1 Q 
 

 
 
 
 LIBRARY 
 
 OF THK 
 
 University of California. 
 
 GIFT OF 
 
 ^AJLA: ^M^.Crf 
 
 Class 
 
 
 
 
 
ELEMENTS OF BUSINESS ARITHMETIC 
 
THE MACMILLAN COMPANY 
 
 NEW YORK • BOSTON • CHICAGO 
 ATLANTA • SAN FRANCISCO 
 
 MACMILLAN & CO., Limited 
 
 LONDON • BOMBAY • CALCUTTA 
 MELBOURNE 
 
 THE MACMILLAN CO. OF CANADA, Ltd. 
 
 TORONTO 
 
MACMILLAN'S COMMERCIAL SERIES 
 
 ELEMENTS OF 
 
 BUSINESS AEITHMETIC 
 
 BY 
 
 ANSON H. BIGELOW 
 
 SUPEBINTBNDENT CITY SCHOOLS, LEAD, 8.1). 
 AND 
 
 W. A. ARNOLD 
 
 DIRECTOR BUSINESS TRAINING, WOODBINE, IOWA 
 NORMAL SCHOOL 
 
 THE MACMILLAN COMPANY 
 1911 
 
 All rights reserved 
 
MAY 5 1911 
 GIFT, 
 
 Copyright, 1911, 
 By the MACMILLAN COMPANY. 
 
 Set up and electrotyped. Published January, 191 1 
 
 NarfaootJ ^regs 
 
 J. 8. Cashing Co. — Berwick & Smith Co. 
 
 Norwood, Mass., U.S.A. 
 
PREFACE 
 
 The preparation of the text which follows was undertaken 
 in the belief that the arithmetic of the schools should teach 
 the methods most in vogue in the business world, and that 
 those methods should be so taught as to form correct habits 
 in those who are to attack the problems of real life. 
 
 The accomplishment of these ends has involved, first, an 
 investigation into the methods of the various fields of business 
 activity, and, second, the writing of the whole subject from 
 the point of view of habit-forming rather than from that of 
 either the conventional or the scientific treatment. 
 
 The methods chosen are believed to have the sanction of 
 usage by those in the business world best qualified to speak. 
 The manner of presentation has been tested by nearly ten 
 years of use, in manuscript form, in the schools and classes 
 of which the authors have had charge. 
 
 The processes presented are strictly arithmetical. No 
 form of domination by higher studies is more insidious or 
 harmful than the attempt to apply the abstractions of algebra 
 and geometry to the problems of arithmetic for the use 
 of children. Immature young people do not comprehend 
 these abstractions and can only memorize them and apply 
 them haltingly. On the other hand, if they fully understand 
 the concrete methods of arithmetic and can understandingly 
 solve its problems, their minds are better equipped with 
 those concrete concepts which alone give meaning to the 
 more abstract forms and truths of the higher branches of 
 mathematics. 
 
 While the methods used in this book are those of the 
 counting room, the shop, or the farm, the pure mathematical 
 
 213230 
 
Vi PREFACE 
 
 element has not been sacrificed. When the mathematical 
 reasoning of available methods is clear, the chief considera- 
 tions have been, just as they are in the business world, short- 
 ness of operation, quickness of solution, and the minimum 
 of opportunity or likelihood of error. These considerations 
 have not always led to formal methods, equally applicable 
 to all contingencies, but have rather pointed to direct ways 
 of solving the kind of problems most frequently met. While 
 the emphasis has not been, therefore, upon the best methods 
 for infrequent and unusual problems, their solution is none 
 the less clearly prepared for. 
 
 Topics admittedly obsolete have been omitted, while others 
 less used than formerly have received correspondingly less 
 emphasiSo In general, only those topics or phases of topics 
 have been treated which are applicable to present-day prob- 
 lems, and in the order of their need, regardless of the tradi- 
 tional arrangement and sequence of subject matter. 
 
 To those who believe that there should be a re-canvassing 
 of the whole field of arithmetic at the close of the grammar 
 school, or in the first high school year, with more attention 
 to the methods used in actual life and with the deliberate 
 purpose of developing an habitual mode of attack which seeks 
 the most direct and accurate methods of solution, we com- 
 mend our book in the hope that it may be of real service. 
 We are firmly convinced that there is a large and growing 
 number of school men and women who believe that this is 
 the only road to adequate and practical results from arith- 
 metic teaching. 
 
 We desire to express appreciation of the sympathetic criti- 
 cism and cooperation of the editor of the series in which 
 the book appears, and of the many courtesies shown us by 
 officers and employees of numerous large business houses 
 whom we consulted in our quest for information as to current 
 arithmetic practices. A. H. B. . 
 
 W. A. A. 
 
EDITOR'S INTRODUCTION 
 
 Two small boys were overheard in conversation as they 
 went out from the morning assembly exercises of a given 
 school, just after the thirteenth chapter of First Corinthians 
 had been read. One's remark to the other fairly expressed 
 the estimate which has been placed on the value of the 
 elementary school studies. The lad's comment was, " Readin, 
 ritin, and rithmetic, and the greatest of these is rithmetic." 
 Arithmetic has claimed a large part of the school's time ; 
 it has presented the very citadel of difficulty to the pupil ; 
 and it has been made the object of first importance by the 
 teacher. 
 
 While the above statements are true there are other facts 
 in striking contrast. Results at present secured from the 
 study of arithmetic are most unsatisfactory. Several inves- 
 tigations in different parts of the country have shown that 
 pupils go out from the schools not understanding the pro- 
 cesses of modern business, and not able to make trustworthy 
 computations. Schools are not realizing an educational 
 result from arithmetic commensurate with the time and 
 effort spent on it, and the present situation is one which 
 calls for careful consideration. 
 
 In the first place arithmetic is of all the elementary school 
 studies the one most in danger of becoming conventionalized. 
 Teacher and textbook tend to perpetuate themselves. 
 Marked changes may occur in practical affairs about which 
 the teacher and the textbook author know little. Thus 
 it was that the mercantile methods of the eighteenth cen- 
 
viii EDITOR'S INTRODUCTION 
 
 tury continued as the basis of arithmetical instruction in the 
 nineteenth century, long after those methods had disap- 
 peared ; and thus there are traces of the eighteenth century 
 still to be discerned in the arithmetics of the present. If 
 the merchant of an earlier generation were to reappear and 
 attempt to do business, he would be dumfounded by the 
 changes in commercial procedure; but he could scarcely be 
 at a greater disadvantage than are pupils who are trained 
 after the methods which he had used and then sent out to 
 take their places in the world of to-day. 
 
 The book herewith presented is an honest attempt to set 
 forth correctly the fundamental operations of modern busi- 
 ness, and to furnish a goodly amount of drill on the kind 
 of computations which make up present-day commercial 
 practice. Messrs. Bigelow and Arnold have spared no 
 pains to inform themselves on current business operations, 
 and the editor believes that they have presented their infor- 
 mation succinctly and logically arranged for purposes of 
 instruction. The book has been the result of much labor 
 in its first preparation, and as first prepared it was duplicated 
 to serve as a text, and modified in class instruction for 
 several years. In addition to this it has been revised and 
 adapted in accordance with suggestions of experienced 
 teachers in different parts of the country. It is believed 
 that all this has resulted in a book of accurate information, 
 of sound mathematical basis, and of high teaching quality. 
 
 Several features of the book will commