawing is used to represent a real length of 4' 9", 
 what is the ratio of real to represented length? 
 
 4. A piece of ore weighs 1.3 kg., while the weight of an equal volume of 
 water is .2 kg. Determine the specific gravity of the ore. 
 
 5. If the circumference of a wheel is to be 16 ft. what must be its 
 diameter? 
 
 6. One of two cubes is 3/4 of the height of the other. Determine the 
 ratio of surfaces and volumes. 
 
 7. Show by ratio, the effect on volume of reducing all dimensions of a 
 bin measuring 16' X 8' X 4', by one-third. 
 
 The Lever. 
 
 If a bar is supported on 
 
 mri 
 
 1 
 
 J, a fulcrum, F, a weight, W, 
 
 attached at the point Y 
 
RATIO AND PROPORTION. 189 
 
 may be balanced by the application of a sufficient pressure at 
 the point X. If the weight is between the pressure point and 
 the fulcrum, pressure 
 
 must be applied upward. 
 
 .k. 
 
 Otherwise, it is applied ^ "T ~~J\ 
 
 downward. If the dis- D3 ^ 
 
 tances x and y are ex- 
 pressed in units of measure, and W and P in units of weight, 
 it is a law of physics that the ratio of the product of the 
 weight by its arm to the product of the pressure by its arm is 
 equal to unity, when the lever balances. In other words. 
 Weight (W) X weight arm (y) 
 
 Pressure (P) X pressure arm (x) 
 
 = 1. 
 
 Illustration. If the fulcrum in the first figure is 4 ft. from Y and 6 ft. 
 from X, what pressure supports a weight of 12 lb.? 
 Solution. 
 
 By the law of levers, p = 1. 
 
 48 = 6P. 
 S = P. 
 .'. the pressure must be 8 lb. 
 
 EXERCISE. 
 
 1. The fulcrum being at one end of a 6 ft. bar, what pressure bal- 
 ances a weight of 80 lb., 1 ft. 6 in. from the end? 
 
 2. A pressure of 20 lb., applied 9 ft. from the fulcrum, supports 
 what weight 2 ft. beyond the fulcrum? 
 
 3. Illustrate how the principle of the lever is applied in the use of a 
 crowbar. Write and solve an illustrative example. 
 
 4. Find, by investigation, six common applications of the lever prin- 
 ciple, and prepare and solve simple problems illustrating them. 
 
 5. Where must one place a support under a 12 ft. bar, in order that it 
 may balance weights of 16 and 40 lb.? 
 
 6. A brake chain is connected with a brake lever 1 ft. below the sup- 
 porting pin. The brake handle is 6 ft. above this support. If the brake 
 lever is pulled with a force of 40 lb., what pressure does the brake exert? 
 What pressure per square inch, if the brake shoe measured 8 in. by 8 in.? 
 
190 BUSINESS ARITHMETIC. 
 
 Proportion. 
 
 An equality of ratios is termed a proportion. Thus 
 6 lb./21 lb. = $12/$42 is a proportion because the abstract 
 value of the ratios is the same. An equality of several 
 ratios, as 3/4 = $9/$12 = 18 lb./24 lb., is termed a continued 
 proportion. An equality of products of ratios, as 1/12 X 5/4 = 
 5/6 X 1/8, is termed a compound proportion. Continued and 
 compound proportion are seldom used in practical arithmetic. 
 
 In any proportion, as 1 : 2 = |8 : $16, the first and fourth 
 terms (1, $16) are the extremes, and the second and third 
 (2, $8) are the means. The proportion is read " 1 is to 2, as 
 $8 is to $16." Since the two ratios are equal in numerical 
 value, and since both the extremes and the means include a 
 numerator of one ratio and a denominator of the other, it is 
 evident that the product of the means equals the product of the 
 extremes. Thus 1 X $16 = $16, and 2 X $8 = $16. 
 
 It is evident, also, that any term of a proportion may be 
 
 found if the other terms are known. The first or third terms 
 
 are always the products of the opposite ratio by the second or 
 
 fourth term respectively. The second term equals the product of 
 
 the first term hy the opposite inverse ratio. To avoid multiplying 
 
 by concrete numbers, each term of either ratio may be made 
 
 abstract. 
 
 X 15 
 Illustrations. (1) In the proportion ^ = j^ find x. 
 
 Solution. 
 
 ^ = ^-^^ = 2. Evidently 1/6 of x = i|, and a; = 6 X ^f . 
 40 40 46 
 
 (2) In the proportion = -^ find x. 
 
 X D 
 
 Solution. 
 
 6 24 
 
 X = $480 X 24 = $120. Evidently $480 = a; X ^ , 
 
 andx = $480 -^ ^ = $480 X ^^ = $120. 
 b z4 
 
 Note. Since the product of the means equals the product of the ex- 
 tremes, it is evident, also, that any mean may be found by dividing the 
 
RATIO AND PROPORTION. 191 
 
 product of the extremes by the other mean; and any extreme may be de- 
 termined by dividing the product of the means by the other extreme. 
 Thus, in illustration (1), writing the proportion in the form a; : 6 = 15 : 45, 
 it is seen that one extreme is missing and that its value is equal to 6 X 15, 
 the product of the means, divided by 45, the other extreme. This principle 
 affords a simple method of solution. 
 
 EXERCISE. 
 
 Find the missing terms: 
 
 1. a; : 12 = $72 : $18. 
 
 2. 5:8:: 14:? 
 
 3. 275 lb. : 125 lb. = $11 : $x. 
 
 4. 3 sq. ft. : 16 sq. ft. : : 48c : ? c. 
 
 5. 1/4 :l/2 ::? :3/8. 
 
 6. a;:12 ::7yd. :480 yd. 
 
 Two quantities are directly proportional if one varies directly 
 as another. They are inversely proportional if one varies 
 inversely as the other. 
 
 Illustrations. (1) If one is traveling on mileage, the cost of his 
 railway passage varies directly as the distance, or is proportional to distance. 
 
 (2) Temperature being constant, the volume of a gas varies inversely 
 as the pressure, that is, grows less at the same rate that the pressure in- 
 creases. Thus doubling the pressure halves the volume. 
 
 The common applications of proportion are those already 
 mentioned under simple ratio. In solutions by proportion, it 
 will be found more convenient to place the unknown quantity 
 in the first term, although any term may be used. Solutions 
 may be checked by unitary analysis, and by fractional 
 methods. 
 
 Illustration. A contractor is to receive $4872 for the construction 
 of 2140 ft. of concrete pipe. What has he earned after he has completed 
 535 ft.? 
 Solution. 
 
 Let X equal the sum earned. 
 
 X : $4872 = the ratio of amount earned to contract price. 
 535 : 2140 = the ratio of quantity done to total quantity. Since 
 the amount earned depends on the work done, the ratios must be 
 equal. 
 
192 BUSINESS ARITHMETIC. 
 
 :.x :$4872 = 535 : 2140. 
 
 X X 2140 = $4872X535. (Products of means and extremes.) 
 
 ^^ 848^X535 =«218. 
 
 Check Solution. 
 
 The part of work done is 535/2140, or 1/4. 
 
 .-. the contractor is entitled to 1/4 of $4872, or $1218. . 
 
 EXERCISE. . 
 
 1. The weight arm of a lever is 16 ft.; the pressure arm is 9 ft.; the 
 weight is 4000 lb. Find the pressure or power to balance it. 
 
 2. A farmer found that he required 7 lb. of a certain seed for a 1/4 
 acre lot. What proportional amount of seed is required for a 3^ acre lot? 
 
 Note, In electricity, the resistance of a wire to the flow of a current 
 varies directly as its length, or inversely as the square of its cross section. 
 
 3. Determine the resistance of If mi. of a certain copper wire, if the 
 resistance of 100 yd. is 5 ohms. (The ohm is an electricity unit of re- 
 sistance.) 
 
 4. If the resistance of iron is seven times that of copper, find the re- 
 sistance in 1000 ft. of iron wire, if 20 yd. of copper wire of same diameter 
 has a resistance of 2.6 ohms. 
 
 5. If a metal sphere of 8 in. diameter weighs 214 lb., what is the weight 
 of a sphere of the same metal having a diameter of 12 in.? 
 
 6. It is a principle of mechanics that the power acting parallel to an 
 inclined plane will support a weight proportional to the ratio of the length 
 of the incline to the distance the weight is raised. If the height is 5/6 
 of the length, what power will support a weight of 840 lb.? If the height 
 is 1/4 of the length? 
 
 Note. For the principle of partitive proportion, see page 387. 
 
CHAPTER XXVIII. 
 
 GRAPHIC ARITHMETIC. 
 
 A simple drawing often does .away with the necessity for a 
 compUcated numerical statement. A line diagram, drawn to 
 scale, will often convey its meaning instantaneously. A num- 
 ber statement, on the other hand, may convey little meaning, 
 or require time for interpretation, or be less likely to attract 
 attention. Diagrams or graphs, therefore, have become a 
 common form of number expression, or language, especially 
 in the presentation of statistical matter. Thus the accom- 
 
 Production of Barley: 1900 
 
 MILLIONS OF BUSHELS 
 8 12 16 
 
 20 
 
 24 
 
 CALIFORNIA 
 
 MINNESOTA 
 
 WISCONSIN 
 
 IOWA 
 
 8. DAKOTA 
 
 N.DAKOTA 
 
 WASHINGTON 
 
 NEW YORK 
 
 NEBRASKA 
 
 OREGON 
 
 KANSAS 
 
 MICHIGAN 
 
 OHIO 
 
 IDAHO 
 
 MONTANA 
 
 ILLINOIS 
 
 COLORADO 
 
 ARIZONA 
 
 VERMONT 
 
 OKLAHOMA 
 
 INDIANA 
 
 MAINE 
 
 UTAH 
 
 NEVADA 
 
 PENNSYLVANIA 
 
 TEXAS 
 
 14 
 
 193 
 
194 
 
 BUSINESS ARITHMETIC. 
 
 panying graph from a Census Report gives the average reader 
 a far clearer impression of the relative production of barley, 
 in different states, than would a column of figures stating the 
 production in bushels. Moreover, if desired, the graph may 
 be read to approximate numerical value by means of its scale. 
 General diagrams, such as building plans, plots of land, 
 maps, etc., commonly are used for reference. It is far easier 
 and surer to say "Find the area of the floor shown in the 
 accompanying diagram" than, without a diagram, to describe 
 the shape and dimensions in such a way as to enable one to 
 compute its area. 
 
 Experiment. Try to describe the plan here shown, in words so that 
 one could compute its area accurately 
 
 I. Scales and Plotting. 
 Ability to read and construct explanatory drawings and 
 statistical graphs depends on a thorough knowledge of the 
 arithmetic of scales, and of the fundamentals of plotting. 
 
GRAPHIC ARITHMETIC. 195 
 
 A linear scale is either the ratio of the plotted length to the 
 true length or distance, or the length that is to represent on 
 paper a fixed quantity of any denomination. If every line in 
 a drawing is one-fourth of the corresponding true length 
 of the object represented, the diagram is said to be drawn on 
 " 1/4 scale. " If a one-inch line is used to represent a quantity 
 of 10,000 bushels, the scale is said to be *' 1 inch = 10,000 
 bushels." 
 
 Linear scales are represented, commonly, in one of three 
 ways: 
 
 (1) As a simple fraction. Illustrations: 1/4, 3/8, 5/4. 
 
 (2) By corresponding dimensions. Illustration: V = T. 
 This means that one inch on paper represents one foot of the 
 object. 
 
 (3) By diagram. Illustration: 
 
 10 5 10 20 30 40 
 
 I I r I I I I I I I I \ I I I 
 
 FEET 
 
 From zero to the right the scale shows diagram lengths 
 corresponding to true lengths of 10 ft., 20 ft., 30 ft., etc. A 
 true distance of 22 ft. is represented by a paper distance from 
 "20" (on the right of zero) to the second sub-division on the 
 left. (Why?) Notice that the paper scale reads to true 
 lengths. In diagram scales decimal sub-divisions are common. 
 
 Linear scales of distance may be expressed in any of the 
 three forms. Thus a scale of 1/3 may be written V = S", 
 or 1 ft. = 3 ft.; or . 
 
 3 INCHES 
 
 QUESTIONS FOR DISCUSSION. 
 
 1. What is the advantage m having the diagram scale read to true 
 lengths? 
 
 2. Why is a linear scale used on a drawing that represents an area ? 
 
196 BUSINESS ARITHMETIC. 
 
 3. How does the "line" scale, if carefully constructed, save compu- 
 tation, in making or reading drawings? 
 
 4. Name a common scale used by architects. 
 
 5. How does the graduated edge of a ruler correspond to a scale? 
 
 EXERCISE. 
 
 (Answer orally, if possible.) 
 Express each of these scales in two other forms : 
 
 11. 1" = 100'. 12. 2' = U". 
 
 13. Which of the above scales could be used to make a diagram larger 
 than the original? 
 
 14. Express the following scales in numerical form. (Use ruler to 
 find ratio.) 
 
 (a) Q>) 
 
 t 5 FT.- J 2p^ ,,,.,,,. 0^ . 2,0' 
 
 9 1 
 
 (c) 
 
 ? 
 
 f 
 
 
 MILES 
 
 
 
 
 id) 
 
 
 -J 
 
 1 // 
 75- 
 
 15. Which of these scales will make the largest diagram: 
 i-- 1" = 4'- 9. 
 
 16' ' FEET 
 
 16. An engineer uses a ruler divided into units, tenths and hundredths 
 of feet. If he uses the smallest subdivision to represent two feet, find 
 his scale. Express it in two ways. 
 
 17. If the smallest subdivision on your ruler is used to represent one 
 foot, what is the scale? 
 
 18. An architect uses a ruler having inches divided into sixteenths. 
 If he uses one of these subdivisions to represent a foot, what is his scale? 
 
GRAPHIC ARITHMETIC. 197 
 
 19. Two towns, known to be twelve miles apart, are represented on the 
 map at an interval of three inches. What is the map scale? 
 
 EXERCISE. 
 
 1. Construct the scale 1/20 in diagram form. 
 
 2. Determine a fractional scale for this (ji_ i^o 
 
 line scale. rods - 
 
 3. Construct a line scale for the scale V = 1200 bbl. 
 
 4. Construct a line scale for the scale 1" = 2500 lb., such that each 
 division shall represent 1000 lb. 
 
 5. Find and illustrate four methods of representing line scales. 
 
 Reduction to Plotted Lengths or Distances. Since a true 
 linear scale shows, in fractional form, the part that the scale 
 length is of the length to be plotted, computations in scale reduc- 
 tion are simple fractional reductions. A scale of one-sixth 
 means that the plotted lengths are one-sixth of the real lengths. 
 Real lengths, in this case, are reduced to plotted lengths by 
 dividing by six. When scales are expressed in other forms, 
 they may be used directly, or in difficult 
 
 n 5 cases, reduced to fractional form. In the 
 
 f'^ET scale shown five feet of true length are repre- 
 
 sented by the given line. Twenty-five feet 
 would be represented, evidently, by a line five times as long. 
 A scale of V = 4' reduces an 8' true dimension to a plotted 
 dimension of 2". (Why?) On a scale of 1" = $600, the 
 amount of $1200 is represented by a 2" line. (Why?) 
 
 EXERCISE. 
 (Answer orally if possible.) 
 
 Find the scale lengths for true dimensions of* 
 
 1. 4' 6", on a scale of 1/2. 
 
 2. 3' 9", on a scale of I" = 1'. 
 
 3. 100 yd., on a scale of 1/600. 
 
 4. 35 ft. on a scale of 1" = 10 ft. 
 
198 BUSINESS ARITHMETIC. 
 
 5. $2000, on a scale of 1" = $400. 
 
 What length lines should be used in graphs to represent these quantities: 
 
 6. 4500 bu. on a scale of 1" = 250 bu. 
 
 7. 4 tons, on a scale of 1/2" = 500 lb. 
 
 8. 1600 cans, on a scale of 1/4" = 200 cans. 
 
 9. 2,400,000 bales, on a scale of 1" = 60,000 bales. 
 
 EXERCISE. 
 
 Find the missing values: 
 No. True dimension. Scale. Scale Dimension. 
 
 1. 4' 6" ' ' ? 
 
 X. t. u 2 FEET * 
 
 2. 2' 3" 1/12 ? 
 
 3. 14' 9" 1/4" = 1' ? 
 
 4. 270' 0" 1" = 50'. ? 
 
 5. 4 mi. 80 rd. ' TmilI ' ? 
 
 6. 8" 5/3 ? 
 
 7. 2 1/2" 1" = 1/3" ? 
 
 8. Which of the above scales will make enlargements? 
 
 PRACTICE EXERCISE. 
 
 1. List twenty true dimensions taken in the class room. In a parallel 
 column, write the corresponding scale lengths. Use the smallest sub- 
 division on your ruler to represent one inch. 
 
 2. Make a freehand sketch of the outline of the class room floor. Take 
 measurements and determine plotted lengths on a scale of 1/16. 
 
 Dimensioning is the process of noting on a drawing the 
 true dimensions of the parts represented (page 194). The 
 symbols (') and {") are commonly used in place of the abbrevi- 
 ations (ft.) and (in.). If the true length contains no fraction 
 of a foot, zero inches is written. For example, 5 ft. is written 
 5' 0". Light dimension lines are drawn parallel to the lines 
 of the drawing to which the dimensions refer, the dimension 
 figures being entered in a break in the center. Dimension 
 
GRAPHIC ARITHMETIC. 199 
 
 lines are limited by arrow points that just touch short 
 perpendiculars erected at the extremes of the line to be 
 dimensioned. Dimensions are generally written up and to the 
 right, in order that all may be read from one position. A circle 
 or arc is dimensioned by locating its center and. giving its 
 radius. "Over all" or "total" dimensions are written as a 
 check on a series of intermediate dimensions. 
 
 EXERCISE. 
 
 1. Point out the different principles of dimensioning illustrated on 
 page 194. 
 
 2. What is the real value of an "over all" dimension? 
 
 3. What is the advantage in writing dimensions outside, rather than 
 inside a small drawing? 
 
 4. Collect from newspapers, periodicals, or other sources, six specimens 
 of dimensioned drawings. Make a list of any unusual details of dimen- 
 sioning. 
 
 Plotting, as related to drawings, is the process of locating 
 the main points of a proposed diagram preliminary to drawing 
 in the details. With the actual mechanical construction of 
 drawings we have little to do, but certain mathematical 
 preliminaries should be noticed. Before plotting, it is neces- 
 sary to choose a paper large enough to contain the drawings 
 of the object at the scale required. The size of paper and 
 the dimensions of the object to be represented will necessarily 
 affect the scale to be chosen. 
 
 EXERCISE. 
 
 1. The greatest dimensions of a tract of land are 1800 ft. by 1450 ft. 
 If plotted on a scale of 1/450, how large a sheet is required, allowing for a 
 two- inch margin all around? 
 
 2. What is the largest scale that could be used in reproducing, on 
 this page, the map of a piece of land measuring 4 miles by 3 miles 40 rods? 
 
 3. What is the largest scale on which the land diagram on page 149 
 could have been drawn? 
 
 4. What is the largest scale on which a diagram of your class room 
 could be drawn, on a paper measuring 17 in. by 24 in.? 
 
200 BUSINESS ARITHMETIC. 
 
 5. A rectangle measures 16 ft. by 5 ft. 2 in. How large a sheet (A 
 paper is required for a drawing of it, on a scale of 1/4" = 1', if a 1^" 
 margin is to be left between the drawing and the edges of the paper? 
 
 Coordinates. While the general principles of scientific 
 projection and plotting, as used in projection drawing, have no 
 
 place in this discussion, the 
 Y system of plotting by coor- 
 
 dinates has a direct use in 
 +3 Pa the graphic phases of ap- 
 
 1+5 p plied arithmetic. If we as- 
 
 ffs 1+4 sume two fixed lines, Y and 
 
 ^^\! j T X, at right angles, any 
 
 I 
 ■iv 
 
 III 
 
 ^. 
 
 T 
 I 
 II I 
 
 _-5 
 
 i 
 
 I 
 
 point, P, may be exactly 
 located if its perpendicular 
 distances from X and Y are 
 known, together with the 
 particular angle, I, II, III, 
 ^ or IV, in which it is situ- 
 
 ated. If P is in angle (quad- 
 rant) I, 4 units from X and 5 from Y, it is located by 
 measuring on X from Y 5 units; then up from X, parallel to X, 
 4 units. The coordinates, or locating distances, are always 
 taken parallel to the axes of coordinates {X and F), so that it 
 makes no fundamental difference whether the axes are at right 
 angles or at a slant. In writing (not plotting), the position 
 of a point with respect to the distance parallel to X is stated 
 first, then the distance parallel to Y. Thus the point P is the 
 point (5, 4); the point Pi is (2, 1); the point P2 is (?, ?); the 
 point Pe is (0, 3). If the point is located in other quadrants, 
 its position is located in the same way, except that the dis- 
 tances parallel to X and to the left of Y; or parallel to Y and 
 below X are marked minus. Thus P3 is the point (—2, 5); 
 P4 is the point (—3, —5), and Ps the point (3, —5). In the 
 simpler applications, with which we have to deal, it is possible 
 
GRAPHIC ARITHMETIC. 
 
 201 
 
 so to locate the axes as to have all our points in the same angle, 
 and thus disregard the plus and minus signs. 
 
 
 EXERCISE. 
 
 
 Draw the axes and locate these points: 
 
 
 1. 
 
 2 in. from X and 3 in. from Y, first quadrant. 
 
 
 2. 
 
 4 in. from Y and 3 in. from X, first quadrant. 
 
 
 3. 
 
 On X 2 m. from Y. 
 
 
 4. 
 
 (3", 3"). 7. (2", -4"). ■ 10. 
 
 (0', 00. 
 
 5. 
 
 (4", 2"). 8. (0", -U"). 11. 
 
 (3/4-, -2"). 
 
 6. 
 
 (2^", 5'0. 9. (5", 0"). 12. 
 
 (1", 2h"). 
 
 13. 
 
 (8", 5"), on 1/2 scale. 
 
 
 14. 
 
 (2 mi., 7 mi.) on a scale of = 2 miles. 
 
 
 If one locates the corners of a field with respect to axes, he 
 may plot the points and, by connecting them, obtain the out- 
 line. In taking measurements 
 of an irregular room, one may 
 find the distances of each cor- 
 ner from any side or sides, 
 taken as axes, and then pre- 
 pare a plot. Points on a curve 
 may be taken at such close 
 intervals as, after plotting, to 
 outline the curve on paper. 
 As a help in plotting, cross- 
 section paper, similar to the il- 
 lustration, may be used, any two lines at right angles being 
 taken as axes. The sides of the sub-squares may then be taken 
 as units of scale, and diagrams plotted speedily. The large 
 squares are usually divided decimally. 
 
 
 
 
 
 "~ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 _J 
 
 _^ 
 
 
 ^ 
 
 
 
 
 
 
 
 EXERCISE. 
 
 1. Plot the points (4, 3), (7, 3) and (6, 6). 
 the resulting figure? 
 
 Connect them. What is 
 
202 BUSINESS ARITHMETIC. 
 
 2. Plot to scale the points (2, 2), (3, 4), (6, 2) and (7, 4). What is 
 the resulting figure? 
 
 3. Make a diagram on a scale of 1" = 6', of a room whose corners, 
 taken in order, are (0', 0'), (0', 11'), (8', 11'), (8', 13'), (13', 13') and (13', 0'). 
 Which sides are taken as axes. 
 
 4. Make a diagram of your class room by the same process. 
 
 5. Plot and find the area of a field whose comers, taken in order, are 
 the points (50 rd., 70 rd.), (200 rd., 70 rd.), (250 rd., 200 rd.) and (50 rd., 
 200 rd.). 
 
 6. Plot one side of a curve of a driveway from the points (0', 10') 
 
 (1, 8), (2i 6), (5, 4), (6, 3), (8, 2), (10, 1). 
 
 Reduction of Plotted Lengths to Real Lengths. Few 
 
 drawings or maps, except draftsmen's construction drawings, 
 are so fully dimensioned as to supply all dimensions desired. 
 It is often necessary to find the real length corresponding to 
 a plotted length. If a plotted length is drawn on a one-fifth 
 scale, the true length is naturally ^ve times that plotted. 
 If the drawing scale is V = 4', a 2'' line on paper represents 
 8' of real length. (Why?) If a scale is 2 feet, a line 
 six and one-half times as long represents a true length of 
 13 feet. (Why?) 
 
 EXERCISE. 
 (Oral and written.) 
 No. Length on Drawing. Scale. True Lengths 
 
 1. 3i" 1" = 1000'. ? 
 
 2. ' 5.3" 1/1200 •? 
 
 3. 2.7" 1/63360 ? 
 
 4. 1', 3.4" 2 INCHES' ^ 
 
 5. 9.2" 1" = 2 mi. ? 
 
 6. 1', 3^' 1/16 ? 
 
 7. 11.45" 1/400 ? 
 
 8. 2^' 1" = $5250 ? 
 
 9. 4i" 1/2" = 10,000 bu. ? 
 10. 1" = 10'. ? 
 
GRAPHIC ARITHMETIC. 
 
 203 
 
 11. Determine the scale of the figure on p. 155. Find the true distance 
 diagonally across the room; and the distances of any point C from each 
 comer, and perpendicularly, from three sides. 
 
 STATE LINE 
 
 Avondale 
 
 Larsliall 
 
 12. This map is drawn at a scale of 1" = ? mi., or what fractional scale? 
 Which towns are in a radius of eight miles of the center of Marshall? 
 By airline, Mason ville is ? miles from Avondale; Norwood is ? miles from 
 the nearest point on a railway; Mason ville is ? miles from the state line, 
 and Norwood is ? miles from the nearest railway junction. An airline 
 electric road from Ashland to Marshall would be ? miles long, approxi- 
 mately ? miles shorter than the nearest railway connection. 
 
 Are distances computed by scale from this map absolutely accurate? 
 What factors tend to affect the results? 
 
204 
 
 BUSINESS ARITHMETIC. 
 
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 BUSINESS ARITHMETIC. 
 
 
 
 
 
 
 
 
 
 
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 13. The owners of the plot here shown desire to buy additional land, 
 fronting 60' on 13th Street and 80' adjoining on Adams Ave., and running 
 
 back in each case so as to make 
 the entire lot a rectangle measur- 
 ing V by ?'. Plot the figure and 
 show the additions. Locate on the 
 plot a building with a frontage par- 
 allel to Adams Ave., of 80', set 
 back 20' from the street line, and 
 running back, parallel to 13th 
 street, to a depth of 60'. The 
 building is 15' from the 13th Street 
 line. The area of the original plot 
 is ? sq. yd.; of the addition, ? sq. 
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 ft. 
 14. Sixth Street, north, meets Ave. A at right angles. On a scale of 
 1/1200, make a plot of the land on the northwest corner, from this de- 
 scription: From the corner, the boundary runs north 180'; thence 
 west, 40' ; thence north, 60' ; thence 
 west, 80'; thence diagonally to a 
 point on Ave. A 100' from corner. 
 The area is — sq. ft., and the valu- 
 ation, at $1.95 per sq. ft. is $ . 
 
 II. Graphs. 
 Reference has been made 
 to the graph as a means of 
 conveying general impres- 
 sions of relative value. It 
 deserves, however, more ex- 
 tended notice, now that the 
 principles of scales are un- 
 derstood. 
 
 t 
 
 What We Have What We Grow What We Use 
 
 OUR TIMBER SUPPLY AND ITS DEPLETION. 
 The large tree represents the amount of timber that we have, the second 
 tree the relative amount of the annual growth, and the third tree the relative 
 amount of annual use. 
 
 From The World^s Work. 
 
GRAPHIC ARITHMETIC. 207 
 
 The graph may be used in the form of: (1) lines drawn to 
 scale in lengths proportional to the quantities of bushels, dol- 
 lars, acres, etc., which they represent; (2) proportional areas 
 of any shape; (3) lines based on the principle of coordinate; or 
 (4) figures, varying in one or more dimensions. xAn illustration 
 of (4) is shown on page 206. 
 
 The Parallel Line Graph is often drawn superposed on a 
 reading scale that enables one to read approximate numerical 
 values at sight. In the graph of the production of corn (p. 204) 
 in the United States, based on Government figures, each scale 
 division represents one hundred million bushels. Thus the crop 
 for 1910 was approximately 3,125,000,000 bushels and for 
 1850 it was just under 600,000,000 bushels. Approximate 
 values are all the general public desires, hence for large values 
 round numbers are sufficient. Such graphs also show at a 
 glance the general increase of crop from period to period and 
 any period of unusual crop or crop shortage. 
 
 By using rectangles of a uniform height so that areas vary 
 with widths, the principle of the line graph is maintained, 
 but changes in the filling of the different sections are made to 
 add details of information. Thus in the figure on p. 205 
 each entire space represents the full arid area of a state, but, 
 by a change of shading, five facts concerning this area are 
 made clear. 
 
 Area graphs are of endless variety but they usually conform 
 to geometrical figures. In the circular graphs (p. 208), areas 
 vary with represented values. Both were constructed to scale, 
 although none is stated. 
 
 The temperature and precipitation chart shown on p. 209 
 illustrate a coordinate graph. The subdivisions of the scale 
 background vertically read represent (1) temperatures in 
 degrees above or below the normal for the periods shown, and 
 (2) the precipitation in tenths of inches. The solid lines show 
 
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GRAPHIC ARITHMETIC. 
 
 209 
 
 the variation in average temperature from week to week, 
 from the normal as shown by the horizontal line. The 
 dotted line shows the amount and variation (departure) in 
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 || TEMPERATURE, IN DEGREES, PRECIPITATION, IN TENTHS OF IN DHE3, ^_ j| 
 
 Temperature (degrees Fahrenheit) and precipitation (inches) depart- 
 ures for the season of 1905 from the normal of many years for the Middle 
 and South Atlantic States and Gulf States. 
 
 From the Year Book of the U. S. Department of Agriculture for the 
 year, 1905. 
 
 EXERCISE. 
 
 1. Read at sight the figure for the production of com, page 206, 
 
 2. Reduce to a number table the graph shown on page 206. 
 
 3. Read the following graph at sight. 
 
 PRODUCTION OF COTTON: 1850-1900. 
 
 10 
 
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 1860 
 1850 
 
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2m BUSINESS ARITHMETIC. 
 
 4, The corresponding 1910 figure for the above cotton graph may be 
 taken as 10,386,000 bales. Determine the length of the proper line on the 
 graph. 
 
 5; A man, receiving a wage of $25.00 per week, estimates his expenses 
 as follows: Rent, $25 per month; food, $360 per year; fuel and Ught $80 
 per .year; clothing $100 per year; amusements, $2 per month; charity $2 
 per month; miscellaneous expense, $6 per month — the balance remaining 
 representing a savings and emergency fund. Draw an outline to represent 
 his $20 bill or weekly wage, and show graphically how it is apportioned to 
 meet his expenses. 
 
 6. The following have been estimated to be the approximate expendi- 
 tures of anthracite coal in New York City in a year: 
 
 Anthracite. Tons. 
 
 Domestic, private houses and small stores 2,500,000 
 
 Flats and apartment houses .' 2,850,000 
 
 Hotels, clubs and theaters 1,250,000 
 
 Gas and electric plants 1,300,000 
 
 Elevated and surface roads 650,000 
 
 Harbor shipping 400,000 
 
 Department stores and office buildings 650,000 
 
 Municipal departments 400,000 
 
 Total 10,000,000 
 
 Represent these figures graphically in some other way than by a line graph. 
 
 7. By means of properly proportioned outline, to suggest " skyscrapers," 
 represent the relative amount of new building operations in different cities, 
 in 1911 based on the following figures: Baltimore, $9,325,000; Buffalo, 
 $10,365,000, Chicago, $105,270,000; Cincinnati, $13,485,000; Cleveland, 
 $17,000,000; Detroit, $19,000,000; New York, $173,500,000; Pittsburgh, 
 $11,700,000. 
 
 8. Represent in some other graphic form the graph on page 211. 
 
 9. Using round numbers, to the nearest "ten thousand," prepare a 
 graph to illustrate these facts concerning immigration: 
 
 Immigration from 1830 to 1906. 
 
 1831-1840 599,125 
 
 1841-1850 1,713,251 
 
 1851-1860 2,598,214 
 
 ■ 1861-1870 2,314,824 
 
 1871-1880 2,812,191 
 
 1881-1890 5,246,613 
 
 1891-1900. ; . . 3,844,420 
 
 1901-1910 .8,795,386 
 
GRAPHIC ARITHMETIC. 
 
 211 
 
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212 BUSINESS ARITHMETIC. 
 
 10. Design a graph to illustrate this statement: "The llama will cany 
 from 50 to 200 pounds; a man from 75 to 150 pounds; the donkey, 100 to 
 200 pounds; an ox, 150 to 200 pounds; a horse from 200 to 250 pounds; 
 the camel from 350 to 500 pounds; the elephant, from 1800 to 2500 pounds." 
 
 11. Construct, on a coordinate or cross-section plan, a graph to rep- 
 resent the recapitulation of daily sales by departments, shown on page 13. 
 Design a form that may be extended day by day. 
 
 Individual Original Work. 
 Prepare a set of statistics of some class of the high school, compiled from 
 office records by direction of Principal. Show same facts graphically. 
 
CHAPTER XXIX. 
 PERCENTAGE. 
 
 In the expression of numerical relations, 100 is a common 
 basis of comparison. The process of percentage, or of com- 
 puting by hundreds, is one of the most important of all the 
 working processes of arithmetic and has most varied applica- 
 tions. 
 
 INTRODUCTORY EXERCISE. 
 
 1. Find 1/100 of 200, 2472, 36, 592 bu., h .006. 
 
 2. Find 3/100 of 400, 70 lb., 9000 tons. 
 
 3. Find (a) 4/100 of 720; (b) 25/100 of 800 sheep. 
 
 Per cent, is a standard term for hundredths, derived from the 
 Latin per centum, meaning " by the hundred." The term is 
 represented by the symbol "%." 
 
 Illustration. 4 per cent, of a number = 4% of it = 4/100 of it = .04 
 
 of it. 
 
 ORAL EXERCISE. 
 
 1. What per cent, of a quantity is one-hundredth of it? 3/100 of it? 
 246 hundredths of it? .03 of it? .92 of it? 3/4 of it? 
 
 2. Express as rates per cent. : 36/100, 270/100, .40, .02, 1.25, 3.205, .6. 
 Note. The rate per cent, is the number of hundredths taken. 
 
 3. Express as hundredths: 8%, 12%, 120%, ^%, 331%, 600%, 
 121%. 
 
 4. i = ? hundredths. ^ of a quantity = ? % of it. 
 
 5. Express as per cents.: 1/3, 3/4, 1/4, 5/8, 3/20, 1/40, 3/5. 
 
 6. 20% of a quantity equal how many himdredths of it? What 
 fractional part of it? 
 
 7. Name simple fractions corresponding to these per cents. : 50%, 5%, 
 25%, 40%, 36%, 1331%, 100%, 106%, 96%.. 
 
 8. Express as decimals: 1/4, 1/3, 5/4, 3/25, 4.3%, 125%, 20%, 12i%, 
 92%. 
 
 9. Express as condmon fractions .24, .4, 3.2, .0125, 90%, 120%, 43%. 
 10. . Express each of these rates per cent, in at least two other forms: 
 
 .26, .8, 1 1/4%, .004, 9/5, 3/4, 1/2%, .50, 48%, 75%. 
 
 213 
 
214 BUSINESS ARITHMETIC. 
 
 11. Express these statements in other arithmetical forms: 
 
 (a) I pay 33^% of a debt. 
 
 (6) The use of wood pulp paper in this country has increased 
 49.9% in five years. 
 
 (c) 94% of the men returned to work yesterday. 
 
 (d) He saves 25% of his salary. 
 
 {e) Thompson's plurality was cut down 37^% at the last elec- 
 tion. 
 
 12. 1/2 = how many per cent.? What is the difference in per cent, 
 between 1/2 and 1/2% of a quantity? 
 
 13. Find the difference in per cent, between: 
 
 1/4 and 1/4%. 1/4 and 1/3%. 33i% and 3/8. .002 and 1/4%. 
 
 EXERCISE. 
 Find the missing values in this table: 
 
 Table of Comparative Value. 
 Decimal. Per Cent. Hundredths (Fraction). Common Fraction. 
 
 .0125 
 
 .02 2% 2/100 
 
 .05 1/20 
 
 .061 
 
 12% 
 
 — m% — — 
 
 15% — 
 
 161% 
 
 20% 
 
 25% 
 
 33i/100 
 37i/100 
 
 .80% 
 
 1.75 
 
 90% 
 
 1/2 
 
 11/20 
 
 3/5 
 
 5/8 
 
 2/3 
 
 7/8 
 9/4 
 
PERCENTAGE. 215 
 
 To Find the Percentage. 
 
 The percentage of a quantity is the value of the hundredths 
 taken. The base is the number on which the percentage is 
 reckoned. 
 
 Illustrations. (1) I collect 45% of a debt of $720, or what sum? 
 Solution. (Decimal method.) 45% = .45 $720 
 
 .45 
 3600 
 2880 
 
 $324.00 the am't collected. 
 
 (2) What profit is made by selling a $42 article at 3% advance on cost? 
 Solution. (1% method.) 1% of $42 = $.42 
 
 3% of $42 = 1.26 (3X1% of $42) 
 The profit = 3 X 1% of $42, or $1.26. 
 
 (3) On a $420 purchase, I am allowed 25% discount, or $ ?. 
 Solution. (Aliquot method.) 
 
 The discount is 25% or 1/4 of $420. 1/4 of $420 = $105, discount. 
 Note. The general tendency is toward the use of the decimal method. 
 
 It is evident that the percentage equals the product of the 
 base by the rate. 
 
 FOR DISCUSSION. 
 
 1. What rates per cent, in the preceding table are aliquot parts? 
 
 2. Is the base concrete or abstract? 
 
 3. Is the 'percentage concrete or abstract? 
 
 ORAL DRILL EXERCISE. 
 
 1. Fmd 1% of 24, 36, 500, 4.65, 1/5, 72.5 yd., 246 lb. ' 
 
 2. By the 1% method, find: 3%, of 2400; 8% of 20 bbl; 12% of $300; 
 4% of 1/3; 15%) of .2. 
 
 Note. 100% of a number = 100/100 of it, or the number itself. 200% 
 of a number is how many times the number? 
 
 3. Find: 200% of 2640; 300% of $48; 400% of 1/3; 250% of 2.4. 
 
 4. Find: 1/4% of 5600; 3f% of 1400; 2/3% of 1/2; 1/4% of 1/4; 
 3/4% of .4. 
 
 5. Find, by aliquot parts: 25% of $3640; 83^% of .006; 33i% of 
 1/4; 12^% of 624 yd.; 90% of $7.20. 
 
 6. Compare 72% of $50 with 50% of $72. Conclusion? 
 
 7. Find:18%of $250;84%of 75T.;19%of66|A. 
 
216 BUSINESS ARITHMETIC. 
 
 EXERCISE. 
 Find: 
 
 , 1. 25% of 27,240 lb. 5. 4^% of 5/36. 
 
 2. U% of $7654.20. 6. 75% of 15.02". 
 
 3. 18% of .00624. 7. 3/8% of $95,240. 
 
 4. 333i%of645T. 8. 62|% of 11,200 shares. 
 
 EXERCISE. 
 
 Each problem illustrates a different use of percentage. State use in 
 each case. Re-word each problem in question form. Find missing values. 
 
 1. The directors of the Rand&ll Mfg. Co. pass a resolution to increase 
 all salaries and wages 15%. My own salary of $1200 is thereby raised to 
 
 $ . The advance will cause an increase of $ in a monthly pay 
 
 roll of $24,600. 
 
 2. I decide to set apart 12^% of my monthly income of $96, or 
 $ , for minor personal expenses, and to deposit in the bank, 25%, or 
 
 3. On a purchase of 40 O. C. stoves, listed at $4.80, a customer is 
 allowed 40% off the price, because of the size of his order. The discount 
 is $ and the customer pays $ each. 
 
 4. The purchaser of the stoves marks them at an advance of 25% on 
 what they cost him. His selling price is $ , of which $ is profit. 
 
 5. A. C. Crane and Thomas Drew form a partnership with a capital of 
 $12,600, Crane supplying 33^%, or $ , and Drew %, or $ . 
 
 6. C. B. Brown fails in business and compromises with his creditors, 
 agreeing to pay at once 66f % of his debts, and a month later 15% more. 
 His debts total $46,240.20. His payments are $ and $ . 
 
 7. It is reckoned that the machinery equipment in a factory decreases 
 5% of its original value in each six months. If worth $26,400 originally, 
 it is worth $ , 12 months later. 
 
 8. An architect charges as a fee for the design of a building, 5% of 
 its cost. His fee on a building costing $11,380 is $ . 
 
 9. Of 700 pupils in a school, 90%, or pupils, are present on Janu- 
 ary 11. Of the number present, 40% are boys and are girls. In 
 
 an arithmetic test, 73% or pupils make a passing average. 
 
 10. "The newspapers predict that Black's majority of 42,400 two 
 years ago will be cut down 75% at the coming election. If this is true 
 he will still win by majority." 
 
PERCENTAGE. 217 
 
 ORAL DRILL EXERCISE. 
 
 (The teacher should prepare similar exercises.) 
 
 1. Base, 240. Find 1%, 3%, 25%, 33i%, 10%, 11%, 16|%, 87^%, 
 1/4%, 20%, 160%, 500%. 
 
 2. Base .6. Find 1/3%, 3/4 of it, 2/3, 2/3%, 150%, 3%, 400%, 2i%. 
 
 3. Base, 3/4. Fmd x% of it, 3%, 20%, 200%, 8%, 12^%, 400%. 
 
 4. 60 decreased 33i% of itself = ? (Called the difference.) 
 320 increased 25% of itself = ? (Called the amount.) 
 
 5. Defins amount and difference. Which implies subtraction and which 
 addition? If the rate is 46%, name the difference per cent, and the amount 
 per cent. 
 
 6. Decrease. 240 bu. 33|%; 1/4 by 1/3 of itself; 3/8, 33^%; 2460, 
 10%; 800 by 1/4 of itself; .6 by 40%; 4.2, 14f %. 
 
 7. Increase: 12, 33^%; 900, 2/3%; 2172 lb., 1%; .04, 125%; 1/4, 
 25%; 3.6 T., 16|% 
 
 In the following, use each intermediate answer as the base for the next 
 step. 
 
 8. Increase 20 by 50% of itself; increase that result 33^%; that 
 result, 25%; that result 20%; that result, 33i%; decrease that result, 
 50%; decrease that 10%; increase 10%. Answer? 
 
 9. 4 increased 50%; that result increased 33^%; increased 12|%; 
 decreased 66f%; increased 100%; decreased 50%; multiplied by 4; 
 decreased 25%; multiphed by 1/3. The final result is ?. 
 
 Finding the Rate Per Cent, 
 introductory exercise. 
 
 1. If 20 = 5% of a given number, 1% of it equals ? and 30=-?%. 
 
 2. 16= what part of 20? What per cent, of 20? 
 
 3. 1% of 400= ? 36= ? % of 400. 924= ? %. 
 Illustrations. (1) What per cent, is 536 of 1200? 
 Solution. (Decimal.) Since it is desired to know how many 
 
 hundredths of 1200 is 536, divide 536 by 1200 
 
 ■44f=44f%. 
 1200)536. 
 (2) By selling for $2.27 an article costing $2.00, what is gained in per 
 cent? 
 
218 BUSINESS ARITHMETIC. 
 
 Solution. (1% Method.) 
 
 A gain of 1% =$.02. $2.27 Sell pr. 
 
 $.27 (total gain) divided by $.02 equals 2.00 Cost 
 
 number of per cent, gained. $.02)$ .27 Gain. 
 
 13|, or per cent, of 
 gain. 
 
 (3) The Charleston basket ball team wins 18 out of 24 games or ■ 
 
 % of its gains. 
 
 Solution. {Fractional.) 
 
 The question is: 18 is how many hundredths of 24? 
 18 3 75 ^^^ , 
 24 = 4 = 100 = ^^^«'^"- 
 
 It is evident that the rate equals the quotient of the percentage 
 divided by the base. 
 
 ORAL DRILL EXERCISE. 
 
 Find missing values: 
 
 1. 1/4= % of 1/2, of 1, of 3/4, of 1/8, of 2, of 1/16, of 3/8. 
 
 2. 4.8 = % of 2.4, of 4, of 1.2, of 9.6, of 48, of 6. 
 
 3. What per cent, of 
 
 2400 is 16? 1.6 is .4? 4 is .8? 2/3 is 1/2? 
 
 300 is 29? .2 is .06? 7^8 1? 4 is 12? 
 
 4. $2.40 are % of $60. 6 qt. are % of 20 qt. 
 
 760 lb are % of 300 lb. 90 sq. yd. are % of 2000 sq. yd. 
 
 5. What per cent, of 
 
 3 is 4? 25 is 30? 2.5 is 5? 1/4 is 1/3? 
 
 4 is 3? 25 is 5? .25 is .5? 3/8 is 5/8? 
 
 6. 4 is what per cent, of 5? 5 of 3? 3 of 4? 4 of 6? 6 of 4? 6 of 8? 
 8 of 6? 8 of 10? 10 of 15? 15 of 20? 20, mcreased 20 %, of 30? 30, 
 decreased 1/5, of 40? 
 
 GENERAL EXERCISE. 
 
 1. What per cent of the cost of a $160 horse is lost by selling it for $120? 
 
 2. I invest $6000 in a partnership capitalized at $27,000, under an 
 agreement to share in profits according to investment. What per cent. 
 of any profits should I receive 
 
 3. If $64 is the cost of the premium on a $4000 fire insurance policy, 
 what is the rate of premium? 
 
 4. What per cent, has a city's population increased in a year, if it now 
 has 240,000 inhabitants, and a year ago had 220,000? 
 
PERCENTAGE. 219 
 
 5. How does a profit of $4 on a $20 investment compare in rate with a 
 profit of $16 on a $60 investment? 
 
 6. What per cent, is made in a class test if 943 credits are obtained 
 out of a possible 1140? 
 
 7. A new printing press has a maximum output of 35,000 copies per 
 hour, compared with an output, for the machine it replaced, of 27,500. 
 What is the per cent, of increase? 
 
 Note. Percentage is frequently employed in technical writing and 
 statistics, to show proportions, rates of increase and decrease and for 
 comparisons of values. 
 
 8. Find the missing rates in this table to three places decimally: 
 
 Where Exports Find Markets. 
 
 Countries. 
 
 Total Imports. 
 
 Share from U. S. 
 
 Per Cent. 
 
 United Kingdom, 
 
 $2,958,289,000. 
 
 $638,006,000. 
 
 
 Germany, 
 
 1,696,660,000. 
 
 236,082,000. 
 
 
 Italy, 
 
 398,463,000. 
 
 45,956,000. 
 
 
 China, 
 
 349,913,000. 
 
 36,304,000. 
 
 
 Canada, 
 
 290,361,000. 
 
 175,862,000. 
 
 
 Australia, 
 
 217,676,000. 
 
 22,549,000. 
 
 
 Japan, 
 
 .208,554,000. 
 
 34,834,000 . 
 
 
 Spain, 
 
 204,401,000. 
 
 23,006,000. 
 
 
 Argentina, 
 
 197,974,000. 
 
 27,908,000. 
 
 
 Denmark, 
 
 166,837,000. 
 
 26,832,000. 
 
 
 Brazil, 
 
 161,587,000. 
 
 18,518,000. 
 
 
 Mexico, 
 
 109,884,000. 
 
 72,509,000. 
 
 
 ORIGINAL EXERCISE. 
 Write and solve original problems illustrating the use of rates per cent.: 
 
 1. In measuring the increase of sales of a business during one year. 
 
 2. In measuring the loss of a firm through bad debts. , 
 
 3. In comparing the expense of repairs on a machine with its original 
 cost. 
 
 4. In comparing the votes of successful and defeated candidates for 
 office, in a political election. 
 
 EXERCISE. 
 Compute missing values in this article. 
 
 BASEBALL AVERAGES. 
 
 Baseball records furnish an excellent illustration of the use of rates 
 per cent, in showing relative standing. Thus, the standing of any team is 
 
220 BUSINESS ARITHMETIC. 
 
 determined by the per cent, of games won out of games played. This 
 rate is computed to three places decimally. For example, the standing of 
 the American League on Sisptember 25, of a recent year was: 
 
 American League Standing. 
 
 Team. Won. Lost. P. C. Team. Won. Lost. P. C. 
 
 Philadelphia. . .84 51 .622 Detroit 70 69 .504 
 
 Chicago 82 54 ? New York 66 66 ? 
 
 Boston 69 66 ? Washington ... 56 80 ? 
 
 Cleveland 70 69 ? St. Louis 48 90 ? 
 
 Owing to bad weather, or other causes, some scheduled games may be 
 postponed, so that on any particular date, the number of "games played" 
 by each team may vary. (Is this true above?) It is evident that per 
 cent, rates offer a better means of comparison than common fractions. 
 (Show why.) 
 
 Of course, averages change every day when teams are playing. On 
 Sept. 26, Washington added a game to its "won" column by defeating 
 Chicago, whose column was increased by 1. Detroit lost to Phila- 
 delphia; New York defeated St. Louis; and Cleveland beat Boston. At 
 the close of September 26, the standing would appear as follows: 
 
 American League Standing. 
 
 Team. Won. Lost. . P. C. Team. Won. Lost. P. C. 
 Philadelphia. .85 51 ? — — — 
 
 Towards the close of the season possible results are often forecasted. 
 On Sept. 26, for example, these possibilities might be considered: 
 
 1. Chicago and Philadelphia have each eight more games to play. 
 If Chicago wins six and loses two, while Philadelphia wins four and loses 
 four, will Chicago be in the lead? 
 
 2. Philadelphia plays the first four of these games in Chicago. If 
 Chicago should win three would Philadelphia lose the lead? 
 
 3. A Philadelphia enthusiast asks " How many of these last eight games 
 must Philadelphia win to be sure of the championship? " 
 
 In a similar way the records of pitchers are determined. Thus a pitcher 
 winning 25 games and losing 18 would have an average of games won of . — . 
 
 In batting, the player who makes 93 safe hits in 240 times at bat is 
 said to have a batting average of . — . 
 
PERCENTAGE. 221 
 
 The average of the players "in the field" are based on three factors: 
 (1) the number of opposing players they "put out"; (2)' the number of 
 plays in which they "assist" in putting out a player; (3) any attempted 
 plays in which the player fails to complete a play within his power (errors). 
 The players' fielding averages are expressed decimally as the per cent. * 
 that (1) plus (2) is of the sum of (1), (2) and (3). 
 
 Fielding Averages. 
 
 Player. 
 
 Outs. 
 
 Assists. 
 
 Errors 
 
 Percent. 
 
 C. Baker (Outfielder) 
 
 217 
 
 109 
 
 21 
 
 ? 
 
 J. Town (1st base) 
 
 824 
 
 103 
 
 37 
 
 ? 
 
 S. Crane (2d base) 
 
 162 
 
 219 
 
 33 
 
 ? 
 
 T. Parker (Shortstop) 
 
 124 
 
 137 
 
 41 
 
 ? 
 
 FINDING THE BASE. 
 INTRODUCTORY EXERCISE. 
 
 1. If 14 = 2% of a number, 1% = ?; 100% or the number = ? 
 
 2. Find 100%, if 2% =4, 32, 420, 6. 
 
 3. Find the numbers of which 33i% =35, 1240, 520, 9, 1/4. 
 
 4. Find 100%, if 300% = 700, 45, 6. 
 
 Illustrations. (1) What is the cost of an article that sells for $618, 
 at 3% profit? 
 
 Analysis and Solution. Let the cost equal 100%. 
 The selling price equals 103% of the cost and the cost equals 100/103 
 of $618. $600, cost. 
 
 1.03)$618 
 (2) 40 pounds of Mocha coffee will form 66 f% of a mixture of how 
 many pounds of Mocha and Java? 
 
 Solution. 40 lb. = 66f % or 2/3 40 = 2/3 of mixture, 
 of the mixture. 20 = 1/3 of mixture. 
 
 60 = 3/3 of mixture, or the total 
 quantity. 
 
 It is evident that the base equals the percentage divided 
 by the rate. 
 
 ORAL DRILL EXERCISE. 
 
 1. When is the "1% method" superior to the aliquot or fractional 
 method? Which is better in the following? 
 40 = 10% of ?; 27=9% of ?; 62=50% of ?; 28=4% of ?; 75=33i of ? 
 
222 BUSINESS ARITHMETIC. 
 
 • 2. 24 =4% of?; 25% of?; .4% of?; 1/3% of?; 150% of? 
 
 3. 420 = 6% of ?; 250% of ?; 40% of ?; 25% of ?; 1/4% of ? 
 
 4. 2/3=25% of ?; 3% of ?; 10% of ?; 400% of ?; 1/2% of ? 
 
 5. 426 = 25% of ?; 6750 = 50% of ?; .06 = 75% of ?; 1/4 = 800% of ? 
 
 6. 16 = 1/4% of ?; 480 = 2/3% of ?; 12,000 = U% of ? .006 = 4/3% of ? 
 
 EXERCISE. 
 
 Find missing values: 
 
 1. 297.6 = 3/5% of — . 4. 45.297 = 7% of — . 
 
 2. 97,485 = 20% of — . 5. 1605 = 120% of — . 
 
 3. 2.0406 = 1/8% of — . 6. 29,623 = 47.2% of — . 
 
 EXERCISE. 
 
 (Each problem illustrates a practical use of this phase of percentage.) 
 Find missing values. 
 
 1. What sum must I invest at a profit of 7^% per year to earn a 
 profit of $900 yearly? 
 
 2. A certain cloth shrinks 6^% in dyeing. In order to have 640 yd. 
 of finished material, one must dye — yd. 
 
 3. "Thus the price has fallen 16% in two months, and the money that 
 would buy 100 shares two months ago, will now buy — shares." 
 
 4. A merchant sends a manufacturer an order for 600 articles, enclosing 
 a draft in payment, and writing, "In case you grant me a discount from 
 your last quotation, send all the goods the money will buy." The manu- 
 facturer allows a discount of 20% from his last quotation. What number of 
 articles should he ship his customer? 
 
 (Note: " grant a discount " means "lower the price.") 
 
 5. For a certain contract a carpenter requires 72,000 ft. of flooring. 
 As he knows the waste in cutting will average 25%, he should order — ft. 
 
 6. If a farmer ofifers to sell his land "at 75% of its real value" for 
 $8250 — in order to dispose of it at once — he places what valuation on it? 
 
 ORAL DRILL EXERCISE. 
 
 All Cases. 
 Find missing values: 
 
 1. 16=4% of — ; that result =20% of — ; that result =200% of — ; 
 that = 500% of — ; that =20 times — ; that =25% of — ; that result =400% 
 of — ; that = l/2%of — . 
 
PERCENTAGE. 
 
 223 
 
 2. 
 
 A percentage study of 36. 
 
 
 
 
 (a) 
 
 (&) 
 
 (c) 
 
 1% 1 
 
 
 " 4%of — 
 
 
 r 1200 
 
 5% 
 
 
 100% of — 
 
 
 200 
 
 20% 
 
 
 80% of — 
 
 
 3000 
 
 150% 
 
 
 7% of — 
 
 
 30 
 
 400% 
 
 ►of 36^ ;36 = - 
 
 3% of— 36 =- 
 
 - % of - 
 
 6 
 
 33^% 
 
 
 33i% of — 
 
 
 1/2 
 
 1/2% 
 
 
 2/3 of — 
 
 
 3.6 
 
 2 3% 
 
 
 2i% of - 
 
 
 .9 
 
 U% 
 
 
 L U%of- • 
 
 
 40000 
 
 EXERCISE. 
 
 1. 121% of 372 = li% of — . 
 
 2. 8% of 33i% of 960 = — % of 1200. 
 
 3. 175% of 36 = 1/3 of 24% of — . 
 
 EXERCISE. 
 
 An Argument for the Panama Canal. 
 
 Trip. 
 
 New York to San Francisco via Panama, 5278 M. ; via Cape Horn, 13,340 M. 
 San Francisco to Liverpool via Panama,. 7907 M.; via Cape Horn, 13,678 M. 
 New York to Manila 
 
 via Panama 11,412 M.; via Cape Good Hope, 13,555 M. 
 New York to Yokohama 
 
 via Panama 9692 M.; via Cape Good Hope, 15,178 M. 
 New York to Sydney 
 
 ■ via Panama, 9560 M.; via Cape Good Hope, 12,218 M. 
 
 Present these figures in tabular form. Express also, the distance via 
 Panama, (o) as a per cent, of other routes; (6) via the other routes as a 
 per cent, of the Panama route. 
 
 ANALYSIS DRILL. 
 MIXTURES AND COMPOUNDS. 
 
 Oral. 
 
 1. 20 lb. of one grade of tea are blended with 30 lb. of a second grade, 
 forming a mixture of ? lb., of which ?% is first grade. The quantity of first 
 grade is ? % of that of second grade. 
 
 2. 30 lb. are blended in the ratio of 5 : 4 with ? lb. of a second grade. 
 The second grade forms ? % of the mixture and equals in quantity ? % 
 of the first grade. 
 
224 BUSINESS ARITHMETIC. 
 
 3. How much witch hazel is required for 8 gal. of a 12% solution? 
 
 4. ? lb. Mocha and ? lb. Java are required for a mixture of 180 lb., 
 if the quantity of Mocha exceeds that of Java 25%. 
 
 Written. 
 
 1. What quantity of each ingredient is required for 25 gal. of a mixture 
 of hard oil diluted 40% with turpentine? 
 
 2. A mixture of tallow and rosin is to contain 15% tallow. How much 
 tallow is required for 40 lb. of rosin? 
 
 3. 22 lb. of standard bulhon, 80% pure, will make — lb. of standard 
 purity (90%). 
 
 4. The products of the distillation of 100 lb. of average gas coal of a 
 certain mine approximate: 64 lb. coke, 6| lb. tar, 12 lb. ammonia liquor, 
 14 lb. purified gas, and — lb. minor products and waste. Express com- 
 position in per cents. What amount of coal (2240 lb. to the ton) must be 
 distilled to manufacture 1,500,000 cu. ft. of gas, if gas measures 10,540 cu. ft. 
 to the ton? 
 
 5. The composition of Babbitt's metal is 3.7% copper, 89% zinc, and 
 7.3% antimony. Determine the quantity of each metal required for an 
 order of 16 T. 
 
 6. The analysis of a metallic paint, per 5000 parts, is: 3943 parts iron 
 peroxide, 165 parts alumina, 598 parts silica, 254 parts water, 16 parts 
 lime, 14 parts manganese, — parts minor elements. Express the paint 
 formula in per cent. form. 
 
 7. A common asphalt varnish contains 3, 4 and 18 parts respectively 
 of asphaltum, boiled oil and turpentine. Give formula in percentage form. 
 What quantity of asphaltum is required to combine with 600 gal. of the 
 proper mixture of oil and turpentine? 
 
 8. A mixture of three grades of material prepared in proportions of 
 20%, 30% and 50%, is found to be markedly improved by reversing the 
 proportions of first and second grades. It is decided to change 120 lb. 
 of the old mixture to the new by adding whatever quantities are necessary. 
 What quantities should be added? What is the quantity of the new 
 mixture? 
 
CHAPTER XXX. 
 
 ELEMENTARY PROFIT AND LOSS. 
 
 INTRODUCTORY. 
 
 1. What do I gain by selling an article that costs me $400 at a price 
 25% higher? 
 
 2. If I sell for $40 a desk that cost me $60 what do I lose? What per 
 cent, of the cost do I lose? 
 
 3. Compare these terms with the corresponding terms of percentage: 
 Cost, selling price at a loss, selling price at a profit, rate of gain or loss. 
 
 Practically all transactions in business are carried on 
 directly or indirectly for profit, or to prevent or reduce loss, 
 or to decrease expenditure. A business man continually faces 
 the question, "Does it pay?'* The question may concern the 
 selling price of an article, the results of some business policy, 
 the comparison of investments, or endless other conditions. 
 In most cases arithmetic is used to determine the answer. 
 Percentage is very commonly used to measure results where 
 profit may be expressed in money. 
 
 ILLUSTRATIVE EXERCISE. 
 
 Find missing values: 
 
 1. Tables bought at $8 are sold at $10, at a profit of — % on cost. 
 
 2. I sell apples costing $2 per barrel at $3 and dispose of 6 bbl. per day. 
 By selling at $2.50 I find my sales average 20 bbl, per day. Lowering my 
 selling price increases my profits $ — or — % per day. 
 
 3. By the installation of new machinery', a manufacturer reduces the 
 force required for the same output from 500 to 420, thus affecting a re- 
 duction of — %. 
 
 4. A manufacturer, by a change of process reduces the cost of manu- 
 facture of a rug from $4.00 to $3.80. If his sellmg price remains $4.80 
 his profits increase — %. 
 
 16 225 
 
226 BUSINESS ARITHMETIC. 
 
 Note. The principle of projit as a saving influences our daily acts as 
 individuals. 
 
 5. I can buy potatoes at an up-town store for $3.00 per bbl., and at a 
 down-town market for $2.65. By buying at the latter, I save — c. 
 
 6. I can buy a working bench for $16.00 or I can buy material for $4.00 
 and make it myself, at a saving of $ — , or — %. If, however, my time 
 is worth $1.50 an hour for the four hours required, I save only $ — . 
 
 Ordinarily, 'profit or loss is reckoned as a percentage of cost. 
 
 The cost, however, is an indefinite quantity. In addition to 
 
 what he pays the manufacturer for goods, the merchant pays 
 
 freight, insurance and storage, the expense of sale, and the 
 
 cost of keeping bookkeeping records, etc. Thus the question, 
 
 often arises as to what incidental charges shall be added to 
 
 first cost, or subtracted from selling price to determine a base. 
 
 Partly to avoid this difficulty, some dealers reckon individual 
 
 profits as a per cent, of sales. 
 
 Illustrations. (1) I buy 6 desks @ $31 each, paying $6 freight. 
 Find the selling price each at 25% profit. 
 Solution. Net cost as hose. 
 
 15% or 1/4 of $31 =$7.75 profit. $31'+$7.75 =$38.75 selling price 
 +$1.00 freight = $39.75, gross selling price. 
 Solution. Gross cost as base. 
 The freight is $1.00 per desk. 
 $31 +$1 =$32, gross cost per desk. 
 5/4 of $32 = $40, sell, price. 
 
 (2) One desk is sold for $36 or ? % profit. 
 
 Solution. Net cost as base. The profit, $36— $31 =$5. 
 
 The rate, 5/31, or 16.1%. 
 Gross cost as base. The profit, $36- $32 = $4. 
 
 The rate, 4/32, or 12.5%. 
 Sell, price as base. The profit, $36 - $32 =$4. 
 
 The rate, 4/36, or 11.1%. 
 Note. In this hook profit ivill be reckoned as a per cent, of net cost 
 unless otherwise stated. Read problems carefully. 
 
 The department store offers a good illustration of profit 
 reckoning. Here the selling price is fixed by the merchandise 
 manager., or by the department buyer. Computations are on 
 a percentage basis. The rate of profit, however, varies with 
 
ELEMENTARY PROFIT AND LOSS. 227 
 
 the article or department. Rapid selling articles yield a lower 
 rate than slow selling goods. The latter may sell at 30% 
 profit, where the rapid seller brings 5%. The department, 
 however, that does $60,000 worth of business annually on 
 $10,000 capital can deal on a smaller margin of profit than one 
 that does $80,000 business on $15,000 capital. By the more 
 rapid "turning over" of capital, goods sold at low profits may 
 bring a department as high an average as others sold at a high 
 rate. Suppose an article sold a month after purchase at 5% 
 profit, the original cost being reinvested in the same goods, 
 and the process continued for a year. The profit on original 
 capital is then 5 X 12, or 60% for the year. The tendency 
 is to make all departments yield the same rate annually by 
 means of higher marking of slow selling goods. 
 
 To the price at which the goods are billed at the store, are 
 added the fixed charges for rent, bookkeeping, selling, delivery, 
 etc. These have been reduced to an average per cent, of cost 
 for each store or department — ranging from 15% to 25%. 
 Many of the stores reckon the profit as a per cent, of sales. 
 It is then a simple matter to approximate any day's profits 
 from the amount of sales. 
 
 I. Gains, Losses and Selling Prices. 
 
 EXERCISE. 
 
 Illustrating simple phases of profit and loss. 
 Answer orally if possible. 
 
 1. Find the loss on desks costing $25 if sold at 10% loss; at 20% loss; 
 at 12^% loss; at 5% loss; at 40% loss. 
 
 2. Find the gain on toys costing $1.60 and sold at 25% gain; at 20% 
 gain; at 5% gain; at 15% gain. 
 
 3. Find the gain or loss if goods cost $1.20 a doz. and are sold at 15 c 
 each; at 20 c each; at $2.00 per doz.; at 5% profit; at 20% loss; at $1.05 
 per doz. 
 
 4. From what known values may the following be computed: (a) 
 Selling price at gain, (6) gain, (c) selling price at loss, (d) loss? 
 
228 BUSINESS ARITHMETIC. 
 
 5. Find the missing values by inspection. 
 
 Cost. Rate Gain or Loss. Selling Price. Gain or Loss. 
 
 a. $56. 20% gain $ $ 
 
 6. 9.60 16f%loss. 
 
 c. 4260.00 25% gain 
 
 d. 3.88 doz. 
 
 e. 1.80 % 39 c 
 
 EXERCISE. 
 
 1. What profit is realized by buying gloves at $9.20 per doz. and selling 
 them at $1.15 per pair? 
 
 2. The goods in the crockery department of a store are marked at 20% 
 profit and 18% store expense. What is the selling price of plates costing 
 $9.60 per doz.? 
 
 3. The book buyer of a department store pays $7.50 for ten copies of a 
 popular novel which he disposes of at $1.08. As each lot is disposed of 
 he re-invests the original sum in a new lot, until he has disposed of 120 
 copies. He has " turned over " his capital — times, and gained a total of $ — . 
 
 4. Apples are bought © $2.40 per bbl. (3 bu.) and are retailed at 90 c. 
 per bu. Determine the final loss or gain, on a lot of 60 bbl., the average 
 loss by decay being 15%. 
 
 5. A dealer retails at $6.50 a grade of flour that costs him $5.00 per 
 bbl. The wholesale price is raised 10%, and the dealer immediately 
 raises his price to $7.50, but his sales fall off 10% in amount. How does 
 this affect his net profits? 
 
 II. Profit as' a Rate of Selling Price. 
 
 (Omit if desired.) 
 ORAL EXERCISE. 
 
 1. If 30% of selling price is profit, how many per cent, is cost? 
 
 2. If $14 is the cost, in ex. 1, find the selling price and profit. 
 
 3. The goods in a grocery department are sold at a profit of 15% on 
 selling price. What are the profits on sales of $6400? 
 
 4. Find the selling price of a table costing $6.00 to gain 25%. To 
 gain 33i%; to lose 20%; to lose $1.40; to gain 20%. 
 
 EXERCISE. 
 
 1. Find the selling price of desks costing $16.00, to allow for 20% 
 store burden (based on cost) and 25% profit. 
 
ELEMENTARY PROFIT AND LOSS. 229 
 
 2. Find the profit on an invoice of $2400 worth of goods, sold at 20% 
 profit, the expense of sale being $98.60. 
 
 3. Find the selling price of silks costing $1.80 per yd., if store burden 
 is 20%, profit allowance 20%, waste in cutting 5%. 
 
 4. Which base for profit, cost or selling price, yields the higher profit 
 at any given rate? Illustrate. 
 
 III. Per Cents of Gain or Loss. 
 
 ORAL EXERCISE. 
 
 1. By selling a $2.00 engraving for $2.40 a profit of — c or — % of 
 the cost is realized. 
 
 2. A profit of 75 c on a $5.00 article is a profit of — %. 
 
 3. A carriage-maker buys a cart for $40.00, expends $15 in repairs 
 and sells it for $69.00, making — % profit. 
 
 4. .Does it pay a dealer to reduce his profit from 15% to 9%, if his 
 sales are thereby doubled? 
 
 5. Compute the rate of gain or loss. 
 
 Cost. Selling Price, 
 
 (a) $3.20 $3.60 (d) 
 
 (6) 4.00 3.98 (e) 
 
 (c) 8.40 9.60 (/) 
 
 Selling Price Gain. 
 
 (jg) $8.00 $2.00 (i) 
 
 (h) 15.00 5.00 (/c) 
 
 7. Find the rates in g-k, ex. 5, if based on selling price. 
 
 EXERCISE. 
 
 1. Skates purchased at $11.40 per doz. are retailed at $1.60 per pair, 
 yielding a profit of — %. 
 
 2. A real estate dealer bought a house at auction for $3600. He 
 expended $160.00 in repairs, $95.00 in painting, $5.00 for advertising and 
 then sold the property for $4250. Compute his gain per cent, on invest- 
 ment. 
 
 3. A dealer purchased 60 doz. straw hats @ $9.00 a doz. He sold 
 42 doz. @ 40% profit, and 6 doz. @ $1.25 each. If he disposes of the balance 
 at cost, what is his net rate of gain or loss? 
 
 4. A novelty, manufactured at a cost of 40c, is to be retailed at a price 
 to yield manufacturer, jobber, wholesaler and retailer each a profit of 10c. 
 What rate of profit is realized by each? 
 
 Cost. 
 
 Gain. 
 
 Loss. 
 
 $600 
 
 $ .90 
 
 
 15.00 
 
 1.20 
 
 
 4000.00 
 
 
 $180.00 
 
 jlling Price. 
 
 
 Loss. 
 
 $ 12.00 
 
 
 $ 3.00 
 
 150.00 
 
 
 30.00 
 
230 BUSINESS ARITHMETIC. 
 
 5. A dealer sells Mocha coffee, costing 25c, @ 32c; and Java, costing 
 28c, @ 36c. A mixture of the two grades containing one-third Mocha, 
 readily sells @ 40c. Compare rates of gain on straight grades and blends. 
 
 6. If a dealer sells at 30% below his regular price of 30% profit, he 
 loses what per cent, of profit? 
 
 IV. The Cost. 
 
 (Answer orally if possible.) 
 
 1. By selling a coat for $6.00 a dealer gains 20% on cost. What is 
 the cost? Suggestion: $6.00 = 120% of cost. 
 
 2. If a profit of 45c each is realized on hats sold for $1.65 what do they 
 cost per dozen? 
 
 3. $40.00 represents 5% profit on what sum? 25% loss on what sum? 
 
 4. Find the cost from the values given: 
 
 Selling Price. Gain. Selling Price. 
 
 (a) $ 5.40 $1.40 (c) $ 2.50 $.18 
 
 (6) $240 25% (d) $960. 33 1/3% 
 
 Gain. Rate. ' Loss. Rate. 
 
 (e) $ .84 7% (g) $ .52 4% 
 
 (/) $30. 20% (h) $84. 25% 
 
 5. Markward offers to sell a piece of land at "40% below cost," for 
 $750. He evidently values the land at what price? 
 
 6. If the retail price of an article is fixed by competition at $5.40 the 
 cost of manufacture must be kept down to what figure to allow for 15% 
 profit to the manufacturer and 20% to the dealer? 
 
 Examples in Savings. 
 
 7. A manufacturer, by a change in process, reduces the cost of office 
 desks, which he sells direct, for $30.00, from $21.00 to $18.60. The saving 
 in cost is ? % and the increase in profits ? %. 
 
 8. By the installation of new machinery, the time required for certain 
 work was reduced from 17 hours to 14| hours. If labor costs 48c per 
 hour this represents what saving on each piece of work? 
 
 9. By a change in the boiler equipment of a factory, the coal con- 
 sumption was reduced from 34 tons per day to 31i tons, or ? %. 
 
 GENERAL EXERCISE. 
 
 1. Does it pay to buy apples @ $2.40 per bbl. (2J bu.) and to sell 
 them @ 32c per pk., if 10% is allowed for decay? 
 
ELEMENTARY PROFIT AND LOSS. 231 
 
 2. What rate of profit is obtained by retailing eight articles at the 
 price they cost per dozen? 
 
 3. A blend of 32c and 18c coffee, mixed in the ratio of 3 to 4, is to be 
 put on the market in two-pound packages, and sold at 25% advance on 
 cost. Find the selling price per package. 
 
 4. Does it pay to lower the selling price of tea (costmg 48c) from 
 80c to 72c, if the change in price leads to a 10% increase in amount of sales? 
 
 5. What is the per cent, of gain on cost if 16% of what is received for 
 an article is gain? 
 
 6. What retail selling price, on an article costing $8.00 to manufacture, 
 will yield profits of 25% to the retailer and 20% to the wholesaler and to 
 the manufacturer? 
 
 7. At a price yielding 6% profit, the sales of a certain article equal 
 3600 pkg. per week; at a price yielding 15% profit, sales average 2500 pkg. 
 per week. Which is the more profitable price and why? 
 
 8. A 60c grade of pepper is blended with a 30c grade in the ratio of 
 3 to 5, and the blend is retailed at 50c By increasing the proportion of 
 the fiirst grade to 50%, sales increase 20%. Does the change of proportion 
 pay, if there is no increase in selling price? 
 
 ORAL DRILL EXERCISE. 
 I. 
 
 The cost of a certain make of chair is $6.00. 
 
 1. What is the selling price each at 10% gain? (Give analysis.) At 
 5% loss? At 8% gain? At 20% loss? At 15% profit? At a profit of 
 36c? At a profit of $1.62 for three chairs (analysis)? What is the selling 
 price of two chairs at 20% gain? Of half a dozen at 20% loss? 
 
 2. What is the selling price of one chair if 20% of the selHng price is 
 profit? If 25% of the selling price is loss? If $1.40 is loss? If 87 cents 
 is gain? 
 
 3. What is gained by selling one chair at 5% profit (analysis)? On 
 three at 10% profit? On one dozen at 20% profit? On three sold for 
 $20.60? What is lost on one at 5%? On six sold at 12|% below cost? 
 By selling three for $17.00? What is the profit if 20% of the selling price 
 is profit? (Analysis.) 
 
 4. What per cent, is gained by selling one chair for $6.50? Three for 
 $24 00? Two for the cost 6f three? Five for what one-half dozen cost? 
 What per cent, is lost by selling at 18c below cost? At $4.73? By selling 
 three for $15.00? By selling three for the cost of two? 
 
 What is the rate of gain or loss, based on selling price, on one chair 
 
232 BUSINESS ARITHMETIC. 
 
 sold for $7.20. (Give analysis.) On three sold for $21.00? On two sold 
 for $10.00. On four sold at price equal to the cost of three? 
 
 5. The selling price of an article is $3.60. 
 
 The cost must be limited to $ ? to gain 20%? To gain 25% based 
 on selling price? To gain 40%. To lose $1.23? What per cent, of profit 
 on cost is realized by buying at $2.40? At $3.00? At $3.20? 
 
 II. 
 Give and solve an origmal example to illustrate each case. 
 
 1. From what known values may cost be computed? (Several cases.) 
 
 2. From what known values may selling price be computed? Rate of 
 loss? Rate of gain? Gain? Loss? 
 
 3. What values can be computed from cost and loss? From gain and 
 rate? From selling price and rate? From cost and selling price? 
 
 4. Cost and ? determine rate of gain. 
 
 5. Selling price and ? determine rate of profit 
 
 FOR DISCUSSION AND ILLUSTRATION. 
 
 1. Why may a dealer lose, who sells an article for the price he paid for 
 it? 
 
 2. Could a dealer lose if he sold an article for more than he paid for it? 
 
 3. What may increase the first or direct cost of an article? 
 
 4. Why may a dealer have less of a certain lot of mdse to sell than he 
 bought? 
 
 5. How may the following affect the selling price of an article: over- 
 stocking; competition; style; fire; depreciation; season of the year, locality? 
 
 6. What may lead to the sale of parts of a lot of mdse. at different 
 rates? 
 
 7. What is meant by "quick sales and small profits"? 
 
 8. May sales at a fixed price increase, and yet profits decrease? 
 
 9. Under what circumstances may profits increase, if selling price 
 decreases? 
 
 10. Under what circumstances may profits decrease, if the selling price 
 increases? 
 
 MARKING GOODS. 
 
 In many businesses it is a common custom to mark goods, 
 directly or by means of a tag, with the selling price and, at 
 times, with the cost. The cost, and sometimes the selling 
 
ELEMENTARY PROFIT AND LOSS. 233 
 
 price, is written in a key or private symbol, so that it may be 
 known only to certain employees of the business. While 
 some keys are exceedingly complicated, consisting of arbitrary 
 characters, the usual form consists of a word or phrase con- 
 taining ten different letters, each of which stands for a par- 
 ticular figure. The letters may be numbered forward, back- 
 ward, or arbitrarily. 
 
 Illustration. Key: Now be quick. 
 
 (1) NOWBEQUICKor (2) NOWBEQUICK 
 123 4567890 098 7654321 
 
 By using different keys, the cost may be concealed from all 
 except marker and manager, and the selling price from all 
 except marker, manager and selling force. There is a tend- 
 ency, however, to mark the selling price in plain figures. 
 
 In writing with a key, the last two letters represent cents. 
 If any figure occurs twice in succession, an arbitrary letter, 
 called a repeater, is used the second time. Arbitrary letters 
 are sometimes used for fractions, but are seldom needed, as 
 fractions of a cent are uncommon in the marking of such goods 
 as are commonly "tagged." 
 
 Illustration. Example. Write the marking tag for canes, no. 221, 
 costing $2.80 and selling for $3.60, using the key (1), as above. 
 
 Solution. Substituting the corresponding letters for the figures, $3.60 
 is written "wqk," and $2.80 "oik." The tag is written 
 
 Note. If the selling price were $4.tX), and s were 
 repeater, the price would be written "bks." 
 
 No. 221 
 wqk 
 oik 
 
 EXERCISE. 
 
 1. Interpret the following, based on the key "hypodermic " : 
 
 peo^ hoc^ iy^ poxc^ opd^ hmr, "x" is repeater, 
 yic hex er ydcx pyc rm 
 
 2. Write, using the key "Mayflower," and any repeater: 
 
 37 4m 675 19.20 1.10 99 4J5 12.32 
 5' 3.80' 5.40' 11.85' 65' 80' 2.60' 9.64* 
 
 3. Repeat (2), writmg sellmg price in figures, and the cost in the key 
 Now be quick." 
 
234 
 
 BUSINESS ARITHMETIC. 
 
 Write the cost and selling price in the following examples, using these 
 keys: 
 
 Cost Key. 
 
 - I \ 
 
 / T + ± =F 1 
 
 n O 
 
 1 2 3 
 
 4 5 6 7 8 9 
 SelUng Key. 
 WASHING TUB M 
 
 R 
 
 ' 
 
 1234567890R 
 
 
 First Cost. 
 
 Charges. 
 
 Gain. x 
 
 1. $2.40 
 
 $ .30 
 
 20% 
 
 2. 3.25 
 
 .45 
 
 25% • 
 
 3. .88 
 
 m% 
 
 15% 
 
 4. 14.56 
 
 .60 
 
 30% 
 
 First Cost. 
 
 Sell. Ex. 
 
 Gain or Loss. 
 
 5. $2.40 
 
 5% of S. P. 
 
 . 20% ofS. P. 
 
 6. 3.82 
 
 10% of S. P. 
 
 25% ofS. P. 
 
 7. 9.60 
 
 10% of S. P. 
 
 20% 
 
 8. 15.00 
 
 $1.00 
 
 25% 
 
 Using the key "M 
 
 ayflowers," write cost and selling 
 
 price per article 
 
 for the following: 
 
 
 
 Cost. Per. 
 
 Gain. Cost. Per. 
 
 Expense. Gain. 
 
 9. $9.00 dz. 
 
 25%. 12. $15.00 dz. 
 
 $.72 20% 
 
 10. 4.80 C. 
 
 20% 13. 7.20 gr. 
 
 5.00 25% 
 
 11. 6.40 cas. (2 dz.) 15%. 14. 1.26 1/2 dz. 
 
 .12c 20% 
 
 15. Using the key "Washing tub," mark the goods in the following 
 invoice at an advance of 25% on gross cost. 
 
 Messrs. Topham and Chase, 
 
 Pittsburgh, Pa. 
 Bought of C. R. Newman & Co. 
 Terms : 60 days. 
 
 Philadelphia, January 29, 19 — . 
 
 1829 
 
 47 
 
 639 
 
 426 
 
 358 
 
 2i 
 
 U 
 
 3 
 
 4 
 
 3 
 
 dz. Neckwear, 
 
 Outing Shirts 
 Caps, 
 Less 10%. 
 
 3.60 
 2.80 
 4.50 
 9.60 
 3.25 
 
 Expressage, $2.80. 
 Selling Expense, 10%. 
 
ELEMENTARY PROFIT AND LOSS. 235 
 
 Individual Original Work. 
 
 Design a marking sheet showing first cost of each class of goods in 
 bulk, cost per article, discounts, gains, terms, selling and list prices, etc. 
 Make entries for at least fifteen articles. 
 
CHAPTER XXXI. . 
 
 COMMERCIAL DISCOUNT. 
 
 One of the most comm9n and important business appli- 
 cations of percentage is the expression and reckoning of trade 
 discounts. 
 
 ILLUSTRATIVE EXERCISE. 
 
 1. On Jan. 6, J. P. Robertson buys 60 bbl. flour @ $5.00. He agrees 
 to pay in 90 days, but is offered 3% off (discount from) the debt if he pays 
 within ten days. Up to Jan. 16, $ ? will settle the bill; after that, and until 
 April'O, $?. 
 
 Note. These terms may be written "90 days net, 3% 10 da." or "Net 
 /90; 3/10." In business, the credit terms are usually 30 da., 60 da., or 
 90 da. Rates of discount vary with firm and customer. Interest is 
 sometimes charged for overdue payments. 
 
 2. A Chicago mail order house allows 2% discount on orders of $10; 
 3% on orders of $20; 5% on $50 orders, etc. What discount should a 
 customer receive on a $20 order? (Why?) 
 
 3. The Virginia Brick Co. allows 5% discount on orders for over 10 M. 
 brick. 26 M. @ $12 per M. cost what sum? 
 
 4. The catalog price of 5" metal piping is $1.40 per foot. The market 
 price is 20% less. 1000 ft. at the market price cost what sum? 
 
 Note. Market prices of some articles vary with demand and with the 
 cost of raw materials used in their manufacture. To avoid constant re- 
 printing of trade catalogs, prices are listed high, and customers are notified 
 of current discounts on goods that they desire. 
 
 5. The Sandford Pub. Co. issues a nature book to retail at $1.80. 
 The book is sold to dealers at 20% discount, enabling them to realize what 
 per cent, of gain? 
 
 Note. Publishers and some manufacturers fix the retail price of their 
 products, selling to dealers and jobbers at a discount, and to private 
 customers at 'list.' 
 
 Solution. $1.80 Retail price. 
 
 20% or 1/5 of $1.80 = $.36, discount. 
 $1.80 - $ .36 = 1.44, cost to dealer. 
 
 $ .36 -^ $1.44 = .25 or 1/4, or 25%, the rate of gain. 
 
 236 
 
COMMERCIAL DISCOUNT. 237 
 
 It is evident that a commercial discount is an allowance or 
 subtraction from the list or marked price of goods for such 
 business reasons as: (1) payment before due; (2) size of order; 
 (3) reduction of list to market price; (4) to meet competition; 
 (5) to allow for profits to middlemen who are expected to 
 sell 'at list,' etc. These discounts commonly are expressed 
 as rates per cent. Many discounts are aliquot rates, thus 
 permitting short methods of computation. 
 
 EXERCISE. 
 (Answer orally if possible.) 
 
 1. To what terms in percentage do the following correspond: Rate, 
 of discount, list price, discount, reduced or net price? 
 
 2. Find the per cent, of list price paid for goods bought at one of the 
 following discounts: 40% (suggestion, 100% - 40%), 30%, 28^%, 63%, 
 75%, 1/3%. 
 
 3. Determine the net cost of: 
 
 200 ft. of piping @ $1.20, less 20%. 
 200 pr. gloves @ $3.60, less 12^%. 
 A $500 piano, less 40%. 
 40 valves @ $.75, less 15%. 
 
 4. How do terms of "90 days, 4/30" affect the settlement of a $950 
 bill of goods, bought February 7 and settled March 1? 
 
 5. Check up the Randall Mfg. Co.'s bill of $552, as follows: 6 sewing 
 machines @ $40, less 25%, $180; 4 washing machines, @ $27, less 30%, 
 $72; 3 farm wagons, @ $85, less 20%, $202. 
 
 EXERCISE. 
 
 A bilhng clerk handled and checked the following transactions on 
 Oct. 24. Find missing values. 
 
 1. Received of S. P. Adams, check for $ in full settlement for 
 
 1200 lb. Japan tea, @ 23c, bought on Oct. 18, on terms of 60 da. net; 
 3/10. 
 
 2. Mailed Waukeska Canning Co. check for $ , for invoice of 
 
 Oct. 22, 300 bbl. Cerota Flour, 1/4 s. (i. e., 1/4 bbl.) @ $5.80 per bbl. 
 Terms, 30 da.; 1%, 10 da. 
 
 3. Checked Parker Bros, invoice of 120 bbl. S. T. Flour @ $4.10; 
 12 sacks Graham Flour, @ $2.80; 40 doz. C. Tomatoes @ $.92; 144 bx. K. 
 
238 
 
 BUSINESS ARITHMETIC. 
 
 Soap, @ $7.00. Discount, 12^%. Terms: 30 day note, $400; cash for 
 balance. Mailed check for $ . 
 
 4. Received check for $ from Franklin Cereal Co., being rebates 
 
 on September sales of $746.20, at 12^%. 
 
 Note. Manufacturers a certain lines of groceries sell to wholesalers 
 at prices at which the latter sell to retailers. At intervals the manu- 
 facturers return wholesalers a discount or rebate, based on the amount 
 of their monthly sales. 
 
 Series of Discounts. Often two or more discounts are 
 allowed on the same item; one, for example, for quantity, 
 and a second for early payment. In such cases, one discount 
 is reckoned on the list price; the second on the list price minus 
 the first discount and so on. 
 
 Illustration. Successive discounts of 33^%, 20%, 5%, are allowed 
 on a shipment of 4000 ft. iron pipe @ $1.50, reducing the cost to $ . 
 
 Solution (a). 
 4000 X $1.50 
 331% = 1/3 
 
 20% = 1/5 
 
 5% = 1/20 
 
 Solution {b). 
 
 1/3 of 6000 
 1/5 of 4000 
 1/20 of 3200 
 
 100% 
 
 = $6000 list price. 
 = 2000 first discount. 
 
 $4000 second price. 
 = 800 second discount. 
 
 $3200 third price. 
 = 160 third discount. 
 
 $3040 net cost. 
 
 5% of 80% 
 33i%of76% = iof76 
 
 =list price. 
 100%, list price. 
 20 one discount. 
 80% second price in per cent. 
 = 4% second discount. 
 
 76% 
 =_25i% third discount. 
 50f % net cost in per cent. 
 501% of $6000 = $3040. 
 Note. 100% — 50|% = 49|%, which is called the single discount 
 equal to the series. 
 
 FOR PROOF OR DISCUSSION. 
 
 1. Why does it pay a dealer to allow discounts for the causes men- 
 tioned? 
 
 2. Proposition. The order in which the discount rates for a series 
 3xe used does not affect the final result. 
 
COMMERCIAL DISCOUNT. 239 
 
 3. Illustrate with the series 33^%, 40%, 10%, the possible arith- 
 metical advantage of changing the order of discounts. 
 
 4. Proposition. The single discount equal to a series is always less 
 than its sum. 
 
 5. Proposition. The single discount equivalent to a series of two 
 discounts, equals the difference between the sum and product of the series. 
 
 ORAL EXERCISE. 
 
 1. Find the cost, in per cent, of list price, if goods are bought at discoimt 
 of: (See solution (6), illustration 2, page 238.) 
 
 40%, 10%, 5%. 5%, 10%. 
 
 80%, 3%. 50%, 20%, 10%. 
 
 161%, 10%, 25%. 10%, 10%, 10%. 
 
 10%, 33i%, 40%. 3 20s (i: e., 20%, 20%, 20%.) 
 
 2. Find the single discoxmts equivalent to the following series: 
 
 40%, 25%. 20%, 25%. 10%, 25%, 33^%. 16|%, 20%. 
 50%, 10%. 50%, 50%. 10%, 15% 20%, 25%, 5%. 
 
 3. Required, the net cost: 
 
 (a) Of 400 lb. castings @ 10c, less 20%, 25%. 
 (6) Of 50 pr. gloves @ $2.00, less 10%, 25%. 
 
 4. Compare these offers for goods of the same quality: 
 
 (a) List price $800. Discounts 25%, 33i%, 10%. 
 (6) List price $600. Discounts 16|%, 25%. 
 
 5. What case of percentage is involved in these examples. 
 
 EXERCISE. 
 
 1. Find the net receipts from the following sales of merchandise: 
 
 $326.50 at discounts of 10%, 20%. 
 $415.10 at discounts of 30%, 5%. 
 $300.00 at discounts of 40%, 20%, 10%. 
 
 2. Bought, Aug. 30, on terms 90 da., 3/30, 5/10, 4 dz. knives @ $8.00, 
 less 15%; 2 dz. sauce-pans @ $3.40, less 10%, 10%; 3i dz. wash boilers 
 
 @ $38.80, less 30%, 5%. Paid, Sept. 16, $ . 
 
 3. A library has an opportunity to purchase books at 40%, 5% off. 
 A purchase fund of $420 per year will buy $ worth of books at list prices. 
 
 4. In selling a bill of 80 rockers @ $8.40, less 25%, 10%, a clerk reckons 
 the equivalent discount at 31%. What overcharge may the customer 
 claim? 
 
 5. Both BrowTi & Co. and A. C. Lewis offer me 8000 ft. of choice cypress 
 @ $80 per M., but Brown & Co. quote discqimts of 33^%, 5%; while 
 
240 BUSINESS ARITHMETIC. 
 
 Lewis quotes 20% and 20%. WTiich is the better ofifer? By accepting it, 
 $ is saved. 
 
 Rates of discount. In wholesaling and manufacturing, there 
 
 is constant necessity to calculate rates of discount — owing to 
 
 fluctuations in market prices and other causes. 
 
 Illustration. An article listed at $9.00 drops to a market value of 
 $4.00. 33i% discount has already been quoted customers. What 
 further discount rate must be allowed? 
 Solution. 
 
 The list price =$9.00 
 
 33i% discount is 1/3 of $9.00 = 3.00 
 
 First discount price 6.00 
 
 Required price 4.00 
 
 Additional discount to be allowed 2-00 
 
 $2.00 dis. must be reckoned on $6. It = 1/3 of $6, or 33|%. Therefore 
 
 the second discount is also 33 i%. 
 
 ORAL EXERCISE. 
 
 1. Discounts of 20% and ? % yield a selling price of 60% of list. 
 
 2. Discounts of ? % and ? % enable one to sell at 50% of list. 
 
 3. Discounts of $20.25% and ? % lower a list price of $380 to a selling 
 price of $180. 
 
 EXERCISE. 
 
 1. What discount in addition to one of 15% will lower a list price of 
 $7.20 to a market price of $5.00? 
 
 2. The Essex chair manufactured at a cost of $5.00 retails at $8.40. 
 The manufacturer lists at retail price. If he is to gain 20%, he can allow 
 the retailer what rate of discount? 
 
 Suggestion. Find the manufacturer's selling price. Compare with list. 
 
 3. The same manufacturer Hsts at $10 an article costing $6.40 to make. 
 If dealers sell at list price, they should buy at what discount in order that 
 they and the manufacturer may make an equal money profit? 
 
 EXERCISE. 
 (Rates of profit and loss.) 
 Find missing values. 
 
 Questions of profits or losses on individual items sometimes arise. 
 Thus a dealer may wish to determine his rate of profit on carpet bought 
 at $2.00 and sold at $4.00 less 20%, 30%. By comparing net selling price 
 
COMMERCIAL DISCOUNT. 241 
 
 with cost, he finds the rate to be %. Or he discovers that a clerk has 
 
 lost him $ by discounting at 45% a $120 sale on a series of 20% and 
 
 25%. 
 
 Sometimes the difference of measure affects the problem, as in com- 
 puting the rate of profit on step-ladders bought @ $12.00 per dz. less 50%, 
 5% and selling at $1.20, findmg it to be %. 
 
 Perhaps a case arises in which part of an invoice has been sold. Say, 
 20 out of 35 bookcases, bought @ $8.00 less 40%, 10% have been sold 
 at list, and the balance at 20% profit. This, he finds, yields him a profit of 
 
 %. 
 
 Questions of comparative sales arise. Goods costing $6.00 may be 
 fisted at $16.00. An increase of discount rate from 40% to 50% results 
 in increasing average sales 30%. "Does it pay?" he asks, and finds that 
 
 . Or, a $20.00 chair, costing $8.40 to make is usually discounted 30% 
 
 to the trade. By experiment over a long period, the dealer finds that his 
 average sales per week increased from 40 to 50 when he allowed 10% ad- 
 ditional discount. By percentage he finds that his present rate of profit 
 
 is %, where formerly it was %. 
 
 Some cases of marking goods arise. 
 
 Illustration. Mark a desk, costing $15.00 to manufacture, at a price 
 to yield the maker 33 §% profit, after allowing discounts of 40% and 
 16f % to the trade. 
 Solution. 
 
 Let 100% =fist price. 
 
 100% - 40% =60%, 1st discount price. 
 
 161%, or 1/6 of 60% =10% 
 
 60% — 10% =?50%, second discount price, or selling 
 
 price per cent. 
 33i% of $15.00 =$ 5.00 profit to maker. 
 
 $15.00 +$5.00 =$20.00 selling price. 
 
 .-. 50% of list =$20.00 
 
 100% =$40.00, fist price. 
 
 EXERCISE. 
 
 1. $8.00 rockers, bought at discounts of 25% and 5%, must be sold 
 at $ to gain 20%. 
 
 2. At what price must articles costing $18.00 be marked to allow for 
 331% profit and 5% bad debts? 
 
 3. A manufacturer's patent extension table cost $8.40 to manufacture. 
 If he is to allow for 20% profit for himself, and for 20% discount to the 
 wholesaler, he must name what list price? 
 
 17 
 
242 
 
 BUSINESS ARITHMETIC. 
 
 4. The public does not like to pay more than $7.20 for a certain article. 
 Allowing for a manufacturer's profit of 20%, and a dealer's discount of 
 20%,- the maker must strive to keep the cost of manufacture within $ — 
 
 EXERCISE. 
 
 1. Sold to Robt. M. Drake, Morristown, N. J., via DLW frt. 30 days 
 2/10; 40 doz. files @ $2.60, less 40%, 10%; 3i doz. hand saws, No. 82, @ 
 $17.50, less 25%, 10%. Write bUl. 
 
 2. Compare bids received for the purchase of 100 C machine bolts 
 3/8" X 6 1/2, needed for stock. 
 
 R. C. Shafer offers them @ $4.00 less 50%, 10%, 5%. 
 
 James Casler offers them @ $3.90 less 60%, 10, 
 
 Newark Bolt Co. offers them @ $4.10 less 60%, 15%, 5%. 
 
 3. Extend this bill. 
 
 Pittsburgh, Pa., , 191— 
 
 The Morris Hardware Co. 
 New York City. 
 Bought of The Randall Iron Co. 
 Terms: 60 days; 3/10 
 
 2 
 
 4^ 
 
 2i 
 
 C Machine bolts, 1/2X5 
 C Machine bolts 7.20 7.52 
 5/8X4" 4^' 
 50%, 15%, 5% 
 
 dz. pr. Hinges, No. 382, 8' 
 Less 60%, 10% 
 
 dz. Steel Squares No. 11 (< 
 Less 30%, 10%, 
 
 5.60 
 
 
 -I 
 
 
 
 
 
 
 
 
 — 
 
 
 4.20 
 
 
 2. 
 
 
 — 
 
 
 
 
 — 
 
 
 * Note. 4^ hundred machme bolts 5/8" X 4" @ $7.20 per 100, and the 
 same quantity 5/8" X4^" @ $7.52. 
 
 EXERCISE. 
 
 (Give an illustration in each case.) 
 
 1. What values must be known to determine: (a) list price, (6) missing 
 discount rates, (c) net cost, (d) discount in money, (e) rate of profit or loss? 
 
 2. What values can be determined from: (a) list price and series of 
 discounts; (h) series of discounts, two as rate, one also as money; (c) cost 
 and series; (d) cost, desired profit, and rate of discount; (e) two put of 
 three rates, cost and list price? 
 
COMMERCIAL DISCOUNT. 243 
 
 It is not at all necessary, in the study of many questions of 
 profits, discounts, and even of questions of business policy 
 involving direct financial results, that the fundamental calcu- 
 lation, or the calculations to determine an advisable course, 
 should be based on money values. Percentage is a valuable 
 means of studying business actions, somewhat abstractly, and 
 of deducing results as rates per cent., that are practically 
 formulas. 
 
 Illustration. A question like this may arise: If merchandise is 
 bought at 40% discount from list, and sold at list, how much of the selling 
 price may be allowed for discounts and selling expense, and yet realize 
 25% profit. The cost equals 60% of Hst. 25% profit on cost equals 
 15% of list. The desired selling price equals 75% of list. 100% list —75% 
 = 25%. Therefore selling expense and discounts must be kept within 25%. 
 
 EXERCISE. 
 
 Each example in this exercise should be solved without assuming any 
 money value — although the resulting answer may be applied as a rate 
 per cent, to an infinite number of prices and money values. Find missing 
 values. 
 
 (a) Questions Primarily of Discounts. 
 
 1. The portion of the list price paid for goods bought at 34 1% discount 
 is -% . 
 
 Note. The answer, 65 |%, is general for the given discount, regard- 
 less of price concerned. 
 
 Apply this answer to finding the cost of articles listed respectively at 
 $4.20, $5.00, 20c, $340. 
 
 2. How do terms of "90 da., 3/60, 5/10" affect the settlement of any 
 bill paid within five days of date of purchase? 
 
 3. % of the list price is paid for goods bought at discounts of 
 
 40% and 10%. Apply, to finding the cost of goods listed at $486, and at 
 $56. 
 
 4. The difference between a series of 10% and 40%, and a single 
 
 discount equal to their sum is %. What is the money difference on a 
 
 list price of $840? 
 
 5. Which is the better *f or a seller to offer — any series of discounts or 
 its sum? 
 
 6. Discounts of 25% and x% will lower a list price 50%. 
 
244 BUSINESS ARITHMETIC. 
 
 7. Determine a series of two discounts to lower a list price 45%. 
 Suggestion. Assume any rate less than the total discount, for example, 
 
 20%. Calculate the other rate. How many solutions are possible? 
 
 8. The separate discounts in a series of 3 20s compare how in real 
 value? Illustrate, after obtaining the proportional value, by application 
 to a list price of $8000. 
 
 9. Determine a series of two discounts, equivalent to a single discount 
 of 40%, which shall be equal in numerical value. 
 
 Suggestion. Divide the single discount into equal parts, and find the 
 equivalent rates. The principle applied in this, and in one or two of the 
 following examples, is applied by producers in allowing for middlemen's 
 profits. 
 
 10. Determine a series of three discounts equivalent to a single discount 
 of 48%, and equal in numerical value. 
 
 11. A series of two discounts is equal to 45% off list, but the first dis- 
 count is twice the second in real value. The series is — % and — %. 
 
 (6) Questions Primarily of Profit and Loss. 
 
 12. % is reaUzed by buying at 10% and 33 i% discount from 
 
 the fist price at which the goods are sold? 
 
 13. When a publisher sells his net books to dealers at 40% discount, 
 the latter reahze ? % profit. 
 
 14. Is it better for a retailer to sell at 30% advance on cost, or to sell 
 at a list price from which his wholesaler has allowed him 30% discount? 
 
 15. Compare these offers, based on the same Ust price: (a) Discount 
 20%, 5%, 10%. (6) Dis. of 3 20s. (c) Dis. of 30%, 10%, 15%. 
 
 16. My stock is bought at 40% discount from the list price at which 
 I sell, but the expenses of sale average 15% of the selling price. What 
 net rate of profit is realized? 
 
 17. A 50% increase in sales, resulting from allowing 20% discount, 
 has what effect on former profits of 30% made by selling at list? 
 
 18. Merchandise is bought at 40% discount and sold at 20% discount 
 from the same list price. What is the effect of interchanging rates? 
 
CHAPTER XXXII. 
 
 AGENCY. 
 
 An agent is one who does business for another. The person 
 
 whom he represents is termed the principal. 
 
 Illustrations. (1) Jones, at your request, buys you a saddle horse. 
 (2) Jefferson buys for Chase 400 shares P. R. R. stock. (3) A farmer ships 
 a carload of potatoes to a city produce dealer who returns the farmer the 
 money proceeds from their sale, less his charge for selling. 
 
 EXERCISE. 
 
 1. Give other illustrations of agency in everyday affairs. 
 
 2. Name the agent and the principal in each illustration. 
 
 3. Why are agents required in business? 
 
 4. State some of the advantages and disadvantages of dealing through 
 agents. 
 
 5. Name common businesses that are agency businesses. 
 
 In certain businesses the agent is called a broker (stocks, 
 bonds, notes, etc.), or a factor (cotton), or a commission 
 merchant (produce). Drummers, buyers, attorneys, auction- 
 eers and salesmen are agents. The agent may act without 
 pay, may receive a salary, or may receive a payment, often 
 termed a commission, based on the quantity or value of goods 
 bought or sold. He may receive both salary and commission. 
 The pay of a broker is termed brokerage and that of a factor, 
 commission. The terms fee, share and allowance are also com- 
 mon. Per cent, commissions are based on the exact amount 
 paid (net cost), or received (gross proceeds) by the agent for 
 the goods — regardless of charges. 
 
 The principal pays his agent for a 'purchase, the net cost 
 plus all charges for freight, drayage, commission, etc. (the 
 
 245 
 
246 BUSINESS ARITHMETIC. 
 
 gross cost). He receives /rom a sale the gross proceeds minus all 
 charges (the net proceeds). 
 
 Guaranty is an additional fee paid the agent for taking the 
 risk of securing payments from customers to whom he sells. 
 Warranty is an additional fee paid the agent for ensuring the 
 quality of goods. The agent pays, but charges to his principal, 
 freight, drayage, storage, insurance, inspection charges, etc. 
 
 The agent acts for his principal. The principal, as a con- 
 signor, consigns or ships goods to the agent, or consignee, to 
 be sold. The principal calls the goods a shipment; the agent 
 calls them a consignment. 
 
 EXERCISE. 
 
 1. After carefully reading the preceding sections, define agent, broker, 
 commission, guaranty, warranty, principal, gross proceeds, gross cost, 
 the sum earned by the agent, the sum the principal receives from a 
 sale. 
 
 2. Compare the following with the terms of percentage: agent's 
 commission, rate of commission, gross proceeds, net proceeds (if com- 
 mission is only charge). 
 
 1. PAYMENT OF AGENTS. 
 
 The rate of commission is expressed as a lump sum, as a rate 
 per cent, based on the price at which the agent buys or sells, 
 as a fractional part of this value, or as a rate per article. 
 
 Illustrations. I pay Scott $25 for selling my horse for me. My 
 agent sells 200 bbl. of my potatoes at $2, and charges me 5% of the selUng 
 price for his services. I offer one-fifth of the sum he obtains to a collector 
 who secures payment of a debt due me. 
 
 Example. What does an agent earn for buying or selling 200 bbl. of 
 flour @ $4.80, on 5% commission? 
 
 Solution. The selling price = 200 X $4.80 = $960. 
 The commission = 5% of $960, or $48. 
 
 It is evident that the agent's per cent, of commission is 
 reckoned on the price he pays or receives for the goods. 
 
AGENCY. 247 
 
 ORAL EXERCISE. 
 
 Find the commission: 
 
 1. On 400 bbl. @ 2c per bbl. 
 
 2. On 2000 bu. @ l/4c. 
 
 3. On a $3600 purchase at 2|% commission; at 4%; at 5%; at 25%= 
 
 4. On 1200 bbl. at 15c per bbl. 
 
 5. On 800 bbl. purchased @ $2 on 5% commission. 
 What is the agent paid for his services? 
 
 6. For the collection of a $1200 debt on 5% commission? 
 
 7. For auctioning off $1800 worth of goods on 3% commission? 
 
 EXERCISE. 
 
 1. Compare a 5% rate of commission with a rate of l/2c per bu. on 
 merchandise costing 30c per bu. 
 
 Suggestion. Reduce the second rate to a rate per cent. 
 
 2. Compare a rate of 3% with a rate of 15c per bx. on goods selling 
 at $2.40 per bx Is it possible to make a comparison in the above case 
 if no money value is given? 
 
 3. C. Johnson. 
 
 $ 40.00 $102.10 
 
 $400.00 
 $206.40 
 An agent succeeds in collecting three-fourths of this debt for me. If 
 his commission is 6%, he earns $ — . 
 
 2. SALES. 
 INTRODUCTORY EXERCISE. 
 
 1. You sell for me 10 T. timothy on 5% commission. The customer 
 pays you what sum? You charge $ ?, sending me what sum? 
 
 Note. The principal receives the selling price (gross proceeds) — 
 charges, or the net proceeds. 
 
 2. An agent secures a purchaser for a house and lot who pays $12,600. 
 The agent pays $30 for survey, $10 for transfer, and charges 5% commission. 
 What does the agent receive and what sum does he pay to other parties? 
 
 3. Whose money does the agent use to pay for charges? Explain. 
 The standpoint of each party to an agency transaction is 
 
 different, and the calculations in which he is directly inter- 
 
248 BUSINESS ARITHMETIC. 
 
 ested vary. Recognition of "viewpoint" is often essential to 
 accurate analysis and solution of agency problems. 
 
 Illustration. A. B. Newton ships to Robert Kent, commission 
 merchant, 200 bbl. apples which the latter sells at $2.40. He pays freight 
 of 15c per bbl., and charges 5% commission. He mails to Newton a 
 check for $ ?, net proceeds. 
 
 From the standpoint of Newton the problem is: "I ship Robert Kent 
 200 bbl. apples which he sells at $2.40, on 5% commission, paying 15c 
 freight per bbl. I should receive $ ? " 
 
 From the standpoint of the freight agent: "What is the freight on 200 
 bbl. at 15c?" 
 
 From the standpoint of the agent (To customer) : "What must I receive 
 for 200 bbl. at $2.40?" (To freight agent): "What freight is due on 200 
 bbl. at 15c?" (For himself) : "What is my commission on this sale at 5%?" 
 (To principal) : " Considering these previous transactions, what do I owe 
 Newton?" 
 
 From the standpoint of the customer: "Two hundred bbl. apples at 
 $2.40 will cost me$?" 
 
 Solution. 200 X $2. 40 = $480 gross proceeds, or sum paid by customer. 
 200 X 15c = $ 30 freight paid to freight agent by agent. 
 6% of $480 = $ 24 commission. 
 $ 54 total charges. 
 Gross proceeds — charges = net proceeds, due principal. 
 $480— $54 = $426, net proceeds. 
 
 ORAL EXERCISE. 
 
 Memoranda of Sales. 
 
 (a) 400 bu. oats, @ 40c, commission l/4c per bu. 
 
 (6) 5 copies Marston's Business Encyclopedia, $14, on 5% commission. 
 
 (c) A horse, $240; wagon $90; and harness $30. Commission 3%; 
 misc. expense $4. 
 
 (d) 20 crates eggs, 30 doz. each, 30c per doz. Commission 5%; express, 
 50c per crate. 
 
 1. What do agent and principal receive in each case? 
 
 2. Any increase in general charges affects what parties? Illustrate. 
 
 EXERCISE. 
 
 Before solving, re-word the transaction from each party's standpoint, 
 and state necessary and unnecessary values for each computation. 
 
 1. The consignor should receive a check for $ , as proceeds of 20 
 
 crates eggs, 30 doz. each, @ 18c. Commission 6%. 
 
AGENCY. 
 
 249 
 
 2. Find the sum due the principal on a shipment of 60 T. timothy 
 hay @ 90c per C. What sum does the agent earn at 4% commission? 
 
 3. S. P. Naylor sells for James Carter 12,000 bu. com at 50 l/4c, 
 commission l/4c per bu. What sum does Naylor receive? 
 
 Note. In the following, name the new factors, and discuss their effect, 
 and find missing values. 
 
 4. I shipped M. C. Parker 360 bbl. apples which he sells at the market 
 quotation of $2.90; on 4% commission and 1% warranty. 
 
 The customer pays $ ; Parker receives $ ; he earns $ ; and 
 
 he pays me $ . 
 
 5. J. C. Jones receives from his agent a check for $60.40 as net proceeds 
 from a sale of three hundred pounds of butter at 27c, commission 53^%, 
 miscellaneous charges $4.75. Check up the returns. The customer paid 
 
 6. 250 bbl. apples, shipped to be sold at the market quotation of $3.20, 
 arrive on a falling market and are sold at $2.90, drayage, 10c per bbl., 
 commission 5%. What effect has the fall in price on the net proceeds? 
 
 7. The consignor should receive $- net proceeds from a shipment 
 
 of 16,000 bu. oats, disposed of at 41c. Freight, 2c per bu.; insurance l/2c; 
 storage, l/2c; commission, l/4c. 
 
 8. Extend 
 
 Account Sales. 
 From John C. Carter, 
 
 Baltimore, Md., Jan. 17, 1913. 
 To Robert G. Mason, 
 
 Hyattsville, Maryland. 
 For Merchandise Received Jan. 2, 1913, via B. & O. 
 Consigment No. 4726 FoHo 88 R. M. Green Clerk. 
 
 Jan. 
 
 Sales 
 
 100 lb. Dressed chicken 
 450 lb. Dressed turkey 
 620 lb. Dressed chicken 
 
 .16 
 .18 
 .17 
 
 Charges. 
 
 Expressage $2.45 Drayage $1.10 Storage $3.15 
 Inspection $ Insurance $ Commission $ (4^%) 
 Other Charges $ Total Charges 
 
 Net Proceeds 
 
250 BUSINESS ARITHMETIC. 
 
 3. PURCHASES. 
 INTRODUCTORY EXERCISE. 
 
 1. You buy through an agent, on l/8c per bu. commission, 6000 bu. 
 
 wheat at 90c. The agent pays $ — for the grain, charges $ commission, 
 
 and you pay $ . What effect have charges on the cost to the prin. 
 
 cipal? 
 
 2. An agent buys for his principal 80 bbl. winesap apples @ $2.50 on 
 5% com., and 50c per bbl. for drayage and storage. The net, or prime 
 
 cost of the good is $ ; the total charges are $ . The principal must 
 
 pay the agent in full settlement, $ . 
 
 3. If buyer and seller had dealt direct, in example 2, who would have 
 saved money? How much? 
 
 Illustration. CD. Rand sends Robert Gates a draft for $3000 with 
 orders to invest in Greening apples, market quotation $2.40. Com. 5%, 
 drayage 20c bbl., storage 10c. Find all values. 
 Solution. 
 
 $2.40 net cost per bbl. , 
 
 (5% of $2.40) .12 com. The number of bbl. = gross cost 
 
 -_ J gross cost per bbl. 
 
 .20 drayage 
 
 .10 storage 1063 = no. bbl. 
 
 $2.82 gross cost per bbl. 2.82)3000 
 
 1063 X $2.40 = $2551.20 prime cost 282 
 
 1063 X .12 = 127.56 com. 1800 
 
 1063 X .20 = 212.60 drayage 1692 
 
 1063 X .10 = 106.30 storage 1080 
 
 $2997.66 gross cost 846 
 
 $3000 - $2.34 (surplus) = $2997.66, gross cost. 2.340 
 
 The principal receives 1063 bbl. and $2.34 surplus. 
 The agent receives $3000, returns $2.34, keeps $127.56 com. and pays 
 $2551.20 for the apples, $212.60 for drayage, and $106.30 storage. 
 
 ORAL EXERCISE. 
 
 (Memoranda of Purchases.) 
 (a) 8000 bu. oats @ 40c, commission l/8c. 
 (5) 200 sh. stock @ $80, brokerage $1/8 per sh. 
 
 (c) 2000 lb. lard @ 10c, com. 2c per 100 lb., misc. charges $5. 
 
 (d) 80 A. land @ $50, com. 4%, transfer and survey $35. 
 
 1. Compute agent's commission in each case. 
 
 2. Determine amounts paid and received by each party. 
 
AGENCY. 251 
 
 Extract from Quotation List. 
 
 Chicago markets yesterday: 
 
 Wheat Opening. Highest. Lowest. Closing. 
 
 July 89i .90i .89 .90^ 
 
 Sept.. 91 .921 .901 .92| 
 
 Dec 94| .961-1 .94f .96^ 
 
 May 991 1.01 .99i 1.00^ 
 
 Com 
 
 July 51| .52^ .511 .52^ 
 
 Sept 51f .521 .51f .52f 
 
 Dec 48^ .491 .48^ .49^ 
 
 May 50i .50|-.51 .50 .50| 
 
 Oats. 
 
 July 42 .43 .41i .43 
 
 Sept 371 .381 .37f .38| 
 
 Dec 38^ .39i .38| .39i 
 
 May 401 .41 .40 .40|-.41 
 
 Pork.. 
 Sept 16.55 16.55 16.52 16.52 
 
 EXERCISE. 
 
 (See Quotation list where necessary.) 
 
 1. Invest $5000 in July wheat, opening quotation, brok. l/8c — no 
 fraction of 100 bu. to be bought. Number of bushels? 
 
 2. What is the purchasing power of $2000 in September com, at 
 highest quotation, brok. l/4c. 
 
 3. Determine gross cost of 500 bbl. pork, lowest quotation, com. 
 2|c a bbl. 
 
 4. An agent, charging 4^% com., draws on me for what sum to 
 cover the full cost of 200 cr. tomatoes — $2.25 expressage and drayage 
 40c each? 
 
 Using account sales form as a model, prepare an accoimt purchase for 
 this transaction. 
 
 5. My draft for $700 will pay for how many crates of eggs, 30 doz. 
 each, @ 18c a doz.; com. 4%; drayage 25c? 
 
 6. A receives from his principal a draft for $12,000 with orders to 
 secure three mgnths' options on as many acres of certain coal lands as 
 possible. A allows $50 for travelling expenses and 4% for commission. 
 He secures options on 80 A. at 15 % of the selling price, which is $420 per A. 
 
252 
 
 BUSINESS ARITHMETIC. 
 
 The balance he spends m the purchase of 12% options on land priced at 
 $210 per A. On how many acres of this latter tract can he secure options? 
 
 7. I bought through an agent 60 cr. choice tomatoes @$2.75; com. 4%; 
 expenses $20. I sold 25 crates direct @ $3.50 and the balance through an 
 agent at $3.80 on 5% com. Find net returns. 
 
 8. In accordance with orders, an agent disposes of 200 A. of a tract 
 of 280 A. @ $68, and the balance @ $43, investing the net proceeds in 
 options on timber lands @ $10 per A. His commission being 4% each way 
 and his expenses $50, he can secure options on how many acres? 
 
 9. An advertising soUcitor agrees to secure advertising for a book- 
 let, on 25% commission. He secures 35 full page advertisements 
 @ $11.50 per inch. The pages are two column width and 10 inches depth 
 of column. His total commission is $ . 
 
 10. Extend this statement of a real estate agent to his principal, charging 
 5% on collections. 
 
 Washington, D. C, Feb. 2, 191-. 
 Mr. James Parker, 
 
 1382 13th Street, N.W. 
 Dear Sir 
 
 Enclosed find my check per statement below. 
 
 Respectfully Robert Crane, Agent. 
 
 Tenant 
 
 Premises 
 
 Explanation 
 
 Receipts 
 
 Charges 
 
 C. P. Lewis 
 
 1491 N.St. N.W. 
 
 Rent to Feb. 1, 19— 
 Repairs to range 
 
 60 
 
 — 
 
 3 
 
 60 
 
 James Farmer 
 
 1280 L St. 
 
 Rent to Feb. 1, 19— 
 Plumbing repairs 
 Commission 5% 
 Balance due 
 
 85 
 
 50 
 
 7 
 
 35 
 
 PROFIT AND LOSS. 
 EXERCISE. 
 (Solve mentally when possible.) 
 Determine the rate of gain or loss on : 
 
 1. 500 lb. Rio coffee bought @ 8c, on 5% com., and sold @ 9c direct. 
 
 2. Quinces bought @ $2 and sold @ $3 on 5% commission. 
 
 3. Determine the profit or loss on 1000 bu. com bought @ 70c and 
 sold @ 71^c; brokerage l/4c each way. 
 
 4. Spring bran bought @ $19.60 per ton, com. 5% and drayage $1.00 
 per ton, must be retailed at what rate to yield 15% profit? 
 
AGENCY. 253 
 
 5. What per cent, of profit is obtained by selling @ $3.20 apples 
 bought through a commission merchant @ $2.50, on 5% commission? 
 Expenses of sale equal 5% of the selling price. 
 
 6. What profit is realized in retailing @ $1.50, through an agent 
 charging 20% commission, a subscription book that costs 92 cents to 
 publish? 
 
 7. An article, costing $1.80 to manufacture, is to be sold at a price 
 to yield the manufacturer 20% profit, after allowing selling agents 25% 
 commission. Find list price. 
 
 8. A sewing machine costs $11.90 to manufacture. Find marked 
 price to allow for 25% profit and 20% commission to agents. 
 
 9. How much must the sales of an agent increase to make it profi- 
 table to lower his rate of commission from 5|% to 4^%? 
 
 10. A book costing 90 cents to publish is sold @ $1.50 through agents 
 who are paid 30% commission. How is the publisher's rate of profit 
 affected by lowering the commission to 20%? 
 
 REVIEW DRILL EXERCISE. 
 
 1. Buy 40 bbl. @ $5, com. 5%. 
 
 Find: gross cost; prime cost; commission; sum agent pays seller; sum 
 agent charges principal; sum agent earns; largest sum involved. Repeat, 
 if commission rate is (1) 8c. bbl.; (2) 1/4%. 
 
 2. Sell 2000 bu. @ 60c, com. l/4c, storage l/2c. 
 
 Find: sum buyer pays; sum agent earns; sum handled by agent; net 
 proceeds; the cost of the sale to the principal. 
 
 3. State the values that may be computed from these facts. For- 
 mulate and solve simple problems involved in them. State whether 
 purchase or sale. 
 
 (a) Commission, $4; gross proceeds, $40. 
 (6) Commission, $12; net proceeds, $188. 
 (c) Commission, $60; prime cost, $2000. 
 {d) Commission, $90; quantity 400 bbl. 
 (e) 5% commission on a sale is $22. 
 
 (/) An agent sends his principal $388 after reserving $12 for 
 himseK. 
 
 4. Give illustrations showing what values can be determined from: 
 
 (a) The commission and a "cent" rate. 
 (6) The commission and a rate per cent. 
 (c) Number of articles and price. 
 (d) The net and gross proceeds. 
 
254 BUSINESS ARITHMETIC. 
 
 (e) The sum for which an agent sells gdods and the sum he remits 
 his principal. 
 
 5. Give illustrations showing from what known values the following 
 may be computed: 
 
 (a) The commission on a sale; on a purchase. 
 
 (6) The sum an agent receives in making a purchase. 
 
 (c) The sum the principal receives from a sale. 
 
 (d) The gross cost. 
 
 (e) The sum paid by an agent in making a purchase; a sale. 
 
CHAPTER XXXIII. 
 
 INSURANCE. 
 
 To Pupils. Collect and bring to class, for reference, old fire, health, 
 accident and life policies, insurance folders, and magazine advertisements 
 that give insurance information. 
 
 INTRODUCTORY EXERCISE. 
 
 1. What is your own definition of "insurance"? 
 
 2. Into what two classes can you divide all the classes of insurance? 
 
 3. Name several kinds of personal and of property insurance. 
 
 4. Give five reasons for taking out life insurance. 
 
 5. Give other uses for life insurance than simply to produce a fixed 
 sum at death. 
 
 6. Read over a life, accident or health policy. Are any conditions 
 imposed on the person insured, as to occupation, place of residence, etc.? 
 Why should such conditions be made? 
 
 7. Give several reasons for insuring one's property. 
 
 8. Distinguish between general and special property insurance. 
 
 9. Study a fire insurance poHcy and name at least eight conditions 
 imposed on the insured. State good reasons for these conditions. 
 
 10. State several ways in which one might "forfeit" his poHcy. 
 
 11. Does property insurance protect the insured property against 
 damage? What does it do? Has an insurance comp ny a right to make 
 repairs rather than pay damages? 
 
 12. How do strict policy conditions tend to prevent losses? 
 
 13. Why do insurance rates vary on different kinds of property? 
 
 14. Why should a person always read a policy carefully before taking 
 out insurance under it? 
 
 A contract is an agreement between two or more parties to 
 do or not to do a particular thing. It is evident that the 
 parties must agree as to the thing desired. It is generally 
 considered, also, that there must be proper consideration — that 
 
 255 
 
256 BUSINESS ARITHMETIC. 
 
 is, a fair return in service or value, for what is received. 
 Naturally, contracts may be of most varied kinds. Among 
 the commonest are those for the sale or rental of property, 
 for the payment of money, for employment and for insurance. 
 Contracts in the form of notes, bonds, etc., have been touched 
 on elsewhere. 
 
 Contracts for insurance are agreements whereby one party 
 agrees to pay another party a specified sum of money in the 
 event of certain happenings, such as the death or injury of 
 specified persons, or the destruction or loss of property through 
 fire, water, theft or accident. 
 
 The written or printed document containing the terms is the 
 insurance policy. The face of the policy is the sum, or limiting 
 sum, which the insurer agrees to pay. The premium is the 
 payment made by the insured to the insurer, for protection. 
 Payments of premiums after the first period are termed 
 renewals. 
 
 Life insurance policies run for a term of years, or for the 
 life of the insured; health and accident insurance poUcies 
 generally for one year; transit insurance (on goods being trans- 
 ported) generally for the period of .travel; fire insurance for 
 terms of one, two, three or five years. 
 
 The rates of premium on personal insurance are usually 
 expressed in dollars per thousand; on property insurance in 
 dollars per $100 or $1000, or as a per cent, of the face of the 
 policy. 
 
 The computations involved in the general offices of insurance 
 companies are exceedingly involved and require a high degree 
 of mathematical knowledge. From the standpoint of the 
 insurance clerk and the insured, the computations are very 
 simple and require no new principles. 
 
 It is evident that insurance is naturally divided into two 
 great groups — personal insurance and property insurance. 
 These will be noticed briefly. 
 
INSURANCE. 257 
 
 PERSONAL INSURANCE. 
 
 Besides general life insurance, there are many classes of 
 personal insurance, of which the following are well known: 
 
 Accident Insurance policies provide for payment to the 
 insured, or to specified beneficiaries, of sums varying with 
 the character and severity of the injury to the insured. There 
 are provisions for payments of surgeons' fees. Payments 
 may be made in a lump sum, or weekly for a specified number 
 of weeks. 
 
 Health Insurance may be written in connection with acci- 
 dent insurance, or as a separate contract. One form pro- 
 vides for weekly or monthly indemnity for illness arising from 
 a limited number of diseases; others are much more strict 
 in limitations. The contract usually provides for payments for 
 a period not greater that twenty-six weeks and for a specified 
 general payment in case of permanent disability. 
 
 Liability Insurance is written for employers to cover settle- 
 ments with employees for injuries for which the employer is 
 responsible. A policy covering all or certain classes of em- 
 ployees in a manufacturing plant, for example, is paid for 
 by a premium based on the total pay roll. The insurance 
 company defends for the insured any damage suits brought 
 by employees on account of injuries, and settles all legal or 
 just claims. 
 
 Automobile liability insurance is also written for any owner 
 of an automobile to indemnify him for liability on account of 
 injuries he inflicts on others while running his machine. 
 
 Fidelity and Surety Insurance. Fidelity insurance is written 
 to protect the insurer against loss through employees who do 
 not faithfully perform their duties. It covers the bonding of 
 employees who handle money, and is often paid for and taken 
 out by the ones bonded. Surety insurance involves also the 
 issuing of general court bonds, and bonds for the faithful 
 performance of any service or contract. Here the insurance 
 18 
 
258 BUSINESS ARITHMETIC. 
 
 company may demand from the person it bonds not only a 
 premium but also certain approved securities on which it can 
 realize funds with which to complete the contract in case of the 
 failure of the bonded person or company. 
 
 Credit Insurance is in a sense, also, a class of property 
 insurance. This is issued to merchants to protect them from 
 excessive loss from bad debts. Merchants who sell on credit 
 and desire such insurance submit their books to the credit 
 insurance company which averages the losses from bad debts 
 for a series of preceding years. A premium rate is then fixed, 
 in return for which the insuring company agrees to make good 
 any loss from bad debts during the year, in excess of the 
 average loss. The credit insurance company places restric- 
 tions on the merchant, however, to prevent the careless grant- 
 ing of credit to unreliable customers. 
 
 EXERCISE. 
 
 1. Compute the cost of a combined accident and health policy for 
 $7500, if the premium is $5.60 per unit of $500. 
 
 2. What should the insurance company pay the holder of the above 
 policy for a serious illness of 9 weeks 3 days, at the rate of $37.50 per week? 
 
 3. Compute the cost of a $75,000 bond for a cashier, secured from a 
 bonding company at a rate of $2.45 per $1000? 
 
 4. The pay roll of a manufacturing company amounts to $175,000 
 per annum. A liability insurance company agrees to assume the em- 
 ployer's liability for this force at an annual premium of 3i%. What is 
 the premium? 
 
 5. What does the insurance company gain in the above case, if it pays 
 damages of $4580 and incurs legal expense in defense and examinations 
 amounting to $547.50? 
 
 6. A credit insurance company examines the books of a mercantile 
 house and finds the normal loss from bad debts to be $4500. It issues 
 a policy for an excess loss of $8000, at a premium of 4%. What is the 
 premium? For the year covered by the policy the loss is $7896. What 
 sum does the insurance company pay? 
 
 7. Compute the cost of a $7500 excess policy at 3J%. Compute 
 the normal loss in this case if the respective losses for the six preceding 
 years was $2512, $2681, $2512, $2800, $3126, and $2107. 
 
INSURANCE. 259 
 
 Life and Endowment Insurance. Insurance on the life of a 
 person is issued under policies of most varied form. The most 
 common classes might be termed straight life, limited payment 
 life, endowment and term. Life policies guarantee the pay- 
 ment of a specified sum at death of the insured, to the bene- 
 ficiary named in the policy, in consideration of premiums paid 
 during the life of the insured, or during a given period. Ten- 
 payment and twenty-payment life policies are common. Term 
 policies limit the guarantee payment in case of death within a 
 fixed number of years. Endowment policies, in consideration 
 of annual premiums paid for a fixed number of years, guar- 
 antee a fixed payment to the beneficiary at death of the in- 
 sured, or to the insured at a certain age. 
 
 Premiums are usually expressed on a basis of $1000 of 
 insurance, and vary with the age of the insured and the 
 character of the insurance agreement. They are most 
 carefully compiled and are based on the chances of life and 
 the earning power of the premiums or investments of the 
 company. The processes of computation are exceedingly 
 complicated. ^ 
 
 The reserve of an insurance company is the portion of income 
 from premiums that it is required^ by law to set aside as a 
 fund for the payment of policies when they fall due. The 
 surplus of a company is the excess of assets over liabilities. 
 In the case of mutual companies, or those issuing participating. 
 policies, a certain portion of the surplus is returned to policy- 
 holders whose contracts entitle them to share in it. 
 
 Many causes may lead or force a policyholder to give up his 
 policy, or to cease paying premiums on it. (Name some.) 
 However, as a rule, if he has paid two or more annual pre- 
 miums, he receives some return for premiums already paid. 
 Thus, he may get cash, on surrendering his policy, equal to 
 practically its reserve value. He may secure a smaller policy 
 in exchange, all "paid up,'' for .whatever amount the reserve 
 
260 BUSINESS ARITHMETIC. 
 
 will purchase; or he may secure extended insurance for a 
 certain number of years. Many companies will also lend 
 money to the insured, on his policy, up to its cash value. 
 Some companies allow a fixed number of days of grace, say 
 thirty, for the payment of premium after due. Some rein- 
 state, after lapse of a payment, on proof of insurability and 
 payment of back premiums with interest. 
 
 A good, modern twenty-year endowment policy, for example, 
 at age 35 years, for $2000, in a certain company, if the 
 insured live, will cost $100.72 per year for 20 years, at the 
 end of which time it will produce to the insured $2000 and 
 its share of surplus or accumulation. 
 
 Statement of Guaranteed Values. 
 
 No. of 
 
 
 Paid up 
 
 Upon Surrender. 
 
 lual Prem. 
 
 Cash or Loan. 
 
 Insurance. 
 
 Participating Term. 
 
 Paid. 
 
 Value. 
 
 (Participating.) 
 
 Insurance for 
 
 2 
 
 $112 
 
 $180 
 
 6yr. 
 
 21 days. 
 
 3 
 
 186 
 
 292 
 
 10 
 
 259 
 
 4 
 
 262 
 
 400 
 
 14 
 
 258 
 
 5 
 
 342 
 
 ■ 512 
 
 15 
 
 
 
 6 
 
 424 
 
 • 623 
 
 14 
 
 
 
 7 
 
 508 
 
 724 
 
 13 
 
 
 
 8 
 
 596 
 
 828 
 
 12 
 
 
 
 9 
 
 688 
 
 932 
 
 11 
 
 
 
 10 
 
 784 
 
 1032 
 
 10 
 
 
 
 11 
 
 884 
 
 1136 
 
 9 
 
 
 
 12 
 
 988 
 
 1236 
 
 8 
 
 
 
 13 
 
 1096 
 
 1336 
 
 7 
 
 
 
 14 
 
 1210 
 
 1432 
 
 6 
 
 
 
 15 
 
 1328 
 
 1528 
 
 5 
 
 
 
 16 
 
 1448 
 
 1620 
 
 4 
 
 
 
 17 
 
 1572 
 
 1722 
 
 3 
 
 
 
 18 
 
 1762 
 
 1804 
 
 2 
 
 
 
 19 
 
 1838 
 
 1892 
 
 1 
 
 
 
 20 
 
 2000 and 
 
 accumulations, 
 
 Bhaxes in 
 
 dividends, ii 
 
 addition. 
 
INSURANCE. 261 
 
 EXERCISE. 
 
 Assume that the above policy was taken out on July 17, 1910. 
 
 1. What sum could be borrowed on the policy at the end of 7 years? 
 On August 18, 1917? 
 
 2. If the insured surrenders the policy on May 14, 1916, what amount 
 of paid up insurance should he receive? 
 
 3. What sum can he borrow on the policy after 12 premiums have been 
 paid? 
 
 4. If the insured surrenders the policy on May 19, 1918, he can secure 
 for it insurance for what term? How much is paid then in case of death 
 6 months later, no allowance being made for accumulations? 
 
 5. In 1912 his dividend was $19.66. He allowed it to go to reduce 
 his premium. What was his net premium in this year? 
 
 6. What is the corresponding premium per annum for a policy of 
 $17,000? 
 
 7. What cash surrender value has this new policy after 9 annual 
 premiums have been paid? 
 
 8. What is the premium on a $12,000 poUcy at same rate? If the 
 insured dies after the payment of five annual premiums, what sum do the 
 heirs receive? What sum in excess of premiums paid in? What sum in 
 case accumulations during the period of insurance have amounted to 
 $1,345.61? 
 
 Annual Premium Rates for Different Types of Policies. 
 Per $1000 of Insurance. , 
 
 Ordinary. 10 Pay. 20 Pay. 10 Year 20 Year 
 
 Age. Life. Life. Life. Endowment. Endowment. 
 
 21 $18.50 $46.40 $28.30 $101.68 $47.70 
 
 25 
 
 20.26 
 
 49.34 
 
 30.17 
 
 102.00 
 
 48.10 
 
 28 
 
 21.73 
 
 51.82 
 
 31.76 
 
 102.29 
 
 48.47 
 
 30 
 
 22.90 
 
 53.62 
 
 32.92 
 
 102.52 
 
 48.78 
 
 35 
 
 26.48 
 
 58.68 
 
 36.27 
 
 103.25 
 
 49.80 
 
 40 
 
 31.12 
 
 64.69 
 
 40.43 
 
 104.33 
 
 51.43 
 
 EXERCISE. 
 
 1. Find the cost of a poHcy of each kind for $4500 issued at age 28. 
 
 2. A man at age 30 takes out a 20-payment life policy. When he has 
 completed payments on it, how much less than the face of the policy has 
 he paid in? 
 
 3* WTiy does the age of the insured affect the amount of the premium? 
 
262 BUSINESS ARITHMETIC. 
 
 4. Give general reasons for the difference in amount of premium for 
 policies of different kinds, at any fixed age. 
 
 5. Suppose a man, at age 21 years, had taken out a $5000 policy of 
 each of the above kinds. Suppose that he died at the age of 50 yeajB. 
 What benefit would he have had from each policy during his life? What 
 benefit would his heirs receive in each case? 
 
 6. Why is an insurance company able to pay large sums at death 
 of the insured when it receives only small premiums? 
 
 7. Name several advantages of insuring at as early an age as possible. 
 
 8. Why are intending poUcyholders required to undergo a strict medical 
 examination? 
 
 Income Insurance. In return for the payment of annual 
 premiums varying as usual with the age of the insured, the 
 insurance company agrees to pay the beneficiary, on the 
 death of the insured, a fixed income per month, for a fixed 
 number of years, or for life, in place of paying a lump sum. 
 Under some policies both a lump sum and an income is pro- 
 vided. If the income is payable during the lifetime of the 
 beneficiary, then the premium rate depends in part on the age 
 of this person. Income insurance can be arranged in income 
 multiples of $10. 
 
 Old Age and Savings Insurance. This insurance is increasing 
 in this country and is common abroad. For the payment of 
 small monthly premiums, ordinary straight life and endow- 
 ment policies are issued. There are also combined insurance 
 and annuity policies, involving payment in full at death before 
 65, years of age, and in full, less annuities paid, for death 
 between, say, 65 and 69. Annuities may be paid after 65 
 years, the annual payment being a certain fraction of the face 
 of the policy. Straight old age pension policies are also issued, 
 wherein a certain annuity is given after the age of 60 or 65, in 
 return for annual premiums paid up to that age. In case of 
 death before the annuity age begins, the amount of the 
 premiums paid in is turned over to the heirs. The maximum 
 annuity is often a small sum, say $200, and the maximum 
 
INSURANCE. 263 
 
 insurance limit is very small. No medical examination is 
 required for old age pensions. 
 
 EXERCISE. 
 
 1. Compute the annual cost of an old age pension policy ensuring an 
 annuity of $200, purchased at an age of 36 yr., at the rate of $3.14 per 
 month. How much will have been paid in in premiums by the time the 
 insured reaches the age of 60? 
 
 2. Anderson takes out an old age policy guaranteeing $400 annuity 
 after. the age of 60 years, and the face of the premiums in case of death 
 at an earher age. He is 34 years old and pays a monthly premium of 
 $1.12 per $100 annuity. Compute his monthly and yearly premiums. 
 In case of the death of the insured at the age of 57 years, what sum should 
 be paid his beneficiaries? 
 
 3. Roberts, aged 26, takes out a $400 savings endowment policy payable 
 in twenty years or at death. What is his monthly premium, if the rate 
 is $2.19 per $500 insurance? What is his annual premium? 
 
 4. Newton, aged 40 years, takes out an income policy, guaranteeing 
 a monthly income of $60 at his death, to his beneficiary for 20 years, at a 
 premium rate of $5.76 per $10 income unit. What is his annual premiimi? 
 
 5. Osgood, aged 30 years, takes out an income policy, calling for the 
 payment to his beneficiary after his death, of a monthly income of $75 
 for 20 years, at a premium rate of $33.47 per $10 income imit. What is 
 his annual and his quarterly premium? In case of death, at age 47, what 
 will he have paid in premiums? What siun will his beneficiaries receive 
 during the following 20 years? Why is the insurance company's loss 
 less than the difference between these two amounts? 
 
 PROPERTY INSURANCE. 
 EXERCISE. 
 
 1. Is death sure, sooner or later, to every insured person? 
 
 2. Is loss by fire certain to occur, sooner or later, to every building 
 insured against it? Is lightning sure to strike every building? 
 
 3. Name differences between life and property insiu-ance, as regards 
 probable losses payable by insurance companies? 
 
 4. Why are property insurance rates lower than personal insurance 
 rates? 
 
 5. Why should fire insurance rates vary with the (a) use of a building; 
 (&) with its surroundings; (c) with the character of material and construction 
 of it? 
 
264 BUSINESS ARITHMETIC. 
 
 6. Why is a brick paper mill a greater "risk" than a brick residence^ 
 
 7. Why do property insurance companies make restrictions as to how 
 an insured building shall be built or used? 
 
 8. Why do fire insurance companies favor and work for strong fire 
 departments? 
 
 9. Why are country dwellings insured at a higher rate than city 
 dwelhngs of the same class? 
 
 10. Read the conditions of a policy and determine some of the duties 
 of the person whose property is insured, in case of damage to it which is 
 covered by insurance. 
 
 As in the case of life insurance companies, property insur- 
 ance companies may be either stock companies, owned and 
 managed for the stockholders' profit, under certain legal 
 restrictions; or mutual companies, in which the policyholders 
 share in gains and losses. Some companies are partially 
 mutual. 
 
 Policies vary greatly in form and conditions. They may be 
 valued policies, stating a fixed amount to be paid in case of loss; 
 or open policies, in which a maximum payment is named but 
 the actual payment is based on evidences of loss. 
 
 Illustrations. Insurance on ships is frequently covered under valued 
 policies Insurance on cargoes, and ordinary fire insurance is generally 
 covered by open policies. In case of total or partial loss under open policies, 
 the insurance company pays that part of the maximum stated in the policy 
 that will indemnify the insured. Thus, if $2500 damage is caused by fire 
 to a building insured for $6000, the entire damage is paid; but if the damage 
 is $6800, only the maximum of the policy, $6000, is paid. 
 
 Some policies contain a co-insurance clause requiring that 
 the property shall be kept insured up to a certain per cent, of 
 its value — otherwise that per cent, only of the loss is paid that 
 the insurance named is of the per cent, required. 
 
 Illustration. If 80% is the per cent, required to be insured then, on 
 a $15,000 building, for example, $12,000 insurance must be carried to 
 receive full benefit. In case of $10,000 insurance, five-sixths of $12,000, 
 only five-sixths of any loss, up to the face of the policy will be paid. 
 
 Some policies contain an average clause, specifying the 
 
INSURANCE. 265 
 
 payment of that part of the loss that the insurance is of the 
 
 value of the property. 
 
 Illustration. If the insurance on a piece of property is four-ninths 
 of its value, four-ninths of any loss will be paid. 
 
 Insurance companies reserve the right of replacing or re- 
 pairing damaged property covered by their policies, in place 
 of paying cash for losses. In case the property is insured in 
 several companies, the loss is usually apportioned among them 
 according to the faces of the policies they issued on it. Some- 
 times a single company issues a policy on a valuable property, 
 for a heavy amount, and re-insures a part of its "risk" in 
 another company. Losses are appraised by special inspectors 
 and often agreements are reached with owners, as to the loss 
 payable. 
 
 The rate of premium varies with the character of the risk 
 and the period of the policy. It may be for a specified 
 number of days, for a year or for a period of years, or in 
 the case of merchandise in transit, for the length of a certain 
 voyage. "Short rates" are often charged for terms under one 
 year, and are relatively higher than the yearly rates. Rates 
 are expressed in cents per $100, or as a rate per cent. 
 
 Many of the computations of premiums and settlements of 
 losses are based on the simple principles of ratio and of per- 
 centage. No new process is involved in any problems in this 
 book. 
 
 SOME CLASSES OF PROPERTY INSURANCE. 
 
 Fire Insurance is insurance of property against loss by fire. 
 The liability covers damage by water in fighting the fire, or 
 destruction of property in the same cause. Sometimes, it 
 covers damage caused by Hghtning, or by wind storms. 
 
 Marine Insurance is insurance of vessels and cargo against 
 the perils of navigation. This includes damage by fire, or 
 storm, shipwreck, etc. Marine policies frequently contain 
 the average clause. 
 
266 BUSINESS ARITHMETIC. 
 
 Transportation Insurance is insurance of goods in transit by 
 land, or by land and water, against loss resulting from the 
 ordinary dangers of such traffic. 
 
 Live Stock Insurance is issued to cover loss by casualties to 
 horses, cattle, etc. 
 
 Miscellaneous Special Types of Insurance. It is said that it 
 is now possible to insure any property against any damage. 
 The following titles illustrate common types, and are practi- 
 cally self-explanatory: Automobile and equipment insurance, 
 tourists' (personal effects) insurance, motor boat and equip- 
 ment insurance, steam boiler insurance, explosion insurance, 
 theft insurance, plate glass insurance, derailment insurance, 
 fly wheel insurance, etc. 
 
 ORAL 
 
 EXERCISE. 
 
 
 Find the cost of insurance: 
 
 
 
 Face of Policy. Premium. Face of Policy. 
 
 Premium. 
 
 1. $ 2,000 U% 
 
 5. $ 4,000 
 
 $1.25 per $100 
 
 2. $ 3,600 f % 
 
 6. $ 3,000 
 
 .80 per $100 
 
 3. $50,000 2i% 
 
 7. $12,000 
 
 1.75 per $100 
 
 4. $ 8,000 11% 
 
 8. $ 7,500 
 
 2.50 per $1000 
 
 Find the rate of premium and express in two ways: 
 
 
 Face. Premium. 
 
 Face. 
 
 Premium. 
 
 9. $4,000 $120 
 
 11. $12,000 
 
 $150 
 
 10. $6,000 45 
 
 12. $ 5,000 
 
 75 
 
 13. $400 will purchase $ ? insurance at 2%. 
 
 , 
 
 14. $600 will purchase $ ? insurance at 1\%. 
 
 
 15. $90 is the premium on a poHcy of $ ? at $1.50 per $1000. 
 
 EXERCISE. 
 
 1. Johnson takes out $2500 insurance on his touring automobile at 
 2% at a cost of $ . 
 
 2. What will it cost Robinson to place $1500 insurance on his stable 
 for one year at 25c per $100, and $800 on his horses and carriages at 40c 
 per $100? 
 
 3. Newton takes out an 80% policy on his brick house, valued at $7500, 
 at a rate of $2.75 per $1000. It costs him $ . 
 
INSURANCE. 267 
 
 4. A factory owner takes out $40,000 insurance on his buildings at 
 3^%; and $3000 special insurance on his fly-wheel, at $12.50 per $1000. 
 His total payment is $ . 
 
 5. James Sanborn insures his property through a broker, as follows: 
 Brick and stone house, valued at $24,000, at 75% of its value at $3.75 per 
 $1000; a broad plate glass window, for $400 at lf%; furniture, valued 
 at $12,500, at $5.75 per $1000 for five years; a brick stable and garage, for 
 $2000, at $2.50 per $1000 one year policy; horses and vehicles, $2000 at 
 $4.25 per $1000; automobile and equipment, $2000 at 3^%; frame boat 
 house $600, at 60c per $100; motor boat and equipment, $1000 at 2f %. 
 Compute his total annual cost of insm'ance. 
 
 6. A shipment of goods to Manila, valued at $85,000, is insured for 
 90% value at 11%. What does the insurance cost? 
 
 EXERCISE. 
 
 1. On a house insured for $7500, damage is caused to the extent of 
 $3748. What should the insurance company pay? 
 
 2. Randall insures his brick house, valued at $9000, at 4/5 value at 2%. 
 The property is damaged to the extent of $8500. What should the insurance 
 company pay? What is his net loss and that of the company, allowing for 
 premiums? 
 
 3. Property worth $12,000 is insured for $7500, under a policy con- 
 taining an 80% clause. In case of proved damage to the extent of $2750, 
 what should the insurance company pay? 
 
 4. On a stock of goods amounting to $30,000, insurance is carried as 
 follows: Company A, $4000; Company B, $6000; Company C, $15,000. 
 The property is damaged by fire to the extent of $16,500. What should 
 each company pay? 
 
 5. A stock bam, insured for $2500 against damage by tornados at 
 80c per $100, is damaged in such a storm to the extent of $3750. What is 
 the real loss of the owner? 
 
 6. Prepare an illustrative example showing how the losses paid to a 
 poUcyholder are met by other poUcy holders in the same company. 
 
 7. Robinson bought merchandise abroad to the value of £3000. 
 (£1 = $4.86). He paid a commission of 2%, and insurance of 3/4 value, 
 
 at 11% (average clause). The entire shipment cost him $ . Fire 
 
 broke out in transit and the goods were damaged to the extent of $5860. 
 What should the insurance company pay? What was Robinson's net loss? 
 
 8. Fire broke out on the estate of James Sanborn (ex. 5 of previous 
 exercise), causing loss of $19,200 on the house; -destroying all but $800 
 
268 BUSINESS ARITHMETIC. 
 
 worth of furniture; destroying the plate glass window; damaging the stable 
 to the extent of $1500; destroying the boat house, and damaging the motor 
 boat to the extent of $350. Xhe insurance company repaired the motor 
 boat, and sent a check for the other losses for which it was responsible to 
 Mr. Sanborn. The face of check was $ . 
 
 9. The "Warfield," freighter, valued at $180,000, is insured under 
 average clause poUcies, for $40,000 in Company A and for $65,000 in 
 Company B — the former at 21% and the latter at 2^%. (Pre- 
 miums?) On one trip, the vessel carried a shipment of 28,000 bushels of 
 corn, valued at 61c per bu. and insured at 1^% to cover goods and 
 charges. (Face of policy and premium?) The vessel also had aboard 
 some general merchandise valued at $85,000 and insured for 3/4 value at 
 lf% — average clause. (Premium?) Fire broke out on this trip. 
 The general merchandise was destroyed or thrown overboard. The 
 insurance company sold the grain that was not destroyed, amounting to 
 7520 bu., at 31c per bu. The damage to the vessel was appraised at 
 $57,800. Compute the payments by each company, assuming that Company 
 B carried the insurance on both grain and merchandise. Compute the 
 net losses of the companies and of the owners, taking premiums into 
 account. 
 
CHAPTER XXXIV. 
 TAXATION AND PUBLIC REVENUE. 
 
 GENERAL INTRODUCTORY EXERCISE. 
 
 1. What protection does a city government give its inhabitants? 
 
 2. What besides protection does it supply? 
 
 3. What does it do for the young people? 
 
 4. How does the local government aid travel and intercourse? 
 
 5. How are your public officers chosen? 
 
 6. Who can vote? Does it cost anything to vote? 
 
 7. How IS the money for local government expenses secured? 
 
 8. What kind of property is taxed? 
 
 9. Why must many business men pay for a license to do business? 
 
 10. What does your state government do for its citizens? 
 
 11. How does it raise money for these objects? 
 
 12. Name twelve public objects for which the United States Govern- 
 ment pays money. How are the people benefited in each case? 
 
 13. How does the National Government aid commerce and industry? 
 
 14. How does the National Government raise its funds? 
 
 15. Why does the cost of government increase steadily? 
 
 16. In what ways are governments doing more than they used to do 
 for their people? 
 
 1. LOCAL AND STATE TAXATION. 
 The cost of local and, to some extent, of state governments 
 is met by a tax or charge levied on real estate, in proportion to 
 its value; and on general personal or movable property — such 
 as furniture, machinery not built in as a part of a building, 
 horses and wagons, and sometimes, stocks and bonds. A 
 further income is derived from fees for licenses, or permits 
 to conduct special businesses; from assessments, or charges 
 
270 BUSINESS ARITHMETIC. 
 
 against private real estate, for improvements in the form of 
 sidewalks, water mains, etc.; and from fines and 'permits. 
 Polls or capitation taxes are levied on all qualified voters. 
 
 EXERCISE. 
 
 Secure a printed statement of your own, or a nearby, city showing total 
 expenditures for a recent year for governmental purposes. Show what 
 are the largest items of expense. Find the average expenditure per person. 
 
 The greatest source of public income is realty. The process 
 of taxing realty in one Eastern city is as follows: A complete 
 assessment (valuation) of real property " at not less than two- 
 thirds of the real value thereof," is made every three years 
 by tax assessors. The chief assessor and his assistants then 
 sit as a Court of Equalization and Review to hear owners and 
 witnesses, readjusting any seemingly unfair assessment. Their 
 decisions are then confirmed by the governing ofiicials. 
 Special assessments are made annually of property that has 
 changed in value $500 or over, owing to its improvements, or 
 to destruction of buildings. The tax, 1|% of assessed value, 
 is payable to the collector of taxes in May for the year ending 
 June 30th. 
 
 Tax rates on real or personal property, expressed as mills • 
 or cents per $1.00 or $100, or as a rate per cent., are based on 
 the assessed value. Thus, 1^ %, 15 mills (1| % of $1.00) 
 or $1.50 (on $100) are equivalent rates. Several taxes, such 
 as local, county and state, may be levied on the same property, 
 and collected at one time. 
 
 ORAL EXERCISE. 
 
 Express each rate in two other forms: 
 
 1. 1.8%. 3. li%. .5. 12 mills ($1.00). 7. 22^ mills. 
 
 2. 3/4%. 4. U%. 6. 16 mills. 8. $1.75 ($100). 
 Find total rates equivalent to these series: 
 
 11. 2/5%, 8 mills, $1.50. ' 13. U%, 75c, 7J mUls. 
 
 12. li%, 3 mills, $1.25. 14. 75c, $1.25, 2\%. 
 
TAXATION AND PUBLIC REVENUE. 271 
 
 The computation of the tax involves simple percentage or 
 multiplication. 
 
 Illustration. Compute state and local tax at 1|% and 8 mills, on 
 property assessed at $4200. 
 
 Solution. State tax = U% of $4200 = $63.00 
 
 Local tax = 8 mills per $100 = 4200 X $.008 = $33.60. 
 
 Note. The total tax may be found directly. 8 mills = 4/5%. 1|% + 
 4/5% =2x\%. 2j%% of $4200 = $96.60. 
 
 ORAL EXERCISE. 
 
 Find total tax. 
 
 
 
 
 Hates. 
 
 Assessed Value. 
 
 
 Rates. Assessed Va 
 
 1. 2/5% 
 
 $4500 
 
 5. 
 
 8 mi. $24,500 
 
 2. U% 
 
 7280 
 
 6. 
 
 1/4% and 6000 
 
 3. $1.25 
 
 14,500 
 
 
 $1.50 
 
 4. .50c 
 
 8420 
 
 7. 
 
 15m., 2^m. 8400 
 
 EXERCISE. 
 
 Find missing values. 
 
 State each example in form of a question, and show how it differs from 
 preceding problems of a similar nature. 
 
 1. John Sampson's property is assessed for $12,600, at 11%. His tax 
 
 is $ . 
 
 2. C. P. Crane has assessed at $1.25 per sq. ft., a lot measuring 240 ft. 
 X 80 ft. His tax, at 1^%, is $ -. 
 
 3. Robert Walter owns three lots, assessed at $8420, $9200 and $14,620. 
 On the latter is a building assessed at $16,500. He has personal property 
 assessed at $4500, but $1000 is exempt by law. His total tax at 12| mills 
 is$ . 
 
 4. A Permanent Improvement Fund tax of 2§ mills, levied by a 
 western city on property assessed at $26,450, costs its owner $ . 
 
 5. A Current Expense Fund is created, in a Minnesota city, by a 
 tax of 6/10%. On a realty assessment of $99,547,484 and a personal 
 property assessment of $29,049,250, this would yield what sum if every 
 person paid? In this city, the delinquent tax averages 4|%. What net 
 tax should the property yield? 
 
 6. A western city recently levied this general tax per $1000. What 
 does it cost C. A. Mansfield whose property is assessed at $8455? 
 
 State revenue S1..50 
 
 State School and University Fund 2.23 
 
272 BUSINESS ARITHMETIC. 
 
 County revenue 2.75 
 
 School revenue 6.00 
 
 City funds 14.37 
 
 ? 
 7. James Field, collector for the town of Mayfield, receives a salary 
 of 3% of his collections. If he collects a $1.25 tax on $364,000 of property, 
 he earns $ — -. 
 
 Note. Penalties are charged for delinquent (unpaid) taxes. In one 
 city, penalty interest of 1% per month is charged. In default of payment, 
 the property is sold, but the purchaser gets simply a tax deed bearing 1% 
 per mo. interest, and the owner may redeem his property within two years. 
 
 8. The property here shown is as- 
 120' sessed at $30 per front foot, and taxed 8 
 
 mills. The tax due on Jan. 1, was not 
 paid until Oct. 1, incurring a penalty, at 
 3/4% per month, of $ . 
 
 9. A tax certificate of $241.60, bear- 
 ing 1% interest per month, dated Dec. 
 10 last, is worth to-day $ . 
 
 10. J. C. Parker leaves his wife by 
 will, $42,000 hi property; his son, $12,- 
 600, and a niece $5200. The state has 
 
 an inheritance tax of 1?% on bequests to Imeal heirs, and 6% to others. 
 The tax on these bequests yields the state $ . 
 
 11. The Randolph Manufacturing Co. is taxed 1% of its legal capital, 
 on issuance of its state charter. If the authorized capital is $240,000, 
 the tax is $ . 
 
 12. An interurban trolley road is taxed 90c per $100 on its equipment 
 and 1/2% on its gross earnings. Last year its equipment was assessed 
 at $8,240,000 and its gross earnings were $6,547,325.60. Its state tax 
 amounted to $ . 
 
 Many problems arise in tax offices when making up advanced 
 estimates. A few are stated below. 
 
 EXERCISE. 
 
 1. What rate in mills and per cent, is sufficient to raise $12,460 on an 
 assessed value of $800,000? 
 
 2. In a city having a fixed legal tax rate of 15 mills, the total assessed 
 value must be fixed at what sum, in order that an income of $172,000 may 
 be secured? 
 
TAXATION AND PUBLIC REVENUE. 
 
 273 
 
 3. Last year, in a Nebraska town, the legal rate of 3| mills yielded 
 $142,500. This year the income required being only $122,000, the assessed 
 value may be reduced to approximately %. 
 
 4. A Pennsylvania town must meet the following appropriations by 
 a tax of ? mills, on an assessed valuation of $1,246,300: salaries $2140; 
 schools, $8720; roads, $2465; general expenses $5840. Assume 5% 
 deUnquent tax. 
 
 5. 520 polls @ $1.50; Ucense fees, $1260; and a $1.25 tax on realty 
 and personalty, assessed at $1,652,000, will yield an annual income of $ . 
 
 • 
 For convenience in calculation, tax computation tables are 
 
 often prepared for the total tax rate of a town or city. The 
 
 following table gives the tax, at the given rate, on any sum 
 
 from $10 to $99. The tax on other sums is found by " pointing 
 
 off " and by addition. The original table must be computed 
 
 with great care, and written, preferably, in six decimal places. 
 
 7§ mills 
 
 Tax Table 
 
 Per $100 
 
 
 
 
 
 
 Unit lines 
 
 
 
 
 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 1 
 
 .075 
 
 ,0825 
 
 .0900 
 
 .0975 
 
 .0150 
 
 .1125 
 
 .1200 
 
 .1275 
 
 .1350 
 
 .1425 
 
 1 
 
 2 
 
 .150 
 
 .1575 
 
 .1650 
 
 .1725 
 
 .1800 
 
 .1875 
 
 .1950 
 
 .2025 
 
 .2100 
 
 .2175 
 
 2 
 
 3 
 
 .225 
 
 .2325 
 
 .2400 
 
 .2475 
 
 .2550 
 
 .2625 
 
 .2700 
 
 .2775 
 
 .2850 
 
 .2925 
 
 3 
 
 4 
 
 .300 
 
 .3075 
 
 .3150 
 
 .3225 
 
 .3300 
 
 .3375 
 
 .3450 
 
 .3525 
 
 .3600 
 
 .3675 
 
 4 
 
 5 
 
 .375 
 
 .3825 
 
 .3900 
 
 .3975 
 
 .4050 
 
 .4125 
 
 .4200 
 
 .4275 . 
 
 4350 
 
 .4425 
 
 5 
 
 6 
 
 .450 
 
 .4575 
 
 .4650 
 
 .4725 
 
 .4800 
 
 .4875 
 
 .4950 
 
 .5025 
 
 .5100 
 
 .5175 
 
 6 
 
 7 
 
 .525 
 
 .5325 
 
 .5400 
 
 .5475 
 
 .5550 
 
 .5625 
 
 .5700 
 
 .5775 
 
 .5850 
 
 .5925 
 
 7 
 
 8 
 
 .600 
 
 .6075 
 
 .6150 
 
 .6225 
 
 .6300 
 
 .6375 
 
 .6450 
 
 .6525 
 
 .6600 
 
 .6675 
 
 8 
 
 9 
 
 .675 
 
 .6825 
 
 .6900 
 
 .6975 
 
 .7050 
 
 7125 
 
 .7200 
 
 .7275 
 
 .7350 
 
 .7425 
 
 9 
 
 0123456789 
 Illustrations. The tax on $82 is found on the horizontal line "8" 
 
 at its intersection with the vertical unit line "2." It is $.615 or 62c. 
 The tax on $8200 (100X82) = $61.50. The tax on $972 = tax on $900 
 
 + tax on $72, etc. 
 
 EXERCISE. 
 
 1. Show how the above table may be constructed by addition, 
 how short methods may be used for construction or checking. 
 
 19 
 
 Show 
 
274 BUSINESS ARITHMETIC. 
 
 2. Read from the table, the tax on assessed values of: 
 
 $840 $24,200 $1,220 $ 815 
 
 1200 36,450 19,600 2,290 
 
 960 8,000 452 42,600 
 
 3. Use the table to read the tax on these same values at 15 mills. 
 
 4. Construct tax tables for these rates: (a) 1^%; (6) 60c; (c) $1.45; 
 id) $27.60 (per $1000). 
 
 Licenses. Many municipalities secure an income from a 
 charge for licenses to conduct, certain businesses. An eastern 
 city received from this source, in one year, $572,473.43. 
 These are a few of its rates: 
 
 Busmess. Rate. Remarks. 
 
 Apothecaries $ 6.00 per annum. 
 
 Auctioneers 100.00 per annum. 
 
 Boarding houses, hotels, public. . 1.00 per guest room. 
 
 Brokers, real estate 50 per annum. 
 
 Carriages, for hire 6.00 one horse. 
 
 Carriages 9.00 more than one horse. 
 
 Circuses 200.00 per day. 
 
 Confectioners 12.00 per annum. 
 
 Livery stables 25.00 per annum 10 stalls. 
 
 Livery stables 2.00 each additional stall. 
 
 Passenger transfer lines 6.00. each vehicle seating ten or less. 
 
 Passenger transfer lines 12.00. . . .each vehicle seating over ten. 
 
 EXERCISE. 
 
 Using these license rates, tabulate the income derived by a city from 
 61 apothecaries, 18 auctioneers, 529 street cars, 63 confectioneries, 17 
 livery stables (under 10 stalls), 8 stables — total of 286 stalls, 282 carriages 
 for hire; 41 brokers; 21 hotels, guest rooms respectively: 149, 217, 300, 84, 
 29, 80, 120, 360, 400, 210, 88, 146, 97, 63, 48, 88, 95, 85, 66, 22, 38. 
 
 Assessments. In some cities a portion of the cost of public 
 improvements and of repairs, such as curbing, sidewalks, water 
 mains, sewers, alleys, etc., is levied by assessment against 
 property that is especially benefited. If an owner asks for 
 improvements, he must deposit his estimated share of the 
 cost in advance. If he does not ask for the improvements, 
 his payments may usually extend over two years. The 
 
TAXATION AND PUBLIC REVENUE. 275 
 
 proportion paid b} private owner and the methods of assess- 
 ment vary greatly in different cities. Several different 
 methods are illustrated in the following problems. 
 
 EXERCISE. 
 
 1. John Evans is assessed $ ? for a water main, at a rate of $1.10 per 
 front foot, on two lots of 85 ft. and 40 ft. frontage respectively. 
 
 2. 320 ft. of curbing are laid at a cost of 90c per foot. The frontage 
 benefited is 300 ft., of which Adams owns 65 ft. He is assessed one-half 
 the pro rata cost, or $ . 
 
 Note. The cost of grading and paving streets is sometimes paid in 
 full by the city or town; sometimes assessed one-half on abutting property, 
 according to frontage, sometimes one-half according to area, sometimes part 
 by frontage and part by area. Owing to the fact that comer lots may be 
 assessed for improvements on two streets, their rates are sometimes reduced. 
 Areas are measured from established building lines. 
 
 3. Rockford Avenue (see page 276) is paved with asphalt from First to 
 Second street at a cost of $2.25 per square yard. The area paved is 60 
 ft. by 360 ft. If one-half of the cost is assessed on property owners, what 
 sum must be met by the owners of frontage on one side of the street? 
 If this cost is shared according to frontage, what hts must be assessed? 
 
 Andrews, who owns lots No. 266 and No. 265, pays $?; Chapman, for No. 
 270, and No. 271, pays $?; Eastman, the owner of No. 267, 268, 269, 
 270, and No. 273 pays $?. 
 
 4. Find assessments (ex. 3) if comer lots are assessed on basis of 1/2 
 frontage. 
 
 5. Determine assessments if 25% is levied by frontage and balance 
 by area. 
 
 6. A curbing is laid from the alley around these lots, 25 ft. from building 
 line, at a cost of 35c per ft. Assess one-half the cost against the lots ac- 
 cording to frontage. 
 
 7. The property fronting on Rockford Avenue is assessed 65c per front 
 foot for water mains. What does each owner pay? 
 
 8. The alley is paved at a cost of $1.20 per sq. yd. from 1st to 2nd street. 
 One-fourth of the cost is assessed by area against lots Nos. 264-273. 
 Determine the "lot" assessments. 
 
 II. NATIONAL GOVERNMENT INCOME AND EXPENSE. 
 
 The main sources of National Government income and 
 expense are shown in the tables that follow. There are many 
 minor sources also, such as profits on coinage, fees, sales of 
 
276 
 
 BUSINESS ARITHMETIC. 
 
 SECOND 
 
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 notice. The other taxes are paid in money. 
 
TAXATION AND PUBLIC REVENUE. 
 
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 EXERCISE. 
 
 1. Complete the tables. 
 
 2. Define the terms in the first column. 
 
 3. Where does authority for expenditures rest? What is an appro' 
 priation? 
 
278 BUSINESS ARITHMETIC. 
 
 Customs. 
 
 The National Government levies duties or customs on 
 many classes of articles, for revenue and for protection of 
 home industry. Congress has adopted a tariff or schedule 
 showing the rates of duties on imports; and a free list, or 
 schedule of articles imported exempt from duty. 
 
 Duties are either ad 'valorem (levied as a per cent, of their 
 net value) or specific (levied by measure). Both classes of 
 duties are imposed on some imports. In general, classes of 
 goods varying markedly in quality, and therefore in value, 
 are taxed '' ad valorem.'' These frequently are manufactures, 
 such as carpets, crockery, knives, etc. Raw materials, and 
 many articles of fairly uniform grade, are taxed specifically, 
 for example, rivets, gold leaf, eggs, wheat. 
 
 Ad valorem duties are not computed on fractions of a dollar, 
 less than 50 cents being discarded, and 50 cents or more 
 being considered an additional dollar. The same rule applies, 
 in specific duties, to fractions of the unit measure. The 
 Government employs the long ton (2240 lb.) in weighing, 
 but makes allowances, in many cases, for tear and breakage. 
 
 Rates from a United States Tariff. 
 
 Camphor, refined, six cents per pound. 
 
 Castor oil, thirty-five cents per gallon. 
 
 Paris green and London purple, fifteen per centum ad valorem. 
 
 Castile soap, one and one-fourth cents per pound; medicinal or medicated 
 soaps, twenty cents per pound; fancy or perfumed toilet soaps, fifty per 
 centum ad valorem. 
 
 Lime, five cents per weight of one hundred pounds, including weight 
 of barrel or package. 
 
 Cast polished plate glass, finished and unfinished, or unsilvered, not 
 exceeding three hundred and eighty -four square inches, ten cents per square 
 foot, above that, and not exceeding seven hundred and twenty square 
 inches twelve and one-half cents per square foot; all above that twenty- 
 two and one-half cents per square foot. 
 
 Pen knives, pocket knives, clasp knives, valued at not more than forty 
 cents a dozen, forty per centum ad valorem; valued at more than forty 
 
TAXATION AND PUBLIC REVENUE. 279 
 
 cents per dozen, and not exceeding fifty cents per dozen, one cent per 
 piece, and forty per centum ad valorem. 
 
 Gold leaf, thirty-five cents per one hundred leaves. 
 
 Shingles, fifty cents per thousand. 
 
 Horses and mules, valued at one hundred and fifty dollars or less per 
 head, thirty dollars per head; if valued at over one hundred and fifty 
 dollars, twenty-five per centum ad valorem. 
 
 Buckwheat, fifteen cents per bushel of forty-eight pounds. 
 
 Carpets, woven whole for rooms, and Oriental, BerHn, Aubusson, 
 Axminster and similar rugs, ten cents per square foot, forty per centum 
 ad valorem. 
 
 Firecrackers of all kinds, eight cents per pound, the weight to include 
 all coverings, wTappers and packing material. 
 
 EXERCISE. 
 
 1. Define: Income, duties, revenue, tariff, free list, miports, ad valorem, 
 and specific. 
 
 2. Which is simpler to levy, a specific or an ad valorem tax? 
 
 3. State the kind of duty specified in each of the given sections of the 
 tariff. Suggest a reason for the kind of duty. 
 
 The country is divided into collection districts, each con- 
 taining a port of entry and one or more ports of delivery. All 
 imports must be brought first to the port of entry at which is 
 situated a custom house for the collection of duties and for the 
 official entry or clearance of vessels. After duties are paid 
 and permission given to unload, goods may be discharged at 
 any port of delivery. Thus the collection district of the City 
 of New York has two ports of entry, New York and Jersey 
 City, which, as usual, are also delivery ports; and has addi- 
 tional ports of delivery at New Windham, Albany, Cold 
 Spring, Port Jefferson and Patchogue. 
 
 In the larger ports, a Collector of the Port is the chief officer, 
 having under him appraisers to determine values, and sur- 
 veyors ^ weighers, gaugers, etc., to determine quantities. 
 
 In brief, the process of importing goods is as follows: The 
 seller or shipper at a foreign port makes out an invoice of the 
 marks, number, quantities, value, etc., of the goods shipped, 
 
280 BUSINESS ARITHMETIC. 
 
 weights and values being stated in the standards of the foreign 
 country. The invoices, if over $100, are certified by the 
 local U. S. consul who retains a copy and sends a second copy 
 to the collector of the port at which the goods are to be 
 entered. On loading the goods aboard ship, the shipper re- 
 ceives duplicate bills of lading, or receipts involving a contract 
 to deliver at a specified port The shipper forwards copies of 
 invoice and of bill of lading to the purchaser in the United 
 States. The latter fills out certain forms and presents them, 
 with the invoice and bill of lading, at the custom house, pays 
 the estimated duties, and receives a permit to land the goods 
 subject to examination and verification. 
 
 The permit goes to the Surveyor who is in direct control of 
 the vessel and its cargo. The goods are then landed and 
 examined and appraised, to determine And check quantities 
 and dutiable value. One package in each invoice, one package 
 out of every ten of the same kind and same price of each class 
 of material, is examined carefully. The master of the vessel 
 also submits to the Collector of the Port a manifest, or state- 
 ment of the name of the vessel, of the names of consignors 
 and consignees, and of the items of his cargo. 
 
 If the importer does not care to use and pay duty on his 
 goods, at the time of receipt, he may give bond to pay the duties, 
 and may store the goods in bonded warehouses, maintained by 
 the Government. All or part of the invoice, within certain 
 minimum limits, may be withdrawn as desired on payment of 
 duties and reasonable charges for storage and incidental labor. 
 Such goods may be withdrawn for export without being subject 
 to duty. If the goods are retained in storage over one year, 
 however, the duty is increased 10%. After three years, the 
 goods are subject to forfeit. If duties have been paid and 
 imports afterwards exported, a draivback or refunding of the 
 duties is granted. 
 
 For the collection of the excise, or internal revenue tax, the 
 
TAXATION AND PUBLIC REVENUE. 281 
 
 country is divided into districts each in charge of a collector. 
 In many cases, the duty is paid by affixing to packages of 
 manufactured goods the requisite amount of tax in the form 
 of adhesive stamps which may be purchased in quantities and 
 used as needed. The stamps must be destroyed when the 
 packages are opened. Drawbacks are allowed on goods made 
 in this country that are exported after the excise tax has been 
 paid. 
 
 As the invoice value of imports is expressed in the coinage 
 of the exporting country, it is necessary to reduce such values 
 to United States standards. Coinage reduction tables are 
 issued periodically by the Director of the United States Mint. 
 
 Values of Foreign Coin in United States Gold. 
 (Foreign gold coins unless otherwise stated.) 
 
 Country. Coin. Abbreviation. U S. Value in Gold 
 
 Argentine Republic Peso p $.965 
 
 Austria Krone kr 203 
 
 Belgium, France, Switzerland . Franc fr • 193 
 
 Brazil Milreis ml 546 
 
 Chili Peso p 365 
 
 German Empire Mark M 238 
 
 Great Britain Pound Sterling . £ 4.8665 
 
 Italy Lira 1 193 
 
 Japan Yen y 498 
 
 Mexico Peso, Dollar (sU.) 498 
 
 Netherlands Florin fl 402 
 
 Philippines Peso p 500 
 
 Russia Ruble r 515 
 
 Computations under this table involve simple multiplication. 
 Illustration. Determine the IT. S. value of a shipment of hides from 
 Argentine, invoiced at 4200 pesos. 
 
 Solution. From the table, 1 peso = $.965. 4200 X $-965 = $4053, the 
 value of 4200 pesos. 
 
 ORAL EXERCISE. 
 .Determine value of these invoices in U. S. gold. 
 
 1. From Argentine, 2000 pesos. 3. From France, 120Q fr. 
 
 2. From England, £ 600. 4. From Germany, 5000 M. 
 
282 BUSINESS ARITHMETIC. 
 
 5. From Chili, 250 p. 8. From Russia, 400 r. 
 
 6. From Mexico, 1000 pesos. 9. From Philippines, 720 p. 
 
 7. From Netherlands, 300 fl. 10. From Japan, 10,000 yen. 
 
 EXERCISE. 
 
 Determine the value in U. S. gold of these invoices: 
 
 1. From Russia, 7246.5 r. 5. From London, £365.82. ■ 
 
 2. From Japan, 864.36 yen. 6. From Amsterdam, 4368.20 fl. 
 
 3. From Switzerland, 15,329 fr 7. From Havre, 32,129.6 fr. 
 
 4. From Germany, 2468 M. 8. From Naples, 673.80 1. 
 9. Of bone and tallow from Argentine, 6754 pesos. 
 
 10. Of nitrate of soda from Chili, 73,854 pesos. 
 
 11. Of coffee from Brazil, 785.25 ml. 
 
 12. Of knives from Sheffield, £542.10. 
 
 13. Of glass from Bohemia, 529.65 krone. 
 
 Specific Duties. 
 
 When specific duties are charged, the reduction of foreign 
 measures to U. S. standards is necessary. Great Britain uses 
 measures similar to our own, and most other exporting 
 countries use the metric system. 
 
 Once the invoice quantity is expressed in U. S. standards, 
 the computation of the specific duty is a matter of direct 
 multiplication, except where the wording of the U. S. tariff 
 necessitates a further computation. 
 
 Illustrations. (1) What is the duty on 50 M. shingles at 50c per M.? 
 Solution. At 50c per M. the duty on 50 M. =50 X 50c = $25.00. 
 (2) What is the duty on 7200 lb. buckwheat at 15c per bu. of 48 lb.? 
 Solution. The number of bu. = 7200 -^ 48 = 150. 150 X 15c = $22.50 
 duty. 
 
 ORAL EXERCISE. 
 
 See Tariff, pp. 278, 279 for missing rates. 
 Compute the duty on: 
 
 1. 2400 bu. wheat @ 25c. 6. 625 lb. camphor. 
 
 2. 60 gr. pens @ 12c. 7. 3000 lb. lime. 
 
 3. 2750 lb. rivets @ Uc. 8. 28 pkg. gold leaf. 
 
 4. 16 T. grindstone @ $1.75 per T. 9. 10 pc. plate glass, 24 in.X 
 6. 4600 lb. Castile Soap. 30 in. 
 
TAXATION AND PUBLIC REVENUE. 
 
 283 
 
 EXERCISE. 
 Compute the duty on: 
 
 1. 246 pc. plate glass, 16" X 24". 
 
 2. 7600 ft. bar iron, weighing 1.2 lb. per foot, @ 6/lOc per ft. 
 
 3. Three eases of fire crackers, weighing respectivelySlS, 129, 142 lb., 
 including all packing material, @ 8c per lb. 
 
 4. 1520 short tons of barley, @ 30c per bu. of 48 lb. 
 
 5. 320 bx. Castile soap, averaging 110 lb. per bx., allowance for tare 
 
 5%, 
 
 6. 22 long tons of lime. 
 
 7. 60 gr. bottles of mineral waters @ 20c per doz. 
 
 8. Find the entire duty on this invoice. 
 
 Sheffield, Nov. 17, 191- 
 Invoice of Hardware Purchased. 
 By James Evanston of Baltimore, Md, 
 From The Peter Mfg. Co. of Sheffield. 
 To he shipped per Cambria. 
 
 Marks. Quantity. Description. 
 
 Price. Amount. Consular 
 Corrections. 
 
 ^68 
 <$>69 
 ^70 
 
 6 
 
 160 
 
 42 
 
 gr. Files, 
 lb. Bolts, 
 lb. Wood screws, 
 
 Total. 
 
 U. S. Money. 
 Duty on files, 25c doz. 
 Duty on bolts, l^c lb. 
 Duty on screws, 8c lb. 
 
 £ 
 
 s 
 3 
 1 
 1 
 
 d 
 
 6 
 9 
 
 £ 
 
 XX 
 
 X 
 
 X 
 
 s 
 
 X 
 X 
 X 
 
 d 
 
 X 
 X 
 X 
 
 
 Ad Valorem Duties. 
 
 The computation of ad valorem duties often involves the 
 reduction of money values to U. S. gold standard, and always 
 involves percentage. 
 
 Illustration. Compute the duty (40%) on forty watch cases invoiced 
 @ 36 fr. 
 
 Solution. The value of forty watch cases = 40X36 fr. = 1440 fr. 1 fr. 
 = $.193. 1440 fr. = 1440 X. 193 = $277.92. The dutiable value is $278. 
 The duty = 40% of $278 = $111.20. 
 
284 
 
 BUSINESS ARITHMETIC. 
 
 EXERCISE. 
 
 Compute the duty on: 
 
 1. An invoice of London purple, valued at £46 8s. 
 
 2. 8 doz. pen knives @ 0/1/6 per doz. 
 
 , 3. 40 head of horses from Canada, @ $135. 
 
 4. An invoice of crockery, 675 fr. Duty 60%. 
 
 5. A collection of orchids from Brazil, @ 526 ml. Duty 25%. 
 
 6. An invoice of swords, valued at $4264.14. Duty 50%. 
 
 7. A set of "standard authors" valued at £64. Duty 25%. 
 
 8. Compute ad valorem duty at 60% on this invoice: 
 
 Mme. S. Ren ant. 
 Robes Manteaux Modes. 62. Place du TrocaMro 
 
 Madame C. Fanning, Dr. 
 
 Paris, le 18 Novembre, 191— Fr. 
 
 
 2 
 
 Robes, garni motif 
 
 775 
 
 
 
 
 Robe, en foulard 
 
 310 
 
 
 
 
 Robe, en draps bronze, Princesse 
 
 400 
 
 
 
 
 Robe, drap gris, brader 
 
 526 
 
 
 
 
 Blouse, en crepe de chene blanc 
 
 92 
 
 
 
 
 Blouse, taffeta bleu 
 
 65 
 
 
 Note. Both ad valorem and specific duties are levied on many imports. 
 9. A rug, 6' X 8', valued at 146 fr. (see tariff). 
 
 10. 86 lb. celluloid manufactures, valued at 756 fr. Duty 65c per lb. 
 and 30%. 
 
 11. 1200 m. oil cloth, 2 yd. wide, @ 2.4 M. per m. Duty 6c sq. yd. 
 and 15%. 
 
 12. Extend this form: 
 
 Manifest No. 868. Invoiced at Amsterdam, Netherlands, Nov. 29, 19 — . 
 
 Inward Foreign Entry of Merchandise. 
 Imported by J. J. Jervis In the steamer Kriedam 
 
 A. B. Roe Master From Amsterdam. Arrived Dec. 15, 19 — 
 
 No. 
 
 920-1 
 922-3 
 
 Packages and contents 
 
 2 cas. Brussels carpet 
 
 3/4 yd. wide. 
 2 cas. Tap. 3/4 yd. 
 
 wide. 
 
 40% of $ 
 
 ? yd. @ 44c 
 ? yd. @ 40c 
 
 Quan- 
 tity 
 
 Free 
 List 
 
 44^8q.yd. 
 and 40^ ad 
 valorem 
 
 40^Bq. yd. 
 and 405t ad 
 Talorem 
 
 Duty 
 
 400m. 
 420m. 
 
 
 600 fl. 
 
 $? 
 
 840 fl. 
 
 $? 
 
 $? 
 ? 
 ? 
 
 
 
 Total 
 
CHAPTER XXXV. 
 
 INTEREST. 
 
 Interest ranks with the fundamental processes, and with 
 simple percentage, as one of the most useful branches of 
 arithmetic. It has more general and varied applications than 
 simple percentage, and its use is being constantly extended. 
 It is the great interpretative branch of arithmetic and has to 
 do, especially, with the study of values. 
 
 INTRODUCTORY EXERCISE. 
 
 1. Give business reasons for borrowing money. 
 
 2. Need a prosperous business ever borrow money? 
 
 3. Why is money paid for the use of others' money? 
 
 4. I owe Smithson $500, due to-day. If I delay payment for a month, 
 does Smithson gain or lose? Why? 
 
 5. Does the time I keep another's money affect the charge for its use? 
 
 Interest has double meaning. Strictly, it is the use of money. 
 Commonly, it is the sum paid for that use. The sum on which 
 interest is charged is the principal. Interest is computed as 
 a per cent, of this principal per year, with relatively less 
 amounts for portions of a year. Each state has a legal rate 
 of interest, used when no rate is stated in interest-bearing 
 obligations. By agreement, a lower rate may be charged; or 
 in some states, a higher rate up to a maximum fixed by law 
 may be charged. A requirement to pay higher than the legal 
 maximum is termed usury. 
 
 Commercially, the interest year is reckoned as 360 days, 
 or 12 months of 80 days each. If, however, the time is a year 
 or more, calendar years are taken for the full years. The 
 United States Government, and some banks, reckon interest 
 
 285 
 
286 BUSINESS ARITHMETIC. 
 
 on a basis of 365 or 366 days. Interest tables commonly are 
 used to save computation. For long periods, interest time is 
 computed by compound subtraction; for short periods the exact 
 number of days is used, as a rule, this depending, however, on 
 the custom of the business or the character of the obligation. 
 
 ORAL EXERCISE. 
 
 1. Determine the interest on $6000 for one year at 6%. At 8%. 
 
 2. Find the interest on $1200 for 1 year at 5% ; for 2 years ; for 6 months; 
 for 2 months. 
 
 3. What part of the interest for a year is the interest for 3 mo.? 2 mo., 
 4 mo., 60 da., 90 da., 80 da., 120 da , 30 da.? 
 
 4. Determine the interest on $800, at 4%, for 1 yr., 3 mo., 4 yr., 2 mo., 
 90 da., 6 mo., 30 da. For a fixed rate and principal, how is interest affected 
 by a change of time? 
 
 5. How does interest at 4% compare with interest at 5%, on the same 
 principal? 
 
 6. If the interest for a fixed time, on a given principal is $32, at 4%, 
 find the interest at 5%, at 6%, 2%, 8%, 1/2%, 4|%. What is the 
 effect of increasing or decreasing the rate, if principal and time remain 
 constant? 
 
 7. The interest on a given principal, at a fixed rate, for 20 days, is 
 $1.60. Find the interest for 5 da., 10 da., 30 da., 40 da., 2 mo., 80 da., 11 da., 
 3 mo. What is the effect of change of time? 
 
 8. If rate and time remain unchanged, how will the increase or decrease 
 of principal affect the interest? If the interest on $1200, for a certain 
 time, is $36, determine the interest on $900, $800, $700, $6000, $720, $400. 
 
 Most practical methods of computing interest depend on the 
 principles just emphasized, are closely related, and are based 
 on a year of 360 days. It is best to understand several 
 methods and to select for general use the one that seems most 
 natural and simple. 
 
 I. CANCELLATION METHOD. 
 
 (Based on 360 or 365 days to the year.) 
 
 Illustration. Example. Compute the interest on $720 for 84 daya 
 at 6%. 
 
INTEREST. 287 
 
 Analysis and Solution. The interest for 1 year is 6/100 of $720. Fol 
 84 davs, it is 84/360 of this amount. 
 
 2 
 
 4720rX. 06X84 
 
 ■360- 
 1 
 
 $2 X. 06X84 = $10.08. 
 
 EXERCISE. 
 
 Compute the simple interest on: 
 
 1. $720 for 36 da. at 8%. 5. $1260 for mo. 15 da. at 6%. 
 
 2. $925 for 2 mo. 6 da. at 6%. 6. $829.60 for 39 da. at 4%. 
 
 3. $456.50 for 8 mo. at 7^%. 7. $3600 for 20 da. at 5%. 
 
 4. $862.85 for 7 mo. 3 da. at 6%. 8. $240 for 18 da. at 9%. 
 9. $1200 from Jan. 5 to Nov. 21 at 6%. (Approximate time.) 
 
 10. $12,900 from Aug. 21 to Dec. 3, at 4%. (Approximate time.) 
 
 11. $270 from Mar. 13 to May 24 at 7%. (Exact time.) 
 
 12. $1200 for 75 da. at 5% 
 
 13. $320 for 3 mo. 5 da. at 8%. 
 
 II. SIXTY-DAY METHOD. 
 The sixty-day, or bankers' method, is one of the most effec- 
 tive in use. By it, interest is first computed at 6% and then 
 reduced to the required rate. 
 
 INTRODUCTORY EXERCISE. 
 
 1. If the interest rate is 6%, what per cent, of the principal is the 
 interest for 6 mo.? 2 mo.? 8 mo.? 1 mo.? 
 
 2. What per cent, for 60 da., 30 da., 90 da., 20 da.? 
 
 3. For what period, in months, is the interest 1% of the principal? 
 For what period in days? 
 
 4. What is the simplest method of finding 1% of a number? 
 
 5. Find the interest at 6%, for 60 days, on $1200, $800, $300. 
 Illustrative Example. Compute the interest on $7200 for 96 days 
 
 at 6%. At 3%. 
 
 Solution. The interest for 60 days at 6% equals 1% of the principal. 
 $7200.00 = the principal. 
 
 72.00 = interest for 60 da. (1% of principal) 
 
 36.00= " " 30 " (30 da. = 1/2 of 60 da. .'. interest equals 1/2 
 
 of interest for 60 da.) 
 7.20= " " 6_ " (6 da. = 1/10 of 60 da.) 
 $115.20= " " 96 " Ans. (6%) 
 The mterest at 3% = ^he interest at 6% or $57.60. 
 
288 
 
 BUSINESS ARITHMETIC. 
 
 Example 2 Compute the interest on $840 for 4 mo. 24 da. at 6%. 
 Solution. 
 
 $840 =the principal. 
 
 8 40 = interest for 2 mo. (Use a line to mark decimal point.) 
 
 8 40= " 
 
 2 80= " " 20 da. (20 da. = 1/3 of 2 mo.) 
 
 56= " " ^^( 4 da. = 1/5 of 20 da.) 
 
 " 4 mo. 24 da. 
 
 $20 
 
 16 = 
 
 $ 8 
 
 40 = 
 
 16 
 
 80 = 
 
 2 
 
 10 = 
 
 
 84 = 
 
 
 42 = 
 
 $20 
 
 16 = 
 
 Check. Examples may be checked by changing time components. 
 Illustrations. (Checks on example 2 ) 
 4 mo. 24 da. = 144 da. 
 (1) 
 
 interest for 60 da. 
 
 " 120 da. (Twice interest for 60 da.) 
 
 " 15 " (1/4 mterest for 60 da.) 
 
 " 6 " (1/10 " '' 60 ." ) 
 
 " 3 '' (i " " 6 da. ) 
 
 (2) 
 $ 8 140 = interest for 60 da. 
 
 21100= " " 150 " (2f times interest for 60 da.) 
 |84= " " 6 " 
 $20116= " " 144 "~ (By subtraction.) 
 Note. This method substitutes simple addition and division for more 
 complicated operations, and offers positive check solutions. If times are 
 large, 600 days may be used as a unit time in place of 60 days; if the time 
 is small, three places may be pointed off in order to find interest for a unit 
 time of 6 days. 
 
 It is generally advisable to select such sub-times that the interest at each 
 step may be found from the previous step, or from the unit. See ex. 1, 
 above. 
 
 ORAL EXERCISE. 
 
 Subdivide these times into parts, and also suggest a "check" series: 
 
 1. 84 da. 5. 5 mo. 7 da. 9. 3 mo. 21 da. 
 
 2. 90 da. 6. 87 da. 10. 48 da. 
 
 3. 11 da. 7. 4 mo. 15 da. 11. 37 da. 
 
 4. 21 da. 8. 7 mo. 3 da. 12. 16 da. 
 
 ORAL EXERCISE. 
 
 Compute the interest at 6% on: 
 
 1. $1200 for 60 da. 
 
 2. $ 840 for 30 da. 
 
 $ 960 for 6 da. 
 $ 120 for 20 da. 
 
5. $ 600 for 3 da. 
 
 9. $ 884 for 30 da. 
 
 6. $640 for 45 da. 
 
 10. $ 390 for 2 da. 
 
 7. $ 900 for 2 mo. 6 da. 
 
 11. $ 900, 2 mo. 20 da. 
 
 8. $ 126 for 600 da. 
 
 12. $1600 for 15 da. 
 
 Compute the interest on: 
 
 
 13. $300, 4 mo., 5%. 
 
 15. $800, 45 da., 4^%. 
 
 14. $240, 30 da., 7%. 
 
 
 EXERCISE. 
 
 Compute the interest on the following, checking by change of time: 
 Principal. Time. Rate. Principal. Time. Rate. 
 
 1. $ 542.50 17 da. 6%. 5. $2164 23 da. 6%. 
 
 2. $3690 97 da. 5%. 6. $ 980 3 mo. 8 da. 6%. 
 
 3. $ 452.80 48 da. 4%. 7. $ 452.75 5 mo. 6 da. 6%. 
 
 4. $ 725 39 da. 6%. 8. $3726. 6 mo. 13 da. 4%. 
 Compute the amount, principal plus interest, on the following: 
 
 9. $426 11 da. 6%. 13. $1268 132 da. 5%. 
 
 10. $ 300 25 da. 8%. 14. $ 282 90 3 mo. 17 da. 6%. 
 
 11. $1285 34 da. 6%. 15. $ 562.80 1 mo. 5 da. 6%. 
 
 12. $ 950 68 da. 6%. 16. $1262.50 7 mo. 15 da. 7§%. 
 Using approximate time, compute the interest on: 
 
 17. $9650 from Jan. 27 to Oct. 19, at 6%. 
 
 18. $248.20 from Nov. 12 to Jan. 13, at 8%. 
 
 19. $3242.90 from Mar. 28 to Dec. 19, at 6%. 
 Using exact time, compute the amount of: 
 
 21. $1220 from Mar. 22 to Apr. 14, at 6%. 
 
 22. $356 from Feb. 12, 1909, to June 14, 1909, at 7%. 
 
 In order to simplify computation, the numerical value of 
 the principal and time (in days) may be interchanged without 
 affecting the resulting interest. In finding the amount, 
 however, it is necessary to remember to add the original 
 principal. 
 
 Illustration. The interest on $120 for 17 days is the same as that 
 of $17 for 120 days. Thus in the first case, the factors are 
 
 120 X 17 X. 06, 
 360 
 
290 ^ BUSINESS ARITHMETIC. 
 
 and in the second case, they are simply interchanged in order, 
 
 17 X 120 X .06. 
 360 
 Example. Compute the interest on $72 for 185 da. 
 Solution. Read the example: Compute the interest on $185 for 72 da. 
 $1.85 = interest for 60 da. 
 .37 = " " 12 " 
 $2 22= " " 72 " on $185, or for 185 da. on $72. 
 
 ORAL EXERCISE. 
 
 By interchanging terms, compute interest at 6%, on: 
 
 1. $60 for 29 da. 4. $120 for 38 da. 7. $600 for 71 da. 
 
 2. $20 for 96 da. 5. $15 for 92 da. 8. $150 for 42 da. 
 
 3. $30 for 47 da. 6. $12 for 132 da. 
 
 EXERCISE. 
 
 Compute interest and amount on: 
 
 1. $126 for 197 da. at 6%. 3. $3000 for 179 da. at 6%. 
 
 2. $540 for 83 da. at 8%. 4. $ 450 for 168 da. at 6%. 
 
 III. SIX PER CENT. METHOD. 
 In case the interest term is greater than one year, the 6% 
 method is frequently used. Some prefer it, also, for short 
 terms. 
 
 INTRODUCTORY EXERCISE. 
 
 1. The interest on $1.00 for one year at 6% = ? 
 
 2. The interest on $1.00 for 2 mo. = ? . Find the interest for 30 da., 
 6 mo., 4 mb. What is the interest on $1.00 for any multiple of 2 mo. or 
 60 da.? 
 
 3. The interest on $1.00 for 6 da. = ? . How may it be found for 2 da., 
 6 da., for any multiple of 6 da.? 
 
 4. The interest on $1 .00 for 4 mo. = ? . From it, determine the interest 
 on $400 for the same time. 
 
 Theoretically, the 6% method consists in applying the 
 60-day method to building up the interest on $1.00, multiply- 
 ing the result by the number of dollars. For practical use, 
 the following values should be kept in mind : 
 
INTEREST. 
 
 291 
 
 $.06 = interest on $1.00 for 1 yr. at 6%. 
 
 .01 = " " " " 2 mo. " " 
 or .005 = " " " " 1 " " " 
 
 .001 = " " " " 6 da. " " 
 or .000J= " " " " 1 " " " 
 
 Illustration. Example. Compute the interest at 6% on $700, 
 for 1 yr. 7 mo. 18 da. 
 
 Solution. $.06 = interest on $1.00 for 1 yr. 
 
 .035= " " " " 7mo. (S^Xint. for2mo.). 
 ■003= " " " " 18da. (3Xint. for6da.). 
 $.098= " " " " 1 yr. 7 mo. 18 da. 
 700 X $.098 = interest on $700 for given time = $68.60. 
 
 Note. Interest at other rates may be obtained by ratio, applied either 
 to the interest on $1.00, or to the .total interest. Thus 2/3 of either $.098, 
 or of the product, $68.60, equals the interest at 4%. 
 
 ORAL EXERCISE 
 
 Find the interest on $1.00, at 6%, for: 
 
 1. 88 da. 5. 29 da. 
 
 2. 92 da. 6. 62 da. 
 
 3. 17 da. 7. 54 da. 
 
 4. 14 da. 8. 2 da. 
 Compute the interest on: 
 
 13. $200 for 24 da. 
 
 14. ♦ $1000 for 3 mo. 6 da. 
 
 9. 
 10. 
 
 1 mo. 9 da. 
 
 2 mo. 27 da. 
 
 11. 3 mo. 14 da. 
 
 12. 1 yr. 4 mo. 11 da. 
 
 15. $500 for 2 yr. 6 mo. 12 da. 
 
 16. $400 for 96 da. 
 
 EXERCISE. 
 
 Compute the interest on: 
 
 
 
 
 Principal, 
 
 Time. 
 
 
 Rate. 
 
 1. $60432 
 
 1 yr. 2 mo. 
 
 8 da. 
 
 6%. 
 
 2. $ 98.25 
 
 2 yr. 5 mo. 
 
 3 da. 
 
 6%. 
 
 3. $ 7294 
 
 3 yr. 1 mo. 
 
 15 da. 
 
 6%. 
 
 4. $ 520 
 
 8 mo. 
 
 18 da. 
 
 3%. 
 
 5. $ 2460 
 
 lyr. 
 
 16 da. 
 
 7%. 
 
 6. $ 590.84 
 
 1 yr. 4 mo. 
 
 12 da. 
 
 5%. 
 
 7. $ 1580 
 
 3 mo. 
 
 16 da. 
 
 5%. 
 
 8. $ 2050.40 
 
 5 mo. 
 
 13 da. 
 
 6%. 
 
292 BUSINESS ARITHMETIC. 
 
 Principal. Time. Rate. 
 
 9. $ 840 7 mo. 24 da. G%. 
 
 10. $ 1292 2 yr. 5 mo. 20 da. 6%. 
 
 11. $36,432, from Jan. 29, 1908, to Oct. 17, 1909, at 6%. 
 
 12-21. Solve by 6% method, examples 1 to 8 and 21-22, page (289). 
 Find the interest on the following by the 6% method and "check" by 
 the 60-day method: 
 
 22. $324 for 1 yr. 2 mo. 8 da. at 6%. 
 
 23. $98.40 for 2 yr. 6 mo. 15 da. at 8%. 
 
 24. $7250 for 3 mo. 21 da. at 4 %. 
 
 Find the amount of the following, using 60-day method and checking 
 by the 6% method: 
 
 25. $76.80 from Jan. 12 to Aug.. 17, at 6%. 
 
 26. $358.25 from Feb. 14 to Dec. 29 at 6%. 
 
 IV. TABLE METHOD. 
 Interest tables of many forms are in common use among 
 bankers and brokers. They usually require decimal division 
 of the principal and ordinary subdivision of time. The 
 following outline illustrates one common form. 
 
 Six Per cent. Interest Table. 
 
 Time. $10 $20 $30 $40 $50 $60 $70 $80 $90 
 
 1 da. .00167 .00333 .00500 .00667 .08333 .01000 .01167 .01333 .01500 
 
 2 da. 
 
 3 da. 
 
 1 mo. 
 
 2 mo. 
 
 Note. The practical tables give interest for each day from 1 to 30. 
 Such tables should be constructed with the utmost care and should read 
 to five or more decimal points, to avoid errors when applied to large 
 amounts. 
 
 Illustration op Error. Suppose an interest table is constructed to 
 show interest at 6% for twenty days, on amounts of $1.00 to $99.00. In 
 a three-place table the interest for $10.00 would be $.033; in a seven-place 
 table, $.0333333. Applying the table to determine interest on $100,000, 
 it is necessary to move the decimal four places to the right. By the small 
 table this would give $330; by the seven place, $333.33 — a diflference of $3.33. 
 
INTEREST. 293 
 
 EXERCISE. 
 
 1. Complete the table (p. 292) by computation, and from known values. 
 
 2. Show how, from the given times, to obtain the interest for any- 
 time. Find the interest on $10 for 17 da. (Suggestion, 17 da. = 10 da. 
 plus 7 da.), 23 da., 4 mo. 6 da. 
 
 3. Show how to use the table in finding interest on any principal, 
 (a) if a decimal of a given principal; (6) if an irregular amount. Find the 
 mterest on $40,000 for 1 da.; on $43 for 1 da. (Suggestion. $43 = $40+ 
 $3.) 
 
 4. Using the table just constructed, determine the interest in ex. 
 1-11 of the preceding exercise. 
 
 Exact Interest. 
 Exact interest, based on 365 days, may be computed by the 
 cancellation method, or it may be obtained from interest 
 computed on a 360-day basis. Thus 1 day's interest, on a 
 business basis, is 1/360 of a year's interest; on an exact basis, 
 it is 1/365. The difference is 1/360 - 1/365, or 1/73. That 
 is, exact interest is 1/73 less than common interest. 
 
 Illustration. Example. Compute exact interest on $450 for 84 da., 
 at 6%. 
 
 Solution. $4.50 = interest for 60 da. 
 1.50 = " " 20 " 
 .30 = " " _4 " 
 $6.30 = " " 84 " " 
 , $6.30 ^ 73 = $.09. 
 
 $6.30 - $.09 = $6.21, the exact interest. 
 
 EXERCISE. 
 Find the exact interest on: 
 
 1. $582.90 for 39 da., at 6%. 5. $2960 from Aug. 17 to Dec. 29, 
 
 2. $1254 for 87 da., at 5%. at 6%. 
 
 3. $632.96 for 215 da. at 6%. 6. $382.50 from Jan. 5 to June 26 
 
 4. $98.25 for 49 da., at 4%. at 6%. 
 
 Interest Problems. 
 
 Interest is a truer measure of profit and loss than simple 
 percentage, and is practically the common instrument for 
 
294 BUSINESS ARITHMETIC. 
 
 the computations incident to financial operations. The 
 problems that follow are selected at random to show common 
 uses of interest. 
 
 EXERCISE. 
 
 1. I lend James Brown $2400 on Jan. 6, 1909, which he repays with 
 interest on Aug. 15, 1910. What does he pay in settlement? What does 
 he pay for the use of the money? 
 
 2. I borrow $1365, on my written promise to pay in 120 days with 
 5% interest. At the end of the period, what is due in settlement? 
 
 3. What is the quarterly income from a $20,000 loan placed at 4|% 
 interest? 
 
 4. $824, if placed in a savings bank paying 3|% interest, will 
 amount to what sum in 6 months? 
 
 5. If money is worth 8% to a buyer, is it better for him to pay $880 
 cash, or to wait 3 months and pay $900? 
 
 6. Determine the per cent, of gain on a four months' investment of 
 $1280, which yields 5% interest. 
 
 7. If goods are bought for $420, is it better to sell them in three months 
 at 5% profit, or at 5% interest on cost? 
 
 8. I am thinking of buying a piece of property for $8750 which, it is 
 claimed, will rent for enough to yield 9% net interest on investment. What 
 sum must be cleared? If repairs, taxes, etc., are estimated at $240 per 
 year, what monthly rental should be charged? 
 
 9. A bill of crockery, $752.80, was bought Jan. 27, on 2 months credit. 
 It was not paid until June 17. What was then due? Interest, 6%. 
 
 10. I lent through a broker, for a term of 6 months : $500 at 5 % interest ; 
 $800 at 4:1%; $1200 at 5h%. The broker charged 1/4% interest as his 
 commission. Determine my net income. 
 
 11. If this account draws 6% interest, what is due on Aug. 10 of the 
 year following? 
 
 S. R. Ramey 
 
 191— 
 
 Jan. 17 
 
 300.— 
 
 
 
 Mar 15 
 
 286.— 
 
 
 
 July 12 
 
 125.— 
 
 
 
 Aug. 10 
 
 85.60 
 
 
 12. The Montrose County School Fund is invested in: 
 $10,000 City of Minneapolis 3^% bonds. 
 $40,000 Morristown Water Works bonds, bearing 4i%. 
 $120,000 Lewiston Gas Co. bonds, bearing 5%. 
 What is the annual income from the fund? 
 
INTEREST. 295 
 
 13. I owe a London exporter £266 5 s. 10 d., with interest at 6%* 
 from Mar. 12 to Jan. 8. What is due? 
 
 CREDIT PRICES. 
 ORAL EXERCISE. 
 
 1. Is it better for a merchant to sell an article for $400 cash, or for 
 $400 credit? 
 
 2. If he gives credit what does he lose? 
 
 3. Should a credit price be greater, or less, than a cash price? 
 
 4. If the worth of money to a merchant be known, how may a credit 
 price, equivalent to a cash price, be determined? 
 
 Note. While many merchants add arbitrary amounts to the cash 
 price when selling on credit, others use a carefully computed rate of interest 
 in determining credit prices. 
 
 Determine credit prices for the following: 
 Cash Price. Credit Term. Money Worth. 
 
 5. $ 8.40 2 mo. 6%. 
 
 6. 80.00 1 mo. 7i%. 
 
 7. 6.50 30 da. 12%. 
 
 8. 5.00 3 mo. 8%. 
 
 9. 240.00 75 da. 9%. 
 
 EXERCISE. 
 
 (Use 6% as a rate unless otherwise stated.) 
 
 1. Does it pay to buy flour on 5 months credit, at $6.50, if it can be 
 bought for $6.00 cash, and money is worth 8%? 
 
 2. What profit results from buying a $16.50 article on 5 months 
 credit, and immediately selUng it at the same price on 1 month credit? 
 
 3. What is a fair settlement for a bill of merchandise amoimting to 
 $1240 paid 3 months before its 4 months credit term is up? 
 
 Note. Subtract interest for period before due. This is the interest- 
 discount method. 
 
 TO DETERMINE RATE OF INTEREST. 
 INTRODUCTORY EXERCISE. 
 
 1. If the interest on $800 for 1 year is $36, what is the rate of interest? 
 
 2. If the interest on $600 for 3 months is $15, what is the interest for 
 a year. The year's interest is what per cent, of the principal? 
 
296 BUSINESS ARITHMETIC. 
 
 3. What is the interest on $1200 for 9 days at 6%? At 1%? $7.20 
 is the interest at what rate for the same time? 
 
 It is evident that if the interest for a year, on any principal, 
 is known, or can easily be obtained, the rate of interest is 
 determined by dividing the known interest by 1% of the 
 principal. If the time is irregular, it is usually easier to 
 measure the given interest by the interest on the given prin- 
 cipal, at 1% for the given time. 
 
 Illustration. At what rate does $720 earn $4 in 40 days? 
 Solution (1). 
 
 $ 4.00 = interest for 40 days. 
 36.00 = " " 360 " (1 yr.) 
 $36.00 -r- $720 = .05. 
 The rate is 5%. 
 Solution (2). The principal is $720. 
 $7.20 = interest at 6% for 60 da. 
 4.80 = " " 6% " 40 " 
 .80 = " " 1% " 40 " 
 $4 (given interest) -i- $.80 (interest at 1%) = 5. 
 The rate is 5%. 
 
 ORAL EXERCISE. 
 
 Principal and time being constant. 
 
 1. If the interest at 6% is $48, $60 is the mterest at ? %. 
 
 2. $3.60 is the interest at ? %, if $9 is interest at 2\%. 
 
 3. 84c is the interest at ? % if 16 c is the interest at 2%. 
 
 Find the rate of interest if 
 
 4. $1200 earns $72 in one year. 6. $400 earns $3 in 2 months. 
 
 5. $800 earns $20 in 6 months.. 7. $600 earns 36c in 12 days. 
 
 EXERCISE. 
 
 1. If the Russian Government sells its last 6% bond issue at 10% 
 below face, what rate of interest does it really pay on the funds that it 
 secures? 
 
 2. Discuss the investment value of a 6% investment running one year 
 with a 5% one running ten years. 
 
 3. The offer of $80 for the use of $1500 for 45 days is equivalent to an 
 offer of ? % interest. 
 
INTEREST. 297 
 
 4. $800 is loaned from January 15 to September 12 at 6%, and from 
 October 1 to January 1, at 5%. What annual rate is earned? 
 
 PROFIT AND LOSS MEASURED BY INTEREST. 
 INTRODUCTORY EXERCISE. 
 
 1. Is it better to allow 3 months or 6 months credit on merchandise 
 sold at a fixed price? 
 
 2. Is it better to earn 5% profit, or 5% interest, on an article that is 
 held 2 years before sale? On one held 6 months? 
 
 3. When does a rate of interest yield a higher profit than the same rate 
 of gain? 
 
 4. Which more clearly shows the real return from an investment — the 
 rate of interest, or the rate of gain? Why? 
 
 The quickness of sale, after purchase, greatly affects the 
 profits of a business. This is the period of "turn-over of 
 capital." The above exercise emphasizes, as in percentage, 
 the value of a rapid turn-over, and shows the mathematical 
 basis for the old business motto "Quick sales and small 
 profits." 
 
 EXERCISE. 
 
 1. A merchant invests in articles costing $2.40, which sell, on an 
 average, in 2 mo. 6 da., at 3% profit. What is his annual rate of income 
 on investment? 
 
 2. Property bought January 29, for $5000, and sold June 26 for $5400, 
 yields what rates of gain and interest? 
 
 3. A house, bought for $9600, is rented for $50 per month, except for 
 2 months of the year. Taxes are lj% on an assessed value of $8000, 
 and repairs cost $118.60. What is the per cent, of income on investment? 
 
 4. Is it better to make 5% interest or 3% profit on a 7 months in- 
 vestment. Illustrate. 
 
 5. A real estate dealer advertises as a 10% investment for $5000, a 
 house that rents for $50 per month. Taxes are 1^% on an assessed 
 value of $4200; msurance is 11% on $3000. Is allowance made for 
 repairs? 
 
298 BUSINESS ARITHMETIC. 
 
 TO FIND THE PRINCIPAL. 
 INTRODUCTORY EXERCISE. 
 
 1. What is the interest on $1 at 6% for 2 months? $4 is the interest 
 on what principal for the same rate and time? 
 
 2. What is the amount of $1 for 6 months at 10%. $525 is the amount 
 of how many dollars, same time and rate? 
 
 3. What is the measure and quantity measured in examples 1 and 2 ? 
 
 . -ri 1 . T» • • 1 Total interest ,. ^. , , v 
 
 4. Explam: Prmcipai = :j— ^ (for same tmie and rate). 
 
 Interest on 3pi 
 
 Illustration. (1) What principal earns $18.30 in 12 days at 6%? 
 Solution. 
 
 Int. on $1 for 12 da. is $.002. If $1 under given conditions, 
 
 $18.30 ^ $.002 = $9150. earns $.002, $18.30 represents the 
 
 The principal is $9150. earning of as many dollars as 
 
 $.002 is contained times in that 
 amount. 
 Illustration (2). $976.32 is the amount of what principal for 
 3 mo. 12 da. at 6%? 
 Solution. 
 
 The amount of $1 for 3 mo. 12 da. at 6% is $1,017.^ 
 $976.32 -^ $1,017 = $960. The principal is $960. 
 
 ORAL EXERCISE. 
 
 Find the principal. 
 
 Int. Time. Rate. 
 
 1. $30. 18 da. 6%. 
 
 2. $48. 6 mo. 8%. 
 
 3. $ 9.60 36 da. 6%. 
 
 7. What principal amounts to $550 in 2 years at 5%? 
 
 8. What principal amounts to $2080 in 6 mo. at 8%? 
 
 EXERCISE. 
 
 1. On January 1, 1911, a corporation must meet a debt of $25,000. 
 What sum set aside on October 15, 1909, at 4% will amoimt to the debt 
 when due? 
 
 2. What sum must be loaned at b\% to earn a quarterly income of 
 $180? 
 
 3. Capitalize at 6%, semi-annual earnings of $84,200. 
 
 Note To capitalize earnings or expense is to determine the capital, 
 or principal that, at the given rate, will produce the stated earnings. 
 
 Int. 
 
 Time. 
 
 Rate. 
 
 4. $3.20 
 
 24 da. 
 
 6%. 
 
 5. $9. 
 
 1 yr. 6 mo. 
 
 3%. 
 
 6. $2.40 
 
 3 mo. 
 
 8%. 
 
INTEREST. 299 
 
 4. Capitalize at 5% annual charges for rented buildings for which a 
 city government pays $18,600. 
 
 5. By borrowing at 4%, a city government might have how much 
 money with which to secure its own buildings, for the $16,500 it now pays 
 annually in rentals? 
 
 The present ivorth, or true value, of a future payment or 
 debt, is the principal that, at the " worth of money," will 
 amount to the payment when due. Present worth is some- 
 times used in place of interest-discount, in reckoning advance 
 payments, in comparing bids and investments. 
 
 Illustration. Compare the present value of a $960 cash ofifer and a 
 $964.25 credit price for 6 months, money being worth 6%. 
 Solution. 
 
 The amount of $1 for 3 months at 6% is $1,015. 
 $964.25 -=- $1,015 = 950. 
 The present value of the credit price is evidently $950. The credit 
 price is better for the buyer than the cash price. 
 
 Question. Why is the present worth method more accurate than the 
 interest-discount method? 
 
 EXERCISE. 
 
 1. Money being worth 8% to me, what do I gain by buying property 
 for $960, on 6 months credit, and immediately disposing of it for cash, 
 at the same price? 
 
 2. By present worth, at 6%, compare these bids for the construction 
 of a building: 
 
 C. Smith & Son, $12,000: 1/4, cash, 1/4, 6 mo.; balance, 1 yr. 
 James Andrews; $11,750: 1/3, 4 mo.; balance, 1 yr. 
 A. B. Norton, $11,500: 1/2, 4 mo.; balance 1 yr. 
 
 3. If money is worth 6%, what sum fairly settles, on June 15, a 
 payment of $1200 not due until August 20? 
 
 4. What sum settles in cash for a purchase of $4284, on terms of 1/4 
 cash, balance 90 days. Money is worth 5%. 
 
 TO FIND THE TIME. 
 INTRODUCTORY EXERCISE. 
 
 1. If 6c is the interest for 1 day, on a certain principal, 54c is the interest 
 for how many days? 
 
300 BUSINESS ARITHMETIC. 
 
 2. Find the interest of $720 for 1 day. 84c is the interest on the same 
 principal for how many days? 
 
 o T7« 1 • rr.- -J Total interest , . , , x 
 
 3. iixplam. Time, m days, = , . — fTT^ (same prm. and rate). 
 
 4. Show how to use one month, or one year, as a measure, in place of 
 one day. 
 
 Illustration. In what time does $840 earn $28 at 6%? 
 
 Solution. 
 
 The interest on $840 for 1 da. = 14c'. 
 
 $28 -h $.14 = 200. 
 
 The interest term is 200 da., or 6 mo. 20 da. 
 
 ORAL EXERCISE. 
 
 Find the time, using 1 day. Find the time using 1 month. 
 
 
 Prm. 
 
 Int. 
 
 Rate. 
 
 Prin. 
 
 Int. 
 
 Rate, 
 
 1. 
 
 $ 600 
 
 $1.50 
 
 6%. 
 
 5. $ 420 
 
 $ 6.30 
 
 6%. 
 
 2. 
 
 $1200 
 
 $6.00 
 
 9%. 
 
 6. $ 800 
 
 $90. 
 
 6%. 
 
 3. 
 
 $720 
 
 $1.32 
 
 6%. 
 
 7. $1200 
 
 $80. 
 
 5%. 
 
 4. 
 
 $3000 
 
 $7.50 
 
 6%. 
 
 8. $2000 
 
 $75. 
 
 9%. 
 
 9. 
 
 In what 
 
 time does 
 
 a princii 
 
 pal double itself 
 
 (simple int€ 
 
 Test) at 
 
 4%, 6%, 6%, 8%, 7%, 10%, 9%, 3%? 
 
 EXERCISE. 
 
 1. On January 25, $7200 is invested at 6% simple interest to remain 
 until it amounts to $7500. Find the interest period. 
 
 2. On what date will $3200, invested July 8, 1909, at 6%, amount to 
 $5000? 
 
 Periodic Interest. 
 
 Periodic interest is the interest on a principal, payable at 
 stated intervals, plus simple interest on overdue interest 
 payments. It can be legally enforced in but few states, but 
 it is collected indirectly by means of non-interest notes, given 
 for the regular payments, which draw simple interest when 
 overdue. 
 
 Illustration. What is due after one year and six months on a $1200 
 loan, at 6% interest, payable quarterly, nothing having been' paid? 
 
INTEREST. 301 
 
 Solution. 
 
 Interest on $1200 for 3 mo. = $18, the quarterly payment. 
 Interest on $1200 for 1 yr. 6 mo. = $108, the total sunple payments. 
 At the end of 1 yr. 6 mo., 
 
 The first payment is overdue 1 yr. 3 mo. 
 " second " " " 1 " " 
 " third ," " " 9 " 
 
 " fourth '" " " 6 " 
 
 u g£^i^ a u u 3 " 
 
 The sum of overdue payments 3 yr. 9 mo. 
 
 The total overdue interest on the several equal payments equals the 
 interest on one payment for the total time, or for 3 yr. 9 mo. 
 Interest on $18 for 3 yr. 9 mo. = $ 4.05 
 Regular interest payments, 108. — 
 
 Principal, 1200. 
 
 Amount due, $1312.05 
 
 EXERCISE. 
 
 1. On June 1, 1908, $3000 was lent at 5%, payable semi-annually. 
 Nothing having been paid, what sum is due in full settlement, on December 
 1, 1909? 
 
 2. Find the amount due in full settlement for a loan of $7000, at 6%, 
 payable quarterly, on which nothing has been paid for a term of four years? 
 
 3. Repeat example 2, on a basis of semi-annual payments. 
 
CHAPTER XXXVI. 
 
 COMPOUND INTEREST. 
 
 Compound interest is interest computed on the sum of a 
 principal and its due and unpaid interest. Interest must be 
 " compounded," or added to the principal, at stated intervals, 
 as quarterly, semi-annually, or annually. Other periods may 
 be fixed by agreement. While not recoverable by law, in 
 most states, the accepting of compound interest does not 
 constitute usury. If a new obligation is given at the end of 
 any of the fixed periods for the amount then due, the obli- 
 gation is binding. 
 
 Compound interest is used by savings banks, in reckoning 
 interest on deposits, by insurance companies and by business 
 men generally, in computing the cumulative value of invest- 
 ments at a fixed rate, where earnings are not withdrawn. 
 
 Illustration. $12,000 invested at 6%, and compounded semi-annually 
 for two years, will amount to what sum? 
 Solution. 
 
 $12,000 = original principal. 
 
 360 = interest for 6 mo. (3%). 
 12,360 = amount, end of 6 mo. 
 
 370.80 = interest for 6 mo. on last amount (3%). 
 
 12,730.80 = amount, end 1 yr. 
 
 381.92 = interest for 6 mo. on last amount (3%). 
 
 13,112.72 = amount, end 1 yr. 6 mo. 
 
 393.38 = interest for 6 mo., on last amount (3%). 
 
 $13,506.10 = final amount. 
 
 ORAL EXERCISE. 
 
 1. The interest for 4 years at 8%, compounded semi-annually, equals 
 the interest for ? semi-annual periods, at ? %. 
 
 2. In compounding quarterly for 8 years at 6%, there are ? interest 
 periods at ? % per period. 
 
 302 
 
COMPOUND INTEREST. 
 
 303 
 
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304 BUSINESS ARITHMETIC. 
 
 3. Compute the amount of a $4000 savings deposit, compounded 
 quarterly for one year, at 4% per annum. 
 
 EXERCISE. 
 
 1. A fund of $16,000 at 6%, compounded semi-annually, amounts in 
 5 years to $ — . , 
 
 2. $5000 invested Jan. 1, 1906, at 4%, compounded quarterly, will 
 amount to $ on March 21, 1907. 
 
 Note. Compute interest for fractional term on last full amount. 
 
 Compound interest is computed, as a rule, from tables similar 
 to the one on page 303. 
 
 Illustration. Find the amount of $2500 for 6 years at 6%, com- 
 pounded quarterly. 
 Solution. 
 
 In 6 years there are 4 X 6, or 24, interest periods. 
 The rate of 6% per year equals 1^% per quarterly period. 
 The problem is to find the amount of $2500 for 24 periods at 1^%. 
 $1.429503 = amount of $1.00 for 24 periods at 1^%. 
 2500 
 $3573.76 = amount of $2500 for 24 periods or 6 years. 
 
 It is evident that time and rate should be reduced to periods 
 and period rates, and that the amount of $1.00 for this number 
 of periods should be multiplied by the number of dollars in 
 the principal. If the number of periods is greater than 
 twenty-five, find the amount for twenty-five periods, and then 
 find the amount of this amount for the remaining periods. 
 
 ORAL EXERCISE. 
 
 Read from the table the amount of $1000 if compounded: 
 
 1. Quarterly at 6% per annum for 6 years. 
 
 2. Semi-annually, at 8% for 5 years. 
 
 3. Bi-monthly, at 6% for 3 years. 
 
 4. Every 4 months, at 9% for 6 years. 
 
 EXERCISE. 
 
 Compute the amount of 
 
 1. $5950, for 3 yr. 6 mo, at 5%, compounded semi-annually. 
 
COMPOUND INTEREST. 305 
 
 2. $12,600 for 6 yr. 6 mo., at 8%, compounded quarterly. 
 
 3. $7200, from July 1, 1904, to October 27, 1909, at 6%, compounded 
 semi-annually. 
 
 Compute the compound interest on 
 
 4. $1200 for 8 yr. 4 mo. at 6%, compounded quarterly. 
 
 5. $2500 for 2 yr. 6 mo., at 9%, compounded every 4 months. 
 
 6. $3250 from October 1, 1909, to November 10, 1913, at 5% com- 
 pounded 33mi-annually. 
 
 What sum must be invested at 
 
 7. 6%, compounded quarterly, to amount to $7500 in 6 years? 
 
 8. 8%, compounded semi-annually, to amount to $12,500 in 4 yr. 6 mo.? 
 
 9. 9%, compounded semi-annually, to amount to $8000 in 3 yr. 6 mo.? 
 
 
 21 
 
CHAPTER XXXVII. 
 
 LOANS AND PAYMENTS. 
 
 INTRODUCTORY EXERCISE. 
 
 1. What may lead one, in his private capacity, to borrow money? 
 
 2. Name objections to such borrowing. 
 
 3. What may cause a business man, or organization, to borrow? 
 
 4. Does a prosperous business ever need a loan? 
 
 5. When might a business borrow in order to pay cash for goods bought 
 on credit? 
 
 6. How may a loan spread the cost of an improvement, for which it 
 pays, over a long period? 
 
 7. What is the danger of lending money on a verbal promise to repay? 
 
 8. What business paper commonly is used as evidence of a loan, or of 
 a payment due in the future? 
 
 9. How else may loans be "secured"? 
 
 ORAL EXERCISE. 
 
 1. I borrow, to-day, $1200 for 3 mo. at 6%. I receive $ ?, but must 
 
 pay back $ ?, paying % ? for the use of the money. 
 Determine interest and amount due on these loans: 
 
 
 Loan. 
 
 Period. 
 
 Rate. 
 
 Loan. 
 
 Period. 
 
 Rate. 
 
 2. 
 
 $ 800 
 
 9 mo. 
 
 6%. 
 
 6. $2500 
 
 24 da. 
 
 6%. 
 
 3. 
 
 1200 
 
 36 da. 
 
 5%. 
 
 7. 540 
 
 30 da. 
 
 6%. 
 
 4. 
 
 400 
 
 5 mo. 
 
 9%. 
 
 8. 250 
 
 2yr. 
 
 4%. 
 
 5. 
 
 1500 
 
 2 mo. 
 
 4%. 
 
 
 
 
 EXERCISE. 
 
 1. What interest has accrued to date, at 4^%, on a loan contracted 
 Feb. 11 of last year, for $820? 
 
 2. On Mar. 18, I must meet a loan of $840, contracted Oct. 21 last, 
 and drawing 6% interest. If I have $520, what additional sum must 
 I raise? 
 
 306 
 
LOANS AND PAYMENTS. 307 
 
 3. Settlement of a 3% loan of $90,000, dated Jan. 20, is demanded 
 Jan. 25. What is due? 
 
 4. A 3% loan of $6000, payable on demand, is contracted on Feb* 
 ruary 16, and renewed on February 25 at 4%. Settlement is made, March 
 2, by paying $— . 
 
 5. My brokers lend for me, on a commission of 1/2% interest: 
 
 Jan. 19, to C. Warren, $900 at 6%. 
 Feb. 3, to M. C. Poston, $1250 at 5%. 
 Feb. 16, to R. Fallon, $1920 at 6%. 
 Determine my net interest on the following January 1. 
 
 PROMISSORY NOTES. 
 A promissory note is a written promise to pay a specified 
 sum of money, on demand, at a specified time, with or without 
 interest. It is a common evidence of debt. 
 
 Illustration. 
 
 %560xx Cincinnati, 0., Mar. 21, 19/5. 
 
 Sixty days after date I promise to pay 
 
 to the order of James Fielding 
 
 at 7S6 Fifth Ave 
 
 Five hundred sixty X2;/100 Dollars 
 
 Value received with Interest at ^ % per annum. 
 
 Robert C. Evans 
 No. 97.... Due. May W.... 
 
 Note. Robert C. Evans, who signs this note, is the maker; James 
 Fielding is the payee. If the note had read "On demand, I promise," 
 etc., it would have been payable whenever presented to Evans. 
 
 Notes are negotiable, if transferable by the payee to other 
 parties by endorsement of his signature on the back. To be 
 
 transferable, notes must read " Pay to , or order," 
 
 " Pay to the order of ," or " Pay to , or bearer." 
 
 If the words " or order," or " or bearer" are omitted the note 
 is not transferable. 
 
 The following are the common forms of endorsement: 
 (1) Endorsement in Blank. (1) James Fielding. 
 This transfers right of owner- 
 ship to the bearer. 
 
308 BUSINESS ARITHMETIC. 
 
 (2) Full Endorsement. (2) Pay to the order oi 
 Here James Fielding transfers Robert Anderson, 
 
 the note to Robert Anderson, James Fielding, 
 
 but since the expression "pay 
 to the order of" is used, 
 Anderson may transfer the 
 note to others. 
 
 (3) Qualified Endorsement. (3) Pay to the order of C. 
 The note is transferred by P. Rankin, without recourse 
 Fielding to Rankin, but the to me 
 
 former by use of the phrase James Fielding. 
 
 "without recourse" refuses to 
 
 agree to pay the note when 
 
 due, in case the maker fails to 
 
 pay. 
 
 The last legal holder of the note secures payment from the 
 maker at maturity. If the maker fails to pay, each endorser 
 in turn, unless his endorsement is qualified, may be held for 
 payment. In case of non-payment, the note may be given to 
 a notary public, who formally demands payment. If payment 
 is refused, he legally protests the note, issuing notice to each 
 indorser and thus binding them to payment. This protest 
 is a basis for legal action to compel payment. The amounts 
 collectible are the face, interest if any, and legal charges. 
 Non-interest-bearing notes draw interest after due. 
 
 Notes signed by two or more parties and reading " we jointly 
 promise," are termed joint notes, each party being held for 
 his proportional part of the note (see p. 314). If the note 
 reads " We jointly and severally promise," each signer is held 
 for full settlement. Some states hold joint notes as similar 
 to Joint and several notes. 
 
 Collateral notes contain a permit to enable the holder, in 
 case of failure to pay at maturity, to dispose of certain property 
 (collateral) of the maker, and placed in possession of the payee, 
 
LOANS AND PAYMENTS. 309 
 
 as security, at the time of giving the note. One form of 
 collateral note is shown on page 346. 
 
 When a loan is made on real estate, notes usually are given, 
 secured by a mortgage or deed of trust, under which the 
 property may be disposed of in order to meet payments. 
 Receipts for goods in warehouse or grain elevator likewise, 
 may be endorsed to the lender as security for loans. 
 
 The bonds of corporations are simply corporate notes, and 
 frequently are secured by mortgages. 
 
 A commercial draft is an order written by one person re- 
 questing a second person to pay a stated sum of money to the 
 order of the former, or to a third person. The parties to a 
 draft are the drawer, or maker, the drawee, or person drawn on, 
 and the payee. When accepted by the drawee, commercial 
 drafts usually are negotiable. Accepted drafts are termed 
 acceptances. 
 
 Illustration (1). Andrews sells Parker $646 worth of merchandise, 
 on terms of "60 day acceptance." He sends Parker an mvoice, accom- 
 panied by the draft shown on page 310. Parker, finding the goods satis- 
 factory, writes "Accepted" across the face of the draft and signs his name, 
 thus binding himself to pay the paper when due. He then returns the draft 
 to Andrews. By writing "Accepted, Payable at the First Nat. Bank, 
 Chas. Parker," Parker could have turned the settlement over to his bank. 
 Practically, the draft is now a note. 
 
 Illustration (2). Andrews might wish to transfer payment to 
 Henry Connors, a creditor of his. In that case he might make out the 
 above draft and transfer it by endorsement, or he might make out the 
 draft in the form shown on page 311. In the latter case he would prob- 
 ably send the draft to Connors and the latter would present it to Parker for 
 acceptance. 
 
 When drafts read "after sight," in place of "after date," the acceptance 
 must be dated, and maturity counts from that date. Drafts reading 
 simply "at sight," are payable at once without acceptance. They are 
 often drawn by merchants against customers, and are sent through a bank 
 for collection. If the terms of a sale read, "subject to draft after 30 days" 
 it is understood that a sight draft will be drawn at the end of the period. 
 
310 
 
 BUSINESS ARITHMETIC. 
 
LOANS AND PAYMENTS. 
 
 311 
 
312 BUSINESS ARITHMETIC. 
 
 Factors Affecting the Maturity of Negotiable Paper. 
 
 The wording of negotiable paper affects the date of maturity. 
 
 On papers reading " — months after date " and " days after 
 
 date" maturity is reckoned from the date of the note, the 
 former by approximate and the latter by exact time. A few 
 states still allow " three days of grace," after date of maturity, 
 before payment must be made. 
 
 If drafts read " after sight," or " at — days sight," the 
 maturity is reckoned from the date of acceptance. Thus the 
 draft on page 311 is payable 60 days from March 20. 
 
 After the calendar date of maturity has been reckoned, the 
 actual payment date must be determined with due regard to 
 the varying state laws concerning Sundays, holidays, etc. 
 Thus, in Pennsylvania, a note coming due Saturday is not 
 payable until the following Monday, while if Monday 
 should happen to be a holiday, it would become payable on 
 Tuesday. 
 
 Summary of Laws Relating to the Maturity op Negotiable 
 Instruments. 
 
 Note. These laws are in process of change, in many states. Ask a 
 banker for the latest regulation. 
 
 For states allowing days of grace, or for local laws, consult a local 
 banker or business man. 
 
 Requirement. State or Country. 
 
 Due Sunday or holiday, pay Ala., Ariz., Cal., Col., D.C., Fla., Hawaii, 
 
 next business day. 111., Ind., la., Kas., Ky., La., Me., Minn., 
 
 •Mo., Mont., N.H., N.M., Okla., S.C, 
 
 S.Dak., Tex., W.Va., Wis., Canada. 
 
 Due Sunday or holiday, pay Alaska, Ark., Del. (Kent and Sussex Co.), 
 
 preceding business day. Ga., Miss., Nev., Philippines, P.R., Cuba, 
 
 Mexico. 
 
 Due Saturday, Sunday or Conn., Idaho, Md., Mass., Neb., N.J., 
 
 holiday, pay next business N.Y., N.C., N.Dak., O., Oreg., Pa., R.I., 
 
 day. Mich., Tenn., Ut., Vt., Va., Wash., Wyom. 
 
LOANS AND PAYMENTS. 313 
 
 Holidays. 
 
 Day^ Country or State. 
 
 January 1. All states (except Mass. and Kansas), 
 
 Canada, Cuba, and U. S. possessions. 
 February 22. All states and U. S. possessions. 
 
 May 30. All states and possessions, except Ala., 
 
 Ark., Fla., Ga., La., Miss., N. C, Okla., S. 
 C. and Tex. 
 July 4. All states and U. S. possessions. 
 
 Labor Day (first Monday All states (except Md., Wis., Wyom.) 
 
 in September). and U. S. possessions, except Philippines. 
 
 Thanksgiving Da}'^ (by proc- All states and possessions. 
 
 lamation, usually last 
 
 Thursday in November). 
 December 25 (Christmas) . All states and U. S. possessions, Canada 
 and Cuba. , 
 
 Many states or counties have local holidays other than those named above. 
 
 EXERCISE. 
 Find the dates of maturity of this commercial paper, consulting the 
 above summary. Use a calendar and allow for hoUdays, etc. 
 
 Maturity in Your Own State 
 
 1. 90 day note, dated January 27. 
 
 2. 60 day note, dated Dec. 24. 
 
 3. "Three months after date" paper, dated Mar. 4. 
 
 4. "Sixty days after date" draft, dated June 14, accepted June 17. 
 
 5. "At 90 days' sight" draft, dated Aug. 31, accepted Sept. 3. 
 
 Maturity in Different States. 
 
 6. "60 days' sight" draft, dated May 4, accepted May 5, payable in 
 Minnesota. 
 
 7. A "90 days after date" draft dated September 3, accepted Sep- 
 tember 4, payable in Ohio. 
 
 8. A 4 months note date Mar. 4, 1909, payable in Texas. 
 
 9. A 120 day note, dated April 24, 1909, payable in Canada. 
 
 QUESTIONS FOR DISCUSSION. 
 
 1. As a rule, which allow a longer time for payment, notes given for 
 months, or for the same number of thirty day periods? 
 
314 BUSINESS ARITHMETIC. 
 
 2. Is thirty day paper ever better for the payer than one month paper? 
 
 3. Give reasons for allowing days of grace. Why are they being 
 abolished? 
 
 4. Ordinarily, do the regulations as to Sundays and holidays favor 
 payer or payee? 
 
 5. Why should the date of payment be fixed beyond question 
 
 EXERCISE. 
 
 1. What yearly saving in interest results from refimding at 3|%, 
 $52,500 worth of 5% bonds? 
 
 Note. Refunding means re-issuing instead of paying at maturity. 
 
 2. What sum should be paid on June 20, in full payment for a 90 day 
 non-interest note for $1200, dated February 11? Interest, 6%. 
 
 3. What sum will settle this note to-day : 
 
 $520xY?r Washington, D. C, August 12, 1909. 
 
 On demand, after date, we jointly promise to pay to 
 
 the order of C. M. Fairley 
 
 Five hundred, twenty ^ Dollars. 
 
 Value received, with interest at 6% per annum. 
 
 A. B. Morton, 
 Henry C. Mason. 
 
 4. I lend C. P. Rogers $2260, at 5%, taking a first mortgage on his 
 farm. What quarterly payment should he make? 
 
 At the end of two years, nothing having been paid, the mortgage is 
 foreclosed, and the property sold for $3675, the charges for the sale being 
 $61.90. What is due Rogers? 
 
 Income and Outgo. 
 
 5. An investment of $19,700 in 5% mortgages yields a semi-annual 
 income of $ — . 
 
 6. A broker lends for a customer $18,000 at 5%, charging a com- 
 mission of 5% of the first year's income. What is the client's net income 
 the first year? 
 
 7. On February 19, I borrowed $12,000, at 3i%, giving my demand 
 note. On March 3, 1 lent the money at 5%. On December 15, my note 
 being presented for payment, I called in my loan. Net gain? 
 
 Partial Payments. 
 Partial payments may be made, by agreement, before or 
 after the maturity of a note or bond. Such payments should 
 
LOANS AND PAYMENTS. 
 
 315 
 
 be endorsed on the back of the paper in one of the forms shown. 
 
 Face. 
 $1800.1^^ Washington, D. C, June 10, 1909. 
 
 One year after date I promise to pay 
 
 to the order of C. P. Crandall 
 
 Eighteen hundred Dollars. 
 
 With interest at 6% per annum. 
 
 Robert Davis. 
 
 Back. 
 Paid, July 12, 1909, $200. 
 Paid, Oct. 12, 1909, $300. 
 Paid, Mar. 10, 1910, $400. 
 
 Other forms of 
 
 Indorsement. 
 
 (1) 
 
 July 12, 1909, paid $200. 
 
 (2) 
 Rec'd, 7/12/09, $200. 
 (3) ^ 
 Rec'd on the within 
 note, July 12, 1909, three 
 hundred dollars. 
 
 Question. Why is no signature to an endorsement of a partial pay- 
 ment necessary? 
 
 The partial payment introduces several factors into the 
 computation of the amount due, the question arising as to 
 whether payments shall be applied to principal or to interest, 
 or to both. State laws govern the matter, but many states 
 have now legalized the United States rule, established by the 
 Supreme Court of the United States. The so-called 
 Merchants' rule is also frequently used. The United States 
 rule is very commonly used for long term paper. 
 
 INTRODUCTORY EXERCISE. 
 
 1. What is due at the end of 6 months on a 6% note for $1000? 
 
 2. If $230 is then paid, what sum will pay the interest due? What is 
 left to reduce the face? How much of the debt is still outstanding? On 
 what sum should interest now be paid? 
 
 3. What interest is due 6 months later? What amount? 
 
 4. If, now, $324 is paid, the new principal becomes $ . 
 
 5. The debt is finally settled a year later, by paying $ . 
 
 Illustration. United States Rule. 
 Example. Find the balance due at maturity, on the note shown above. 
 
316 BUSINESS ARITHMETIC. 
 
 Solution. 
 
 Face of note $1800. — 
 
 Interest, June 10 to July 12, 2 mo. 2 da 18.60 
 
 Amount due, July 12 1818.60 
 
 Payment, July 12 200.— 
 
 New Principal 1618.60 
 
 Interest on new principal, July 12 to Oct. 12, 3 mo 24.28 
 
 Amount due, Oct. 12 1642.88 
 
 Payment, October 12 300.— 
 
 New principal, balance due 1342.88 
 
 Interest on new principal, Oct. 12 to Mar 10, 1910, 4 mo. 28 da. . . 33.12 
 
 Amount due. Mar. 10 1376. — 
 
 Payment, Mar. 10 400.— 
 
 New principal, balance due 976. — 
 
 Interest from Mar. 10 to maturity, Jime 10, 3 mo 14.64 
 
 Amount due at maturity $ 990.64 
 
 Note. Under this rule, a payment is first applied to pay off interest, 
 and then to reduce principal. In case any payment is less than the interest 
 accrued, the principal remains unaltered untU such a date that the sum of 
 
 Eayments exceeds the interest accrued. For example, if the first payment 
 ad been $10, it would have been added to the following $300 payment, 
 and interest found on the original face from June 10 to October 12. 
 
 Question. What would be the result of using a payment jess than the 
 accrued interest? 
 
 EXERCISE. 
 Use approximate time for ex. 1-3; exact time for ex. 4-5. 
 
 1. Date of loan, June 26, 1908; amount, $3000; interest 6%. Pay- 
 ments: Aug. 15, 1908, $450; June 21, 1909, $800; Aug. 21, $300; July 16, 
 1909, $425. Fmd the balance due Oct. 21, 1909. 
 
 2. A two year note for $3760, dated Oct. 15, Paid, 11/8/11, $400 
 1911, and bearing 6% interest, has these endorsed " 12/9/11, $200 
 payments. What is due at maturity? " 7/3/12, $600 
 
 " 11/6/12, $ 20 
 " 1/7/13, $520 
 
 3. On Oct. 27, Robert Evans paid $1200 on a demand note for $3000, 
 dated Jan. 17, and bearing 6% interest. On Nov. 5, he paid $200, and 
 gave a new 3 mo. note for the balance due. Determine the face of the 
 new note. 
 
 4. Determine the amount due on this note at maturity. 
 
 $3500.— Baltimore, Md., July 15, 1911. 
 
 Fifteen months after date, I promise to pay to the order of 
 
 Elton Howells, thirty-five hundred Dollars. 
 
 Value received, with interest at 5%. Jambs Elwbll. 
 
LOANS AND PAYMENTS 317 
 
 Payments endorsed as follows: 
 
 Paid, Nov. 5, '11, $ 800. 
 " Mar. 17, '12, $ 250. 
 " Nov. 12, '12, $1500. 
 5. From this Loan Book record, find the balance due, Oct. 1, 1912, 
 at 3%: 
 
 Loan to C. P. Davidson, 
 NoJ Date. Amount. Date Payment. Amount Payment- 
 
 136 Oct. 1, '10 $40,000 Jan. 15, '11 $5,000. 
 
 Mar. 20, '11 10,000. 
 
 Aug. 21, '11 10,500. 
 
 Jan. 1, '12 3,000. 
 
 INTRODUCTORY EXERCISE. 
 
 1. What is due at maturity on a 6 months note for $1200, dated June 1, 
 and bearing 6% interest? 
 
 2. On Oct. 1, a payment of $600 was made. If this payment was 
 allowed to draw interest, to what would it amount when the note was due? 
 
 3. What is the difference, at maturity, between the value of the note 
 and the value of the payment? 
 
 The Merchant's rule is commonly used in banks and com- 
 mercial houses, especially for short term notes. 
 
 Illustration. Merchant's Rule. 
 
 Example. Determine the balance due on the note, page 315. 
 
 Solution. 
 
 Face of note $1800. — 
 
 Interest to maturity — 1 year 108. — 
 
 Amount due at maturity $1908. — 
 
 First pajonent $200. — 
 
 Interest on payment to maturity, July 12, 1909, to 
 
 June 10, 1910, 10 mo. 28 da 10.93 
 
 Second payment 300. — 
 
 Interest, October 12 to June 10, 1910, 7 mo. 28 da. 1 1 .90 
 
 Third payment 400.— 
 
 Interest, Mar. 10 to Jime 10, 3 mo 6. — 
 
 Total value of payments at maturity $928.83 928.83 
 
 Difference between value of note and of payments $ 979.17 
 
 Note. Under this method, the face draws interest until the date of 
 settlement, and each payment is allowed interest until the date of settle- 
 ment or maturity. The balance due is the difference at time of settlement, 
 between the value of the face and the value of the payments. 
 
318 
 
 BUSINESS ARITHMETIC. 
 
 EXERCISE, 
 Compute by both rules, the balance due March 29, 1912, on this 
 
 1. 
 
 note. 
 
 $2400.— Cincinnati, Ohio, Sept. 7, 1911. 
 
 On demand, after date, I promise to pay- 
 to the order of S. P. Campbell 
 
 Twenty-four hundred Dollars 
 
 with interest at 6% per annum until paid. 
 A. B. Norton. 
 
 Solve the following, using exact time. 
 
 Rec'd, Nov. 
 Rec'd, Jan. 
 Rec'd, Mar. 
 
 5/11, $300. 
 16/12, $580. 
 3/12, $627.50 
 
 Loan No. 
 
 Date 
 
 Name 
 
 Amount 
 
 Rate 
 
 Date Pay. 
 
 Am't 
 
 Settled 
 
 2 
 
 1911 
 Oct. 8 
 
 James Baker 
 
 $36,000 
 
 3% 
 6% 
 6% 
 
 Nov. 5/11 
 Dec. 21/11 
 Mar. 20/12 
 
 $4000 
 6000 
 8000 
 
 June 
 21/12 
 
 3 
 
 Oct. 11 
 
 Robert Jones 
 
 $4,500 
 
 Jan. 15/12 
 Mar. 12/12 
 Apr. 21/12 
 
 $ 400 
 1150 
 1200 
 
 July 
 17, '12 
 
 4 
 
 Oct. 15 
 
 Henry Sanders 
 
 $1850 
 
 Dec. 20/11 
 Jan. 15/12 
 Feb. 20/12 
 Apr. 14/12 
 
 $200 
 185 
 300 
 400 
 
 May 
 21/12 
 
 5-9. Solve all the examples given under the U. S. rule by the Mer- 
 chants' rule. 
 
 QUESTIONS FOR DISCUSSION. 
 
 1. Compare the illustrative examples for the two rules and describe 
 the different treatment of payments. 
 
 2. How are interest periods obtained under the U. S. rule? Under 
 the Merchants' rule? 
 
 3. Why does it make no difference, under the Merchants' rule, whether 
 the payments are greater or less than the interest? 
 
 4. Explain the reason for the difference in the final amount due under 
 the two rules. 
 
 Bank Discount. 
 A dealer buys "on credit" in order that he may dispose of 
 his goods, in part at least, before the bill for them becomes due. 
 He thus makes them "pay for themselves" and lessens the 
 amount of his necessary capital. The manufacturer, however, 
 pays cash for his heavy labor expense, and for some supplies. 
 
LOANS AND PAYMENTS. 319 
 
 Were he obliged to wait for payments from customers, while 
 paying out cash, he might be forced to carry extra heavy capi- 
 tal, on which he could not earn a fair return on investment. 
 To avoid this, he takes notes from his customers, or drafts 
 on them, and sells these to a commercial bank, at a slight 
 discount. This discount is computed as interest for the time 
 the note has to run. The bank collects the full amount at 
 maturity, from the maker, the discount representing its profit, 
 while the manufacturer, although selling on credit, gets the 
 immediate use of most of his money. 
 
 Commercial banks are chartered by law to receive and lend 
 money and to make collections. A large part of the income 
 of many banks is obtained by loans through discounts. The 
 bank's directors, or some designated officials, examine all 
 paper offered for discount, as to its makers, endorsers, or 
 security. Some banks discount only for their regular custo- 
 mers. In such cases, the loan, or proceeds, is placed to the 
 credit of the discounter, who "checks out" the money as 
 needed. The bank gains slightly from the extra use of the 
 deposited money. 
 
 QUESTIONS FOR DISCUSSION. 
 
 1. Of what advantage is it to a dealer to buy on credit, rather than for 
 cash? 
 
 2. What are the objections to selling "on credit"? 
 
 3. What is the disadvantage of buying goods for cash and selling on 
 time? 
 
 4. What is meant by "tying up" one's capital? 
 
 Discount Terms and Rules. 
 
 The term of discount is the time between date of maturity 
 and date of discount. A note or draft is discounted for its 
 "full term" when it is discounted for the time stated in it. 
 
 The amount or value at maturity is the face, plus interest, 
 if any. 
 
320 BUSINESS ARITHMETIC. 
 
 The discount is the sum deducted by the bank. 
 
 The proceeds is the sum credited, or paid, to the person 
 discounting. 
 
 Commercial banks vary greatly in their rules for discounting. 
 Some use exact time, others approximate time; some have a 
 minimum discount of 25c, etc. The general rules given below 
 may be modified to conform to local custom: 
 
 1. Discount is reckoned on the amount due at maturity. 
 
 2. Notes discounted for full terms are discounted for the 
 time stated in them. 
 
 3. Notes discounted for partial terms are discounted for the 
 exact number of days between the date of discount and date 
 of maturity. 
 
 Exception. When interest-bearing notes are discounted, at the same 
 rate as the interest rate, for a full term, the proceeds equal the face. 
 
 Illustrations. (1) NonAnterest note. Discounted for full term at 6%. 
 $840^ji Philadelphia, Pa., Jan. 27, 190— 
 
 Three months after date . . / . . promise to pay to the 
 
 order of James C. Leonard 
 
 at The Cross National Bank 
 
 Eight hundred forty Dollars 
 
 Value received with Interest at — % per annum. 
 
 Chas. R. CarleUm. 
 ^o. 89 . . . .Bue Apr. 27 
 Solution. 
 
 Discount term, 3 mo. $840.00= face. 
 
 Amount due at maturity, $840. — $ 8.40= discount, 2 mo. 
 Discoimt, for 3 mo. 12.60 4.20= discount, 1 mo. 
 
 Proceeds, $827.40 iri2.60= discount, 3 mo. 
 
 (2) The same note, discounted February 9. 
 Date of maturity, April 27. 
 Discount term, February 9 to April 27, 77 days. 
 
 Amount due, $840. — $840. — =base, for discount. 
 
 Discount, 10.78 $ 8.40 = discount for 60 da. 
 
 Proceeds, 
 
 $829.22 
 
 $ 1.40 = 
 
 (( 
 
 " 10 " 
 
 
 .84 = 
 
 « 
 
 a g « 
 
 
 .14 = 
 $ 10.78 = 
 
 
 (( -t ft 
 
 "Y7 " 
 
LOANS AND PAYMENTS. 
 
 321 
 
 (3) Interest note. Discounted for full term at 6%. 
 
 $784^f^ Cleveland, 0. 
 
 Sixty days after date ..../. 
 
 to the order of .... Cha^. Morgan & Co. 
 
 at. . . .1685 Payne Ave. 
 
 Seven hundred eighty-Jour 
 
 October U, 190— 
 . . .promise to pay 
 
 .Dollars 
 
 Value received with Interest at 6% per annum. 
 
 Henry T. Carmody 
 No. 9 Due Dec. IS 1685 Payne Ave. 
 
 Solution. See rule 3, exception. The proceeds equal $784. — 
 (4) The same note, discounted for full term at 8%. 
 Solution. 
 
 Face of note $784. — 
 
 Interest, 60 da., 6% 7.84 
 
 Discount term, 60 da. 
 $791.84 =base for discount. 
 
 Amount at maturity, $791.84 
 
 $ 7.9184 = 
 
 discount. 
 
 60 da. 6%. 
 
 Discount, 10.56 
 
 2.6394 
 
 (< 
 
 " 2%. 
 
 Proceeds, $781.28 
 
 $ 10.56 = 
 
 (( 
 
 " 8%. 
 
 (5) The same note. Discounted, 
 
 November 10, 
 
 at 6%. 
 
 
 Solution. 
 
 
 
 
 Date of maturity, December 12 
 
 >. 
 
 
 
 Discount term, Nov. 10 to Dec 
 
 :. 13 = 33da. 
 
 
 
 Amount due at ma- 
 
 $791.84 = 
 
 base for discount. 
 
 turity, as above, $791.84 
 
 $ 7.9184 = 
 
 discount, 
 
 60 da., 6%. 
 
 Discount, 4.36 
 
 3.9592 = 
 
 (( 
 
 30 " 
 
 Proceeds, $787.48 
 
 .3959 = 
 
 li 
 
 3 " 
 
 
 $4.36 = 
 
 tt 
 
 33 " 
 
 ORAL EXERCISE. 
 
 
 
 Find date of maturity. 
 
 Find time of discount. 
 
 Date 
 
 Date of Note. Term. 
 
 Date. 
 
 Term. 
 
 Discount. 
 
 1. Jan. 27. 2 mo. 
 
 6. Apr. 9. 
 
 60 da. 
 
 
 2. Jan. 31. 1 mo. 
 
 7. Feb. 17 
 
 1 mo. 
 
 , Mar. 6 
 
 3. Jan. 30. 60 da. 
 
 8. Mar. 12. 
 
 30 da. 
 
 Mar. 26. 
 
 4. Feb. 28. 4 mo. 
 
 9. Apr. 19. 
 
 60 da. 
 
 May 7. 
 
 5. Jan. 30. 6 mo. 
 
 10. July 14. 
 
 6 mo, 
 
 , July 29. 
 
 Find the discount and proceeds of these non-interest notes for full term 
 Face. Term of Note. Rate Discount. 
 
 11. $960 60 da. 6%. 
 
 12. $600 2 mo. 5%. 
 
 13. $420 30 da. 6%. 
 
 14. $840 60 da. 7%. 
 
322 BUSINESS ARITHMETIC. 
 
 Face. 
 
 Term of Note. 
 
 Rate Dis. 
 
 4 mo. 
 
 
 6%. 
 
 30 da. 
 
 
 6%. 
 
 12 da. 
 
 
 6%. 
 
 20 da. 
 
 
 6%. 
 
 Term of Note. 
 
 Date Dis. 
 
 8. 
 
 90 da. 
 
 (6%.) Mar. 17. 
 
 9. 
 
 90 da. 
 
 (6%.) July 6. 
 
 15. 
 
 16. $1200, discounted for 
 
 17. $ 840. 
 
 18. $ 920. 
 
 Face. Date 
 
 19. $ 600 Mar. 
 
 20. $1200 June 19 
 
 EXERCISE. 
 
 1. Discount 5% for full term, the draft on page 310. 
 
 2. Discount, at 7%, for full term, the draft on page 311. 
 
 3. Discount at 5%, for full term, the. note on page 307. 
 
 4. Discount, April 14, at 5%, the acceptance on page 311. 
 
 5. Discount February 19, 1912, at 6%, the note on page 316. 
 
 6. A merchant discounts on May 16, at 6%, the following notes and 
 acceptances. What sum should be placed to the credit of his account? 
 
 (a) A 3 mo. note for $725.60, dated Apr. 12. 
 
 (6) A 60 day sight draft, for $852, dated Apr. 20 and accepted 
 Apr. 23. 
 
 (c) A 4 mo. note for $192.68, dated May 16. 
 
 (d) A 30 day acceptance, for $365.42, dated May 15. i 
 
 ORAL EXERCISE. 
 
 Find the amount due at maturity on the following notes: 
 
 Face. Term. Rate Int. 
 
 2 mo. 3%. 
 
 60 da. 6%. 
 
 90 da. 4%. 
 
 30 da. 6%. 
 
 4 mo. 6%. 
 
 20 da. 6%. 
 
 90 da. 6%. 
 
 60 da. 5%. 
 
 Find the discount and proceeds of the following notes, bearing interest 
 
 at 6% and discounted at the same rate: 
 
 1. 
 
 $ 246. 
 
 2. 
 
 $ 88.: 
 
 3. 
 
 $ 900. 
 
 4. 
 
 $1260. 
 
 5. 
 
 $ 425. 
 
 6. 
 
 $ 960. 
 
 7. 
 
 $ 528. 
 
 8. 
 
 $ 720. 
 
LOANS AND PAYMENTS. 323 
 
 
 Face. 
 
 Term. 
 
 
 Term. Dia 
 
 9. 
 
 $ 600. 
 
 4 mo. 
 
 
 60 da. 
 
 10. 
 
 $1200. 
 
 60 da. 
 
 
 20 da. 
 
 11. 
 
 $ 900. 
 
 30 da. 
 
 
 30 da. 
 
 
 Face. 
 
 Date. 
 
 Term. 
 
 Date Dis. 
 
 12. 
 
 $ 840. 
 
 Jan. 6. 
 
 60 da. 
 
 Feb. 5. 
 
 13. 
 
 $1200. 
 
 June 5. 
 
 30 da. 
 
 June 25. 
 
 14. 
 
 $ 600. 
 
 Aug. 10. 
 
 EXERCISE. 
 
 60 da. 
 
 Aug. 30. 
 
 1. Find the discount and proceeds on the note on page 321 , if discounted 
 November 4, at 6%. If discounted October 27, at 6%. 
 
 2. Determine the proceeds of the same note, if discounted at 5% on 
 Nov. 3. 
 
 3. Find the proceeds of the following firm note, if discounted March 17, 
 
 at 5%: 
 
 $582.tV7 Kansas City, Mo., Mar. 5, 1913 
 
 Ninety days after date we. , . .promise to pay to 
 
 the order of_ _John Sheldon 
 
 at our office 
 
 Five hundred, eighty-tvjo y^^Dollars. 
 
 Value received with Interest at 6% per annum. 
 
 J. M. Brovmson & Bro. 
 No. 189 Due 
 
 4. James Devlin is obliged to pay his 90 day, 5% interest-bearing note, 
 for $960, on Nov. 19. He has only $564.30 to his credit in the bank 
 where his note becomes payable. On Nov. 18, therefore, he discounts 
 the note shown below at 6%. He also deposits checks for $82.67 and $91.80. 
 After his own note has been paid, what balance remains to his credit? 
 
 $468.tTnr Baltimore, Md., Nov. 18, 190— 
 
 Eighty days. after date I promise to pay 
 
 to the order of James Devlin 
 
 at ... . Third Nat. Bank 
 
 . . . .Four hundred sixty-eight xViyDollars 
 
 Value received with Interest at 5% per annum. 
 
 Chas. D. Warrenton. 
 No. 82 Due.... 
 
324 
 
 BUSINESS ARITHMETIC. 
 
 5. Extend this form: 
 
 Discount Ticket. 
 
 Third National Bank. 
 
 March 16,190. 
 
 NOTES AND DRAFTS LEFT FOR DISCOUNT. 
 
 By James Fielding 
 
 Note or 
 Draft. 
 
 Maker or Drawer. 
 
 Face. 
 
 840 — 
 
 1765 
 
 882 
 
 1206 
 
 Int. to 
 Maturity, 
 
 20 
 
 (6%) 
 (5%) 
 
 Amount. 
 
 30 d. n. S. P. Norton 
 
 60 d. dft. Johnson Bros. 
 
 4 mo. n. Roberts and Bro. 
 
 3 mo. n. Champlain Mfg. Co. . 
 
 Total, 
 
 Discount, @ 6% (full term) 
 
 Proceeds, 
 
 7. Complete the extension of the bank discount register on page 325. 
 ORAL EXERCISE. 
 
 1. If a non-interest note is discounted for its full term of 4 mo., at 
 6%, the proceeds equal what per cent, of the face? If the proceeds are 
 $980, what is the face? 
 
 2. A 60 day, 6%, interest-bearing note is discounted for 30 days at 
 6%. The proceeds equal what per cent, of the face? 
 
 3. Knowing the proceeds of a note, how may its face be determined? 
 
 EXERCISE. 
 
 Note. The per cent, method, suggested above, may be used, or the 
 proceeds of a dollar note, under the given conditions ,may be determined. 
 The proceeds divided by the proceeds of a dollar note will determine the 
 face. 
 
 1. What must be the face of a non-interest-bearing note for 3 mo., 
 that will yield $750 proceeds when discounted for its full term at 6%? 
 
 2. On January 17, 1 sold A. B. Norton a bill of merchandise amount- 
 ing to $762.50, less 20%. Terms, 10 days. At the end of the period, 
 Norton, being temporarily short of cash, offers his non-interest, 30 day 
 note for such an amount that when discounted for its full term at 6%, the 
 proceeds equal the face. This offer being accepted, the note should be 
 made out for what sum? 
 
 Individual Original Work. 
 Report on local business methods of computing discounts on notes, 
 drafts and business paper. Give local process of discounting. 
 
LOANS AND PAYMENTS. 
 
 325 
 
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CHAPTER XXXVIII. 
 
 SAVINGS ACCOUNTS. 
 
 "The best way to accumulate money is to resolutely save and bark a 
 fixed portion of your income, no matter how small the amount,' — The 
 WorWs Work. 
 
 INTRODUCTORY EXERCISE. 
 
 1. Name three good reasons for saving money from one's income. 
 
 2. If a journeyman carpenter saw a chance to become a contractor 
 on a small scale, he would have what reason for saving money? 
 
 3. Direct savings increase rapidly. A salesman, setting aside ten 
 
 cents per day, saves $ per month. An iron-worker, saving $3.50 per 
 
 week, has $ at the end of one year. It takes an office clerk, who is 
 
 saving 20% of his weekly salary of $12, years to save the $600 necessary 
 
 to secure an interest in a small independent business. 
 
 4. What are the dangers of simply "hoarding" one's money? 
 
 5. What difficulties lie in the way of investing small sums as they are 
 saved? 
 
 6. What advantages result from placing those small savings in a bank? 
 
 7. What rates of interest do your local banks and trust companies pay? 
 
 8. What is the smallest sum accepted by them to open an account? 
 What is the smallest sum accepted thereafter as a deposit? 
 
 9. Give reasons for one's withdrawing his deposit — (a) in prosperous 
 times; (6) in times of panic; (c) in times of business depression. 
 
 Practically all investment by wage-earners starts in the 
 savings bank, an institution chartered by the State for receiving 
 savings deposits, on which it pays interest. Savings banks 
 receive enormous deposits of individually small amounts. 
 A New York bank, in a poor district, has over $100,000,000 on 
 deposit. 
 
 The savings banks usually have the primary quality of 
 safety, and are more carefully regulated and guarded by law 
 than are other banking institutions. Most savings banks are 
 
 326 
 
SAVINGS ACCOUNTS. 327 
 
 partly or entirely mutual and are run, in such cases, for the 
 benefit of the depositors. A few are stock companies. Unless 
 run for the benefit of stock holders, the banks place the major 
 portion of their funds in safe mortgages, or in government, 
 or other "gilt edge" bonds. Often the state laws specify very 
 definitely the class of investments that may be made. It is 
 well to examine the laws, or to get reliable business advice 
 before depositing in any bank. If good banks are not within 
 easy reach, deposits may be sent to them by mail. 
 
 Many trust companies also accept savings deposits. They 
 generally hold deposits '^ subject to check," but are not 
 always as closely safeguarded as savings banks. 
 
 EXERCISE. 
 
 1. Eleven savings banks of Pennsylvania recently had 398,885 de- 
 positors, an average of ? per bank. 
 
 2. Complete this tabulation concerning the savings banks of the 
 country : 
 
 Year. No. Banks. Deposits. Depositors. Av. per Bank. Av. Depos. 
 
 1905 ' 1237 $3,093,077,357 7,693,229 $ $ 
 
 1910 1759. 4,070,486,247 9,142,908 $ $ 
 
 5 yr. increase ? ? ? ? 
 
 % of increase ? ? ? ? 
 
 3. Account, in some way, for these increases. 
 
 Most deposits are for long periods. Withdrawals are at 
 infrequent intervals, and are further restricted by the fact 
 that a bank may require, if it desires, 10 days to 60 days notice 
 before making a payment. This rule is seldom enforced except 
 in times of money stringency or sudden panic. On notice 
 of expected withdrawal, the bank begins to save incoming 
 deposits to meet this sum. It thus leaves practically its entire 
 savings receipts free for investments. Money that it cannot 
 immediately invest, it may lend "on call" (see p. 345), or lend 
 to commercial banks at 1% to 2% interest. 
 
 Whether mutual or stock organizations, savings banks 
 
328 
 
 BUSINESS ARITHMETIC. 
 
 usually pay from 3% to 5%, the average for the country 
 being 3f %. Interest is added to deposits at periodic in- 
 tervals. Since the period of compounding has some effect 
 on income, it is well to note this, as well as the method of 
 computing interest, before selecting a bank. The method is 
 generally stated in the rules of the bank. 
 
 EXERCISE. 
 
 (Use compound interest table, page 303.) 
 Find the amount for one year: 
 
 Of $400 compounded semi-annually at 4%. 
 
 Of the same sum compounded annually. 
 
 Of $364 compounded quarterly at 4%. 
 
 Of $942 compounded semi-annually at 3|%. 
 
 Find the amount of $368 for three years if compoimded annually 
 
 1. 
 2. 
 3. 
 4. 
 5. 
 at 3%. 
 
 Since, in most cases, the amount on deposit is constantly 
 changing, a special method is necessary for computing the 
 simple interest for each term. Many savings banks allow 
 interest on the smallest balance during any interest period. 
 Sums deposited early in a period draw no interest until the 
 next period, unless deposited the first day of the period. 
 Ffactions of a dollar do not draw interest. 
 
 ILLUSTRATION I. 
 Northern Savings Bank. 
 In account with James P. Norman. 
 
 Date. 
 
 Withdrawn. 
 
 Deposited. 
 
 Interest. 
 
 Balance. 
 
 190— 
 
 
 
 
 
 
 
 
 1 
 
 Jan. 
 
 1 
 
 29 
 
 10 
 
 
 
 
 
 
 320 
 310 
 
 Feb. 
 
 17 
 
 
 
 462 
 
 
 
 
 772 
 
 Mar. 
 
 2 
 
 90 
 
 50 
 
 
 
 1 
 
 68150 
 
 Apr. 
 
 1 
 
 
 
 
 
 3,10 
 
 684 
 
 60 
 
 
 4 
 
 
 
 40 
 
 
 
 
 724 
 
 60 
 
 June 
 
 28 
 
 160 
 
 
 
 
 
 
 564 
 
 60 
 
 July 
 
 1 
 
 
 
 
 
 5164 
 
 570 
 
 24 
 
SAVINGS ACCOUNTS. 
 
 329 
 
 Explanation. The deposits and withdrawals are entered in order of 
 dates. The rate in this bank is 4%, compounded quarterly, or 1% per 
 quarter. The first interest period extends from Jan. 1 to March 31. The 
 smallest balance on hand during the period is S3 10, from Jan. 29 to Feb. 17. 
 The interest is 1% of $310, or $3.10, which is added to the balance, as a 
 new deposit, on Apr. 1. The next interest period extends from Apr. 1 
 to June 30. The smallest balance is $564.60, on June 28, and the interest 
 is 1% of that balance, or $5.64. This interest is added July 1. 
 
 If compounded semi-annually, the smallest balance for the entire period 
 is $310, and the interest is 2%, or $6.20, a loss to the depositor, by change 
 of compounding, of $2.54. 
 
 Some savings banks reckon interest on monthly or quarterly 
 balances, but compound it, or add it to the deposit, only at 
 semi-annual, or annual, intervals. 
 
 ILLUSTRATION II. 
 
 Washington Savings Bank. 
 In account with Henry C. Brewster. 
 
 Date. 
 
 Withdrawn. 
 
 Deposits. 
 
 Interest. 
 
 Balances. 
 
 191— 
 
 
 
 
 
 
 
 
 
 
 July 
 
 1 
 
 
 
 
 
 
 
 560 
 
 
 Aug. 
 
 5 
 
 
 
 246 
 
 10 
 
 
 
 806 
 
 10 
 
 Sept. 
 
 8 
 
 26 
 
 
 
 
 
 
 780 
 
 10 
 
 
 19 
 
 
 
 129 
 
 80 
 
 
 
 909 
 
 90 
 
 Oct. 
 
 31 
 
 40 
 
 
 
 
 
 
 869 
 
 90 
 
 Dec. 
 
 14 
 
 
 
 37 
 
 43 
 
 
 
 907 
 
 33 
 
 Jan. 
 
 1 
 
 
 
 
 
 14 
 
 29 
 
 921 
 
 62 
 
 Explanation. Here assume interest at 4%, on quarterly balances, 
 compounded semi-annually. The lowest balance in the quarter ending 
 Sept. 30 is $560, and the interest is 1%, or $5.60. The lowest balance in 
 the fourth quarter, ending Dec. 31, is $869.90 and the interest is $8.69. 
 The total interest, compounded on Jan. 1, is $5.60-}- $8.69, or $14.29. 
 
 EXERCISE. 
 
 1. What is the effect of a heavy withdrawal just before the end of an 
 interest period? 
 
 2. Would Norman (illus. 1) have gained by delaying his withdrawal 
 of June 28, until July 1? 
 
 3. When would an additional withdrawal of $400 have caused the 
 greatest "interest" loss in Brewster's account? 
 
330 
 
 BUSINESS ARITHMETIC. 
 
 4. Determine the interest and final balance of Brewster's account, if 
 the interest is reckoned on monthly balances. 
 
 5. Balance this account. Rate 3%, reckoned and compounded 
 quarterly : 
 
 Thornton Savings Bank. 
 In account with John Kensington. 
 
 Date. 
 
 Withdrawals. 
 
 Deposits. 
 
 Interest. 
 
 Balances. 
 
 190— 
 
 
 
 
 
 
 
 
 
 
 Jan. 
 
 1 
 
 8 
 
 31 
 
 40 
 
 
 72 
 
 50 
 
 
 
 362 
 
 40 
 
 Mar. 
 
 3 
 
 
 
 80 
 
 45 
 
 
 
 
 
 May 
 
 19 
 
 162 
 
 65 
 
 
 
 
 
 
 
 June 
 
 12 
 30 
 
 40 
 
 
 321 
 
 
 
 
 
 
 6. Balance this account. Rate, 4%, reckoned on quarterly balances, 
 and compounded semi-annually. 
 
 Illinois Center Savings Bank. 
 In account with C R. Martin. 
 
 Date. 
 
 Withdrawn. 
 
 Deposited. 
 
 Interest. 
 
 Balances. 
 
 July 
 
 5 
 29 
 
 102 
 
 10 
 
 328 
 
 45 
 
 
 
 
 
 Aug. 
 
 15 
 
 
 
 107 
 
 38 
 
 
 
 
 
 Sept. 
 
 29 
 
 
 
 54 
 
 13 
 
 
 
 
 
 Nov. 
 
 10 
 
 28 
 
 127 
 
 69 
 
 24 
 
 63 
 
 
 
 
 
 Dec. 
 
 12- 
 18 
 
 
 
 132 
 
 46 
 
 31 
 50 
 
 
 
 
 
 7. Extend this account. Rate, 4%, reckoned on monthly balances 
 and compounded semi-annually. 
 
 Exchange Savings Bank. 
 In account with Robert S. Fenwick. 
 
 Date. 
 
 Deposits. 
 
 Interest. 
 
 Payments. 
 
 Balance. 
 
 Jan. 
 
 1 
 
 
 
 
 
 
 
 582 
 
 61 
 
 Mar. 
 
 3 
 
 45 
 
 
 
 
 
 
 
 
 May 
 
 16 
 
 312 
 
 62 
 
 
 
 
 
 
 
 June 
 
 21 
 
 
 
 
 
 46 
 
 58 
 
 
 
 Sept. 
 
 17 
 
 
 
 
 
 92 
 
 46 
 
 
 
 Oct. 
 
 20 
 
 545 
 
 20 
 
 
 
 
 
 
 
 Nov. 
 
 10 
 
 
 
 
 
 416 
 
 92 
 
 
 
 Dec. 
 
 21 
 
 38 
 
 80 
 
 
 
 
 
 
 
SAVINGS ACCOUNTS. 
 
 331 
 
 Note. Notice change in column headings. 
 
 8. Balance this "subject to check" trust company account. Interest, 
 2%, compounded semi-annually. $240 is on hand January 1, 1912. 
 
 Name c^oie^t ^lo. ^/uyj^itkyn 
 
 No. 
 
 72M^6 Address '/M29 Min^lmj^ S^^e. 
 
 Occupation ^C^2vtyucto7( 
 
 DATE 
 
 WITHDRAWN | 
 
 DEPOSITED 
 
 BALANCE 
 
 '/9i/2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ja^. 
 
 /7 
 
 
 
 
 
 
 
 
 2 
 
 3 
 
 2 
 
 4 
 
 M 
 
 
 
 
 
 
 
 
 2S 
 
 
 
 6 
 
 5 
 
 3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 JW. 
 
 /7 
 
 
 
 
 
 
 
 
 3 
 
 4 
 
 4 
 
 5 
 
 
 
 
 
 
 
 
 
 ^yna-tt. 
 
 s 
 
 
 / 
 
 M 
 
 2 
 
 5 
 
 4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2e 
 
 
 
 
 
 
 
 4 
 
 2 
 
 
 
 7 
 
 3 
 
 8 
 
 
 
 
 
 
 
 S^/c^. 
 
 A 
 
 
 
 2 
 
 9 
 
 6 
 
 V 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 44 
 
 
 / 
 
 3 
 
 2 
 
 5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 t^Ma^ 
 
 // 
 
 
 
 
 
 
 
 
 4 
 
 2 
 
 6 
 
 Jj- 
 
 3 
 
 
 
 
 
 
 
 ^. 
 
 8 
 
 
 
 
 
 
 
 
 
 Jf 
 
 6 
 
 5 
 
 3 
 
 
 
 
 
 
 
 
 1/6 
 
 
 
 M 
 
 s 
 
 5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 29 
 
 
 
 
 
 
 
 
 
 3 
 
 6 
 
 6 
 
 
 
 
 
 
 
 
 
 €cc. 
 
 47 
 
 
 
 
 
 
 
 
 2 
 
 5 
 
 9 
 
 / 
 
 
 
 
 
 
 
 
 
 
 23 
 
 
 / 
 
 / 
 
 4 
 
 6 
 
 3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 jVo^. 
 
 43 
 
 
 
 M 
 
 / 
 
 6 
 
 9 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 24 
 
 
 
 
 
 
 
 
 2 
 
 
 
 
 
 V 
 
 
 
 
 
 
 
 
 
 ^ec. 
 
 44 
 
 
 
 
 
 
 
 
 3 
 
 4 
 
 
 
 5 
 
 
 
 
 
 
 
 
 
 
 46 
 
 
 
 
 
 
 
 
 
 A 
 
 2 
 
 5 
 
 
 
 
 
 
 
 
 
 
 2S 
 
 
 
 8 
 
 / 
 
 / 
 
 6 
 
 
 3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 30 
 
 
 
 
 
 
 
 
 
 6 
 
 8 
 
 8 
 
 2 
 
 
 
 
 
 
 
 ^. 
 
 4 
 
 
 J. 
 
 t^ 
 
 ^« 
 
 i<r 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
CHAPTER XXXIX. 
 
 STOCKS AND BONDS. 
 
 The proprietor of a business who finds a need for increased 
 capital may secure it by taking into partnership the one who 
 suppHes it. But partnerships are often unsatisfactory, because 
 each partner is usually responsible for the acts of the others 
 and in most cases is liable for all debts. Each change of 
 partners, moreover, causes a new adjustment of interests and 
 often interrupts business. These objections increase with the 
 enlargement of a business and often seriously affect ease and 
 continuity of management. 
 
 The corporation meets many of the objections to a partner- 
 ship. It is an association of individuals recognized under the 
 law as a single artificial person. The capital, or capital stock, 
 of such an organization is divided into small equal amounts 
 called shares. A single person may hold one or more shares. 
 Many individuals may be interested in large companies. 
 A single, well-known railway corporation has 63,000 share- 
 holders or stockholders. 
 
 The steps in the organization of a corporation are: 
 
 (1) A specified number of people (at least three, in New York) come 
 together and draw up a certificate, stating (a) the name of the proposed 
 corporation, usually including the word "Company"; (6) the place of 
 business; (c) the nature of business; {d) the amount and face (par) value of 
 shares; (e) duration of corporation, (/) other information required by law. 
 
 (2) This certificate is filed with the Secretary of State, who issues a 
 license to these people to open books for subscriptions to the capital stock. 
 
 (3) When the legal amount of subscriptions have been made, and in 
 some instances paid in, a meeting of subscribers is called, officers are chosen 
 and by-laws adopted. Usually the stockholders elect directors, each 
 stockholder Having as many votes as he holds shares, and the directors 
 elect officers. 
 
 332 
 
STOCKS AND BONDS. 333 
 
 (4) A record of these proceedings is filed with the Secretary of State 
 who then issues a certificate to the effect that the corporation is legally 
 organized and ready for business. ^ 
 
 Certificates of stock are issued to subscribers when sub- 
 scriptions are paid, and these certificates may be transferred 
 at will thereafter, unless the stockholder is indebted to the 
 corporation. In such cases, transfer may be permitted by 
 special vote. 
 
 Advantages. The principal advantages of corporations to 
 their partners (stockholders) are: (1) Financial responsibility 
 for only the par value of stock (twice the value in bank stock) 
 until paid; (2) small shares, permitting the purchase of interests 
 in many prosperous businesses where a partnership interest 
 could not have been bought; (3) a simple method of transfer 
 of ownership ; (4) possibility of holding an interest in a business 
 for which one has neither time nor qualifications for direct 
 control; (5) the value of stock as an investment for surplus 
 funds. The advantages to the business are (1) possibility of 
 indefinite expansion as to capital and organization; (2) unity 
 of direction and management, enabling the employment of 
 experts. 
 
 Shares. The par, or face value of shares in the larger 
 corporations is usually $100, but may be any simple multiple 
 or ahquot part of $100, as $1, $5, $10, $500, etc. A majority 
 of shares outstanding control the company. Thus, a few large 
 shareholders may control a company in which a host of small 
 holders are interested. 
 
 EXERCISE. 
 
 Find the missing values: 
 
 Capital. Par. Val. per Sh. No. Shares. 
 
 1. $ 200,000 $100 
 
 2. $4,000,000 $ 50 
 
 3. $ 80,000 $ 10 
 
 4. $ 250,000 500 
 
334 BUSINESS ARITHMETIC. 
 
 5. $ 600,000 $200 
 
 6. $400 1500 
 
 7. Andrews holds 1200 shares ($100) m a $250,000 company. He 
 must buy shares, in order to control it. 
 
 8. Name the number of shares necessary to control each of the com- 
 panies in ex. 1-7. 
 
 9. In the reorganization of a $50,000 company, the par value of shares 
 is changed from $20 to $100. Snell, who owns 400 $20 shares, should 
 receive $100 shares. 
 
 Preferred and Common Stock. Usually the original issue 
 of stock is called common stock and has full voting power. 
 Sometimes ^preferred stock is issued at time of organization 
 and may or may not have voting power. Preferred stock- 
 holders must be paid a certain stated proportion of profits 
 before common stockholders receive anything. Thus, "4% 
 preferred stock" entitles the holders to 4% of the face value in 
 profits, annually, hut no more. The preferred stock may be 
 cumulative, carrying over to a succeeding year its right to its 
 share for a previous year, even if in those years there was no 
 profit. Preferred stock is sometimes non-cumulative. 
 
 Preferred stock is issued, sometimes, to meet special needs 
 for additional capital, and is in the nature of a perpetual loan, 
 the holders having no right to vote. If the common and 
 preferred stockholders agree, a second issue of preferred stock 
 may be made that will have rights over previous issues. 
 
 Raising the Capital. Some states require that the original 
 capital stock shall be paid in full; in other states it may be 
 sold for what it will bring. The market price is termed a 
 quotation. If the stock is sold at face value, quotation is 
 "at par"; if below, it is ''below par," or at a discount; if in 
 advance of par it is "above par," or at a premium. 
 
 At times, private bankers will take over a large block of 
 stock, not originally subscribed for, at a fixed price and sell it 
 to the public. Often a commission is paid, in money or stock. 
 
STOCKS AND BONDS. ' 335 
 
 EXERCISE. 
 Find missing values. 
 
 Actual 
 
 Capital Stock. Shares. Orig. Selling Quotation. Cash Cap. 
 
 1. $200,000 $100 $100 $ 
 
 2. $500,000 $100 $98 $ 
 
 3. $2,000,000 $50 $47 $ 
 
 4. $500,000 $100 $103 $ 
 
 5. C. P. Jacobs has subscribed for 500 $100 shares in a new company, 
 payable in full. He turns over a small factory, which is accepted at a 
 valuation of $36,492. What must he pay in cash? 
 
 6. The partnership firm of Adams and White is reorganized is a cor- 
 poration of $60,000 capital, in $100 shares. Adams' present worth in 
 the old firm was $18,500 and W^hite's $32,000. What number of shares 
 should each receive for his present w orth, and what nuinber of shares should 
 go into the treasury for future sales? 
 
 7. Suppose the full number of shares is divided between Adams and 
 White in proportion to their present worth, Adams paying White in cash 
 for the fraction of a share, what number of shares should each receive? 
 The capital stock is watered, or in excess of the real capital by $ . 
 
 8. Martin agrees to take 80 shares ($100), Brown 450 and Baker the 
 balance, in an $80,000 company. They decide to pay for their stock at 
 15% premium, in order to create a surplus or extra working capital. 
 Determine the capital, surplus and payment of each subscriber. 
 
 Distribution of Profits and Losses. In case of losses, if stock 
 is assessable, stockholders, by vote of directors, may be re- 
 quired to pay a certain assessment, levied as an amount per 
 share, or as a certain per cent, of par value. The profits, 
 on the other hand, are held as undivided profits or surplus, 
 until dimdends are declared by the directors. Dividends are 
 more commonly expressed in dollars per share, or as a per 
 cent, of par value of stock. Dividends must be paid, first, on 
 preferred stock, and then on common stock. The directors 
 are not forced to dispose of all profits as dividends. 
 
 ORAL EXERCISE. 
 
 1. The Crown Manufacturing Co., capitaHzed at $1,000,000 ($100 
 shares) declares a dividend of 5% ($5 per share). The total dividend 
 
336 BUSINESS ARITHMETIC. 
 
 in money, is $ ; and the dividend of Jamison who owns 60 shares is 
 
 60X$5, or$ . 
 
 Find the missing values. 
 
 Capital. No. Sh. Par Val. Div. % 
 
 2. $200,000 1000 ? 4% 
 
 3. $500,000 ? $100 5% 
 
 4. $80,000 ? $50 ? 
 
 5. $2,400,000 ? $100 ? 
 
 6. $60,000 ? ? 4i% 
 
 EXERCISE. 
 
 Note. Assume par value as $100 per share, unless otherwise stated. 
 
 1. What sum of money is required to pay a 3j% dividend on a com- 
 pany capitaUzed at $4,000,000? 
 
 2. A 7% dividend on 670 shares of P. R. R. stock will yield the owner 
 
 3. A quarterly dividend rate of 1|% will yield the owner of 250 shares 
 $ per year. 
 
 4. What rate of dividend, in dollars per share, must be declared in 
 order to distribute $42,000 to stockholders in a $500,000 company? 
 
 5. An assessment to meet a loss of $36,000 must be levied against a 
 capital stock of $400,000 ($50 shares). Determine the assessment rate 
 A must pay $ on his holdings of 120 shares. 
 
 6. An industrial corporation, capitalized at $2,000,000, has net earnings 
 of $172, 541.50, for the year. Of this sum, 20% is transferred to surplus. 
 What is the largest whole per cent, of dividend that can be declared from 
 the balance? If this is declared and paid, what is the balance of undivided 
 profits? 
 
 7. The Concklin Wheel Co. has a capital of $200,000, one-fourth con- 
 sisting of 5% preferred stock. Its gross income for one year is $729,342, 
 and its gross expense $512,860. The directors set aside 10% of profits 
 for surplus, and after paying preferred dividends, declare the largest possible 
 dividend (in per cents and fourths). The rate declared is %. 
 
 Note. At times, when profits are very high or when, for any reason, a 
 cash payment is not advisable, stock dividends are sometimes declared. 
 Additional stock, thus issued, participates in all future dividends. When 
 business is profitable, but there is a temporary money stringency, dividends 
 are sometimes paid in scrip, or certificates cashable after a certain date. 
 
 8. The Lamberton Electric Co , capitalized at $500,000, has decided 
 to distribute $80,000 in dividends. The highest rate it pays in cash is 
 
STOCKS AND BONDS. 337 
 
 8%, the balance being paid in new stock. Harper, who owns 400 shares, 
 
 receives $ cash and new shares. The new capital stock of the 
 
 company is $ . 
 
 Bond Issues. When a government or corporation borrows 
 large sums of money it usually issues a formal promissory note 
 called a bond. The bonds usually are issued in series of like 
 tenor and for equal amounts, with interest payable annually, 
 semi-annually or quarterly. The face value is usually $1000, 
 but $500 bonds are common, and smaller ones are issued. 
 
 The payment of industrial bonds is secured by a mortgage 
 on the property of the corporation. Minor municipal bonds 
 are sometimes paid from a sinking fund raised by taxation. 
 Government bonds and securities, and corporation debenture 
 bonds have no security. First mortgage bonds usually have a 
 first right to lien on property; succeeding issues, second, third, 
 etc., have rights only after the first issue is paid. 
 
 Bonds are issued commonly in registered or coupon form. 
 If registered, the names of the owners are recorded in the books 
 of the company, as in the case of stock; the bonds are made 
 payable to the specified buyer, and are transferable by endorse- 
 ment and office record. Interest is sent by check to the 
 recorded owner. Coupon bonds have attached coupons or 
 vouchers, one for each interest period during the life of the 
 bond. These are detached as they become due, and are 
 presented for payment, or collected through a bank. 
 
 Many bonds derive their name from their rate of interest, 
 and from their term and time of maturity. " D. C. 4s of 1925 " 
 are District of Columbia 4% bonds due in 1925. 
 
 While bonds are usually payable at a fixed time, provision 
 is sometimes made to pay off a certain number of an issue each 
 year. Sometimes the company reserves the option of paying 
 off between certain dates, as "after ten years but not later 
 than twenty years." 
 
 Bonds are similar to preferred stock in that they pay a fixed 
 23 
 
338 BUSINESS ARITHMETIC. 
 
 rate of dividend or interest. They have a Hmited life and 
 they confer no voting power, but they have a right to the 
 payment of interest ahead of the preferred stock. 
 
 EXERCISE. 
 
 1. The Selford Electric Co. sells a $40,000 6% bond issue at $1050 
 per $1000 bond, receiving $— . The interest on the issue will cost the com- 
 pany $ — annually. 
 
 2. The Crane Hardware Co. issues 5% bonds running five years for a 
 face value of $120,000. The company creates a 4% sinking fund to meet 
 the issue at maturity. If it sets aside equal amounts at the end of each 
 year and also pays interest, what is the annual expense? 
 
 . 3. The Western Banking Corporation underwrites the bonds of the 
 West River Lumber Co., par value $50,000, at 90% of face value. The 
 banking corporation disposes of the issue at 93% of face value, thus making 
 a profit of $ — , and paying the lumber company $ — . 
 
 Buying and Selling. The market quotations are affected 
 by many varied factors. Primarily, the price should depend 
 on earning power, present and prospective. Thus, price is 
 influenced indirectly by the capital of the company, for over- 
 capitalization means les^ income per share. The price may 
 be seriously, though temporarily, influenced by speculation, 
 or by an attempt of parties to gain control of a company. The 
 general prosperity or depression of industry affects quotations 
 seriously. A financial depression may cause liquidation or the 
 selling of stock to meet financial debts — sales often made at 
 a heavy loss. A sudden loss of confidence in one stock may 
 affect others. The fear of injurious legislation, the threat 
 of war, the falling off of exports, the closing down of a great 
 factory are other illustrations of the many influences affecting 
 market price. 
 
 Quotations. General purchases and sales are governed 
 largely, as to price, by the quotations of the stock exchanges. 
 The quotation lists, as printed, often vary in form, but usually 
 show the price in dollars per share, at several times of the 
 
STOCKS AND BONDS. 339 
 
 business day. For bonds, the price is quoted per $100 of par 
 value. 
 
 Stock Quotation List. 
 (From newspaper lists. Consult newspapers for current quotations.) 
 
 Open. High. Low. Close. 
 
 Mo., Kan. & T., com 26 26 25^ 25^ 
 
 M.S. P. &S. S. M 1411 142 141 142 
 
 Missouri Pacific 44 44^ 43| 43| 
 
 N. Y. Air Brake 82f 82f 82t . 82f • 
 
 New York Central 109^ 110 107 107i 
 
 N. Y., Ont. & West 33 33^ 32f 32^ 
 
 Norfolk & Western 113 113 113 113 
 
 Northern Pacific 126| 127| 125 f 126 
 
 Pennsylvania R. R 124 124i 123i 123^ 
 
 Pressed Steel Car 35 35 35 35 
 
 Rail's Steel Springs 36 36 35 35 
 
 Reading 164^ 165^ 162| 163J- 
 
 Rep. S. & I., pfd 89 89 89 89 
 
 Rock Island, com 24i 24^ 24 24 
 
 Rock Island, pfd 47| 47| 47| 47f 
 
 Bonds. 
 
 C. R. I. & P. 4s 69 69 68^ 68| 
 
 Atcn. 4s. con 104^ 104^ 104^ 104| 
 
 Nor. & West 48 98^ 98 98^ 98f 
 
 U.S. Steel 5s 101| 102 lOlf 102 
 
 Government Bonds. 
 
 United States 38, coupon 104f 104f 104| 104f 5 
 
 United States 4s, coupon 114^ 114^ 114^ 114^ 15 
 
 Imp. Japanese Gov. 6s ctfs. full paid. . 90f 92 91 f 92 678 
 
 Republic of Cuba 5s, full m. paid 103 103 102^ 102 1 232 
 
 In the above stock list, N. Y. C. stock, for example, is 
 quoted at the opening of the day at $109.50. During the day, 
 the highest price was $110 and the lowest, $107. At the close 
 of the day, the price was $107,125. Such stock is usually 
 bought or sold through brokers. The common charge is 
 1/8% of par value, or $1/8 on a $100 share. Small brokers, 
 handling small lots, sometimes charge 1/4%. 
 
340 BUSINESS ARITHMETIC. 
 
 Illustrations. (1) Determine the cost of 300 shares of P. R. R. stock, 
 at closing quotation, brokerage 1/8. 
 
 Solution. Bj'^ reference to list, the closing quotation is $123.25. Broker- 
 age is $1/8 or $.125. 
 
 The entire cost of 1 share is $123.25 + $.125, or $123,375. 300 shares 
 cost 300 X $123,375, or $37,012.50. 
 
 (2) What should I receive from the sale of 400 shares Reading at opening 
 quotation, brokerage 1/8? 
 
 Solution. Opening quotation, 1641. 
 
 Net proceeds of 1 share, $164. | - $| = $164f. 
 Net proceeds of 400 shares = 400 X $164 1 = $65,750. 
 
 ORAL EXERCISE. 
 
 1. Which of the stocks in the quotation list are jyref erred? 
 
 2. Was the general tendency of prices, during the day, upward or 
 downward? 
 
 3. If I buy Reading at highest quotation, brokerage 1/8, I pay $ — 
 per share. If I had sold at the same price I would have received $ — net 
 proceeds. 
 
 4. Brokerage being 1/8, determine the cost of 
 
 No. shares. Stock or Bond. Quotation, 
 
 (a) 100 N.Y. Air Brake Closing. 
 
 • (h) 1000 Mo. Pac. Open. 
 
 (c) 10 Penn. R. R. High. 
 
 (d) 2 No. Pac. Low. 
 (c) $10,000, par. val. Nor. & Western High. 
 (/) $10,000, par val. U. S. 4s, Open. 
 
 5. Determine the net proceeds, brok. 1/8, from the sale of 
 
 Shares or Val. Stock or Bond. Quotation 
 
 (a) 200 sh. U. S. Steel, 68S. 
 
 (6) $ 1,000. Southern Pacific, 106f. 
 
 (c) $10,000. S. L. & S. F., com., 16. 
 
 (d) $10,000. U. S. ?s., Closing. 
 
 (e) 100 sh. Reading, Closing. 
 
 6. Determine the brokerage at 1/4 on lots of: 80 sh., 25 sh., 1000 sh., 
 250 sh. 
 
 7. If brokerage is 1/8, one way only, find the net profit or loss per share 
 on: 
 
 Buying Quot. Selling Quot. Buying Quot. Selling Quot. 
 
 (a) 79i 81 i (c) 8 prem. 6 prem. 
 
 (Jb) lllf 112 id) 51- dis. 1 prem. 
 
 8. Express these discounts in other forms: 105 (Ans. 5 prem. or 5 
 above par), 96, 110^, 8 dis., 7 J prem.; 5^ below par, 100. 
 
STOCKS AND BONDS. 341 
 
 EXERCISE. 
 
 1. Determine the .cost at opening quotation, brokerage 1/8, of 200 sh. 
 M.K.&T.com.; 350sh. Mo. Pac; 3000 sh. Reading; $10,000 par of C, 
 R. I. & P. 4s; $20,000 par U. S. Steel 5s. 
 
 2. Determine the net proceeds received from the sale, at high quo- 
 tation, brokerage 1/8, of: 3000 sh. N. Y. C; 2000 sh. Nor. & West.; 800 sh. 
 Rock Is. com. ; 3000 sh. P. R. R. ; 18,000 sh. Pressed Steel Car; $500 Japanese 
 Gov't 6s. 
 
 Note. In determining the gain or loss from a purchase or sale, it is 
 advisable to find the net gain or loss per share, and to multiply by the 
 number of shares. 
 
 Illustration. Determine the profit or loss from 800 sh. bought at 
 98f and sold at 101, brok. | each way. 
 Solutwn. The cost = 98| + i = 98|. 
 
 The net proceeds = 101 - i = 100 1. 
 The net profit per sh. = 100 1- - 98| = 2f. 
 The total profit = 800 X $2f = $2100. 
 
 3. Find the profit or loss in the following, brokerage \ one way: 
 
 No. Sh. or Val. 
 
 Stock. 
 
 Buy. Quot. 
 
 Sell. Quot. 
 
 (a) 200 sh. 
 
 Penn. R. R. 
 
 Open 
 
 1131 
 
 (6) $8000 
 
 Japanese 6s. 
 
 High 
 
 Close. 
 
 (c) 500 sh. 
 
 No. & West. 
 
 Open 
 
 Close. 
 
 {d) 1200 sh. 
 
 No. Pac. 
 
 Open 
 
 Close. 
 
 (e) 3000 sh. 
 
 Reading. 
 
 1041 
 
 Close. 
 
 (J) On $5000 of each of the bonds in the quotation list, buying at 
 opening and selling at closing price. 
 
 4. I directed my broker to sell $20,000 U. S. Steel 5s and invest proceeds 
 in Reading. Buying quotation, 104 1; selling quotation, 88 1; brokerage 
 1/8 each way. Number of shades bought? Surplus? 
 
 Note. No fraction of a share is bought. In purchases through 
 exchanges, lots of hundreds of shares are usually dealt in. 
 
 5. I ordered my broker to sell 2600. shares N. Y. Ont. & W. (32|) 
 and to buy 2000 shares N. P. (125 f). He should draw on me for what 
 sum to settle the balance? 
 
 Find missing values: 
 
 6. Carroll County issues ten-year coupon school bonds, bearing 6% 
 interest, payable semi-annually, which are disposed of at 102^, brok. \. 
 
 The bonds produce a fund of $ , but the County must meet interest 
 
 charges of $ every six months, or $ in ten years. At the close of 
 
 the final year it must pay out $ . The bonds have coupons 
 
 attached, each worth $ , if the face value is $500 each. Abbott, who 
 
342 BUSINESS ARITHMETIC. 
 
 buys $12,000 worth, pays $ , and gets an annual income of $ , or 
 
 — % on his original investment. 
 
 Stock Exchanges. Stock exchanges are organizations, 
 formed for the purpose of supplying a regulated market for 
 the purchase and sale of stocks and bonds. All cities of any 
 size have such organizations, but the New York Stock Ex- 
 change is by far the best known, and is national in scope. 
 This has 1000 members. Membership is called a seat, and 
 may be sold. The buyer of a seat must be elected by the 
 Committee on Admissions. 
 
 The strictest regulations are in force in regard to methods 
 of doing business, and the organization maintains the highest 
 standard of commercial integrity among its members. Trans- 
 actions involving hundreds of thousands of dollars are con- 
 tracted verbally and are carried out to the letter. Severe 
 penalties are inflicted for any breach of rules. 
 
 An Exchange handles listed and unlisted securities. It 
 does not guarantee the value of securities that it admits or 
 lists, but before listing, it makes examinations of the com- 
 panies whose stock is offered for listing. Railway companies, 
 for example, must file statements giving location and detailed 
 description of property, and a financial statement of liabihties, 
 assets, earnings, mortgages, etc. An industrial corporation 
 must prove that it is legally organized, and must also make a 
 statement of its financial and busii ^ss affairs. An Exchange 
 makes further requirements in regard to regular reports, notices 
 to stockholders, etc., thus forcing the company to make public 
 its actual condition, before listing is granted. 
 
 For corporations of standing that are unable, or unwilling, 
 for some reason, to conform to the strict requirements of 
 "listing,'' an "unlisted department" is made. Less strict 
 statements are required, but the stocks are dealt in in prac- 
 tically the same way. Unlisted securities are not considered 
 of so high a rank as listed securities, and are not so highly 
 esteemed by bankers as collateral for loans. 
 
STOCKS AND BONDS. 343 
 
 On one Exchange business is done by members for outsiders 
 at 1/8% commission; by members for named members, at 
 1/50% and by members for unnamed members at 1/32%. 
 Lots of 100 shares, or $10,000 par of bonds, are usually dealt in. 
 Trading must be done between 10:00 A.M. and 3:00 P.M. 
 Brokers often meet ''on the curb," or in the street in the 
 neighborhood of the exchange, and buy or sell small lots, 
 unlisted securities, etc. This is the '' Curb Market." 
 
 All transactions are by word of mouth. There are posts in 
 places in the Hall of the Exchange where specified stocks are 
 traded in. Brokers desiring to buy or sell the given security 
 congregate at these posts and shout their orders or sales. A 
 nod, a sign with the finger, or the word '' sold," may be the 
 sign of a transaction involving thousands of dollars. Memo- 
 randa are roughly made on paper, at the first possible moment, 
 and later compared. Prices rise or fall by 1/8%. If a broker 
 bids 98 for a stock and no one offers to sell, his next higher bid 
 is 98|. During the times a corporation's books are closed, 
 its stock sells "ex-dividend'' — that is, the last holder gets the 
 dividend. 
 
 Stock is sold "cash" for deHvery the same day; "regular," 
 to 'be delivered the day following; "at three days"; or at 
 "buyer's option" or "seller's option" — usually 4 to 60 days. 
 Most sales are "regular." 
 
 Practically all speculative exchange business is on " margin." 
 Thus, on January 21, C. Brown may order his broker to buy 
 100 shares D. & M., whenever securable at 186. He deposits 
 10% of the money, or $1860 (called the margin). The 
 broker supplies the rest of the money needed. The stock is 
 bought the same day. On January 29, when the stock is quoted 
 at 189, he orders the broker to sell. The broker charges 1/8% 
 brokerage each way, and 6% interest on the money lent to 
 complete the transaction, rendering a statement as follows: 
 
344 
 
 BUSINESS ARITHMETIC. 
 
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STOCKS AND BONDS. 345 
 
 EXERCISE. 
 
 1. On March 17, I purchased through a broker 600 sh. Reading, at 
 163 1 — buyer's option at twenty days. I called for the stock March 26, 
 and directed ita sale at 164j. Brokerage 1/8. Determine my net profit. 
 
 2. On May 9, my broker bought for me 800 sh. P. R. R. at 123| on 
 10% margin, selling it May 14 at 12o|. Interest, 6%; profit, $ ? . 
 
 3. On June 9 a broker sold short, on 10% margin, 500 sh. N. P. at 
 125 f. The stock declined so that on June 16, his principal ordered his 
 sale covered (121 1). The principal's net gain was $ ?. 
 
 4. Prepare a broker's statement for his principal, A. C. Benson, from 
 the following memoranda: Margin 10%; interest, 6%. Bought: Jan. 11, 
 200 sh. N. Y. C, regular, 106|; Jan. 18, 500 sh. N. P., regular, 127 1; 
 Sold: Jan. 22, 500 sh. N. P., 131f ; Jan. 23, 200 sh. N. Y. C, 105^. 
 
 Collateral Loans. Many business men invest surplus funds 
 in dividend-paying stocks and bonds. In times of financial 
 stringency, they borrow of their banks on time or demand, 
 depositing this stock as security or collateral. These stocks 
 and bonds are accepted, if reliable, on a margin of 10% to 
 20% (i. e., at 10% or 20% below market quotation). No 
 margin is required on government bonds. In case of failure 
 to pay the loan, the securities are sold by the bank, the debt 
 is cancelled and any balance is returned to the borrower. 
 If the market value of the securities falls while they are being 
 held as collateral, additional collateral must be deposited or a 
 portion of the loan repaid. 
 
 Stock brokers use the certificates of stocks and the bonds 
 they buy for customers as temporary collateral for loans of 
 this order — termed call loans. The banks lend at the pre- 
 vailing money rate and have the privilege of changing the 
 rate from day to day as the financial market varies. They 
 may also "call'' for payment at any time. Rates are "easy" 
 at 1% to 3%; "firm" at 6% to 8%, and "stringent" at higher 
 rates. During a marked panic some money was lent at the 
 rate of 100%. 
 
346 
 
 BUSINESS ARITHMETIC. 
 
 Face of a Collateral Note. 
 
 $ Washington, D. C 190 . . . 
 
 promise to pay to 
 
 The National Bank of Washington City, or order 
 
 at said Bank Dollars 
 
 for value received with interest at per centum per annum, having 
 
 deposited with said Bank, as collateral security for the payment of the 
 full sum of principal, with interest and cost due on this note, and also as 
 collateral security for all other present or future demands, of any and all 
 kinds, of the said Bank, against the undersigned, due or not due, the 
 following to- wit : 
 
 with full power and authority to said bank to sell the whole or any part of 
 said security, or any substitutes therefor, or any additions' thereto, at 
 public or private sale, at any time, and at the option of said Bank or its 
 assigns, on the non-performance of this promise or any part thereof, or 
 the non-payment of any other present or future demands of said bank as 
 aforesaid, and without advertisement or notice to the undersigned; and 
 upon such sale the holder thereof may purchase all or any part of said 
 securities, discharged from any right of redemption. After deducting all 
 proper costs and expenses, the residue of the proceeds of sale shall be applied 
 to the payment of the note, and of any other present or future demands of 
 said Bank as aforesaid, and the undersigned agrees to remain liable for any 
 deficiency then remaining. In case of depreciation in market value of said 
 security at any time pledged for this loan a payment shall be made on 
 account, or additional security added, as required by said Bank. 
 
 WRITTEN EXERCISE. 
 1. Does this record of a call loan show sufficient security, if the margin 
 must be 10%? 
 
 Loan No. 86739. 
 FoHo No. 472. 
 Demand loan for $300,000. 
 To Robert Harlenan. 
 
 The National Bank. 
 
 New York, February 17, 190— 
 
 400 
 1000 
 
 200 
 400 
 500 
 200 
 800 
 100 
 
 111. Central 
 B. &0. 
 D. L. & W. 
 Pennsylvania 
 Reading 
 S. Pac. 
 
 127 
 106 
 
 388 
 127 
 167 
 108 
 
STOCKS AND BONDS. 347 
 
 Note. In reckoning value, fractions of quotations are omitted, and 
 final value written to thousands and hundreds of dollars. The first item 
 of value is $25,400. 
 
 Alterations of collateral are shown by quantities in italics. 
 
 2. Could any line of above stock be withdrawn without destroying 
 margin? 
 
 3. In lots of 100 sh., how much N. Y. C, at 106, might be substituted 
 for the D. L. & W.? 
 
 4. Compute the interest on this demand loan at 3%, Feb. 17-24; 
 3|%, Feb. 25-26; 4%, to Mar. 3 when ''called." 
 
 5. On Mar. 11, $70,000 is borrowed at 3%, on 400 sh. N. Y. N. H. & H., 
 at 202, as collateral. During a violent break in the market stock falls 
 to 154. How much additional collateral should be demanded in order to 
 preserve a 10% margin? 
 
 6. A call loan of $200,000 is secured by collateral of $80,000 U. S. 
 bonds (103); 100 sh. 111. C. (127); 200 sh. Penn. (128); 500 sh. U. P. (160), 
 200 sh. N. Y. C. (106) and 300 sh. C. P. (265). If the borrower pays 
 $50,000 of the loan, what collateral may be released? Allow 10% margin 
 except on Government bonds. 
 
 Incomes and Investments. The investor in stocks and 
 bonds, as distinct from the speculator, is looking for a steady 
 and safe return from his money. He must be familiar with 
 factors that affect quotations, but he does not look at them 
 in the same light. As a rule, he desires a fair interest on his 
 money, security against loss, and a market for disposal in 
 case of necessity. In all cases, it is well to have the advice of 
 a reliable banker or broker. 
 
 Government securities, in the form of national or state 
 bonds, are usually safe investments and are easily marketable, 
 but they yield relatively low returns. Many municipal and 
 county bonds are also reliable, but before buying it is wise to 
 see what security is back of them and how regular has been 
 the payment of past interest. Railway and industrial bonds 
 should be examined even more strictly along the same lines. 
 
 In buying, one should note the date of maturity of the issue, 
 especially if he desires a long term investment. If the quo- 
 
348 BUSINESS ARITHMETIC. 
 
 tation is at a premium, he should remember that the premium 
 represents a portion of his investment which will not be 
 repaid him when, at maturity, the bonds are settled. The 
 nearer the maturity of the bonds, the greater is the effect of 
 this premium. 
 
 In buying stocks, one should invest either in local organi- 
 zations, whose policy, standing and resources he can examine, 
 or in organizations of national reputation, whose stock has 
 shown steadiness through a long period, and which is readily 
 marketable. Before buying, one should look up quotations 
 for months past to be sure that the current quotation is not 
 far above the ruUng price because of some temporary market 
 influence. Times of depression are excellent periods for the 
 purchase of sound stocks. One should be careful not to be 
 misled by high rates of dividends. The safest investments on 
 the market seldom pay over 5% or 6%. It is well to investi- 
 gate and determine the average dividend for years past, to 
 discover whether the rate is steady, steadily increasing, or 
 decreasing, etc. 
 
 ORAL EXERCISE. 
 
 1. I can buy one stock, paying 6% dividen'Is, at 120, and another, 
 paying the same rate, for 150. How many dollars per share income in 
 each case? Based on these rates, which is the better investment? Suggest 
 a method of comparing the value of stock investments. 
 
 2. I have $20,000 to invest. A good 8% stock can be purchased at 
 200 and a 5% stock at 80. The money will buy how many shares of either 
 kind? What is the income in each case? How may one compare invest- 
 ments, knowing the sum to be invested? 
 
 3. Investments in bonds are to be made to secure a yearly income of 
 $720. What par value of 6% bonds will produce it? Of 8% bonds? 
 The 6% bonds can be bought at par; the 8% bonds at 120. Security 
 being equal, which is the better investment? How is the comparison 
 made in this case? 
 
 Illustration. Compare the value of C. B.& M. 4s at 101 f, and 
 P. R. R. 4J-S at 1071, for an investment of $20,000. 
 
STOCKS AND BONDS. 349 
 
 Solviion. 
 
 C. B. & M. costs 101 f + 1/8, or lOU. 
 
 A $1000 bond will cost 10 X $101^, or $1015 
 
 P. R. R. costs 1071 + i, or $108. 
 
 A $1000 bond costs 10 X $108, or $1080. 
 
 $20,000 ^ $1015 = 19 and $715 remaining. 
 
 Therefore 19 C. B. & M. bonds could be purchased. 
 
 $20,000 ^ $1080 = 18 and $560 remaming. 
 
 Therefore 18 P. R. R. bonds could be purchased. 
 
 19 X $40 = $760, income from C. B. & M. 4s. 
 
 18 X $45 = $810, income from P. R. R. 4|s. 
 
 $4 -^ $101.5 = .0394 (3.94%), rate income on C. B. & M. bonds. 
 
 $4.5 ^ 108 = .0416 (4.16%), rate of income on P. R. R. bonds. 
 The P. R. R. bonds are evidently the better investment if long term. 
 
 EXERCISE. 
 
 1. What income will be received from the investment of so much as is 
 possible of $60,000 in U. S. 4s at 104^, brok. i? 
 
 2. Erie 4s, at 108, brok. 1/8, earn ? % on investment. 
 
 3. Compare the investment values of L. & N. stock at 145, yielding 
 6%, and Pullman Palace Car Co. stock, at 181, earning 8%. 
 
 4. Brown has $60,000 to invest. His choice narrows down to T. R. R. 
 at 141 (7%); B. & O. 4% pfd. at 80, and D. L. & W. (10%,) at 485. Com- 
 pare investment values. 
 
 5. In the latter part of 1907 a serious financial stringency caused heavy 
 breaks in stock quotations. This break was finally stayed on Oct. 26. 
 From the following quotations, find the rates of income on investments 
 based on each of the quotations (high, low and Oct. 26). Show that 
 times of depression are good periods for investment in sound stocks. 
 
 1907. 
 
 High. Low. Stock. Oct. 26. Reg. Dividend. 
 
 145i 92f L. & N 95 6%. 
 
 75i 40 Mackay Co 48 4 
 
 14G 1001 Manhattan Elev 103^ 7 
 
 92| 48 Mo. Pac 51| 5 
 
 147 108 Nash. & Chat 115 6 
 
 117f 103 Nat. Bis. pfd 103 7 
 
 1341 961 N. Y. C 991 6 
 
 921 56 Nor. & Western 61| 5 
 
 141-1 1131 Penn. R. R 115 7 
 
 98| 72^ People's Gas & Coal 73f 6 
 
350 BUSINESS ARITHMETIC. 
 
 Convertible Bonds. Some railroad corporations, and a few 
 other organizations, have issued, of late, series of bonds that 
 are convertible, at a fixed ratio, into shares of stock. Thus a 
 certain company issues 4% bonds that may be exchanged for 
 stock at a quotation of 150 (par 100). The bonds offer a safe 
 but low rate of income. If the business gains in prosperity 
 the dividends on the stock may increase to such a point that a 
 greater rate of income, though less secure, may be obtained 
 by conversion. Questions of control, speculation, etc., may 
 lead, also, to conversion. The market price of both stocks 
 and bonds also has its effect. 
 
 Illustration. Mansfield R. R. 5% convertible bonds are exchangeable 
 for stock at 120. The bonds were bought at 108, the stock is quoted at 
 136 and yields 8% dividends. Study the efifect of converting $6000 par 
 of the bonds. 
 
 Solution. 
 
 $6000 -v- 120 = 50, number of shares by conversion. 
 5% of $6000 = $300, income from bonds. 
 8% of $5000 = $400, income from stock. 
 
 Increase of income by conversion, $100, or 33^%. 
 
 60 X $108 = $6480, original cost of bond investment. 
 
 50 X $136 = $6800, present selling value of equivalent stock. 
 
 Direct gain by converting and selling, $6800 - $6480 = $320. 
 
 Note. This does not take into account the effect of the time bonds 
 have to run before maturing. Bankers have "bond tables" to show the 
 real investment value of bonds. 
 
 EXERCISE. 
 
 1. $12,000 (par) of 5% bonds may be converted into ? shares of stock, 
 if exchange quotation of stock is 150. 
 
 2. $30,000 of 4% bonds, bought at 92, are converted into stock at a 
 quotation of 120. The stock is sold at the market quotation of 146 J. 
 Compute net gain or loss. 
 
 3. M. T. R. R. Conv. Gold 3^'s, due 1925, are convertible into 
 stock at 75 (par 50). Bought at 90, the bonds yield ? % on investment. 
 Compare market value by conversion, if stock quotation is 122f. 
 
STOCKS AND BONDS. 351 
 
 4. N. P. Conv. 4s, of 1927, at 86, yield ? % on investment. $80,000 
 (par) of bonds are convertible into ? shares at 175. Compare investment 
 values if 9% dividend is earned by stock. 
 
 5. R. M. Tr. Conv. 5s at 98 are convertible at par. The last stock 
 quotation is 85 f and the last dividend 4%. Discuss advisability of ex- 
 change. 
 
 Building Loan Associations. Building loan associations 
 are a form of bank organized under state laws to encourage 
 the accumulation of money among people of limited means, 
 with the object, primarily, of their making real estate invest- 
 ments and building homes. They originated in England in 
 1789. The first association of the kind in the United States 
 was organized in 1831, at Frankford, Pa. Owing to many 
 abuses in the past, laws are now in force in many states, 
 restricting the methods and operations of associations. One 
 state, for example, limits operating expenses to 2|% of in- 
 debtedness. It is well to study the laws of the incorporating 
 state, before entering any association. 
 
 The capital of loan associations, consisting of the accumu- 
 lated savings of members, is divided into shares of $50 to $250, 
 usually $200. Each share runs until the sum of installments 
 paid and dividends earned equals its maturing value. The 
 shares are issued in series, at periodic intervals, say six months 
 or a year. A member may buy any number of shares up to 
 a fixed limit paying by small weekly installments. He receives 
 the face value of the shares at maturity. 
 
 In most organizations, borrowers must be shareholders. 
 They may borrow, on their shares as security, up to 90% of the 
 sum paid in. The intending builder, who borrows, usually has 
 a piece of land, which is pledged with hi§ shares for his building 
 loan. Here the maturing value of his shares is commonly 
 taken. Usually a shareholder may withdraw his money at 
 any time, although he is penalized by having his share of 
 earnings reduced. 
 
 Often there are demands from members for more funds than 
 
352 BUSINESS ARITHMETIC. 
 
 are available for lending. It has become common, therefore, 
 to auction off the money in the treasury. There may be a 
 fixed minimum of interest, say 5%, and the one who bids the 
 highest rate over 5% gets the money. In other cases the 
 rate is fixed, the highest bidder being the one who offers the 
 largest number of months interest in advance. This advance 
 interest is subtracted from the face of the loan when the money 
 is paid over. Rates are often higher than 5%, but since the 
 stockholder has a share in the profits, the net rate is reason- 
 able. 
 
 The repayment of loans, as in the case of other features, 
 varies in different associations. Often the payments are 
 monthly, and for equal amounts, but the proportion going to 
 reduce the principal steadily increases and the interest de- 
 creases. Fully paid shares are accepted as cancelling the loan. 
 
 EXERCISE. 
 
 1. Recently, there were 2,029,927 persons said to be sharing in the 
 benefits of 5737 associations in the United States, and the total assets 
 were $618,795,414. Find the average membership and assets per associ- 
 ation. 
 
 2. The paid in value of Arnold's ten shares is $746.80. He borrows 
 90% of this value on his personal note, receiving $ . 
 
 3. Norton subscribes for fifteen one-hundred-dollar shares in an 
 association where the rate is 25c per week per share. What is his weekly 
 payment? Ignoring dividends, how long will it take him to pay for his 
 stock? To what sum is he entitled at maturity? 
 
 4. Norton borrows, on real estate security, the face of his shares, at 
 6|%. What is the quarterly interest on his loan? 
 
 5. Mason borrows $1,000, to be repaid at the rate of $10 per week, 
 for 138 weeks, and a final jfeyment of $7.79. What sum does he repay in 
 all? What is the average rate of simple interest on his loan? 
 
 6. Randolph borrows $2,000 at 6%, on a bid of 32 weeks' advance 
 interest. What proceeds of his loan are paid him? 
 
 7. Anderson wishes to obtain $4,000. He applies for twenty shares 
 (par, $200), agreeing to pay the usual dues of $1.00 per month per share, 
 
STOCKS AND BONDS. ^3 
 
 and 1^% of dues additional, as premium. If the rate of loan is 6%, what 
 is his total monthly payment? 
 
 8. An association issues one thousand $200 shares on payments of 
 $1.00 per month. What sum has it to lend the first month? Including 
 8% interest on the first month's loans, what has it to lend the second 
 month? Construct a table showing compounded funds to the end of the 
 sixth month, and the book value and profits of each share in the company. 
 
 Distribution of Profits. When only one series of shares is 
 issued, the distribution of profits for any one year consists 
 simply in dividing the profits by the number of shares to 
 determine the profits per share. If, however, one series 
 follows another, at short intervals, the problem is complicated. 
 Many methods are followed, the most common being that of 
 averaging. 
 
 Illustration. An association, at the end of its third year, has in 
 force three annual series of 1,000, 500 and 1,000 shares respectively. The 
 payments are $1.00 per month per share. Its net profits to the time of 
 issue of the fourth series are $4860. What profits are due each share? 
 
 Solution. 
 
 The first dollar on a share, in series (1) has run 36 months, the second 
 35 months, and the last 1 month, an average time of 18^ months. In a 
 similar way, the average dollar in series (2) has run 12^ months, and in 
 series (3), 6^ months. $36, $24, and $12 have been paid in respectively, 
 on shares of the three issues. 
 
 Series (1) 1000 shares, @ $36 yield $36,000, on an 
 
 average of 18^ mo. = 18| X 36,000 = $666,000 for 1 mo. 
 
 (2) 500 shares, @ 24 yield 72,000, on an 
 
 average of 12| mo. 12| X 24,000 = $150,000 for 1 mo. 
 
 (3) 1000 shares, @ 12 yield $12,000, on an 
 
 average of 6^ mo. = 6^ X 12,000 = $ 78,000 for 1 mo. 
 
 Equivalent for 1 mo. $894,000. 
 
 a • /IN -A ^^ u 666,000. „, 666 , 
 
 Senes (1) evidently shares m profits, or gqT-QQQ per share. 
 
 /ON -A .1 V, 1 50,000 . ^^ 300 , 
 
 (2) evidently shares gg^QQQ m profits, or gg^QQQ per share. 
 
 78 000 78 
 
 (3) evidently shares gQ^QQ^ in profits, or gg^Q^Q per share. 
 
 ^ of $4860 = $.0054. 
 
 894,000 
 
354 BUSINESS ARITHMETIC. 
 
 Profits per share: 
 
 Series (1) 666 X $.0054, or S3. 596-4. 
 Series (2) 300 X .0054, or 1.6200. 
 Series (3), 78 X .0054, or .4212. 
 
 EXERCISE. 
 
 1. At jhe end of one year, the book value of one series of an association 
 is $12.39, and of a second series, $6.10. Series (1) has 2,000 shares and 
 series (2) 1200 shares. The year's profits are $1860. Apportion these 
 profits according to book values. 
 
 2. The Martinsburg Building and Loan Asso. has issued five annual 
 series as follows: (1) 2,000 shares; (2) 5,000 shares; (3) 3500 shares; (4) 
 1600 shares; (5), just out, 1800 shares. Dues $1.00 per month. Ap- 
 portion profits to date of $11,456. 
 
CHAPTER XL. 
 
 DOMESTIC AND FOREIGN EXCHANGE. 
 
 Domestic Exchange. 
 
 INTRODUCTORY EXERCISE. 
 
 1. Name common reasons for sending money to persons in other 
 localities. 
 
 2. In what different ways may a person send the actual money? 
 
 3. Name objections to sending the actual money. 
 
 4. To what extent are the postal authorities hable for money sent by 
 registered mail? 
 
 5. Who is liable for losses of money sent by express or messenger? 
 
 6. In what ways, to your knowledge, may payment be made without 
 sending the actual mone}^? Descnbe such as you are familiar with, and 
 state advantages and disadvantages of each. 
 
 7. Is it cheaper to send the actual money? Is it safer? 
 
 The process of making payments, or of settling accounts 
 due at other localities, without sending the actual money, is 
 termed exchange. Exchange within a country is termed 
 domestic or inland exchange. 
 
 The credit papers commonly used in effecting exchange are 
 money orders, checks and drafts. 
 
 Money Orders. Money orders are issued by the National 
 Government and by express companies. They are really 
 drafts or orders, issued at one office, drawn on the office in 
 the locality in which it is desired to make payment, and pay- 
 able, as a rule, on presentation. Identification before payment 
 may be required. 
 
 The funds paid by the purchaser of the money order are not 
 transferred by government or express company to the office 
 of payment, but payment at that office is made from any 
 
 355 
 
356 
 
 BUSINESS ARITHMETIC. 
 
 funds on hand. Proper bookkeeping records of the transfer 
 are kept. Both express and postal orders may be deposited 
 Hke checks for collection. Many hotels accept and cash them. 
 They may be transferred by endorsement — a single endorse- 
 ment being allowed on postal orders, while express orders are 
 freely negotiable. 
 
 If more than one hundred dollars is to be sent by postal 
 order, additional orders must be taken out. The individual 
 express money order is limited to fifty dollars. 
 
 Express Money Order. 
 
 6-0000000 
 
 WhenCountersioked i WiTf+y.il'IV EEE^^ 
 
 dVAOCNT AT POINT OF l&SUt 
 
 Pay to twf nanrR n FO'T'^^::^ 'tS-:?^A,Svf >^'"^ ^ ^£ 
 
 . *&REE.S-IOtP»NSMIT AND 
 
 ^ 
 
 6-0000000- 
 
 AaCRICAN EXPRESS CO. 
 
 Mm- 
 
 Note. The cost of an order equals the face plus the fee. 
 
 Fees for Money Orders Drawn on Domestic Form. 
 Payable in the United States and its possessions; also for orders payable 
 in Bermuda, British Guiana, British Honduras, Canada, Cuba, Mexico, 
 Newfoundland, the United States Postal Agency at Shanghai (China), 
 the Bahama Islands, and certain other Islands in the West Indies. 
 For Orders From $ 0.01 to $ 2.50 3 cents. 
 
 From % 2.51 to $ 5.00 5 cents. 
 
 From $ 5.01 to $ 10.00 8 cents. 
 
 From $10.01 to $ 20.00 10 cents. 
 
 From $20.01 to $ 30.00 12 cents. 
 
 From $30.01 to $ 40.00 15 cents. 
 
 From $40.01 to $ 50.00 18 cents. 
 
 From $50.01 to $ 60.00 20 cents. 
 
 From $60.01 to $ 75.00 25 cents. 
 
 From $75.01 to $100.00 30 cents. 
 
DOMESTIC AND FOREIGN EXCHANGE. 357 
 
 Telegraph Orders. 
 Telegraph orders are obtainable in the larger cities. The 
 money is paid in at the sending office, and a telegram is sent to 
 the telegraph company's agent at the place of payment, direct- 
 ing him to pay the sum, from any funds on hand, to the person 
 specified, on identification. The rates are usually one per 
 cent, of the face of the order plus twice the cost of a ten word 
 message. The message is frequently sent in the Company's 
 private cypher or code, each word standing for a phrase. 
 
 EXERCISE. 
 
 1. Let some pupil describe, from personal experience, the purchase of 
 a money order, giving the steps to the process. 
 
 2. Let some pupil describe, from personal experience, the cashing of 
 a money order. 
 
 3. Compute the total cost of money orders to cover each of the following 
 amounts. Give the denominations of money orders where more than one 
 required to make a payment. Amounts: $1.64, $5.00, $7.00, $L50, $25.00, 
 $48.00, $65.50, $200.00, $165.00, $78.50, $2.50, $37.28, $96.84, $299.00. 
 
 4. An express agent issues money orders on a certain day for $34.16, 
 $345., $23.69, $50., $76.15, $20.25 and $107.10. He started the day with 
 a balance in his money order drawer of $124.00. During the day he cashed 
 orders for $23.90, $45.50, $120.00, $13.00, $112.00, $3.26 and $1.56. What 
 should be his balance in the drawer at the close of the day? 
 
 5. Determine the individual and total cost of these telegraph orders 
 from Washington, the rate being 1%: (a) To Philadelphia $600.00, plus 
 message cost of 50 cents; (6) To New York, $850, message 50 cents; (c) 
 To Newark, Del., $386.50, message 40 cents; (d) To Indianapolis, $590; 
 message 80 cents and to New Orleans, $2500, message 75 cents. 
 
 Checks. It is customary for business men and others to 
 keep the major portion of their cash funds on deposit in banks 
 or trust companies. They then make practically all payments 
 by checks, or orders on the bank to pay to the person named in 
 the check the sum specified. 
 
358 BUSINESS ARITHMETIC. 
 
 Form op Check. 
 
 GIRARD XRUSTT CONIRANY 
 
 Note. In this instance, John Doe desires to make a payment of $400 
 to Henry Newton. Doe draws a check on the bank, in which he has 
 money deposited, ordering it to make the payment to Newton. Newton 
 may either take the check to the bank and cash it, after being identified, 
 or he may deposit it with his own bank, after endorsing it, and the bank 
 will collect payment from the drawer's bank. The latter bank will charge 
 the amount against Doe's deposit. 
 
 The checks of others, received in the course of business are 
 deposited to the credit of the depositor's account, as in the 
 case of actual cash. It is estimated that ninety per cent, of 
 all business transactions involving money settlements are 
 settled by checks. 
 
 In commercial banks, checks may be drawn against a 
 deposit at any time, but they must not exceed, at any time, 
 the depositor's balance. Checks are drawn payable (1) '^To 
 
 , or order" — in this case,- being negotiable; (2) "To 
 
 Cash" — as a rule payable to the depositor himself; (3) "To 
 Bearer" — payable to anyone presenting the check; or (4) 
 
 "To ," payable only to the person named. 
 
 Depositors desiring to draw money for a specific use, as for 
 example, to pay wages, may draw checks to the order of the 
 particular expense, thus " Pay to the order of Pay Roll." 
 Such checks are cashed by the depositor, or after endorsement, 
 by his special representative. In some localities this form of 
 check is now forbidden. 
 
 As in the case of notes (pp. 307, 308) checks may be trans- 
 ferred by endorsement to others before being finally cashed or 
 deposited. 
 
 Business men deposit in their own banks, practically in the 
 
DOMESTIC AND FOREIGN EXCHANGE. 359 
 
 same manner as cash, the checks received by them from others 
 and drawn on local or out-of-town banks. The banks re- 
 ceiving these checks on deposit must collect payment from 
 the banks on which they are drawn. To avoid the necessity 
 of sending messengers daily to all other local banks on which 
 they, hold checks, to make collections, banks form local 
 clearing houses at w^hich representatives of each bank meet 
 daily at a specified hour. Here, in a systematic way, all 
 checks drawn against a given bank, and brought by repre- 
 sentatives of other banks are charged against it, and all 
 checks it presents against other banks are credited to it. 
 Balances are then struck. In each case where the debits 
 exceed the credits, the bank concerned pays the debit balance 
 to the clearing house officials before a certain time. Immedi- 
 ately after the receipt of this money from the banks that are 
 said to owe the clearing house, it is paid out to the banks 
 that have a credit balance. The clearing house thus begins 
 and ends the day without money. By means of this insti- 
 tution balances of millions are sometimes settled by the 
 actual payment of only a few thousand dollars. 
 
 Illustration. Rosslyn has three banks. On a certain day their 
 representatives meet. The First National Bank presents $12,000 in 
 checks against the Second National Bank, and $18,756.42 against the 
 Third National Bank. The Second National Bank presents $19,540 in 
 checks against the First National Bank and $15,126.10 against the Third 
 National Bank. The Third National Bank presents $14,1.18 in checks 
 against the First National Bank and $11,121.70 against the Second National 
 Bank. The Clearing House credits each bank with the checks it presents 
 and charges it with the checks drawn on it and presented by other banks, 
 thus: 
 
 Cr. Bal. 
 
 $11,544.40 
 
 Bank. 
 
 Charges. 
 
 Credits. 
 
 Dr. Bal. 
 
 First Nat. Bank 
 
 $33,658.00 
 
 $30,756.42 
 
 $2,901.58 
 
 Second Nat. Bank 
 
 23,121.70 
 
 34,666.10 
 
 
 Third Nat. Bank 
 
 33,882.52 
 
 25,239.70 
 
 8,642.82 
 
 $11,544.40 $11,544.40 
 The Clearing House collects the amounts of the debit balances against 
 the First and Third National and pays them over to the Second National 
 Bank. 
 
360 BUSINESS ARITHMETIC. 
 
 The checks that each bank receives from the clearing 
 house are taken back by its messenger and charged against 
 the accounts of the persons drawing them. 
 
 Each bank receives, with other deposits, many checks 
 drawn on out-of-town banks. It may collect these by mail, 
 direct from the banks on which they are drawn, or, as is the 
 common custom, it may collect them through its correspondent 
 bank or banks in other localities, commonly in the larger cities, 
 with which it keeps accounts. In the same way, banks in the 
 villages and towns of its neighborhood, and banks of other 
 cities, may keep accounts with it. 
 
 Illustration. Peter Morton draws a check on his bank, The Idaho Na- 
 tional Bank of Washington, in favor of James Rawson of Newark, N. J., and 
 sends it to the latter in settlement for a purchase of merchandise. James 
 Rawson deposits the check to the credit of his account in his home bank, 
 the First National. The First National Bank credits Rawson and sends 
 the check to its correspondent bank nearest Washington, the Fortieth 
 National Bank of Baltimore, charging that bank. The Fortieth National 
 Bank credits the Newark bank, and sends on the check, properly endorsed, 
 to its correspondent bank in Washington, say the Twentieth National 
 Bank. The Twentieth National Bank credits the Fortieth National Bank 
 of Baltimore, and exchanges the check with the Idaho National Bank 
 through the Clearing House. The Idaho National Bank charges the 
 check against Morton's account. 
 
 Thus it is evident that all the checks a depositor draws 
 against his account, and gives out in payment, either locally 
 or at other points, find their way back to his own bank and 
 are charged against his account. Sometimes they take very 
 roundabout paths back, but the whole history is shown by the 
 endorsements on the back. It will be found interesting to 
 study the endorsements of cancelled checks. 
 
 Of course, any two correspondent banks are continually 
 exchanging checks for collection, and their debit or credit 
 balances with each other are constantly changing. At more 
 or less periodic intervals, however, these balances are settled 
 in full or in part by transfers of actual currency. 
 
DOMESTIC AND FOREIGN EXCHANGE. 361 
 
 Collection Fees. Some banks charge for out-of-town col- 
 lections made for other than their depositors, and sometimes 
 for collections for their depositors. The fee (rate of exchange) 
 for such collections, while still more or less variable, is becoming 
 steadily more uniform, especially in the East, being frequently 
 only 1/10%. Sometimes there is a minimum charge of 10 
 cents or 25 cents. 
 
 Bank Drafts and Cashiers' Checks. It frequently happens, 
 owing to the risk of loss, that objection is made to receiving 
 the check of a private depositor. In such cases the payer 
 may have his check certified by an ofiicial of his bank on which 
 the draft is drawn. The certification is stamped across the 
 face of the check and signed by the cashier or other bank 
 oJBScer. The bank thus guarantees the check and protects 
 itself by immediately charging the check against the depositor's 
 account. Such checks are more readily accepted. In case, 
 however, a certified check is not used, it should not be de- 
 stroyed, but should be deposited to the credit of the account 
 against which it was originally drawn. 
 
 The payer may also make settlement by means of a hank 
 draft, purchased at his own bank, or at any other that will 
 accommodate him, and drawn on some correspondent bank 
 either in the neighborhood of the payee, or in some large 
 central city. Exchange on New York or Chicago is acceptable 
 anywhere. These drafts are often drawn to the order of their 
 purchaser in order that he may endorse them over to the 
 intended payee and thus secure proof of payment. The payee 
 may deposit them or transfer them by endorsement, the same 
 as checks. The charge (rate of exchange) is usually the 
 face plus a slight premium, generally 1/10%, although a bank 
 will often issue drafts to its regular customers at a cost equal 
 to the face. 
 
 If the correspondent or paying bank has funds to the credit 
 or issuing bank, the drafts are charged against these. If not, 
 
362 BUSINESS ARITHMETIC. 
 
 the issuing bank will send bank checks to its correspondent 
 for collection, or will give the bank credit and balance up later. 
 As a rule, no charge is made for the collection of bank drafts 
 when deposited for that purpose, if they are drawn on banks 
 at the large money centers. 
 
 Form of Bank Draft. 
 No. 6129 Seatde, Wash., Mar. 2, 1912. 
 
 Johnson, Morton and Co. 
 Bankers 
 
 Pay to the order of James F. Porter $826^*/^ 
 
 Eight hundred, twenty-six j%\ Dollars. 
 
 To Fifteenth National Bank, Andrew Norton, 
 
 St. Paul, Minn. Cashier. 
 
 Form of Bank Draft. 
 
 In place of a bank draft, an ordinary cashier's check on his 
 own bank may be obtained by a depositor, its cost being 
 charged against his account. Such checks, being official in 
 character, and having the weight of a bank back of them, 
 pass more readily among strangers than do private checks. 
 
 Certificate of Deposit. Instead of opening a regular checking 
 account, one may deposit money in a bank and receive therefor 
 a certificate of deposit, transferable by endorsement. This 
 form is often used when some fixed sum of money is to be 
 deposited for safe-keeping. Some savings banks and trust 
 companies will issue certificates for fixed periods, such as one 
 year, bearing interest at a certain rate, usually higher than that 
 allowed on deposits subject to check. 
 
 A Certificate of Deposit. 
 
 $6000 T^iy Chicago, Illinois, June 16, 191 — 
 
 National Bank 
 
 This certifies that Thomas Medway has deposited 
 
 Six Thousand j%% Dollars, 
 
 payable to the order of C. T. Robinson 
 
 on the return of this certificate properly indorsed. ,^ . _ ^ 
 
 Martin Kentj 
 
 Cashier. 
 
DOMESTIC AND FOREIGN EXCHANGE. 363 
 
 Commercial Drafts. Ordinary commercial drafts are fre- 
 quently used for settlements and collections. 
 
 $825tV7 Columbus, Ohio, Sept. 19, 191— 
 
 At sight Pay to the order of 
 
 Ourselves 
 
 . . . .Eight hundred twenty-five ^^^ Dollars. . 
 
 Value received and charge to the account of 
 
 To Thompson Bros. Lewis Mark & Son. 
 
 No. 67 Baltimore, Md. 
 
 Note. Thompson Bros, owe Lewis Mark & Son $825.67. When the 
 account is due Lewis Mark & Son draw a sight draft on Thompson Bros, 
 and give it to their home bank to collect. The Columbus bank sends 
 the draft to its correspondent in Baltimore. The bank in Baltimore 
 presents the draft to Thompson Bros, for settlement. When payment Is 
 received the Columbus bank is notified and it credits Lewis Mark & Son's 
 account for the amount less a small fee (collection and exchange). 
 
 Time or sight drafts are used frequently to collect for 
 merchandise at time of delivery. Goods are shipped to a 
 customer and a through bill of lading is taken from the trans- 
 portation company. This is a receipt for the goods and an 
 agreement to deliver them to the consignee, or to his order. The 
 selling firm also makes out a sight draft on the customer, if 
 the terms are cash on delivery, or a time draft, if credit for a 
 certain period is to be allowed. The draft and bill of lading 
 are fastened together and given to the seller's bank. The bank 
 endorses the draft and sends it to its correspondent bank in 
 the neighborhood of the purchaser. This bank presents the 
 draft to the customer for payment, if a sight draft, or for 
 acceptance if a time draft. When payment or acceptance has 
 been made, the bank surrenders the bill of lading and the 
 customer may secure his goods. Time drafts frequently are 
 left in the bank for presentation again when payment becomes 
 due. 
 
 Rates of Exchange. Postal and express money orders cost 
 more than their face value. Bank drafts frequently sell 
 at, or above, their face value. Bank and commercial drafts 
 
364 BUSINESS ARITHMETIC. 
 
 may sell for less than their face value. Exchange is at par 
 if the cost of a draft equals its face; it is above par, or at a pre- 
 mium, if the cost exceeds the face; it is below par, or at a dis- 
 count, if it sells for less. Exchange on small sums is usually 
 at a premium to repay cost of handling. The rate of exchange 
 varies with the demand for money at certain centers of trade 
 and with the cost of shipment of actual currency. 
 
 Illustration. In the harvest season for grain, eastern purchasers 
 are sending checks west in settlement for purchases. Suppose, for example, 
 that large orders are being received in Chicago from New York customers. 
 Checks and drafts drawn on New York banks and presented or deposited 
 in Chicago banks will tend to seriously decrease, or to more than exhaust, 
 the balances to the credit of the New York banks. ^ At such times exchange 
 on Chicago will be selling at a premium in New York, because the New York 
 banks selling such exchange will be under the necessity of shipping actual 
 currency to Chicago to meet the drafts. 
 
 At the same time, however, Chicago purchasers will find exchange on 
 New York below par, because Chicago banks have so much due them from 
 New York, that drafts on the latter city may be met easily without the 
 transfer between the cities of the actual currency. The sale of such drafts 
 really benefits Chicago by lessening the balance due from New York to 
 Chicago. 
 
 In spring, when manufactured and imported goods are going out from 
 New York all over the country, and merchants all over the country owe 
 New York dealers and manufacturers, conditions are reversed. There 
 is such a demand for exchange on New York that the rate of exchange is 
 almost sure to go above par. 
 
 The rate of exchange seldom if ever goes above the rate 
 for shipping the actual money. If the cost of shipping money 
 between two given cities is $10 per $10,000 the rate of premium 
 will not go above 1/10%. 
 
 Collection rates for drafts are somewhat arbitrary, being gov- 
 erned by general trade conditions. As a rule, however, bank 
 drafts on the large money centers are collected without charge. 
 
 EXERCISE. 
 1. Secure a bank deposit slip. Find out how to make a deposit. 
 Explain the process of opening an account and making the original deposit. 
 
DOMESTIC AND FOREIGN EXCHANGE. 
 
 365 
 
 2. If you owed a man in Albany a payment of $760, which would 
 probably be the cheapest safe way of making settlement? 
 
 3. C. Y. Portner collects through his bank, at a charge of 1/10% for 
 collection, checks on New York for $4.56.89 and $887.50 and on Phila- 
 delphia for $1,456. What is the net sum to be credited to his bank account? 
 
 4. Extend the following form showing collections made by one bank 
 for its correspondent bank. Collection 1/10%. 
 
 Fortieth National Bank. 
 
 , N. Y., September 15, 191—. 
 
 Mr. Robert T, Brown, Cashier. 
 
 Washington, D. C, Ninth National Bank, 
 Dear Sir: 
 
 We have credited your account this day for the proceeds of the collections 
 stated below. j^^^ Fadelet, Cashier. 
 
 Your No. 
 
 Payer. 
 
 Amount Collection. Proceeds. 
 
 456 
 
 Robert Chase 
 A. B. Nortins 
 Bruce & Co. 
 Henry Ames 
 Besley Bros. 
 
 480 
 1239 
 2311 
 
 477 
 5465 
 
 00 
 
 80 
 
 45 
 
 62 
 
 00 
 
 - 
 
 
 461 
 
 
 
 511 
 
 
 
 512 
 
 
 
 589 
 
 .... 
 
 .... 
 
 5. Banks A, B, C and D form a Clearing House. On Sept. 16, 
 Bank A presents checks against Bank B for $12,178.80; against Bank C 
 for $4566.75; against Bank D for $56,124.59. Bank B presents checks 
 against Bank A for $45,000; against Bank C for $34,123; against Bank D for 
 $19,234.59. Bank C presents checks against Bank A for $13,543; against 
 Bank B for $24, 118.25 5 and against Bank D for $7,188.55. Bank D pre- 
 sents checks against Bank A for $12,325; against Bank B for $35,455.60 
 and against Bank C for $41,157.20. Find the debit or credit balance of 
 each bank with the Clearing House and determine what payments must 
 be made to settle all balances. 
 
 6. What is the net sum placed to the credit of John Grandfield on 
 deposits of checks for $435, $350, $527.50 and $56.88, on which a collection 
 fee of 1/8% is charged? 
 
 7. Compute the cost of a bank draft on Richmond, Va., for $723.90, 
 at 1/10%. 
 
 8. How large a draft may be purchased for $1127.13 at 1/10% pre^ 
 mium? 
 
 9. Determine the proceeds from the collection through an express 
 company of an account of $2,356, at a rate of $1.75 per $1,000. 
 
366 BUSINESS ARITHMETIC. 
 
 10. Draw up a check on an assumed bank to pay for a bank draft of 
 $2264 at 1/5% premium. 
 
 11. Robert Gaines sold H. M. Chilton a bill of goods amounting to 
 $1764.80. Gaines drew a draft in settlement and sent it, with the bill of 
 lading for the goods, through his bank. What sum was added to the credit 
 of his account when the bank had collected the draft, its charge being 1/5%? 
 
 12. A. C. Crandall sold a bill of goods for $7500 to George Grimes, the 
 terms being "30 day acceptance." • Crandall drew the draft and sent it 
 through his bank for acceptance. After acceptance, the bank discounted 
 the draft for Crandall at 1/10% and 6% interest. What proceeds did 
 Crandall receive? 
 
 13. A commercial draft for $475 cost $474.05. Was exchange at a 
 premium or at a discount? At what rate? 
 
 14. A. M. Beardsley drew a draft on John Evans, for $467.56 and 
 collected it through his bank at a rate of collection of 1/20%. What sum 
 should be credited to his account? 
 
 15. Was exchange at a premium or at a discount, and at what rate, if 
 a draft for $23,450 cost $23,567.25? 
 
 16. Boston is selling drafts on New Orleans at 3/4% premium. Which 
 way is the balance of trade? What is the cost of a draft on New Orleans 
 face value of $1245? 
 
 17. Suppose the balance of trade between Indianapolis and Pittsburg 
 is largely in favor of the latter. What docs this mean? In which city is 
 it likely that exchange will be at a premium? Why? At a discount? 
 Why? 
 
 18. Name several cities that might have a balance of trade in their 
 favor in the spring. In the fall. Give your reasons. 
 
 FOREIGN EXCHANGE. 
 
 All forms of payment possible under domestic exchange are 
 available for payments abroad. Theoretically, foreign ex- 
 change is based on the same general principles as domestic 
 exchange. In practice, however, all computations are affected 
 by differences in currency denominations and standards, and 
 by methods of quoting exchange. From the banker's stand- 
 point, foreign exchange is, in many ways, the most intricate 
 of all forms of banking. 
 
DOMESTIC AND FOREIGN EXCHANGE. 367 
 
 Rates of Exchange. Foreign rates are variously expressed, 
 but may be broadly classified under three heads. 
 
 1. Rates per foreign coin. Thus a London exchange rate of "4.875" 
 means that one pound sterling can be bought for $4,875. A rate of "40f " 
 on Amsterdam means that one guilder exchange on that city costs $.40 f. 
 
 2. Rates on a set number of coin. Exchange on German cities is usually- 
 quoted "per four marks." Thus a quotation of "96," on Berlin, means 
 that a four marks bill of exchange costs $.96. 
 
 3. Rates in "coin per dollar." This form of exchange is often quoted 
 for France and other Latin countries, as Spain, Italy, Switzerland, etc. 
 Thus a rate of 5.15 f on Paris means that a draft on Paris for 5.15 f francs 
 may be purchased for one dollar. 
 
 Note. Form (1) is sometimes used for exchange on any country 
 especially in connection with some forms of money orders. 
 Illustrations. 
 
 (1) Compute the cost of a £450 draft on London, purchased at 4.875. 
 Solution. 
 
 $4,875 Since £1 cost $4,875, £450 will cost 
 
 450 450 X $4,875. 
 
 243750 
 19500 
 
 $2193.75, Ans. 
 
 (2) Compute the cost of a draft on Berlin for 1200 M., purchased at 
 
 96. 
 Solution. 
 
 4)1200 In 1200 M. there are 300 sets of 4 M. 
 
 ~~300, no. sets 4 M. If one set costs 96c, 300 sets cost 300 X 96c. 
 $0.96 
 300 
 $288.00 cost. 
 
 (3) Compute the cost of a 12,000 fr. draft on Paris, purchased at 5.15. 
 Solution. 
 
 $ 2330.09 Since 5.15 francs cost $1.00, 12,000 francs 
 
 5.15)12000. will cost as many dollars as 5.15 is contained 
 
 1030 tmies in 12,000. 
 
 1700 
 1545 
 1550 
 1545 
 5000 
 4635 
 
368 
 
 BUSINESS ARITHMETIC. 
 
 Foreign Money Orders. The International Postal Money- 
 Orders and the express orders, which are similar to domestic 
 money orders, are issued for amounts not exceeding one 
 hundred dollars, and are usually payable in the money of the 
 country on which they are drawn. Fees for this service are 
 small. 
 
 1. What will an express money order for £5 6s. cost at $4.87? 
 
 2. How large a draft on London will $50 buy, at the same quotation? 
 
 3. Robbins wishes to send a friend in Rome an order for 290.5 lira. 
 At 19.5c per lira, the order costs $ ? 
 
 4. For $40 one may buy an order on Berlin for ? M., if the rate is 24c 
 per M. 
 
 Note. Money payments may be made by cable in the same way in 
 which they are made by telegraph. In cable orders, a charge is made for 
 each word of the address. 
 
 5. At 20c per word, and 1% of the amount, find the cost of a 24-word 
 cable order on London for £600, at a rate of exchange of $4.8725. 
 
 Travelers' Checks. Travelers may secure from express 
 companies and banks travelers' checks, which can be cashed 
 
 Form of Traveler's Check. 
 
 at hotels and banks abroad. They are issued for fixed amounts 
 and state on their face their value in the coinage of the differ- 
 ent countries in which they may be presented for "cashing." 
 
DOMESTIC AND FOREIGN EXCHANGE. 369 
 
 The purchaser signs his name on the face at the time of 
 purchase, and again on face or back, for purposes of identi- 
 fication, at time of cashing. The checks are sometimes issued 
 in books or series. 
 
 Letters of Credit. A traveler's letter of credit is an 
 instrument issued by a bank to its representatives abroad, 
 authorizing them to honor drafts of the person named in 
 the body of the letter, to total not exceeding an amount 
 stated. 
 
 The person desiring a letter of credit deposits with the 
 issuing bank money or bonds as security. On receiving the 
 letter of credit, he places his .signature on the face, and also 
 gives the issuing banker other specimens of his signature, 
 which are forwarded to the correspondent banks with the 
 formal notice of the issuance of the letter. Once abroad, 
 whenever the holder of the letter desires money, he stops at 
 any correspondent bank named in the letter and presents his 
 personal draft for the amount desired. After the signature 
 has been verified, he is paid the amount in the coinage of the 
 country. This amount is indorsed, also, on the letter of credit. 
 When the "credit" called for has been exhausted, the letter 
 is taken up by the bank making the last payment, cancelled, 
 and returned to the issuing bank. 
 
 EXERCISE. 
 
 1. Determine the cost of a 12,000 franc letter of credit, at 5.18 and 1% 
 commission. 
 
 2. On a £600 letter of credit, are endorsed payments of £120, Nov. 10; 
 £50, Dec. 5; £180, Jan. 16; £125, Feb. 12. Purchased at 4.86 and 1% 
 commission, and unused portion redeemed at same rate, what repayment 
 should be made to the holder? 
 
 Commercial Letters of Credit. These letters are issued 
 to mercantile houses desiring: to import goods to enable them 
 25 
 
370 BUSINESS ARITHMETIC. 
 
 to get credit from exporters for the usual periods of 30, 60 or 
 90 days. On proper evidence of the shipment of goods, a 
 foreign bank at point of shipment, will buy the shipper's 
 draft against the person named in the letter, at the prevailing 
 rate of commercial exchange, reimbursing itself by sending 
 the draft to party drawn on for collection, or by selling the draft 
 in the open market. Credits are issued usually in pounds 
 sterling because such exchange is easily negotiated in any 
 part of the world. They are sometimes issued payable in the 
 money of the country to which they are sent. Commercial 
 credits are issued in four parts, one being sent to the exporter, 
 one to the bank named in the credit, and one to the importer, 
 or purchaser, the fourth being retained. Prices vary with 
 credit term, financial responsibility of applicant, and many 
 other conditions. The problems of computation are the same 
 as those involved in other draft transactions. 
 
 Bills of Exchange. Drafts drawn by a person or bank in 
 one country on a person in another country are termed bills 
 of exchange. They include bankers', commercial and docu- 
 mentary bills. Bankers' bills are a bank's checks, drawn 
 against its deposits in foreign banks. Commercial bills are 
 those drawn by one merchant against another. Documentary 
 bills are commercial bills, secured by bills of lading, insurance 
 policies, or other documents that control the merchandise 
 in transit. Bills of exchange were formerly issued in triplicate. 
 They are now issued, as a rule, in duplicate, so worded that 
 the payment of either one makes void the other. Sometimes 
 the ordinary single draft is used. Owing to delays and dangers 
 of transit, the duplicate drafts were formerly sent by different 
 routes or mails. Now the tendency is to keep the duplicate 
 on file. 
 
2 
 
 
 o S 
 ii ^ 
 a, « 
 
 I" 
 
 DOMESTIC AND FOREIGN EXCHANGE. 371 
 
 A Duplicate Bill. 
 
 PHiLADBLPmA, Pa., April 19, 19 — 
 2450 M. 
 
 At sight of this second of Exchange (first unpaid), pay to 
 
 the order of Ourselves 
 
 Twenty-four hundred and fifty Marks, 
 
 Value received and charge to the account of 
 To K. R. Schmidt 
 
 Berhn S. T. Coles & Co. 
 
 Simple Foreign Bill. 
 £1000— Philadelphia, March 12th, 190— 
 
 On demand Pay to the order of 
 
 John J. Doe 
 
 One thousand Pounds sterling 
 
 Value received and charge to account of 
 
 To J. S. Morgan & Co. 
 
 No. 16 London, England. John Smith. 
 
 Fluctuation of Exchange. The rate of exchange is affected 
 by numerous causes. If exported goods greatly exceed imports 
 in value, the rate of exchange tends to lower because there is 
 less demand for remittances abroad. If more money is due 
 abroad than is due the home country, the rate tends to rise. 
 When money is in demand here, and rates rise, rates of 
 exchange may fall because there is less demand for bills. If 
 money rates are high abroad, exchange rates may rise because 
 of the demand for bills in order to send money to a place of 
 good investment. In general, the question of supply and 
 demand seriously affects the quotations. 
 
 The discount rates on commercial bills, at foreign money 
 centers, also affect the rate of exchange. If rates of discount 
 are low, commercial drafts may "cash'' for a greater sum than 
 could be realized by selling them as exchange. The question 
 of the parity of units of foreign coinage, as compared with 
 the units of our own coinage, also has its effect. The exchange 
 rate will seldom go beyond the limits of the cost of buying 
 
372 BUSINESS ARITHMETIC. 
 
 and shipping the actual gold — these costs including commis- 
 sion, freight, insurance, and often interest lost while money 
 is in transit. That is, the rate of exchange does not go beyond 
 the point where it is cheaper to send the actual money. 
 
 Newspaper quotations are given, usually, for both sight 
 and time paper. Par of exchange is either (1) mint par — the 
 equality of actual value of pure metal in units of money in the 
 countries concerned ; or (2) commercial par — a rarely used term 
 referring to a practical equality of debt between countries that 
 reduces the rate of exchange to the face value plus the cost 
 of shipping. 
 
 ILLUSTRATIONS OF EXCHANGE. 
 
 1. ^, in New York, sells B in London merchandise, $600. A draws his 
 draft on 5, in pounds sterling. He may then discount the draft at a bank; 
 or he may sell it in open market to some party who wishes to make a pay- 
 ment in England. In the former case, the bank either sends the draft to 
 England for collection — in which case the money is probably held by the 
 correspondent bank in England, and the home bank draws and sells its 
 bank exchange against the sum, and like sums, to any purchaser — or it 
 may sell the original draft to a customer. In case A sells the original draft 
 direct, or through a broker, the party buying it, C, who may owe D in 
 England, sends the draft, properly endorsed, to the latter, who presents 
 it to B for payment. By such a general process, it is evident that A gets 
 his payment at once, and that sooner or later B pays the draft on pres- 
 entation and thus settles for the goods that he bought. Incidentally, the bill 
 has served the further purpose of making settlements between other parties. 
 
 2. In case A had been the buyer and B the seller, B might carry through 
 the same process shown above. The settlement might start with A, 
 however, who might go to a bank or broker, and buy a draft on England 
 for the amount due, sending it to B who would cash it. In this case A 
 corresponds to C, in ex. (1). 
 
 Notice, in each case, that this process permits each person to make or 
 to receive settlement in the money of his own country. 
 
 ORAL EXERCISE. 
 
 1. In what coinage would an American importer order of a French 
 exporter? If he bought exchange, with which to make settlement, in 
 what coinage would it read? With what money would he make settlement 
 for the draft? 
 
DOMESTIC AND FOREIGN EXCHANGE. 373 
 
 2. An English dealer orders goods from the catalogue of an American 
 manufacturer. He finds prices quoted in what coinage? If he pays by 
 bank draft, the draft is made out in what money? If he is drawn upon, 
 in settlement, the draft is made out in what money? If the manufacturer 
 sells the draft in New York he will receive money of what country? 
 
 How would you determine: 
 
 3. The cost of a draft on Paris, at 5.19? 
 
 4. The cost of a draft on London at 4.88? 
 
 5. The cost of a draft on Berlm at .958? 
 
 6. The proceeds from the sale of a draft on Lyons at 5.185? 
 
 7. The proceeds from the sale of a draft on Amsterdam at 40$%? 
 
 8. The proceeds of a draft on Berlin at 96i? 
 
 9. Which are the higher rates, 5.19 or 5.20? 96 or 95^? 
 
 EXERCISE. 
 
 (For Mint exchange rates, see page 281.) 
 
 1. What is the face of a draft that an English exporter should draw 
 on a New York customer, at mint par, to cover the purchase of 240 yd. 
 broadcloth at 0/14/6 per yd.? If the exporter sells this draft in the market 
 for 4.87, what proceeds does it yield him? 
 
 Compute the cost of drafts on: 
 
 2. Paris, for 9700 francs at 5.18 
 
 3. Berlin, for 12,500 M. at .962. 
 
 4. Glasgow, for £161 4s., at 4.87|. 
 
 5. Amsterdam, for 752 g., at .405. 
 
 6. Rome, for 768 L. at .238. 
 
 7. Geneva, for 1284.6 fr., at 5.18|, brok. 1/8%. 
 
 8. Is the balance of trade probably for or against this country if ex- 
 change is quoted at 5.2, 4.88, .412? 
 
 Exchange Rates. 
 (From a daily newspaper.) 
 
 Sight. 60 days 
 
 Cable transfers 4.87? 
 
 Bankers' sterling 4.87 4.85^ 
 
 Commercial bills 4.84^ 
 
 Francs 5.161 5.171 
 
 Reichsmarks ; " 95| .95i 
 
 Guilders 40| .40i 
 
374 BUSINESS ARITHMETIC. 
 
 Note. Use the newspaper quotations on page 373 in the following: 
 
 9. James Bros. & Co. purchase an invoice of £2000 worth of woolens, 
 paying by 60-day bank draft. Cost of draft? 
 
 10. How large drafts would $500 buy at each of the above quotations? 
 
 11. A. B. Norton drew on a Paris customer a draft for 12,500 fr. and 
 sold it at 1/8% commission, receiving what amount? 
 
 12. What does it cost to cable an agent £120? 
 
 13. B. C Baker shipped S. Tournier, Paris, 5000 bu. wheat, invoiced 
 at 4.8 fr. drawing a 30 day draft on the purchaser. This draft, accom- 
 panied by bill of lading and insurance certificate, he sold to Mason and 
 Dean, private bankers, at 5.19, and should receive what sum? 
 
 14. Mason & Dean sold the draft just mentioned to Robert Bros., at 
 5.18, receiving $ — , and making a profit of $ — 
 
 15. Robert Bros, mailed the same draft, in settlement of account, to a 
 Paris customer, who discounted it at a bank at 1%, receiving — fr. The 
 bank presented the draft to Tournier(ex. 13) who was allowed to settle for 
 it instead of accepting it, at 1/2% discount. He then received the bill 
 of lading that entitled him to the wheat. 
 
CHAPTER XLI. 
 
 DEPRECIATION. 
 
 Depreciation is the loss of value of any property due to weaj*, 
 new inventions, exhaustion of product, or other causes. It 
 is the difference between original value and marketable value 
 at time of disuse. 
 
 Illustrations. (1) In spite of repairs, a factory or piece of machinery- 
 deteriorates to the point where it must be discarded or rebuilt. This 
 deterioration is depreciation. (2) New inventions or improvements may- 
 make old machinery valueless. (3) A lessening of demand for a product 
 may cause the disuse of machinery, and a consequent lessening in value. 
 (4) A mine deteriorates from exhaustion of its ore, etc. 
 
 To avoid the necessity of meeting the entire cost or replace- 
 ment in any one year, it is customary in large concerns, to 
 allow for depreciation, setting aside each year during the 
 probable useful life of the property, sufficient funds to equal 
 the cost of replacement when necessary. The sum set aside 
 is largely a matter of scientific guess-work based on past 
 experience. It is considered a running expense and may 
 amount to 30% of such expenses. Allowing for individual 
 variation, the following table shows common rates. 
 
 Annual Rates op Depreciation. 
 
 Modern fireproof steel and tile factories 4% to 5%. 
 
 Steel construction, partly fireproof 6%. 
 
 Warehouses, modem, fireproof 2^% to 4%. 
 
 General freehold buildings, office and general use 1|% to 3%. 
 
 Cheaply constructed building, according to use 8% to 15%. 
 
 Fixtures of power and lighting plants 12% to 15%. 
 
 General fixtures of buildings, other than machinery 6% to 15%. 
 
 Machinery (average 11§%) 8% to 15%. 
 
 Horses, work 15% to 25%. 
 
 375 
 
376 BUSINESS ARITHMETIC. 
 
 Annual amounts of depreciation are computed (1) as a fixed 
 proportion of the original value; (2) as a fixed per cent, of 
 reduced values; or (3) by the sinking fund method. 
 
 1. Based on Original Value. 
 
 Illustration. A machine in a cotton mill, costing $560, has an 
 average working life of 10 years, when it is disposed of for $120. Determine 
 the annual allowance for depreciation. 
 
 Solution. Depreciation = Original cost — disuse value = $560 — $120 
 = $440. Since this must be covered in 10 years the annual depreciation is 
 1/10 of $440, or $44. 
 
 EXERCISE. 
 
 1. Compute the annual depreciation at 2f %, on a fireproof warehouse 
 costing originally $364,200. 
 
 2. Compute the annual depreciation on $340,200 worth of fixed ma- 
 chinery, at 12%, and on tools originally worth $3640, at 18%. 
 
 3. A department store owner has 40 dehvery horses originally costing 
 $5680. He reckons the depreciation, at 22^%, as $ . 
 
 4. A boiler, costing $520, is estimated to have a life of 12 years, and 
 then to be worth $60. Estimate annual depreciation and rate of depre- 
 ciation. 
 
 5. A contractor employs a $14,500 dredge in contract work. In com- 
 puting cost of work, as a basis for bids, he includes interest on the cost of 
 the dredge and depreciation. He reckons interest at 6% per annum and 
 depreciation 1|% per month. What should he" allow for interest and 
 depreciation on a contract that he estimates will take 7 5 months? 
 
 II. Based on Reduced Values. 
 
 With machinery, for example, repairs grow heavier with age. Hence, 
 with uniform depreciation and increasing repairs, the working cost is 
 heaviest when the machine is least useful. To equalize the yearly cost 
 therefore, depreciation is often heavier when repairs are light, decreasing 
 as repairs grow heavy. Depreciation of this type is often expressed as 
 a fixed per cent, of the net value of the previous year. 
 
 Illustration. Compute the third year's depreciation, at 10% on 
 reduced values, of machinery costing $900. 
 
DEPRECIATION. 377 
 
 Solution, 
 
 $900 Orig. value 
 
 90 Depreciation at 10%. 
 810 Net value, end 1st year. 
 81 Dep. 10%. 
 729 Net value 2d year. 
 72.90 Dep. 10%, for 3d year. Ans. 
 Note. If term and rate of depreciation are known, tables similar to the 
 following may be computed for shop use. 
 
 Table of Depreciation per $100 op Original Value. 
 
 Term Year. 
 
 2% 
 
 3% 
 
 4% 
 
 5% 
 
 10% 
 
 15% 
 
 20% 
 
 lyr. 
 
 2.00 
 
 3.00 
 
 4.00 
 
 5.00 
 
 10.00 
 
 15.00 
 
 20.00 
 
 2 
 
 1.96 
 
 2.91 
 
 3.84 
 
 4.75 
 
 9.00 
 
 12.75 
 
 16.00 
 
 3 
 
 1.92 
 
 2.82 
 
 3.69 
 
 4.51 
 
 8.10 
 
 10.84 
 
 12.80 
 
 4 
 
 1.88 
 
 2.74 
 
 3.54 
 
 4.29 
 
 7.29 
 
 9.21 
 
 10.24 
 
 5 
 
 1.85 
 
 2.66 
 
 3.40 
 
 4.07 
 
 6.56 
 
 7.83 
 
 8.19 
 
 6 
 
 1.81 
 
 2.58 
 
 3.26 
 
 3.87 
 
 5.90 
 
 6.66 
 
 6.55 
 
 7 
 
 1.77 
 
 2.50 
 
 3.14 
 
 3.68 
 
 5.31 
 
 5.66 
 
 5.24 
 
 8 
 
 1.73 
 
 2.42 
 
 3.00 
 
 3.49 
 
 4.78 
 
 4.81 
 
 4.19 
 
 EXERCISE. 
 
 1. Compute the depreciation for the sixth year, at 8% on reduced 
 balances, on fixtures originally worth $8240. 
 
 2. A factory costing $82,600.00 is five years old. What is its inventory 
 value, if depreciation is reckoned on reduced balances at 4|%? 
 
 By Table. 
 
 3. Reckon the depreciation for the eighth year, on machinery worth 
 originally $6525, if the depreciation rate is 10%. 
 
 4. What is the total loss of value in seven years, of a boiler costing 
 $1800, and depreciated 15% annually? 
 
 5. Compute the inventory value at the end of 8 years, of an equipment 
 of cotton mill machinery, costing originally $38,250, and depreciated at 
 20% annually. 
 
 6. Determine a rate of depreciation for machinery that is expected to 
 depreciate from $900 to $630 in 7 years. 
 
 III. Sinking Funds. 
 Depreciation is sometimes provided for by establishing a sinking fund, 
 that is a fund which is invested at compound interest at the rate necessary 
 
378 
 
 BUSINESS ARITHMETIC. 
 
 to provide for renewals. Frequently this fund is based on annual invest- 
 ments of equal amounts. Business establishments seldom establish a sink- 
 ing fund, since, when prosperous, funds earn a higher rate by being con- 
 tinued in the business, than by outside investment. 
 
 Sinking Fund Table. 
 Shxywing Accumulations oi Compound Interest of Annual Investments of 
 
 One Dollar. 
 
 Year 
 
 At 4^ 
 
 At 5^ 
 
 At 6^ 
 
 Year 
 
 At45< 
 
 At6?t 
 
 At6)t 
 
 1 
 
 1.040000 
 
 1.050000 
 
 1.060000 
 
 11 
 
 14.025805 
 
 14.917127 
 
 15.869941 
 
 2 
 
 2.121600 
 
 2.152500 
 
 2.183600 
 
 12 
 
 15.626838 
 
 16.712983 
 
 17.882138 
 
 3 
 
 3.246464 
 
 3.310125 
 
 3.374616 
 
 13 
 
 17.291911 
 
 18.598632 
 
 20.015066 
 
 4 
 
 4.416323 
 
 4.525631 
 
 4.637093 
 
 14 
 
 19.023588 
 
 20.578564 
 
 22.275970 
 
 5 
 
 5.632975 
 
 5.801931 
 
 5.975319 
 
 15 
 
 20.824531 
 
 22.657492 
 
 24.672528 
 
 6 
 
 6.898294 
 
 7.142008 
 
 7.393838 
 
 16 
 
 22.697512 
 
 24.840366 
 
 27.212880 
 
 7 
 
 8.214226 
 
 8.549109 
 
 8.897468 
 
 17 
 
 24.645413 
 
 27.132385 
 
 29.905653 
 
 8 
 
 9.582795 
 
 10.026564 
 
 10.491316 
 
 18 
 
 26.671229 
 
 29.539004 
 
 32.759992 
 
 9 
 
 11.006107 
 
 11.577893 
 
 12.180795 
 
 19 
 
 28.778079 
 
 32.065954 
 
 35.785591 
 
 10 
 
 12.486351 
 
 13.206787 
 
 13.971643 
 
 20 
 
 30.969202 
 
 34.719252 
 
 38.992727 
 
 EXERCISE. 
 
 1. What is the sum that must be invested annually in a 4% sinking 
 fund to provide for replacing a $12,000 equipment of machinery in 10 
 years, if the estimated value of the equipment at the end of the period is 
 $1950? 
 
 2. What allowance for depreciation should be made annually for 
 machinery that is expected to deteriorate in value $6750 in 5 years, when 
 it must be replaced? The sinking fund is invested at 6%. What at 5%? 
 
 3. A certain factory building is estimated to have a life of 20 years. 
 It is built at a cost of $46,000. Estimate uniform annual depreciation. 
 Compute annual allowance for sinking fund at 6%. 
 
CHAPTER XLII. 
 COST-KEEPING. 
 
 Knowledge of the exact cost of the articles that he makes is 
 of vital importance to a manufacturer. Owing to slight margins 
 of profit, carelessness in estimating materials and labor, or 
 needless waste, may lead to large losses. 
 
 The determining of the exact cost of an article, or piece of 
 work, is termed cost-keeping. It necessitates (1) the exact 
 measurement of the material used and wasted, (2) the com- 
 puting of the time spent on the work by each workman, 
 and (3) the computation of the exact cost of material and 
 labor. The labor cost is often greater and more variable than 
 the material cost, and is considered the vital element in cost- 
 keeping. 
 
 Note. The computation of the cost of production is often exceedingly- 
 involved, and varies with each class of work, but the fundamental principles 
 are very simple and are worth considering, since the subject is becoming 
 a matter of general discussion, and of frequent mention in current pul> 
 lications. 
 
 Illustration. A printing office receives an order for 2000 booklets 
 of a specified character. At once a dupHcate docket (record sheet) is 
 prepared (page 380) to show full particulars of cost. The original docket 
 accompanies the work from start to finish, quantities and costs being 
 entered at each stage, as determined, the form being completed when 
 the work is finished. Each workman keeps his separate ticket, and marks 
 on it the hours that he spends on each job. These tickets are turned into 
 the office at night, and the time and labor costs transferred to the dupli- 
 cate docket here on file. Labor costs are also summed up and entered on 
 the original docket as it passes from hand to hand. The duplicates thus 
 serve as checks. Each job, however, must bear its share of the expense 
 of the office force, foremen, etc. (the non-productive labor), and its share of 
 rent, wear on machinery, lighting, etc. (biu-dens) . This expense, after long 
 experience, is expressed as a per cent, of direct cost, or of labor cost only. 
 To this, if no advance price has been quoted, is added the per cent, of 
 profit. The completed cost cards are filed to serve as a basis for estimating 
 on future work. Errors, carelessness, and excessive cost of labor, may also 
 
 379 
 
380 
 
 BUSINESS ARITHMETIC. 
 
 be traced from them. Many improvements are suggested by a study of 
 cost cards. 
 
 1 
 
 JOB WORK DOCKET 
 
 
 
 
 
 fa/m^ ^ jV<yyl(yn ^om. ^9, i/9i/2 
 
 oue ^'^/26 
 
 726 ^^t.. jv. ir. 
 
 
 
 DESCRIPTION 
 
 CHARGFD ^^<^U. 2 G , /^/i* ^MOMK-T S 75.00 
 
 STOCK SUPPLIED WITH DOCKET 
 
 ITEMS 
 
 TOTALS 
 
 '/O 
 
 OS 
 
 
 
 i^00J2S ,^. /^ ^jc^^.^ %^0.00 
 
 F 
 
 
 F 
 
 
 R Q S $ 
 
 
 
 
 
 R Q S $ 
 
 
 
 
 
 COVERS $ 
 
 
 
 
 
 PRINTING 
 t COST 
 
 1- 
 
 COMPOSITION 
 
 /J? 
 
 08 
 
 
 
 PRESSWORK 
 
 7 
 
 24 
 
 
 
 INK 
 
 / 
 
 €0 
 
 F 
 
 
 ROM WORK 
 
 § i 
 
 H Z 
 
 LABOR 
 
 A 
 
 48 
 
 
 
 MAIL AND POSTAGE 
 
 5 
 
 
 
 
 STOCK 
 
 2 
 
 00 
 
 F 
 
 
 OUTSIDE WORK 
 COST 
 
 2 ^Ua^oi. l/.AO 
 
 2 
 
 80 
 
 
 
 
 TOTAL COST 
 
 F 
 
 
 
 FOREMAN PRINTERV FOREMAN BINDERY 
 
 EXERCISE. 
 
 1. Extend the job work docket, shown above. 
 
 2. Compute the cost of a box requiring 2 pc. wood, 1X6X8; 2 pc. 
 1/2X6X12, and 2 pc. 1/2X8X12; waste 4%. Cost of material, 2fc per 
 ft.; labor, sawing, etc., $.006 per ft.; assembling $.025 per bx.; indirect 
 
 3, 20% of labor. (Reckon values to four decimal places.) 
 
COST-KEEPING. 
 
 381 
 
 3. Complete the following card. 
 
 
 
 
 
 
 
 NO. PIECES NAME OF ARTICLE 
 
 ff^O ^_ 41= €78 
 
 SHOP ORDER 
 
 730e 
 
 ORDERED ^^^^ 
 
 J^a9i. S7. 1/9/2 Ja<n.. S9 
 
 FINISHED 
 
 DEPARTMENT 
 
 OPER. 
 
 HOURS 
 
 RATE 
 
 COST 
 
 MATERIAL 
 
 COST 
 
 
 
 9 
 
 ■/S 
 
 ? 
 
 
 \450'WrJn/' 9.00 
 
 /3 
 
 05 
 
 // 
 
 C/ncr^iin^ 
 
 /2 
 
 ./2 
 
 :? 
 
 
 .^//e^ 
 
 4 
 
 3S 
 
 // 
 
 CUc^icU'na. 
 
 /e 
 
 .09 
 
 y 
 
 
 ^/J^ ^^^^ 
 
 5 
 
 99 
 
 ^nA/iin^ 
 
 ^aic^g. 
 
 ^2^/2 
 
 .20 
 
 p 
 
 
 iiUe 
 
 
 87 
 
 
 Jcc^ftiiA. 'a 
 
 1/3 
 
 .25 
 
 ? 
 
 
 
 
 
 
 0>o/is/,'a, 
 
 /5% 
 
 .30 
 
 .^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 MATERIAL COST 
 
 :? 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 *SHOP BURDEN 
 
 ? 
 
 
 
 
 
 
 
 
 COST PER PC. MAT. 
 
 ? 
 
 
 
 
 
 
 
 
 COST PER PC. LABOR 
 
 F 
 
 
 LABOR COST — 630 PC. 
 
 
 p 
 
 
 / 
 
 
 COST PER PIECE 
 
 F 
 
 - 
 
 '/2 
 
 Woo/ /a^^o-^c 
 
 4. Complete the following card. 
 
 
 
 
 
 
 
 ARTICL. 3000 
 
 FORSBERG STEEL 
 
 DATE 
 
 CO. 
 
 
 t/f^ <ffOcAels „„ 
 
 ^— 673^ 
 
 
 QUANTITY 
 
 OPERATION 
 
 TIME 
 
 RATE 
 PER 100 
 
 NO. OF 
 OPERATOR 
 
 
 3000 
 
 PMoc/t- 
 
 20A 
 
 .45 
 
 72/ 
 
 F 
 
 
 3000 
 
 PJi/^^l 
 
 32A 
 
 .52 
 
 483 
 
 F 
 
 
 3000 
 
 .■ri/yi/A/l. 
 
 ^5A 
 
 .90 
 
 52/ 
 
 p 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^oal /8 i /teiK A^. 
 
 p /x. 
 
 F 
 
 
 
 7000* Alee^ 
 
 2^/ji^ 
 
 9 
 
 
 
 ^^t^^ntn.&it , ■/23 A,^. 
 
 /./8 
 
 F 
 
 
 
 0>,.^.J. 7 ^^. 
 
 .80 
 
 F 
 
 
 
 .^... 7 A*. 
 
 .25 
 
 F 
 
 
 .^/^y 
 
 F 
 
 
 -»- 35 Vo 
 
 p 
 
 
 ^^^ 
 
 F 
 
 
 
 F 
 
 
382 BUSINESS ARITHMETIC. 
 
 Note. In the form just shown, operations axe paid for on the "piece" 
 plan, and coal and use of machinery are charged by the hour, previous ex- 
 periments having determined what is a fair charge. 
 
 The construction engineer, or contractor, estimates costs 
 per unit of finished work, or per day, and holds these unit costs 
 as a basis for future estimating. 
 
 EXERCISE. 
 
 1. A gang of seventeen men, working on a concrete construction, 
 averaged 46 cu. yd. of concrete per day. Estimate labor costs per day 
 and per cubic yard, based on the following organization: 
 
 Per day. Per cu. yd. 
 
 5 men loading barrows, at $1.75 per day, $ . — $ . — 
 
 9 men mixing and placing, at $1.75 per day, . — . — 
 
 2 men tamping, at $1.75 per day, . — . — 
 
 1 foreman, 3.00 . — 
 
 2. A gang of men averaged 42 lin. ft. of concrete conduit (3/4 cu. yd. 
 per lin. ft.) per day. Estimate the labor cost per day and per cu. yd. 
 
 12 men mixing, at $1.80 
 
 4 helpers ". 1.50 
 
 6 loading stone and sand 1.50 
 
 6 wheeling 1.50 
 
 8 pouring concrete 1.80 
 
 2 supplying water 1.50 
 
 2 placing metal 2.35 
 
 1 sub-foreman 2.75 
 
 1 foreman ' 3.50 
 
 3. Estimate the unit cost per cubic yard for 260 cu. yd. of concrete, 
 laid in seven 8-hour days. 
 
 Materials and Labor. 
 0.85 bbl. cement, per cu. yd., at $2.90 per bbl. 
 
 .25 cu. yd. sand, per cu. yd., at $1.12. 
 
 .50 cu. yd. broken stone, per cu. yd., at $.80. 
 
 .40 cu. yd. rubble, at $.55. 
 Water, for the entire quantity, $8.00. 
 Labor, 9 men, at 18c per hour, each. 
 Foreman, $3.60 per day. 
 
CHAPTER XLIII. 
 BIDS AND ESTIMATES. 
 
 Those desiring work done, or material supplied, often 
 advertise for seale'd proposals, or bids, from such contractors 
 as care to meet their requirements. These bids are based on 
 specifications supplied by the advertiser, which state in detail 
 the work to be accomplished, liow it is to be done, the time of 
 completion, the character of material to be used, the conditions 
 as to the acceptance of the completed work, payments, etc. 
 
 The prospective bidder makes an estimate of the cost of the 
 work, allows for his profit and prepares his bid in due form, 
 usually on a blank supplied with the specifications. The 
 sealed proposals are opened, frequently in the presence of all 
 bidders who care to attend, on a fixed date, and the contract 
 awarded. While the lowest bidder, if .reliable, frequently 
 secures the contract for the work, parts of the work may be 
 given to different contractors. The proposal often names 
 prices for parts of the work, or for the work as a whole. 
 Usually the bid must be accompanied by a certified check 
 for a certain percentage of the amount bid, as security for the 
 acceptance and performance of contract, if tendered; or a 
 bond must be given as security. 
 
 Model Proposal for a Gravel Road, 
 proposal. 
 
 To THE Board of Chosen Freeholders, County of 
 
 AND State of New Jersey: 
 Gentlemen — The undersigned hereby declare. . .that. . . .he. . . .ha. . . . 
 carefully examined the annexed specifications and the drawings therein 
 referred to, and will provide all necessary machinery, tools, apparatus 
 and other means of construction, and do all the work and furnish all the 
 material called for by said specifications in the manner prescribed by the 
 
 383 
 
384 BUSINESS ARITHMETIC. 
 
 specifications and the requirements of the Engineer and Supervisor under 
 them, for the following prices: 
 
 (1) Price per cubic yard for earth excavation, without classification, as 
 
 per plans and cross-sections, throughout the length and width of 
 the road per cubic yard. 
 
 (2) Price per acre for grubbing and removing objectionable material from 
 
 sidewalks per acre. 
 
 (3) Price per lineal foot for completed tile drain per lineal foot. 
 
 (4) Price per cubic yard for compacted gravel as specified 
 
 per cubic yard. 
 
 (5) Price per cubic yard for carting gravel more than one half mile and 
 
 each additional half-mile or fraction thereof 
 
 per cubic yard. 
 
 (6) Price per cubic yard for stripping or removing earth from top of 
 
 gravel bed per cubic yard. 
 
 {7) Price per square yard for each ordered inch in depth in excess of 
 
 thickness named per square yard. 
 
 (8) Price (lump) for the whole road complete, according to the specifica- 
 tions and plans prepared by the Engineer 
 
 Accompanying this proposal is a certified check for the sum of one 
 thousand ($1,000) dollars, payable to the order of the Director of the Board 
 
 of Chosen Freeholders of .County; which 
 
 check is to be forfeited as hquidated damages, if, in case this proposal is 
 accepted, the undersigned shall fail to execute a contract with said Board 
 of Chosen Freeholders, under the conditions of this proposal, within the 
 time provided for by the foregoing advertisement for proposals; other- 
 wise said check is to be returned to the undersigned. 
 
 Signed 
 
 Address 
 
 N. J., 
 
 In preparing his bid the contractor must allow for such 
 
 things as: 
 
 1. Preparatory expense. Cost of preparing for the real work, such as 
 setting up working machinery, clearing ground, etc. 
 
 2. Plant expense and supplies. Cost of maintaining tools, machinery, 
 worksheds, etc. Supphes for equipment. Depreciation allowance. 
 
 3. Materials. Those used in the real construction. Freight and 
 cartage. 
 
 4. Labor. Skilled and unskilled workmen. 
 
BIDS AND ESTIMATES. 385 
 
 5. Superintendence and general expense. Wages of firemen, time- 
 keepers, superintendent, clerical force. Supplies for their use. This 
 expense varies from 5% to 15% of labor. 
 
 6. Contingencies. Allowance for unforeseen expense, such as delays 
 from bad weather, labor troubles, accidents, rise in price of materials- 
 Often in excess of 10% of cost. 
 
 7. Profit allowance. At times a fixed per cent, of cost. At times, 
 one rate on cost of materials, say 10% to 15%, and a second rate, say 15% 
 to 25% on labor. These rates are often made high enough to cover probable 
 contingencies. 
 
 Note. Estimates of cost are based on current prices of labor, freight, 
 materials, etc. : estimates of quantities, largely on past experience. Often 
 the unit cost, as obtained by cost-keeping methods, is the basis for an 
 estimate. The exercises that follow illustrate a few principles of the 
 subject. 
 
 ORAL EXERCISE. 
 
 1. A bidder for a contract for office supplies believes that he can secure 
 the material for $3800. He reckons freight at $87.50; labor at $32.50. 
 Estimate his bid on a profit allowance of 20%. 
 
 2. Compute a bidder's profit of 15% on labor estimated to cost $38o. 
 
 3. Compute a bidder's allowance for 12^% profits on materials costing 
 him $1840. 
 
 4. A contractor estimates the gross cost of a prospective structure at 
 $28,500. If he allows 20% for profits and contingencies, what should his 
 bid be? 
 
 EXERCISE. 
 
 1. A contractor estimates the cost of materials, plant, etc., for a 
 certain work, at $46,758, and of labor at $39,200. He adds 10% to labor 
 for contingencies. He then reckons a profit of 20% on labor and 12^% 
 on materials. What is the amount of his bid? 
 
 2. Gross estimated cost of materials, $15,286.50; plant expense, $3495; 
 labor, $7296.40. Profit on materials, 15%; on labor, 25%. Amount of 
 bid? 
 
 3. Reckon quantities of material, cost of labor and arnqjint of bid, for 
 a contract of 260 squares of slate roofing, basing values on these figures: 
 (1) Unit cost per square, 214 pc. slate, $5.60; freight, $1.40; loading and 
 hauling, 35c; slate waste, 2%; 16 lb. sheathing paper, 50c; 1 lb. nails, 5c; 
 21 lb. nails, galvanized 9c; labor, $1.70. (2) Superintendence, 10%. 
 (3) Labor contingency, 10%. (4) Profit, on net cost of labor, 20%; on 
 materials, 15%. 
 
 26 
 
386 BUSINESS ARITHMETIC. 
 
 4. Compute the face of bid on 14,365 cu. yd. concrete foundation, 
 based on a unit cost per cu. yd., as follows: Stone, $1.15; gravel S.14; 
 sand, $.08; cement, $1.36; labor, $1.05; superintendence, $.05. Profits 
 on labor, 25%; on materials, 15%. 
 
 5. Compute bid (ex. 4), allowing a straight profit of 20% on cost per 
 unit. 
 
 • 6. A contractor agrees to cut and supply 60,000 ft., B. M., of 3" 
 plank, basing this bid on the following cost per unit M ft.: stumpage, 
 $1.75; cutting, $1 40; skidding, $.75; sawing, $2.40. Compute his bid, 
 allowing 25% for profit and contingencies. 
 
CHAPTER XLIV. 
 
 PARTITIVE PROPORTION AND PARTNERSHIP. 
 
 Partitive proportion, one phase of ratio, is the process of 
 dividing a number into parts proportional to two or more 
 given numbers. 
 
 ORAL EXERCISE. 
 
 1. Compare 4 and 6. 
 
 2. A's 12 shares compare how with B's 4 shares? Together they 
 have how many shares? 
 
 3. Divide $320 into parts proportional to 2 and 3. 
 
 Suggestion. 2+3=5 parts. 
 $320 = 5 parts. 
 
 1 part = $64. 
 
 2 " = ? 
 
 3 " = ? 
 
 4. Divide $7200 in the ratio of 5 to 7. 
 
 5. Divide 1200 yd. into two lots having the ratio 3/5. 
 
 6. Divide 6300 lb. into parts proportional to 1, 2 and 3. 
 
 7. Divide $1800 into parts proportional to 1/4, 1/3, and 1/6. 
 
 8. Divide $600 between A and B, so that A receives 3 times as much 
 as B. 
 
 9. Two dealers unite in ordering a carload of 40 tons of soft coal. 
 The freight bill is $24. If the first dealer, who bought 15 tons, settles the 
 bill, what should the second dealer pay him? 
 
 10. A piece of freight is carried 200 miles on one railway, and 150 
 miles on a second, for a total charge of $35.00. What should each railway 
 receive? 
 
 A partnership is an association of two or more persons for 
 the purpose of conducting a business in common, and of 
 sharing in its profits or losses. Partnerships are formed under 
 verbal or written agreements. The latter usually include a 
 
 387 
 
388 BUSINESS ARITHMETIC. 
 
 statement of the conditions concerning investments, with- 
 drawals, and the distribution of profits. 
 
 Collectively, the persons forming a partnership are termed 
 a firm, a house, or a company. Individually, they are known 
 as partners. Partnerships may be conducted under fictitious 
 names, under the names of the partners, or under the name 
 of one or more partners " and Co. " Partners are real or ostens- 
 ible, if they are really partners and are known to the public as 
 such. They are silent partners if their connection with the 
 firm is not known to the public; nominal, if they make no 
 investment and have no share in the proceeds of the business; 
 or limited if they have only a partial legal responsibility for 
 the debts of the firm. Real and nominal partners are re- 
 sponsible for all liabilities of their firm. Silent partners when 
 their connection becomes known must assume their share of 
 responsibility for the liabilities of the firm. Limited and silent 
 partnerships are not allowed in some states; in others it is 
 often required that at least one partner shall be real. 
 
 The capital of the partnership is the property invested in 
 the business. It includes money, any tangible form of real 
 or personal property, labor, good will, etc. 
 
 Note. Good will is the term applied to the value of past custom 
 of an old established place of business, in bringing future trade, or of 
 confidence in a partner who "lends his name," etc. Such a partner is 
 often credited with good will as an investment, at some money value, on 
 which his share of profits may be based. 
 
 The net capital or present worth of a firm at any time is the 
 
 excess of resources or assets (property owned and debts due) 
 
 over liabilities (debts owed). If liabilities are in excess there 
 
 is net insolvency, 
 
 FOR DISCUSSION. 
 
 Compare partnerships and stock companies as to: 
 
 1. Method of organization. 5. Effect of death of partner or 
 
 2. Method of investing. stockholder. 
 
 3. Withdrawal from business. 6. Government and management. 
 
 4. Sale of interest in business. 7. Individual responsibility for 
 
 debts. 
 
PARTITIVE PROPORTION AND PARTNERSHIP. 389 
 
 ORAL EXERCISE. 
 
 1. Total gains — total losses = ? 
 
 2. Total losses — total gains = ? 
 
 3. Resources =fc ? = net capital. 
 
 4. Original capital + net gain = ? 
 
 5. Original capital — net loss = ? 
 
 6. Resources — liabilities = ? . How check? 
 
 7. May a firm gain during a period and be insolvent at its close? 
 
 EXERCISE. 
 Supply missing words or values. Solve mentally if possible. 
 
 1. Gains $2400; losses $560; net -, $ . 
 
 2. Gains, $2150; losses $2680; net , $ . 
 
 3. Capital at commencing $16000; at close, $18500; net = $- 
 
 4. Original capital, $5000; gains, $1250; losses $640; net capital at 
 close, $ . 
 
 5. Net insolvency, Jan. 1, 1907, $2500; Jan. 1, 1908— Resources, $5260; 
 liabilities $5000. Result of business for the year a of $ . 
 
 6. A & Co. invested $12,000, and a year later had cash, $2000; mdse., 
 $10,500; real estate $2500, but owed debts of $2000. A & Co.'s net 
 capital then was $ . 
 
 7. Jan. 1, 1912. Net capital $5000. Jan. 1, 1913, gross gains during 
 year $3000; gross losses, $1200. Present resources $8000; liabiUties $900. 
 Prove the existence of an error in these figiu-es. 
 
 Distribution of Profits and Assessment* of Losses. 
 
 In partnerships, profits are distributed and losses met in 
 proportions agreed upon. These may be arbitrary or graded 
 for special service, or based on amounts of investments; or a 
 combination of these factors. 
 
 I. EQUAL DIVISION OF PROFITS. 
 ORAL EXERCISE. 
 
 1. Divide $726 profits equally between Adams and Brown. 
 
 2. A's investment was $4000 and D's $6000. They share equally a 
 profit of $800, but D withdraws $100 of his share. The net capital of the 
 firm is then $ ?, and that of D, $ ? 
 
390 BUSINESS ARITHMETIC. 
 
 3. A invests $2000, B $4000, and C $3000. Determine their present 
 worths, after they have shared equally a loss of $960. 
 
 EXERCISE. 
 
 1. July 1, 1908. Investments: A, $12,600; B, $14,800; C, $9,650. 
 Jan. 1, 1909. Withdrawals; A, $1200; B, $400. 
 
 Jan. 1, 1909. Gains: On mdse., $7260.50; on real estate $1260. 
 Losses: Expense, $3126.50; fuel and light $568.40; taxes $165. 
 Find the present worth, if profits are shared equally. 
 
 2. On Jan. 1, Evans invested $12,000, Brown $8000 and Thompson 
 $15,000 in the real estate business. A year later their resources were: 
 Cash $7290; accounts and notes receivable, $11,360; real estate $26,750, 
 while they owed notes for $3296.50 and taxes of $561.30. Find the present 
 worth of each member. 
 
 II. PROFITS SHARED IN ARBITRARY PROPORTIONS. 
 ORAL EXERCISE. 
 
 1. Divide $792 between A and B in the ratio of 3 to 1. 
 
 2. A receives 2/3 of the profits of a business in which he invested 1/4 
 of the capital of $12,000. Find his present worth if the profits are $900. 
 
 EXERCISE. 
 
 1. Crown and Briggs entered into partnership on Jan. 1, each investing 
 $8000. Owing to Crown's greater experience, he receives 2/3 of the profits. 
 On July 1, their profit and loss account stands as follows. 
 
 Profit and Loss. 
 
 Expense $429.60 
 
 Rent 600. 
 
 Discounts 86. 
 
 Merchandise $3129.65 
 
 Discounts 128.46 
 
 Stocks and Bonds. 82.50 
 
 Find the net gain and present worth of each partner. 
 
 2. A invests $9000 and B his services, under an agreement that the latter 
 shall receive $900 yearly salary, to be paid from any profits, and 1/4 of the 
 profits remaining. The profits at the end of a year amount to $3126.50. 
 Find each partner's present worth, B having withdrawn only one-half of 
 his salary. 
 
 3. A, B and C enter into partnership to manufacture advertising 
 novelties. A invests $6000, and the other partners $4500 each. The 
 partners agree: (1) To share losses equally; (2) To pay A $1200 salary 
 
PARTITIVE PROPORTION AND PARTNERSHIP. 391 
 
 out of any profits, for giving all his time to managing the business; (3) To 
 divide any remaining profits in parts the proportion of 3 (A), 2 (B), and 
 2 (C). At the end of the year the profits are found to be; on merchandise, 
 $9720; discounts, $112.50; and the expenses of the business exclusive of A's 
 salary, $4326.50. Find the present worth of each partner. 
 
 III. PROFITS SHARED ACCORDING TO INVESTMENTS. 
 ORAL EXERCISE. 
 
 1. A invests $5000 and B $7000. What is the partnership capital? 
 
 2. Each partner invests what part of the capital? Each should receive 
 what part of any profits? 
 
 3. What should each receive of $1400 profits? What shall each pay 
 of $350 loss? 
 
 What is each partner's share, according to the following investments? 
 
 (a) A, $6000; B, $10,000. (6) A, $5000; B, $4000; C, $3000. 
 Determine individual gain, loss and present worth in the following : 
 
 Investment. Gain. Investment. Loss. 
 
 4. A, $2000; B, $10,000 $1500. 6. E, $3000; H, $8000 $3300 
 
 6. C, $6000; D, $3000 $1860. 7. G, $6000; K, $1500 $4200 
 
 EXERCISE. 
 
 1. The net profits of the firm of Thompson and Harris for the last 
 year are $3584.64. This is to be apportioned to the partners' investments. 
 Make the apportionments if Thompson's investment was $6000 and that 
 of Harris, $1700. 
 
 2. Brown, Kann and Eastman engaged in a retail business, investing 
 respectively $12,000, $8000 and $19,000. Brown, who was the only 
 active partner, was permitted to take $1500 from the profits, annually 
 for his services, the balance of the profits being divided according to in- 
 vestment. Net profits for the past year were $7843.12. Make the dis- 
 tribution. 
 
 At the time of apportioning profits or losses a business 
 statement is often prepared, showing a summary of the 
 resources and liabilities, the investments and present worth, 
 and, at times, the losses and gains. 
 
392 
 
 BUSINESS ARITHMETIC. 
 
 EXERCISE. 
 1. Complete the following business statement. 
 Partnership Statement. 
 Harris and Davidson, 
 
 December 31, 19- 
 
 Resources. 
 Cash 
 
 5280 
 
 15212 
 
 5000 
 
 8240 
 
 45 
 20 
 
 60 
 
 3204 
 2500 
 ? 
 
 
 Merchandise on hand and unsold 
 
 Notes Receivable 
 
 
 Accounts Receivable 
 
 
 Liabilities. 
 Accounts Payable 
 
 50 
 
 Notes Payable 
 
 
 Present worth, Harris & Davidson 
 
 
 
 ? 
 
 
 ? 
 
 
 Gains. 
 Merchandise 
 
 3640 
 1815 
 2116 
 
 ? 
 
 ? 
 
 60 
 45 
 
 12760 
 521 
 
 ?0 
 
 Interest and Discount 
 
 60 
 
 Losses. 
 Labor 
 
 
 Rentals 
 
 
 Expense 
 
 
 Harris, Net Gain 
 
 
 Davidson, Net Gain 
 
 
 
 
 
 
 
 Check. 
 Harris, Investments 
 
 10000 
 ? 
 
 13000 
 681 
 
 — 
 
 ? 
 ? 
 
 
 Harris, Net Gain 
 
 
 Harris, Present Worth 
 
 
 Davidson Investment 
 
 
 Davidson Withdrawal 
 
 
 Davidson Net Investment 
 
 ? 
 ? 
 
 
 
 Davidson Net Gain 
 
 Davidson Present Worth 
 
 
 Harris and Davidson Present Worth 
 
 
 
 ? 
 
 
 Note. Share profits according to investment. 
 
 
 ^ 
 
 IV. INTEREST ON INVESTMENTS. 
 
 Sometimes interest on investments is paid from the profits, 
 the balance of profits being apportioned otherwise. 
 
 Illustration. Example. A invests $6000 and B $9000 in a business, 
 realizing $3450 profits the first year. Six per cent, interest is allowed on 
 investments, the balance of profit being apportioned in the ratio of 1 : 2. 
 Find present worth of each partner. 
 
PARTITIVE PROPORTION AND PARTNERSHIP- 393 
 
 Solution. 
 
 Interest on A's investment, 1 yr. on $6000 = $360 
 Interest on B's investment, 1 yr. on $9000 = 540 
 
 900 
 
 $3450 - $900 = $2550, net profits. 
 
 . A's share, 1/3 of $2550 $ 850 
 
 " B's share, 2/3 1700 
 
 A's present worth, $6000 +$360 +$850 = 7210 
 
 B's present worth, 9000+540 + 1700= $11240 
 
 EXERCISE. 
 
 1. Andrews entered into partnership with Gates on Jan. 1, 1913, 
 investing $6000, while Gates invested $11,000. They agreed to allow each 
 other 6% interest on investments, and to share the balance of profits in 
 the ratio of 2 to 3. Find the present worth of each member on Jan. 1, 
 1914, after the apportionment of profits of $6585. 
 
 2. Jan. 1, 1913, Carleton invested $12,000; Johnson, $8500. Oct. 1, 
 1913, Carleton withdrew $3000. Jan. 1, 1014, profits for past year 
 $4216.50. Interest allowed at 6%, and balance of profits shared equally. 
 Apportionment? 
 
 The average interest per partner is sometimes computed, 
 and each partner is credited with the excess of his investment 
 interest over the average, or charged with any deficit. The 
 entire profits are then divided by agreement. 
 
 Illustration. Last illustrative example. 
 Solution. Interest on A's investment $360 
 Interest on B's investment 540 
 2 )900 
 Average interest. 450 
 
 $450 - $360 = $90 A's deficit (Debit). 
 540 - 450 = 90 B's excess (Credit). 
 A's 1/3 of profits = 1/3 of 3450 = $1150 
 B's 2/3 = 2300 
 
 A's present worth = $6000 + $1150 - $90 (deficit) = $7,060 
 B's present worth = $9000 + $2300 + $90 = $11,390 
 
 EXERCISE. 
 
 1. April 1, 1912, A. M. Gates and J. B. Farquhar entered into partnership 
 
 for the purpose of carrying on a manufacturing business. Gates invested 
 
 $16,000 and Farquhar $12,500. It was agreed that interest should be 
 
 allowed and charged at 6%, and that gains and losses should be divided 
 
394 BUSINESS ARITHMETIC. 
 
 equally. On Aug. 1, 1912, Gates withdrew $1000, and on Oct. 1, Farquhar 
 withdrew $500. On Apr. 1, 1913, the books were closed and showed the 
 following: 
 
 Gains. Losses. 
 
 Mdse $15,289 Labor $7240 
 
 Stocks and Bonds 986.40 Equipment 1645.20 
 
 Mdse. discounts 560 Discounts 300 
 
 Expense 1298.75 
 
 Find the present worth of each partner after apportionment of net gain. 
 
 2, Oct. 1, 1912, M. Andrews, B. T- Chase, and T. C. Lewis formed a 
 partnership. Lewis invested $8500 and the other partners $7500 each. 
 They agreed to allow and charge interest at 6% and to divide profits 
 equally. On Dec. 1, Lewis withdrew $1000 and Chase $500. At the close 
 of the year, assets and liabilities were as follows : 
 
 Assets. Liabilities. 
 
 Cash $5486.50 Bills pay $1250 
 
 Mdse 6798.15 Accts. pay 1670.60 
 
 Real estate 9725. 
 
 Bills Rec 3000. 
 
 Accts. Rec 4520. 
 
 Make statement showing condition of business. 
 
 VI. APPORTIONMENT BASED ON AVERAGE INVESTMENT. 
 
 The sum that has the same earning power for a stated 
 period as two or more sums invested for different periods, is 
 termed an average investment. Profits and losses are often 
 apportioned by average investment. The apportionment 
 may be made by ratio or by interest. 
 
 Illustration. Example. At organization, A invests $12,000, but 
 withdraws $2000 at the end qf 8 months; B invests $6000 withdrawing 
 $3000 after 6 months. How should the first year's gains of $4750 be 
 apportioned? 
 
 Solution (1). 
 
 A's investment of $12,000 for 8 mo. = $96,000 for 1 mo. 
 A's investment of $10,000 for 4 mo. = 40,000 for 1 mo. 
 A's average investment of $136,000 for 1 mo. 
 
 B's investment of 6,000 for 6 mo. = 36,000 for 1 mo. 
 
 B's investment of 3,000 for 6 mo. = 18,000 for 1 mo. 
 
 B's average investment = 54,000 for 1 mo. 
 
 A's average + B's average = 190,000 for 1 mo. 
 
PARTITIVE PROPORTION AND PARTNERSHIP. 395 
 
 A's share = 136/190 = 68/95. 
 B's share = 27/95. 
 
 1/95 of profit of $4750 = $50 68/95 = 68 X 50 or $3400 = A's profit. 
 
 27/95 = 27 X 50 or 1350 = B's profit. 
 Solution (2). Using any rate, say 6%. 
 Interest on $12,000 for 8 mo. = $480 
 Interest on 10,000 for 4 mo. = 200 
 A's invest, earning power = $680 
 Interest on $6000 for 6 mo. = $180 
 Interest on 3000 for 6 mo. = 90 
 
 "$270 
 Total earning power $680 + $270 = $950, of which A's mvestment 
 represents 68/95 and B's 27/95. 
 
 Final step — as in previous solution. 
 
 EXERCISE. 
 
 1. M. C. Barton and James Thompson form a partnership to carry on 
 an agricultural supply store. Barton invests $6000 for 9 mo., and then 
 adds $2000. Thompson invests $14,000, but withdraws $4000 at the end 
 of 4 months. • At the end of a year their summary accounts stand as 
 follows: 
 
 Farm Implements. Fertilizers. 
 
 Cost $6240 Sales $5268.50 Cost $5692 Sales $7218.20 
 
 On hand 3650. On hand 965. 
 
 Seeds. General Expense. 
 
 Cost $2132 Sales $3586.40 Cost $1296.18 
 
 On hand 1125. 
 Apportion the profits in accordance with average investment. 
 
 2. Chas. Parsons and T. P. Newton form a partnership • to carry on 
 the grocery business. Complete the proprietors' accounts, apportioning 
 profits by average investment. 
 
 Chas. Parsons. 
 Mar. 1, Mdse. $2000 Jan. 1, Invest $8000. 
 
 Jan. 1, Pres. worth? Jan. 1, Net gain? 
 
 T. P. Newton. 
 July 1, Withdraw $6000 Jan. 1, 1908, Investment $18,000 
 
 Jan. 1, Pres. worth? Jan. 1, 1909, Net gain ? 
 
 Profits and Losses. 
 Losses. Profits. 
 
 General Expense $1503.14 Merchandise $9290.68 
 
 Rentals 900 Discounts 587.40 
 
 Labor 1650 
 
 Delivery 2160 
 
 Parson's net gain ? 
 
 Newton's net gain ? 
 
396 BUSINESS ARITHMETIC. 
 
 3, 4, Solve the problems of the last exercise by average investment. 
 
 GENERAL EXERCISE. 
 
 1. Frank T. Morris, R. P. Norton and Henry Collins form a partnership 
 on April 1, to carry on a furniture business, investing respectively, $12,000, 
 $6000 and $20,000. On June 15, Morris withdraws $500 and on Dec. 1, 
 Collins withdraws $2500. At the close of the first business year, the 
 books showed the following facts: Cost of mdse. purchased, $21,690; sales 
 of mdse, $32,169.40; mdse. on hand unsold, $6,742.10. Cost of real estate, 
 $12,000; alterations and repairs, $3167.50; estimated value at close of 
 year, $14,500; notes in favor of firm, $5640; accounts due the firm, $5680; 
 accounts owed by the firm, $1520; profits on discounts, $750; notes owed 
 by the firm, $1200, mortgage on real estate, $6000; general expense for 
 the year, $9650 45; cash, $18,811.45. 
 
 Prepare partnership statements at close of year under following con- 
 ditions: 
 
 1. Share respectively in proportions of 2, 1 and 3. 
 
 2. Interest allowed and charged at 5% and profits divided equally. 
 
 3. $1200 salary allowed Morris, balance of profits being divided ac- 
 cording to original investment. 
 
 4. Profits shared according to average investment. 
 
 FOR DISCUSSION. 
 
 1. Which system of apportionment is most fair, considering simply 
 investments? 
 
 2. Which permits recognition of special service? 
 
 3. When is it impossible to use the method of "average investment"? 
 
CHAPTER XLV. 
 
 EQUATION OF PAYMENTS AND OF ACCOUNTS. 
 
 INTRODUCTORY EXERCISE. 
 
 1. Which party to a debt loses by its delayed payment? 
 
 2. Who gains by payment before due? 
 
 3. How may one measure, arithmetically, this gain or loss? 
 
 4. $600, which I owe, is overdue 1 mo. Money being worth 6%, what 
 ought I to pay the creditor for the delay? Why? 
 
 5. $200 worth of mdse. bought Jan. 10, on 2 mo. credit, should be 
 paid for on what date? The creditor is entitled to how many dollars 
 extra, ff payment is made May 10? If paid Jan. 10 would the creditor 
 gain or lose? What might he allow off? 
 
 Debts draw interest from the time they are due until paid. 
 Discounts, sometimes equal to interest on the debt, may be 
 allowed on payments before due. If several payments are 
 due the same party on different dates, a single payment equal 
 to their total may be made on such a date that the overdue 
 interest may equal the discount on advance payments. This 
 date is termed the average or equated date. The process of 
 determining it is termed averaging or equating. (Why is the 
 settlement on the equated date fair to both parties?) 
 
 EQUATION OF PAYMENTS. 
 ORAL EXERCISE. 
 
 1. The interest on $400 for 6 mo. = interest on $200 for ? mo. 
 
 2. I owe $400 due in 2 mo. By paying $200 to-day for what extra 
 time might I fairly retain the balance? 
 
 3. I owe $600 in 3 mo., and $600 in 5 mo. When might the total debt 
 be paid without gain or loss to either party? 
 
 Illustration (1). On Jan. 10, A. C. Brown bought $900 worth of 
 mdse., payable, $200, Jan. 20; $400, Feb. 9; the balance, Mar. 10. (1) He 
 
 397 
 
398 BUSINESS ARITHMETIC. 
 
 settles in full Mar. 20. What does he owe? (2) On what date would the 
 face of debt be a fair settlement? Interest 6%. 
 Solution (1). Debt paid Mar. 20. 
 
 Date Due. 
 
 
 Payment. 
 
 Overdue. 
 
 Overdue Interest. 
 
 Jan. 20 
 
 
 $200 
 
 6'd da. 
 
 $1.97 
 
 Feb. 9 
 
 
 400 
 
 39 " 
 
 2.60 
 
 Mar. 10 
 
 
 300 
 
 10 " 
 
 .50 
 
 
 
 Debt $900 
 
 , 
 
 $5.07 
 
 
 Int 
 
 . due 5.07 
 
 
 
 Due $905.07 
 
 Note. Use exact time. 
 
 (2) By paying Mar. 20, Brown pays $5.07 extra. The earlier he pays 
 the less interest he owes. The interest on $900 for 1 da. is 15c. .". 5.07 
 represents 507/15 days loss of interest, or 34 days. .'. The equated date 
 for payment of $900, is 34 days before Mar. 20, or Feb. 14. 
 
 It is evident that interest is found on each payment from due 
 date to date of assumed settlement. The total interest, divided 
 by the interest on the total debt, gives the number of days of 
 delayed payment. If there is discount in place of interest, the 
 real payment should be made later. 
 
 Illustration. Find the equated date for pajnments of $600 due Mar. 
 10; $300, April 8; $600, May 12. 
 
 Solution. Assume a date of settlement, called a focal date. 
 
 Select as focal date. May 12, date of last payment. 
 
 Date. Payment. Overdue. Interest. 
 
 Mar. 10. $600 63 da. (to May 12) $6.30 
 
 Apr. 8. 300 34 1.70 
 
 May 12 600 _0 
 
 Due $1500 .25) $8.00 due 
 
 Interest on $1500 for 1 da. = 25c. 32, no. of 
 
 days. 
 
 .'. By paying May 12, the debt is overdue 32 days. 32 days back from 
 May 12 = Apr. 10, the equated date. 
 
 Check Solution. If Apr. 10 is correct, there should be no gain or loss 
 by paying face on that date. Take Apr. 10 as focal date. 
 
 Date 
 
 Payment Overdue Before due 
 
 Interest Discount 
 
 Mar. 10 
 
 $600 (to Apr. 10) 31 da. 
 
 3.10 
 
 Apr. 8 
 
 300 2 
 
 .10 
 
 May 12 
 
 600 32 da. 
 
 3.20 
 
 
 check 
 
 3.20 = 3.20 
 
EQUATION OF PAYMENTS AND OF ACCOUNTS. 399 
 
 Note. This means that $1500 may be paid in full settlement on Apr. 10. 
 On any date thereafter the amount due is $1500 + interest from Apr. 10. 
 Remember: 1. Any date may be used as focal date or trial date. 
 
 2. By using first or last date, one interest computation is saved. Why? 
 
 3. The face of debt is always due on the equated date. 
 
 EXERCISE. 
 Find the equated date, and check by using equated date or another 
 focal date. 
 
 1. A. B. Chase, Dr. 2. S. P. Randall, Dr. 
 
 1912 1912 
 
 Jan. 16 To mdse. $300. Mar. 12 To mdse. $420. 
 
 Jan. 29 To mdse. 900. May 16 To mdse. 500. 
 
 Mar. 12 To mdse. 400. Aug. 15 To mdse. 320. 
 
 What is due Dec. 1? 
 
 3. James Field, Dr. 
 1912 
 
 Oct. 16 To mdse. $520. 
 Oct. 20 To cash 120. 
 Dec. 15 To mdse. 800. 
 1913 
 
 Jan. 3 To mdse. 320. 
 
 In the following, re-write the accounts by substituting the true dates 
 when due, obtained from the credit terms. 
 
 4. Robert Everhart, Dr. 5. James Donnelly, Dr. 
 1912 1912 
 
 Feb. 21 30 da. credit $300 Mar. 3 To mdse. $240. 
 
 Feb. 25 2 mo. credit 600 Mar. 18 To mdse., 30 da. 380. 
 
 Mar. 16 1 mo. credit 900 Mar. 30 To mdse., 2 mo. 456.40 
 
 Apr. 4 To mdse., 60 da. 820.60 
 Apr. 16 To mdse. 900. 
 
 6. Jones & Co., Dr. 
 ■ 1912 
 
 Apr. 16 Mdse. 60 da. $456.10 
 
 Apr. 29 Mdse. 30 da. 829.30 
 
 May 5 Mdse. 3 mo. 420.00 
 
 June 16 Mdse. 36.50 
 
 EQUATION OF ACCOUNTS. 
 Accounts having debits and credits are equated to determine 
 the due date of the balance. The most convenient focal date 
 is the latest named in the account. 
 
400 
 
 BUSINESS ARITHMETIC. 
 
 Illustration. 
 
 James C. Westcott. 
 
 1909 
 
 
 
 
 
 
 
 
 
 Mar. 
 
 12 
 
 Mdse. 
 
 $360 
 
 
 Mar. 
 
 30 
 
 By cash 
 
 $300 
 
 
 29 
 
 
 300 
 
 
 Apr. 
 
 10 
 
 By note 
 
 120 
 
 May 
 
 4 
 
 
 240 
 
 
 
 
 (Balance 
 
 $480 
 
 Solution. Here we have both debit and credit interest, the former 
 being charged on debts overdue, and the latter being credited on pay- 
 ments made to focal date. 
 
 Focal date. May 4. 
 
 
 
 
 Date. 
 
 Debit Amounts. 
 
 Period (to May 4). 
 
 Dr 
 
 . Interest. 
 
 Mar. 16 
 
 $360 
 
 49 da. 
 
 
 $2.94 
 
 Mar. 29 
 
 300 
 
 36 da. 
 
 
 1.80 
 
 May 4 
 
 240 
 
 
 
 
 
 $900 
 
 
 
 $4.74 
 
 
 Credit Amounts. 
 
 
 Cr 
 
 . Interest. 
 
 Mar. 30 
 
 $300 
 
 35 da. 
 
 
 $1.75 
 
 Apr. 10 
 
 120 
 
 24 da. 
 
 
 .48 
 
 $2.23 
 
 $420 
 
 Balance = $900 - $420 = $480. 
 
 Net interest = Dr. int. - cr. int. = $4.74 - $2.23 = $2.51, dr. int. 
 Since the interest is debit, payment is overdue. 
 Interest on $480 for 1 day = 8c. 
 
 2.51 -^ .08 = 31 +, the number of days overdue. 
 
 May 4 — 31 days = Apr. 3, equated date. 
 
 Check. Assume Apr. 3 as focal date. 
 
 Date. 
 
 Amount. 
 
 Period. 
 
 Dr. Int. 
 
 
 Cr. Int. 
 
 Mar. 16 
 
 $360 
 
 18 da. 
 
 $1.08 
 
 
 
 Mar. 29 
 
 300 
 
 5 da. 
 
 .25 
 
 
 
 May 4 
 
 240 
 
 31 da. (before due) 
 
 
 
 $1.24 
 
 Mar. 30 
 
 300 
 
 4 da. (before due) 
 
 
 
 .20 
 
 Apr. 10 
 
 120 
 
 7 da. (delayed) 
 
 1.14 
 
 
 
 
 
 
 $1.47 
 
 check 
 
 $1.44 
 
 Note. The difference between dr. and cr. interest = the remainder 
 in first solution, when dividing total interest by interest for one day. 
 
 EXERCISE. 
 
 Find and check the equated date. 
 
 1. C. P. Newton. 
 
 1912 
 
 
 1912 
 
 Mar. 16 To mdse. 
 
 $1400 
 
 Apr. 4 
 
 May 28 
 
 2200 
 
 June 6 
 
 By cash 
 
 $ 200 
 1000 
 
EQUATION OF PAYMENTS AND OF ACCOUNTS. 401 
 
 2. 
 
 The Newbold Co. 
 
 1912 
 
 Nov. 19 To mdse. $26,580 
 
 30 " 41,590 
 
 f)ec. 12 " 16,250 
 
 30 " 38,845 
 1913 
 
 Jan. 20 " 268 
 
 1912 
 Nov. 
 Dec. 
 Dec. 
 
 30 
 15 
 31 
 
 By cash 
 
 Note 
 
 $200 
 300 
 500 
 
 3. 
 
 Chas. p. 
 
 Randall. 
 
 
 
 1912 
 
 
 1912 
 
 
 
 July 15 
 
 To mdse., 30 da. $360 
 
 Aug. 5 
 
 By cash 
 
 $400 
 
 31 
 
 15 da. 180 
 
 Sept. 10 
 
 Note 
 
 300 
 
 Aug. 12 
 
 30 da. 420 
 
 10 
 
 Cash 
 
 300 
 
 Sept. 6 
 
 2 mo. 620 
 
 
 
 
 4. 
 
 
 James Brownet.t. & Bro. 
 
 1912 
 
 
 
 1912 
 
 Feb. 28 
 
 By note 
 
 $ 300 
 
 Feb. 16 By mdse. 60 da. $325 
 
 Mar. 16 
 
 Mdse. 
 
 120 
 
 20 " 30 da. 244.60 
 
 Apr. 2 
 
 Cash 
 
 400 
 
 Mar. 18 " 3 mo. 840 
 
 May 10 
 
 (( 
 
 600 
 
 29 " 2 mo. 628 
 
 Note. 
 accounts. 
 
 Account sales are often averaged m the same manner as general 
 
 Washington, D. C, May 16, 1913. 
 Account Sales op Potatoes. 
 
 Sold for account of James Parker, 
 
 Hyattsville, Md. 
 By Robert Harper, 
 
 Commission Merchant. 
 
 1913 
 
 Sales. 
 
 
 
 Apr. 30 
 
 20 bbl. potatoes. 
 
 $2.40 
 
 
 May 6 
 
 50 " " 
 
 3.00 
 
 
 8 
 
 120 " 
 
 2.60 
 
 
 12 
 
 10 " 
 
 Charges. 
 
 2.00 
 
 
 Apr. 28 
 
 Freight and cartage 
 
 
 .46 
 
 May 12 
 
 Storage 
 
 
 .30 
 
 12 
 
 Commission, 5% of sales 
 
 
 ? 
 
 ? 
 
 Net proceeds, due by equation 
 
 
 ? 
 
 27 
 
 
 
 
402 BUSINESS ARITHMETIC. 
 
 CASH BALANCE. 
 
 The cash balance of an account on any particular date is the 
 amount due on that date. If interest and discount are not 
 allowed, the cash balance is the same as the account balance; 
 if allowed, it equals the regular balance plus or minus the net 
 interest or discount; or it equals the balance of the account 
 plus interest for the time between equated date and average 
 date, if later than equated date; or minus interest, if given date 
 is earlier. The first illustrative example, page 398, shows the 
 cash balance on March 20. 
 
 EXERCISE. 
 
 Find the cash balance directly for each of the accounts in the preceding 
 exercise, checking by using equated date. 
 Ac. 1. What is due July 1? 
 
 2. What is due Jan. 8? 
 
 3. What is due Oct. 31? 
 
 4. What is due Jan. 1, 1913? 
 6. What is due.July 1? 
 
CHAPTER XLVI. 
 
 BILLING. 
 
 INTRODUCTORY EXERCISE. 
 
 1. Bring to class several bills for goods purchased. State the infor- 
 mation that each heading gives. 
 
 2. What information is given concerning the articles bought? 
 
 3. Who makes out a bill for goods? What becomes of it? Trace 
 its life. 
 
 4. Of what value is the bill to the purchaser? What might happen in 
 case no bill was given? 
 
 The hill of merchandise is the statement in detail of mer- 
 chandise sold in the course of trade, and is prepared by the 
 seller for the buyer. Formerly the term invoice was applied 
 to statements of merchandise sold wholesale, and of incoming 
 goods from abroad. Now the terms are frequently used 
 interchangeably. 
 
 I. SIMPLE BILL. 
 
 Washington, January 11, 1913. 
 C. M. Brown, 
 Groceries and Provisions. 
 Sold to Henry Eastman, 
 
 1426 13th Street. 
 
 1 
 
 1 
 
 sk. Flour 
 
 
 
 85 
 
 2 
 
 6 
 
 lb. Lard 
 
 @.17 
 
 1 
 
 02 
 
 3 
 
 U 
 
 pk. Apples 
 
 .72 
 
 
 
 4 
 
 10 
 
 b. Gran. Sugar. 
 
 .06^ 
 
 
 
 5 
 
 2 
 
 qt. Molasses 
 lb. Rice 
 
 .18 
 
 
 
 6 
 
 3 
 
 .09 
 
 
 
 7 
 
 2 
 
 lb. Starch 
 
 .05 
 
 
 
 8 
 
 
 Yeast 
 
 
 
 02 
 
 9 
 
 
 ? 
 
 10 
 
 
 
 
 
 
 11 
 
 
 Paid, 
 
 
 
 
 12 
 
 
 C. M. Brown 
 
 
 
 
 13 
 
 
 
 
 
 
 14 
 
 
 
 
 
 
 403 
 
404 BUSINESS ARITHMETIC. 
 
 EXERCISE. 
 
 1. State in your own words just what took place according to the 
 bill on page 403. 
 
 2. Are there any non-essentials in this bill? Add two. 
 
 3. If settlement is not made, how is the bill affected? How, if sent 
 C. O. D.? 
 
 4. What is the advantage of numbering the items on the bill? 
 
 5. Rule up three bill forms and prepare threo original bills for a Satur- 
 day's marketing. Secure reliable information as to quantities and prices. 
 
 Receipting for Payment. 
 When payment is made, bills are receipted by the creditor 
 or by the party representing him, who writes the word " Paid," 
 or some equivalent expression, across the face of the bill over 
 his signature. 
 
 EXERCISE. 
 
 1. Why is the creditor's signature necessary? 
 
 2. Examine the forms of receipt of the model bills in this section, and 
 throughout the book, and determine, in each case: 
 
 (a) What facts are stated in the receipt. 
 
 (6) By whom receipted and under what probable conditions. 
 
 (c) What part of the receipt may be printed. 
 
 (d) Under which of the following classes the receipt belongs: 
 
 (1) Receipt by the direct creditor. 
 
 (2) Receipt in the firm name by a partner. 
 
 (3) Receipt of a corporation by an officer. 
 
 (4) Receipt for the direct creditor by a third party. 
 
BILLING. 
 
 405 
 
 11. SIMPLE TWO COLUMN BILL. 
 
 January 31, 191 
 
 Chestnut Grove Diary, 
 
 1421 Oak Street. 
 
 John C. Norton, Jr. 
 
 Sanitary Cream and Milk. 
 
 Sold to James Fielding 
 
 646 Pine Street. 
 
 
 
 
 Acc't Rendered 
 
 132 Qts. Milk .09 
 6 Pts. " .05 
 Pts. Gilt-Edge Cream 
 7^ Pts. " " .19 
 Gills " 
 Pts. Choice " 
 Hf. Pts. Choice " 
 Gills " 
 
 1 A Z^+o, 'Dii4^4^y>.««^i1U AO 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Received payment 
 
 
 
 
 John C. Norton, Jr. 
 
 EXERCISE. 
 
 1. Extend the above bill. How should the form be changed if $3.00 
 is paid on account? How, if nothing is paid? 
 
 2. You are directed by James Kent to deliver daily 2 qt. 1 pt. of milk 
 and 1 pt. of cream. In addition, you deliver: Jan. 5, 3 qt. milk; Jan. 8, 
 2 qt. milk; 1 pt. cream; Jan. 21, 2 qt. 1 pt. milk; 1 pt. cream. Prices: 9c 
 per qt. for milk; 16c per pint for cream. Render a bill and have your clerk 
 receipt for payment. 
 
 3. Design an original bill for similar use for an ice company. 
 
 4. In what other businesses could similar bills be used? 
 
 Terms of Sale. 
 
 As is evident from the problems and illustrations given, the 
 terms of payment on bills of goods may vary markedly with 
 the character and quantity of sales. The list of terms that 
 follow are selected from actual bills. 
 
 Cash. 
 Net cash. 
 
 List of Terms op Sale. 
 
406 BUSINESS ARITHMETIC. 
 
 3. Cash or exchange on New York. 
 
 4. Account. 
 
 5. Interest after 30 days. 
 
 6. 30 days; 2/10. 
 
 7. Net 30 days; note to your own order, payable at a Philadelphia 
 
 bank. 
 
 8. Net 90 days; 5% discount for exchange on New York in 30 days. 
 
 9. 30 days. If paid in 10 days, 2% discount. 
 
 10. 30 days. 
 
 11. 30 days. If not promptly paid interest will be charged from date of 
 
 sale. 
 
 12. Cash, 5%; 30 days, 2%; 60 days, net. 
 
 13. Ten day note. 
 
 14. 60 days from Aug. 10; 2% 10 days. 
 
 15. 5% 30 days, or 6% 10 days. Payable in New York Gty funds. 
 
 16. Due, October 2. Less 2% if paid in 10 days. 
 
 17. C. O. D. 
 
 18. 5/60; 6/30; 7/10. 
 
 19. Best discount allowed for unexpired time is at the rate of 6%. 
 
 20. Net 30 days, or 1% discount in 30 days, or 2% discount in 10 days. 
 
 New York funds. 
 
 21. Net after 60 days. 
 
 22. Subject to sight draft after 60 days. Payable in gold or its 
 
 equivalent. 
 
 23. Net 30 days; 2% discount 10 days. No discount allowed after 
 
 , 191—. 
 
 24. Accounts due the first of the month. 
 
 25. 2% discount if paid , 191—; Due net, , 191—. 
 
 EXERCISE. 
 
 1. Give your understanding of the meaning of each of the terms stated 
 above. 
 
 2. Examine any bills that you can secure and see whether you can add 
 to the above list. 
 
 3. State the effect of each set of terms on the settlement of a bill of 
 merchandise amounting to $1600, purchased August 1. 
 
BILLING. 
 
 407 
 
 III. FORM OF MANUFACTURER'S INVOICE AND LETTER OP 
 TRANSMISSION. 
 
 Address Albany, N. Y. 
 
 Invoice. 
 Name John Smithson 
 Order No. 17296 
 
 Bought of WASHINGTON FLOUR CO. 
 Manufacturers 
 Fancy Patent Flours 
 Buffalo, N. Y. 
 Ship to Order Washington Flour Co. Date 12/30/191- 
 
 Destination Albany, N. Y. 
 Via. N. Y. C. R. R. 
 
 Bbl8. 
 
 H. 
 
 Bbls. 
 
 No.Sks. 
 
 Sz.Sks. 
 
 Kind. 
 
 Brand. 
 
 Marks. 
 
 Pricfi 
 PerCwt. 
 
 Amount. 
 
 180 
 
 50 
 
 200 
 80 
 40 
 
 260 
 
 24 
 
 49 
 
 98 
 
 140 
 
 Paper 
 Cott'n 
 Grain 
 Jute 
 
 Ames 
 
 Buffalo Prime 
 Conrad 
 Golden Queen 
 Harvest 
 Bakers Pride 
 
 
 5.65 
 4.90 
 4.60 
 4.80 
 4.40 
 4.45 
 
 ? 
 ? 
 ? 
 ? 
 ? 
 ? 
 
 
 
 ? 
 
 
 Car 
 47329 
 
 Initials 
 L V. R. R. 
 
 Date 
 12/30/191— 
 
 The above is an invoice of goods shipped this date, in accordance with 
 your order of 12/17/191— 
 
 Washington Flour Co. 
 
 per Temple. 
 EXERCISE. 
 
 1. Extend the bill and compare with previous bill, as to columns, entry 
 of items, etc. 
 
 2. Explain the advantages and disadvantages of the use of many 
 columns. 
 
 3. From the standpoint of the customer, what portion of the heading 
 might be dispensed with? 
 
 4. Prepare a similar original invoice for a wholesale order, using news- 
 paper market quotations. Add terms. Receipt the bill, signing your 
 name as officer of the selUng company. 
 
408 
 
 BUSINESS ARITHMETIC. 
 
 IV. CLOTH MANUFACTURER'S INVOICE. 
 
 Goods Purchased will not be Taken Back Except for Damage or Imperfection. 
 No Claims allowed unless reported within 10 days after receipt of goods. 
 
 MORRIS & NORTON 
 
 DRESS GOODS 
 
 New York, May 21, 191— 
 John Harris & Co. Terms 
 
 Register 1337 N. Y. City. S per cent. 30 das., or 
 
 Book 45 Folio 14 6 per cent. 10 das. 
 
 THIS INVOICE PAYABLE IN NEW YORK niTY FUNDS 
 
 19 Pes NORI^OLK LT PERCALE 
 45-i- 45^ 45^ 45^ 45^ 45^ 46^ 
 46^ 45^ 46^ 46^ 45^ 45^ 45^ 
 46^ 45-^ 46^ 46^ 45^ 
 
 16 Pes E-QREOLK PK PERCALE 
 45 45-3^ 45^ 46^ 45-^ 45^ 45^ 
 45^ 46^ 45-^ 46^ 45^ 45^ 46^ 
 45-^ 45^ 
 
 17 Pes -\^ PEPPERELL BROWN 
 45 45^ 46^ 45-^ 45^ 45-^ 45^ 
 45^ 45-i- 45^ 46^ 45^ 45^ 45^ 
 45^ 45-^ 45^ 
 
 10* 
 
 10 
 
 09 
 
 EXERCISE. 
 
 1. Extend the bill. What amount will settle it June 19? June 24? 
 August 23? October 16? On what date is the face value due? 
 
 2. On October 19, the Fall River Manufacturing Co. sells you, on 
 terms of 4/60; 5/30; 6/10; 36 pc. C. K. Checks, 45, 46, 47S 48, 45^, 46S 
 42', 412, 452^ 46^ 44^ 453^ 442^ 46^ 47^ 431, 45', 42^, 43^, 44S 42', 46, 47, 48«, 
 45, 41', 42, 44, 47', 45^, 43^, 44S 45^, 43', 42', 44 @ 23c. 
 
 Also 12 pc. Ginghams: 45S 47, 46^, 43, 45, 49^, 50, 48', 47, 49, 51», 49« 
 @9f. 
 
 Draw up the bill and receipt for the company for settlement 
 on November 16. 
 
BILLING. 
 
 409 
 
 3. Ascertain by inquiry the names, common piece lengths and prices 
 of four standard cloths and draw up an original bill. Name your own 
 terms and receipt for payment. 
 
 V. DISCOUNT BILLS. 
 See form on page 242. 
 Claims for errors and over- 
 charges must be made 
 within ten days of delivery. 
 Terms, 30 days net; 5% cash. Louisville, October 25, 191- 
 
 THE KENAWHA SUPPLY COMPANY. 
 
 GENERAL CONSTRUCTION AND RAILWAY SUPPLIES. 
 
 1468 River Street. 
 
 Your Order No. 6657 
 
 Your Req. No 
 
 Shipped L L. Frt. 10/25/1- 
 Our No. 'D-S456 
 Salesman Roberts. 
 
 Sold to John P. Carey, 
 
 Evanston, IlL 
 
 Quan- 
 tity 
 Fur- 
 nished. 
 
 Description. 
 
 List. 
 
 Disc, i 
 
 Extension. 
 
 Total. 
 
 16 
 
 sq. 10X14 Gal. Sheet Roofing 
 ft. Gal. Valley 
 ft. Ga . Climax 
 
 lb. W. P. R. Wire 
 
 C. Machine Bolts, \ X3^, 
 C. Machine Bolts, 5/8X4 
 C. Machine Bolts, fX8 
 
 6 
 
 Le 
 
 4 
 
 7 
 
 13 
 
 87 
 11 
 14 
 
 ss 
 12 
 
 72 
 15 
 85 
 
 20% 
 30% 
 
 50, 10 
 
 5% 
 
 
 
 
 
 164 
 
 
 
 
 48§ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 IfiO 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 
 
 
 1^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Trade 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 Received payment, October 17, 191 — 
 
 THE KENAWHA SUPPLY CO. 
 
 J. Towne, Treas. 
 Extend and explain bill. 
 
 How does it differ in arrangement from the bill on page 242? 
 
 Prepare discount bills for examples on pages 237, 239 and 242. 
 
 Supply names of customers and dealers where necessary. 
 
410 
 
 BUSINESS ARITHMETIC. 
 
 VI. MONTHLY STATEMENT. 
 
 Retail Statement. 
 
 Samuel Henderson, 
 
 The Morrison. 
 
 Chicago, 111., Sept, 30, Wl- 
 
 IN ACCOUNT WITH 
 
 HENRY NORTON & COMPANY. 
 
 
 
 Items. 
 
 Chargea. 
 
 Credits. 
 
 Sept. 
 1 
 2 
 
 To Balance (Bill rendered) 
 12 yd. Lawn .18 
 
 8 Lawn Mull .30 
 
 3 iDt. Ribbon 1.15 
 6 yd. Embroidery .28 
 2 yd. Elastic .12 
 5 Binding .15 
 
 4 yd. Linen .65 
 1 set Purs, 
 
 1 Ermine Cape 
 12 yd. Silk 1.25 
 1 Set Furs 
 1 rem. Gingham 
 
 9 yd. Calico .10 
 
 5 C. Thread .10 
 
 1 rem. Gingham 
 
 2 Shirtwaists 
 
 3 pr. Blankets 4.00 
 2 pr. • Pillows 2.25 
 2 Bolsters 2.50 
 
 
 — 
 
 6Q 
 
 40 
 
 165 
 
 
 5 
 
 
 
 
 
 6 
 
 
 — 
 
 
 12 
 
 165 
 240 
 
 00 
 00 
 
 
 *Cr. 
 24 
 
 
 65 
 
 00 
 
 Cr. 
 27 
 
 9 
 
 60 
 
 65 
 
 
 Le 
 
 ss 
 
 
 
 
 — 
 
 *NoTE 1. Credit items are usually entered in red. 
 
 Note 2. Where individual purchases have covered many items on 
 any one date, and a bill has been rendered at the time, details are not 
 entered on the statement, but the date of each purchase, the words "to 
 Bill Rendered" and the amount of individual bill are entered. This ia 
 common in wholesale statements. 
 
BILLING. 
 
 411 
 
 EXERCISE. 
 
 1. How does this statement differ from an ordinary bill of goods? 
 
 2. Extend the statement. 
 
 3. Explain just what has taken place according to the statement 
 How does it happen that credits occur? 
 
 4. Should the statement be receipted in case of partial settlement? 
 
 5. Prepare an original statement of a month's purchase of groceries. 
 Receipt for payment. 
 
 EXERCISE. 
 
 1. Draw up a " to bill rendered " statement from the following account: 
 
 Robert H. Lawrence, 
 
 1356 18th Street, N. W. 
 
 191— 
 
 — 
 
 
 
 
 191— 
 
 
 
 1 
 
 Mar. 3 
 
 
 Mdse. 
 
 1 42 
 
 16 
 
 Mar. 8 
 
 
 Cash 
 
 12 00 
 
 18 
 
 
 tt 
 
 121 
 
 15 
 
 21 
 
 
 Mdse. ret. 
 
 540 
 
 23 
 
 
 Debit memo. 
 
 4 
 
 19 
 
 23 
 
 
 Overcharge 
 
 48 
 
 27 
 
 
 Mdse. 
 
 16 
 
 35 
 
 30 
 
 
 10 da. dft. 
 
 150 00 
 
 28 
 
 
 a 
 
 39 
 
 12 
 
 31 
 
 
 Cash 
 
 50 00 
 
 30 
 
 
 (I 
 
 125 
 
 04 
 
 
 
 
 
 
 31 
 
 
 11 
 
 11 
 
 85 
 
 
 
 
 
 
 2. Draw up an original statement to cover one quarter of a year, 
 containing at least fifteen debit items and six credit items of different 
 character. 
 
 VIII. REQUISITION. 
 The requisition is a combined order and voucher or bill, 
 used in ordering supplies for offices and workshops, when the 
 approval of a higher official is necessary. Sometimes it does 
 not involve prices, but is simply a request from an official or 
 foreman, drawn on the stock clerk, for supplies needed, which 
 the latter keeps in stock ready for use as required. In the 
 illustration (page 412), however, the articles required must 
 be purchased. The superintendent of a certain department 
 asks for the supplies, and the general manager approves the 
 request and directs the purchase. Requisitions vary greatly 
 in different businesses and government offices. 
 
412 
 
 BUSINESS ARITHMETIC. 
 
 Requisition Book. 
 
 March 10, 191— 
 
 THE RANKIN MANUFACTURING COMPANY. 
 
 Charge to the Repair Shop; Model Department. 
 
 Please furnish for use in this shop the materials specified below: 
 
 Henry C. Carter, 
 Superintendent. 
 
 Items. 
 
 Quantity. 
 
 400 b. f. 
 
 2000 b. f. 
 
 600 b. f. 
 
 5qt. 
 
 9qt. 
 
 2 1b. 
 
 101b. 
 
 181b. 
 
 Material. 
 
 W.Pine, 2X3-6', 
 Oak, 1X9, per M. 
 Chestnut, per M. 
 M. Y. Varnish 
 H. O. Finish 
 Indian Red 
 Vermilion Red 
 Yellow Murdock 
 
 perM. 
 
 $42.00 
 50.00 
 36.00 
 .21 
 .24 
 .10 
 .24 
 .26 
 
 Estimated 
 Cost. 
 
 March 11, 191— 
 Approved and ordered purchased. 
 
 Jas. M. Dean, 
 Gen'l Man. 
 
 Received the above mentioned 
 material on March 16, 191 — 
 
 Henry C Carter, 
 Supenntendent. 
 
 EXERCISE. 
 
 1. Extend the form. 
 
 2. State exactly what has taken place. 
 
 3. Prepare an original requisition for furniture and material required 
 for the equipment of some business office. Sign as subordinate and have 
 some other pupil sign as a superior officer. 
 
 FOR INVESTIGATION AND REPORT. 
 
 EXERCISE. 
 
 1. Collect specimen bills and statements from at least thirty distinct 
 businesses. Classify these {a) in order of complexity; (6) according to 
 the difficulty of the arithmetical computations involved. 
 
 2. Write a brief on the information furnished by the bill headings. 
 Compare headings of the bills of retail, wholesale and manufacturing houses. 
 
 3. From your study of the bills discover how the "terms of payment" 
 vary with the character of the business. 
 
BILLING. 413 
 
 4. What knowledge of arithmetic should a bilhng clerk have? 
 
 5. Study and report on the use and value of special columns. Give 
 illustrations. 
 
 6. Collect, and submit an illustrated report on forms which, while 
 not bills, are worked out on the same principle. 
 
 7. Show how each bill form is designed to meet the needs of the business 
 in which it is used. 
 
 8. Examine bills and see how many different ways of entering items 
 you can discover. 
 
 9. Report on the different forms of receipt for payment that you can 
 discover. Explain the exact meaning of each 
 
CHAPTER XLVII. 
 
 STORAGE. 
 
 INTRODUCTORY EXERCISE. 
 
 1. What circumstances may lead a housekeeper to store her fumitiu'e? 
 
 2. Why does a commission merchant place certain merchandise in 
 "cold storage"? Why does he not own a storage plant? 
 
 3. Why is grain "stored" in grain elevators? 
 
 4. If I rent a room in a furniture warehouse from July 15 to October 25, 
 at $9.00 per 30 day month, or fraction thereof, the storage bill is $ . 
 
 5. The cold storage of 40 cases of eggs for 50 days, at 40c per month, 
 or fraction, costs what amount? 
 
 Storage is the charge for storing goods in an elevator or 
 warehouse. It is computed on quantity rather than on value. 
 The term of storage is the time for which the storage rate is 
 quoted. The rate may be for the day, week, month, or period 
 of 30 days, etc. Fractions of terms are usually considered as 
 full terms. 
 
 Storage warehouses and grain elevators are closely regu- 
 lated by law. Often rates are fixed by law, by Chambers 
 of Commerce, associations of warehousemen, etc. In the 
 storage of grain, and of certain other produce, there are strict 
 regulations for grading receipts of merchandise, since the 
 person storing cannot expect to withdraw the identical 
 property he deposited. 
 
 SIMPLE STORAGE. 
 Simple storage is storage computed at the time of withdrawal 
 of goods. 
 
 EXERCISE. 
 
 (Solve mentally if possible.) 
 Compute the storage on: 
 
 414 
 
STORAGE. 
 
 415 
 
 1. 200 cs. eggs, for 3 mo. at 10c per mo. 
 
 2. 500 cs. eggs, for 4 mo., at 10c per mo. for 2 mo., and 8c per mo. 
 for succeeding months. 
 
 3. 800 lb. of cheese from August 17 to October 26, at 8c per 100 lb. 
 per month of 30 days. 
 
 4. The storage rate on the following bill is 9c per 100 lb. per 30 da. 
 Extend the bill, explaining each item. 
 
 March 31, 191— 
 Messrs. James Brown & Son, 
 
 To The Morton Storage Co., Dr. 
 
 
 Article. 
 
 Quantity. 
 
 
 
 
 
 Quantity. 
 
 Time. 
 
 Kate. 
 
 
 Lot 
 No. 
 
 In. 
 
 Out. 
 
 
 i 
 
 Amt. 
 
 6746 
 
 Cheese 
 
 20000 lbs. 
 
 Jan. 
 
 16 
 
 Feb. 
 Feb. 
 Mar. 
 Mar. » 
 
 15 
 
 28 
 
 6 
 
 28 
 
 5000 lbs. 
 8000 
 2000 
 5000 
 
 30 
 
 43 
 
 49 
 
 ? 
 
 1 
 
 2 
 2 
 ? 
 
 9c 
 18c 
 
 ? 
 ? 
 
 4.50 
 ? 
 ? 
 ? 
 
 
 ? 
 
 5. Prepare a bill for the storage of 160 cs. eggs on Jan. 15, by James 
 Quick, at the rate of 10c per month of 30 days. Deliveries: Jan. 28, 40 cs.; 
 Feb. 19, 50 cs.; the balance. Mar. 13. 
 
 6. Compute the storage on 5000 lb. poultry, stored Mar. 15 at l/4c 
 per lb. per 30 da. DeUveries: Mar. 30, 1500 lb.; Apr. 4, 800 lb.; May 12, 
 2000 lb.; June 16, the balance. 
 
 7. Extend this bill. Rates: 10c per bale, 15c per cs., per 30 da. 
 
 October 31, 19— 
 Messrs. Cates & Co., 
 
 To The Commercial Storage Co., Dr. 
 
 Quan- 
 tity. 
 
 Marks and Nos. 
 
 Rec'd. 
 
 Deliv. 
 
 Bate. 
 
 Amount. 
 
 8 
 
 4 
 
 12 
 
 6 
 
 bales No. 675-83 A 
 Cs. c D No. 721-24 
 bales A. C. O No. 829-41 
 bales BC <S> No. 726-31 
 
 June 
 July 
 
 8 
 16 
 27 
 
 5 
 
 Oct. 
 Aug. 
 Sept. 
 Oct. 
 
 20 
 19 
 30 
 
 4 
 
 ? 
 ? 
 ? 
 ? 
 
 
 
 
 
 
 In cases where goods are being constantly received and 
 delivered, and simple storage is charged, it is assumed that all 
 deliveries are made from goods longest in storage. 
 
416 BUSINESS ARITHMETIC. 
 
 Illustration. Robert Connor stores 500 bbl. Apr. 16; 300 bbl. 
 May 1; 200 bbl. May 19. He withdraws 400 bbl., Apr. 30; 200 bbl. 
 May 18; 400 bbl., May 27. Compute his storage at the rate of 6c per 
 month, or fraction. 
 Solution. 
 Date Rec'd Deliv. Rate. 
 
 Apr. 16 500 bbl. 
 
 30 400 bbl. Stored, Apr. 16-30, 14 da. 6c 
 
 May 1 300 
 
 1R rtrjn f 100 Stored, Apr. 16-May 18, 32 da. 12c 
 
 ^° "^^^ \ 100 Stored, May 1-18, 17 da. 6c 
 
 19 200 
 
 97 .^n J 200 Stored, May 1-27, 26 da. 6c 
 
 —- ^^ 1200 Stored, May 19-27, 8 da. 6c 
 
 1000 1000 Total storage, 
 
 Cost. 
 S24.00 
 
 12.00 
 6.00 
 
 12.00 
 12.00 
 
 $66.00 
 
 EXERCISE. 
 
 1. A. C. Bronson's storage memorandum is as follows: Stored, Nov. 12, 
 2000 lb. poultry; Dec. 5, 2000 lb.; Jan. 18, 400 lb. Withdrew, Dec. 24, 
 2200 lb.; Jan. 10, 1500 lb.; Mar. 17, 700 lb. Compute the storage at the 
 rate of l/4c per lb. per month. 
 
 2. John T. Bartlett's memorandum is: Stored, Aug. 18, 50 bbl. 
 potatoes; Sept. 15, 200 bbl.; Oct. 17, 300 bbl.; Withdrawn, Nov. 5, 200 
 bbl.; Nov. 20, 100 bbl.; Dec. 5, 100 bbl.; Dec. 20, the balance. Compute 
 his storage bill at the rate of 6c per bbl. per month. 
 
 3. On Oct. 5, Brown stored 100 bbl. apples; Oct. 20, 50 bbl.; Nov. 16, 
 180 bbl. Nov. 1 he withdrew 80 bbl.; Dec. 1, 50 bbl.; Jan. 5, 100 bbl.; 
 Feb. 9, the balance. Compute the storage at the rate of 12c per bbl. per 
 month. 
 
 2. AVERAGE STORAGE. 
 When receipts and deliveries are frequent, it is a custom to 
 average the time, and to charge average storage. Usually 
 exact time, with 30 day periods, is used. 
 
 1. The storage of 200 bbl. for one month equals the storage of 1 bbl. 
 for ? months. 
 
 2. The storage of 100 bbl. for 2 months equals the storage of 1 bbl. for 
 ? months. 
 
 3. The storage of 500 bu. for 20 days equals the storage of 1 bu. for 
 ? days. 
 
 Illustrative Example. The following is a memorandum of flour 
 stored by C. P. Dean with the Commercial Storage Co., at a rate of 4c 
 
STORAGE. 417 
 
 average storage. Receipts: Feb. 6, 200 bbl.; Feb. 21, 150 bbl.; Mar. 8, 
 400 bbl.; Mar. 29, 200 bbl. DeUveries: Feb. 12, 100 bbl.; Mar. 9, 150 bbl.; 
 Mar. 21, 300 bbl.; Apr. 4, 400 bbl. 
 
 Solution. Arrange entries in order of dates : 
 Date. Rec'd. Deliv. Balance Time. Eqiiiv. to 1 bbl. 
 
 Stored for 
 Feb. 6 200 bbl. 200 bbl. 6 da. 1200 da. 
 
 12 100 bbl. 100 9 900 
 
 21 150 250 15 3750 
 
 Mar. 8 400 650 1 650 
 
 9 150 500 12 6000 
 
 21 300 200 8 1600 
 
 29 200 400 6 2400 
 
 Apr. 4 400 
 
 The storage items are equivalent to the storage of one 
 
 barrel for 16,500 days. 
 
 16,500 days = 550 terms of 30 days each 
 At 4c per term, the storage = 550 X 4c = $22.00. Ans. 
 Note. Notice that each "one bbl." equivalent is the product of the 
 balance on hand by the number of days that balance remains unchanged. 
 
 EXERCISE. 
 
 1. Solve the examples in the last written exercises by average storage. 
 
 2. Are charges lower by average, or by simple storage? Why? 
 
 3. Compute average storage at the rate of l^c per bu. per 30 days, 
 from the following memorandum: Received: Oct. 10, 12,000 bu.; Oct. 25, 
 14,000 bu.; Nov. 1, 3500 bu. Deliveries: Dec. 5, 2500 bu.; Dec. 8, 3000 bu.; 
 Dec. 21, 5,000 bu.; Dec. 30, 8,000 bu.; Jan. 5, 11,000 bu. 
 
 4. On Jan. 16, Robert Osbom bought 2,000 bbl. of flour at $4.25 and 
 placed it in storage at an average rate of 5^c per barrel. On Jan. 29, 
 he bought and stored 200 bbl. at $4.30; on Feb. 16, he withdrew 300 bbl. 
 for sale at $4.65; on Mar. 11, 500 bbl. for sale at $4.80, and disposed of the 
 balance, on Apr. 3, at $5.10. Drayage charges amounted to 15c per bbl. 
 What was his per cent, of profit? 
 
 FOR INDIVIDUAL REPORT. 
 
 1. Prepare a brief on the knowledge of arithmetic required by the 
 average retail clerk. 
 
 2. Prepare a brief on the knowledge of arithmetic required by the 
 average mechanic. 
 
 28 
 
418 BUSINESS ARITHMETIC. 
 
 3. Report on "What Knowledge of Arithmetic is Most Necessary." 
 Secure the opinion of business men and manufacturers. 
 
 4. Report on "Arithmetic Computation Tables in General Use." 
 Make clear their value and range of use. Give sections of typical tables 
 illustrating their use and the principles of their constructions. 
 
 5. Write a brief on "Machines for Arithmetical Computation." 
 Cover range, usefulness, speed, accuracy, cost and extent of use. 
 
APPENDIX I. 
 
 SIGNS AND SYMBOLS. 
 
 a/c . . . .account. 
 
 a/s .... account sales. 
 
 + addition. 
 
 O.K. . . all correct. 
 
 & and. 
 
 @ at, each, to. 
 
 and so on. 
 
 B/L ...bill of lading. 
 
 c/o care of. 
 
 V check mark, correct. 
 
 circle. 
 
 c cent. 
 
 ° degree. 
 
 -^ division. 
 
 $ dollar. 
 
 = equal, equals. 
 
 o equivalent. 
 
 ' foot, minutes. 
 
 .fourths (written, as 
 exponents: 2^ = 2i) 
 
 1. 2. 3 
 
 > greater than. 
 
 C hundred. 
 
 " inches, seconds. 
 
 < less than. 
 
 X multiplication; incor- 
 rect. 
 
 # number (written before 
 
 a figure). 
 
 o/d .... on demand. 
 
 % per centum. 
 
 # pounds (written after 
 
 a figure). 
 
 £ pound sterling. 
 
 : ratio. 
 
 *.' since. 
 
 >/". . . .square root of. 
 
 — subtraction. 
 
 .' therefore. 
 
 M thousand. 
 
 A triangle. 
 
 419 
 
APPENDIX II. 
 
 STANDARD ABBRJSVIATIONS. 
 
 Note. The singular form is now commonly used for singu- 
 lar and plural, unless otherwise noted. 
 
 Ac Account. 
 
 A acre. 
 
 agt agent. 
 
 ans answer. 
 
 B.O back order. 
 
 bg bag. 
 
 bal balance. 
 
 bl bale. 
 
 bk. ...... bank, book. 
 
 bbl barrel. 
 
 bkt basket. 
 
 bot bought. 
 
 bx box. 
 
 bu bushel. 
 
 en can. 
 
 cd card, cord. 
 
 car carton. 
 
 cs case. 
 
 C, C.B. . . cash, cash book. 
 
 Cash cashier. 
 
 csk cask. 
 
 c, ct cent. 
 
 eg centigram. 
 
 cm. centimeter. 
 
 ck check. 
 
 ch chest. 
 
 c.o.d collect on delivery. 
 
 cml .commercial. 
 
 com commission. 
 
 Co company. 
 
 consgt consignment. 
 
 cr crate, credit, 
 
 creditor. 
 
 cu cubic. 
 
 cwt hundredweight. 
 
 da day. 
 
 dr debt, debtor, debit. 
 
 dept department. 
 
 do ditto (the same). 
 
 dol dollar. 
 
 dz., doz. . . dozen. 
 
 dft draft. 
 
 ea each. 
 
 E.O.E errors and omis- 
 sion excepted. 
 
 etc et cetera (and so 
 
 forth). 
 
 ex example, express. 
 
 exch exchange. 
 
 e. g exempli gratia (for 
 
 example). 
 
 exp expense. 
 
 420 
 
STANDARD ABBREVIATIONS. 
 
 421 
 
 far farthings. 
 
 fir firkins. 
 
 f ol folio, page. 
 
 ft foot. 
 
 for foreign. 
 
 fw., fwd. . .forward. 
 
 f r franc. 
 
 f.o.b. . . . . .free on board. 
 
 gal gallon. 
 
 gi gill- 
 
 g gram. 
 
 gr gross, grain. 
 
 guar guaranty, guar- 
 antee. 
 
 hf half. 
 
 hhd hogshead. 
 
 hr hour. 
 
 ewt hundredweight. 
 
 in inch. 
 
 int interest. 
 
 I, inv invoice. 
 
 invt inventory. 
 
 ins insurance. 
 
 inst instant (the pres- 
 ent month). 
 
 kg keg. 
 
 km kilometer. 
 
 L.F ledger folio. 
 
 l.p list price. 
 
 If., Iv loaf, loaves. 
 
 lb pound. 
 
 m.p marked price. 
 
 mdse merchandise. 
 
 m meter. 
 
 Messrs. . . . Messieurs (gentle- 
 men. Sirs). 
 
 mi mile. 
 
 min minute. 
 
 Mr Mister. 
 
 Mrs Mistress. 
 
 mo month. 
 
 no number. 
 
 O.B order book. 
 
 oz ounce. 
 
 p., PP page, pages. 
 
 pkg package. 
 
 pd paid. 
 
 pi pail. 
 
 pr pair. 
 
 pay payment. 
 
 d pence. 
 
 per per (by the). 
 
 p.c per centum, 
 
 pk peck. 
 
 pwt pennyweight. 
 
 pc piece. 
 
 pt pint. 
 
 poc, pkt. . pocket. 
 
 lb pound. 
 
 Pres President. 
 
 prox proximo (the fol- 
 lowing month). 
 
 qt quart. 
 
 qr quire. 
 
 R.R railroad. 
 
 rm ream. 
 
 recM received. 
 
 M Reischsmark. 
 
422 
 
 BUSINESS ARITHMETIC. 
 
 rd rod. 
 
 sk sack. 
 
 S sales. 
 
 sec second. 
 
 sec'y- .... secretary 
 
 s shilling. 
 
 set settlement. 
 
 ship shipment. 
 
 shipt shipped. 
 
 sig signed, signature. 
 
 st street. 
 
 stk stock. 
 
 sund sundries. 
 
 trc .tierce. 
 
 T ton. 
 
 tr., trans. . transfer. 
 
 treas treasurer, treasury. 
 
 tb tub. 
 
 ult ultimate (last 
 
 month). 
 
 via via (by way of). 
 
 viz videlicet (namely, to 
 
 wit). 
 
 wk week. 
 
 wt weight. 
 
 yd yard. 
 
 yr year. 
 
INDEX. 
 
 Abbreviations, 420-422. 
 Acceptance, 309-311. 
 Accident insurance, 257. 
 Account, 72. 
 
 averaging, 397. 
 
 cash, 73. 
 
 expense, 74. 
 
 merchandise, 73. 
 
 personal, 74. 
 
 proprietor's, 75. 
 Account sales, 249. 
 Accuracy, 7. 
 Accurate interest, 293. 
 Acre foot, 177. 
 Acute angle, 143. 
 Addition, 7. 
 
 checking, 11. 
 
 complements, 10. 
 
 dictation, 7. 
 
 grouping, 9. 
 
 horizontal, 12. 
 
 of decimals, 15. 
 
 of denominate numbers, 138. 
 
 of fractions, 95. 
 
 principles of, 8-12. 
 Ad valorem duties, 278, 283. 
 Advertising, 77. 
 
 newspaper, periodical, 79, 80. 
 
 billboard, 79. 
 
 poster, 78, 79. 
 
 records of replies, 83, 84, 85. 
 Agency, 245. 
 Agent, 245. 
 
 Agent, payment of, 246. 
 Aliquot parts, 110. 
 
 division by^ 115. 
 
 multiplication by, 110. 
 Altitude, 145. 
 Analysis of problems, 118. 
 Angle, 143. 
 
 Angular measure, 134. 
 Antecedent, 186. 
 Appraiser, 279. 
 
 Approximate results, 40. 
 
 time, 169. 
 Arabic notation, 3. 
 Arc, 145. 
 Area of circle, 147. 
 
 of parallelogram, 146. 
 
 of triangle, 146. 
 Assav office, 51. 
 Assessment, 269, 270, 274, 389. 
 Assessors, 270. 
 Automobile insurance, 257. 
 Average, 43. 
 Average clause, 264. 
 Average investment, 394. 
 Average storage, 416. 
 Averaging, 43, 397. 
 Avoirdupois weight, 133. 
 Axioms, 46. 
 
 B 
 
 Bank discount, 318. 
 
 register, 325. 
 
 rules for, 320 
 
 terms of, 319. 
 
 ticket, 324. 
 Bank draft, 361, 362. 
 Banker's 60-day method, 287. 
 Base, 145. 
 
 Base for percentage, 215, 221. 
 Base for profit and loss, 226. 
 Base ball averages, 219. 
 Bear, 344. 
 
 Beneficiary, 257, 259. 
 Bids and estimates, 383. 
 Bill, discount, 409. 
 
 manufacturers, 407, 409. 
 
 monthly statement, 410. 
 
 professional, 69. 
 
 simple, 403, 405. 
 
 storage, 415. 
 Billing, 403. 
 
 Bill of exchange, 370, 371. 
 Bill of lading, 280. 
 Bin, capacity of, 162. 
 
 423 
 
424 
 
 INDEX. 
 
 Blank endorsement, 307. 
 Board foot, 161. 
 Boiling point, 175. 
 Bond, 309, 337. 
 
 compared with stock, 337. 
 
 coupon, 337. 
 
 quotations, 339. 
 
 registered, 337. 
 Bonded warehouse, 280. 
 Bonus, 69. 
 Broker, 245. 
 Brokerage, 245. 
 Building and loan associations, 351. 
 
 distribution of profits, 353. 
 
 earnings, 351. 
 
 loans, 351. 
 Bull, 344. 
 
 Business accounts, 72. 
 Business terms, 72. 
 
 Calculation of time table, 170-171. 
 Call loans, 345. 
 Canadian money, 135. 
 Cancellation, 88, 286. 
 Capacity, measures of, 133. 
 
 of bins, tanks, etc., 162. 
 Capital, 334. 
 Capital stock, 332. 
 Carat, 133. 
 Carpeting, 155. 
 Cash balance, 402. 
 Cashier's check, 361. 
 Casting out nines, 11. 
 Centigrade scale, 175. 
 Certificates, gold and silver, 50. 
 Certificates of deposit, 362. 
 Change schedule, 66. 
 Check, 357, 358. 
 Checking the work, 11, 118. 
 Cu-cle, 145. 
 
 Circular measure, 134. 
 Circulation of money, 62. 
 Circulation statement, 12. 
 Circumference, 145. 
 Coal consumption, 210. 
 Coins of the United States, 51. 
 Cold storage temperatures, 176. 
 Collateral, 345. 
 
 loan, 345. 
 
 note, 346. 
 Collection fee, 361. 
 
 Collector of the port, 279. 
 Commercial discount, 236. 
 Commercial draft, 363. 
 Commission and brokerage, 245. 
 Commission merchant, 245. 
 Common stock, 334. 
 Complements, 10. 
 Composite measures, 177. 
 Composite numbers, 86. 
 Compound interest, 302. 
 
 tables, 303. 
 Compound proportion, 190. 
 Compounds, 223. 
 Cone, 156. 
 Conical surface, 156. 
 Consequent, 186. 
 Consideration, 255. 
 Consignee, 246. 
 Consignment, 246. 
 Consignor, 246. 
 Contract, 255. 
 
 Conversion of fractions, 103. 
 Conversion tables, 184. 
 Coordinates, 200. 
 Copying numbers, 8. 
 Cord measure, 160. 
 Com, production of, 205. 
 Corporation, 332. 
 
 advantages of, 333. 
 Correspondent bank, 360. 
 Cost, 226, 230. 
 
 Cost by hundred and thousand, 116. 
 Cost per inquiry, 83. 
 Cost-keeping, 379. 
 
 ticket, 380, 381. 
 
 unit of, 382. 
 Cotton, production of, 209. 
 Counting table, 134. 
 Coupon, 337. 
 Credit insurance, 258. 
 Credit prices, 295. 
 Cube, 156. 
 
 Cube of numbers, 121. 
 Cube root, 125. 
 
 graphic illustration, 129. 
 Cubic measure, 135. 
 
 metric, 183. 
 Curb market, 343. 
 Currency memorandum, 66. 
 Currency of U. S., 50. 
 Custom house, 279. 
 Customs, 277. 
 Cylinder, 156. 
 
INDEX. 
 
 425 
 
 Date line, 167. 
 
 Date of maturity, 172. 
 
 Decimals, 
 
 addition of, 15. 
 
 division of, 39. 
 
 multiplication of, 31. 
 
 subtraction of, 22. 
 Degree, circular measure, 165. 
 Denominate numbers, 132. 
 
 addition, 138. 
 
 division, 141. 
 
 multiplication, 140. 
 
 reduction ascending, 137. 
 
 reduction descending, 136. 
 
 subtraction, 139. 
 
 tables of, 53, 133-135. 
 Denominator, 91. 
 Deposit, 359. 
 Depreciation, 375. 
 
 on original values, 375. 
 
 on reduced values, 375. 
 
 table of, 377. 
 Diagram, 193. 
 Diameter, 145. 
 Dictation, 7. 
 Difiference column, 20. 
 Dimensioning, 195. 
 Discount, 236. 
 
 bank, 318. 
 
 rates, 240. 
 
 series, 238. 
 Discussion — See Questions for dis- 
 cussion. 
 Distribution of profits, 389-395. 
 Dividend, 25, 335. 
 Divisibility, tests, 86, 87. 
 Division, 35. 
 
 by continued subtraction, 35. 
 
 long, 37. 
 
 of decimals, 39. 
 
 of denominate numbers, 141. 
 
 of fractions, 101. 
 
 short, 36. 
 
 short methods of, 38. 
 Domestic exchange, 355. 
 Draft, 309. 
 Drawee, 309. 
 Drawer, 309. 
 Dry measure, 133. 
 Duties, ad valorem, 277, 283. 
 
 specific, 277, 282. 
 
 £ 
 
 Endowment policy, 259. 
 Endorsement, 
 
 blank, 307. 
 
 fuU, 308. 
 
 quahfied, 308. 
 English money, 135. 
 Equality of expressions, 46. 
 Equated date, 397. 
 Equation, 46. 
 
 Equation of accounts, 397, 399. 
 Equation of payments, 397. 
 Equilateral triangle, 144. 
 Estimates, 383. 
 Evolution, 122. 
 Exact interest, 293. 
 Exact results, 40. 
 Exact time, 169. 
 Exchange, 355. 
 
 domestic, 355. 
 
 fluctuation of, 364, 371. 
 
 foreign, 366. 
 
 inland, 355. 
 
 rates, 364, 366, 373. 
 Ex-dividend, 343. 
 Expenditures of U. S. Gov't, 277. 
 Exponent, 121. 
 Exports of U. S., 208, 219. 
 Express money order, 356. 
 Expressions, equality of, 46. 
 Extension, measures of, 134. 
 Extremes, 190. 
 
 F 
 
 Face, 156. 
 
 Factor, agent, 245. 
 
 Factors and multiples, 86. 
 
 Factor, highest common, 88. 
 
 Factoring, 87. 
 
 Factory expense, 68. 
 
 Fahrenheit (temperature), 175. 
 
 Fee, 269. 
 
 Fidelity insurance, 257. 
 
 Figures, geometric, 143. 
 
 Fire insurance, 265. 
 
 Flooring, 151. 
 
 Floor plan, 152. 
 
 Focal date, 398. 
 
 Foot pound, 177. 
 
 Foreign coin, gold value of, 281. 
 
 Foreign exchange, 366. 
 
426 
 
 INDEX. 
 
 Foreign money order, 368. 
 Formula, 179. 
 Fractions, 91-109. 
 
 addition, 95. 
 
 compound, 92. 
 
 conversion of, 103. 
 
 division of, 101. 
 
 improper, 92. 
 
 lowest terms, 93. 
 
 multiplication of, 97. 
 
 proper, 92. 
 
 reduction of, 91. 
 
 similar, 95. 
 
 simplification of, 93. 
 
 subtraction of, 96. 
 
 terms, 91. 
 Fractional relations, 106. 
 Fractional units, 91. 
 Free list, 278. 
 Freezing point, 175. 
 French money, 135. 
 
 Gable, 151. 
 
 Gain, 72. 
 
 Geometric conceptions, 143-147: 
 
 156-160. 
 Geometric line, 143. 
 German money, 135. 
 Gold value of coin, 281. 
 Good will, 388. 
 Graph, 193; 204-211. 
 Graphic arithmetic, 193. 
 Graphic treatment of squares and 
 
 cubes, 128. 
 Greatest common divisor — See 
 
 Highest common factor. 
 Gross cost, 245. 
 Gross proceeds, 246. 
 Grouping, in addition, 9 
 Guaranty, 246. 
 
 Hundreds, cost of, 116. 
 Hypotenuse, 149. 
 
 Immigration, table of, 210. 
 
 Import duties, 277. 
 
 Imports of U. S., 211. 
 
 Improper fraction, 92. 
 
 Income, 347. 
 
 Income insurance, 262. 
 
 Individual work, 42, 142, 163, 174, 
 
 185, 212, 219, 235, 324, 412, 
 
 417, 418. 
 Indorsement — See Endorsement. 
 Insolvency, 388. 
 Insurance, 255. 
 
 companv, 259, 264. 
 
 personal, 256-263. 
 
 poUcy, 256, 259. 
 
 property, 256; 263-268. 
 Interest, 
 
 accurate, 293. 
 
 banker's 60-day method, 287. 
 
 cancellation method, 285. 
 
 compound, 302. 
 
 simple, 285. 
 
 six per cent, method, 290. 
 
 tables, 292, 303. 
 Interest on investments, 392. 
 Investment, 347. 
 Invoices, 279, 282, 283. 
 Involution, 121. 
 Irrigation, 206. 
 Isosceles triangle, 144. 
 
 Job work ticket, 380. 
 
 Knot, 134. 
 
 Health insurance, 257. 
 Hemisphere, 157. 
 Highest common factor, 88. 
 Holidays, 313. 
 Horizontal addition, 12. 
 Horsepower, 177. 
 Household expense, 21. 
 How to read numbers, 4. 
 
 Land measure, 
 
 metric, 183. 
 
 surveyor's, 134. 
 Lateral area, 159. 
 Lawful money, 50. 
 Least common multiple, 89. 
 Legal rate of interest, 285. 
 Letter of credit, 369, 370. 
 Lever, 188. 
 
INDEX. 
 
 427 
 
 Liabilities, 72. 
 Liability insurance, 257. 
 License, 269, 274. 
 Life insurance, 259. 
 
 annual premiums, 261. 
 
 values, 260. 
 Limited partner, 388. 
 Linear measure, 134. 
 Linear metric measure, 182. 
 Linear scale, 195. 
 Lines, 143. 
 Liquidation, 338. 
 Liquid measure, 133. 
 Loans, 306, 345. 
 Long division, 37. 
 Longitude, 165. 
 Loss, 72. 
 Lumber, 160. 
 
 Maker of note, 307. 
 Making change, 54. 
 Map length, 202, 203. 
 Margin, 343. 
 Marking goods, 232. 
 Marine insurance, 265. 
 Maturity, 172. 
 
 amount due at, 319. 
 
 of negotiable instruments, 312. 
 Mean, 43. 
 Means, 190. 
 Measure, 
 
 angular, 134. 
 
 apothecaries', 133. 
 
 avoirdupois, 133. 
 
 circular, 134. 
 
 comparative, 133. 
 
 composite, 177. 
 
 cord, 160. 
 
 counting, 134. 
 
 cubic, 135. 
 
 dry, 133. 
 
 extension, 134. 
 
 linear, 135. 
 
 liquid, 133. 
 
 long, 134. 
 
 metric, 182. 
 
 money, 135. 
 
 sea, 134. 
 
 square, 134. 
 
 surveyor's, 134. 
 
 time, 135, 162. 
 
 Measure, 
 
 temperature, 175. 
 
 Troy, 133. 
 
 value, 135. 
 
 weight, 133. 
 Measuring, 136. 
 Mensuration, 143-163. 
 Merchant's rule, 317. 
 Meridian, 164, 
 Metric equivalents, 184. 
 Metric system, 182. 
 Mexican money, 135. 
 Mints, 51. 
 Minuend, 18. 
 Mixed numbers, 92. 
 
 division of, 101. 
 
 multiplication of, 99. 
 Mixtures, 223. 
 Money, 
 
 circulation of U. S., 52. 
 
 recoinage of U. S., 52. 
 
 table of foreign, 135. 
 
 table of U. S., 53. 
 
 United States, 49. 
 Money order, 355. 
 
 form of express, 356. 
 
 rates, 356, 368. 
 Multiple, 86. 
 
 common, 89. 
 Multiplication, 25. 
 
 aliquot parts, 25. 
 
 commutative law, 25. 
 
 decimal, 31. 
 
 denominate, 140. 
 
 fractional, 97. 
 
 shortening, 28-30. 
 
 tables, 26. 
 Multiplier, 25. 
 
 Negotiable paper, 307. 
 
 bank draft, 361, 362. 
 
 certificate of deposit, 362. 
 
 check, 357. 
 
 maturity, 312. 
 
 promissory note, 307. 
 
 sight draft, 309. 
 
 time draft, 310, 311. 
 Net capital, 72, 388. 
 Net cost, 246. 
 Net gain, 72. 
 Net insolvency, 388. 
 
428 
 
 INDEX. 
 
 Net loss, 72. 
 
 Net proceeds, 246. 
 
 Newspaper, 
 
 advertising, 79. 
 
 advertising rates, 81. 
 
 circulation, 12, 80. 
 Nines, casting out, 11. 
 Nominal partner, 388. 
 Normal, 43. 
 Notation, 
 
 Arabic, 3. 
 
 Roman, 5. 
 Notes, 307, 320. 
 Notice of protest, 308. 
 Numbers, 1. 
 
 abstract, 1. 
 
 composite, 86. 
 
 compound, 2. 
 
 concrete, 1. 
 
 copying, 8. 
 
 denominate, 1, 132. 
 
 dictation of, 7. 
 
 like, 1. 
 
 powers of, 121. 
 
 prime, 86. 
 
 roots of, 122. 
 
 simple, 2. 
 
 unlike, 1. 
 Numeration, 4, 5. 
 Numerator, 91. 
 
 Obtuse angle, 143. 
 Old age insurance, 262. 
 Open policy, 264. 
 
 Panama canal, 223. 
 Papering, 153. 
 Paper table, 135. 
 Parallel line graphs, 207. 
 Parallel lines, 143. 
 Parallelogram, 144. 
 Parcels post, 57, 58. 
 Partial payments, 314. 
 
 indorsements, 315. 
 
 merchant's rule, 317. 
 
 United States rule, 315. 
 Partition, 36. 
 Partitive proportion, 387. 
 Partners, 388. 
 Partnership, 387. 
 
 Partnership statement, 392. 
 Par value, 333. 
 Passenger mile, 178. 
 Payee, 307. 
 Payments, 61, 306. 
 Pay roll, 63, 65. 
 Pay roll check, 358. 
 Percentage, 213. 
 
 comparative table, 214. 
 
 to find base, 221. 
 
 to find percentage, 215. 
 
 to find rate, 217. 
 Perimeter, 145. 
 Periodic interest, 300. 
 Perpendicular line, 143. 
 Pitch of a roof, 151. 
 Plane figures, 143. 
 Plane surfaces, 143. 
 Plastering, 153. 
 Plotting, 194, 197, 199. 
 Policy of insurance, 256. 
 
 open, 264. 
 
 valued, 264. 
 Poll tax, 270. 
 Polyhedron, 156. 
 Port of delivery, 279. 
 Port of entry, 279. . 
 Postage, 56. 
 
 report, 59, 60. 
 
 stamps, 57. 
 Postal information, 57. 
 Postal money order, 356. 
 Post oflBces and routes, 56. 
 Posters, 78, 79. 
 Power of a number, 121. 
 Practical measurements, 143-175. 
 Precipitation chart, 209. 
 Preferred stock, 334. 
 Premium, 256. 
 
 rates of insurance, 261. 
 
 wage, 69. 
 Present worth and true discount, 
 
 299. 
 Present worth, 
 
 of a debt, 299. 
 
 of a firm, 72, 388. 
 Price per hundred or thousand, 116, 
 Prime cost — See Net cost. 
 Prime factor^ 86. 
 Prime meridian, 164. 
 Prime number, 86. 
 Prmcipal, 285. 
 Prism, 156. 
 
INDEX. 
 
 429 
 
 Problem analysis and solution, 118. 
 Proceeds of a note, 320. 
 Proceeds of a sale, 246. 
 Production of barley, 193. 
 Profit as rate of selling price, 228. 
 Profit and loss, 225. 
 Profit and loss account, 390 
 Profit sharing, SSO-391. 
 Promissory note, 307. 
 Proper fraction, 92. 
 Property tax, 269. 
 Proportion, 190. 
 
 compound, 190. 
 
 continued, 190. 
 
 direct, 191. 
 
 indirect, 191. 
 
 partitive, 387. 
 
 principles of, 190. 
 
 simple, 190. 
 
 terms, 190. 
 Proprietor's account, 75. 
 Proposal, 383. 
 Protest, 388. 
 Purchase, 249. 
 Pyramid, 156. 
 
 Quadrilateral, 144. 
 Qualified endorsement, 308. 
 Questions for discussion, 55, 90, 105, 
 
 195, 215, 232, 238, 313, 318, 
 
 319, 388, 396. 
 Quotation, 334, 338. 
 Quotation list, 251, 339. 
 Quotient, 35. 
 
 Radius, 145. 
 
 Rate, 213, 217. 
 
 Rate card, 81. 
 
 Rate of exchange, 364, 366. 
 
 fluctuation of, 364, 371. 
 Rate of interest, 285. 
 Ratio, 186. 
 
 direct, 186. 
 
 inverse, 186. 
 
 signs of, 186. 
 
 terms, 186. 
 
 value, 186. 
 Ration list, 27. 
 
 Reading and writing numbers, 3-5. 
 Reaumur thermometer, 75. 
 
 Receipts of U. S. gov't, 277. 
 Receipts for payment, 74, 404. 
 Reciprocal of a fraction, 101. 
 Record, 
 
 of advertising, 84, 85. 
 
 of postage, 59, 60. 
 
 of professional visits, 71. 
 Rectangle, 145. 
 Rectangular solid, 156. 
 Reduction of denominate numbers, 
 
 136, 137. 
 Reduction of plotted lengths, 197. 
 Registered bond, 337. 
 Relations, fractional, 106. 
 Remainders, 35. 
 Requisition, 411. 
 Requisition book, 412. 
 Resistance, 192. 
 
 Results, exact and approximate, 40. 
 Right angle, 143. 
 Right angled triangle, 144. 
 Roman notation, 5. 
 Root of numbers, 122. 
 
 S 
 
 Salary, 62. 
 
 Salary pay roll, 63. 
 
 Sales (agent), 247. 
 
 Savings accounts, 326, 328, 329, 331., 
 
 interest on, 328. 
 Savings banks, 326. 
 Scalene triangle, 144. 
 Scales, 194. 
 Sea measure, 134. 
 Selling price, 226. 
 Services, 
 
 biU for, 69. 
 
 payment for, 61. 
 Shareholder, 332. 
 Short division, 36. 
 Short methods, 
 
 multiphcation, 28, 30, 100. 
 
 division, 38. 
 Sight draft, 309, 361. 
 Signs, algebraic and numerical, 47- 
 Simple interest, 285. 
 Sinking fund, 337, 377, 378. 
 Six per cent, method, 290. 
 Sixty-day method, 287. 
 Solids, 156. 
 
 Solution of problems, 118 
 Solving equations, 45. 
 
430 
 
 INDEX. 
 
 Specifications, 383. 
 Specific duties, 277, 282. 
 Specific gravity, 41. 
 Sphere, 157. 
 Spherical surface, 156. 
 Square, 144. 
 
 Square (lumber measure), 151. 
 Square measure, 134. 
 
 metric, 183. 
 Square of a number, 121. 
 
 graphic illustration, 128. 
 Square root, 123. 
 
 graphic, 128. 
 Standard time, 167. 
 Standard measures, 133-135. 
 Standard weights, 133-135. 
 Statement, monthly, 410. 
 
 Statistics! 14^ 193-211. 
 Stock broker, 343. 
 Stock certificate, 333. 
 Stockholder, 332. 
 Stock dividends, 336. 
 Stock record, 20. 
 Stock, 332. 
 
 buying and selling, 342. 
 
 classes of, 334. 
 
 quotations, 334, 338. 
 
 shares, 333. 
 Storage, 414. 
 
 average, 416. 
 
 bill, 415. 
 
 term, 414. 
 Subtraction, 18. 
 
 checking, 18. 
 
 complement method, 19. 
 
 of decimals, 22. 
 
 of denominate numbers, 139. 
 
 of fractions, 96. 
 Subtrahend, 18. 
 Surplus, 335. 
 Surveyor, 280. 
 Surveyor's measure, 134. 
 Symbols, 419. 
 
 Tables, 
 
 apothecaries' weight, 133. 
 avoirdupois weight, 133. 
 Canadian money, 135. 
 circular measure, 134. 
 circular measure and time, 165. 
 conversion, 133. 
 
 Tables, 
 
 counting, 133. 
 
 cubic measure, 135. 
 
 denominate numbers, 133-35. 
 
 dry measure, 133. 
 
 English money, 135. 
 
 French money, 135. 
 
 German money, 135. 
 
 import duties, 278. 
 
 insurance premiums, 261. 
 
 interest, 292. 
 
 linear measure, 134. 
 
 liquid measure, 133. 
 . metric, 182. 
 
 metrical equivalents, 184. 
 
 Mexican money, 135. 
 
 multiplication, 26. 
 
 paper, 134. 
 
 siiJcing fund, 378. 
 
 square measure, 135. 
 
 tax, 273. 
 
 time, 137, 170, 171. 
 
 Troy weight, 133. 
 
 United States money, 53. 
 
 values, insurance, 260. 
 
 wage, 67. 
 Tabulations, 14. 
 Tanks, capacity of, 162. 
 Tariff, 277. 
 Tariff act, 277. 
 Taxation, 269. 
 Tax table, 273. 
 Telegraph money order, 357. 
 Temperature, 175. 
 
 scale reduction, 176. 
 Temperature chart, 209. 
 Term of discount, 319. 
 Term of note, 319. 
 Terms of a fraction, 91. 
 Terms of a sale. 405. 
 Tests of divisibihty, 86, 87. 
 Textile manufactures, 208. 
 Thousands, to find cost of, 116. 
 Timber supply, 207. 
 Time, 
 
 calculation table, 170-171. 
 
 commercial, 168. 
 
 difference in, 166. 
 
 standard, 167. 
 Time card, 68. 
 Time draft, 309-311. 
 Time of discount, 119 
 Time sheet, 68. 
 
INDEX. 
 
 431 
 
 Ton mile, 178. 
 Train mile, 178. 
 Transit insurance, 266. 
 Transportation, 47. 
 Travelers' check, 368. 
 Triangle, 144. 
 Troy weight, 133. 
 True discount, 299. 
 Turn-over of capital, 227. 
 Types of problems, 118, 119. 
 
 Value of foreign coin, 281. 
 Valued policy, 264. 
 Visits, record of, 71. 
 
 Volume, 
 
 cylinder, 159. 
 prism, 157. 
 
 Undivided profits, 335. 
 Unit, 
 
 of freight traffic, 178. 
 
 of irrigation, 177. 
 
 of passenger traffic, 178. 
 
 of power, 177. 
 
 of train traffic, 178. 
 
 of work, 177. 
 United States customs, 275. 
 United States money, 49, 50. 
 United States rule, 315. 
 Units, 1. 
 
 fractional, 91. 
 Unity, 1. 
 Usury, 285. 
 
 Wages, 62, 64. 
 Wage earners, 61. 
 Wage pay roll, 65. 
 Wage table, 67. 
 Wall signs, 78. 
 Warehouse, 280. 
 Watered stock, 325. 
 Weight, 
 
 apothecaries', 133. 
 
 avoirdupois, 133. 
 
 comparison of, 133. 
 
 long ton, 133. 
 
 miscellaneous, 133. 
 
 metric, 183. 
 
 Troy, 133. 
 Whole life policies, 259. 
 
 Printed in the United States of America. 
 
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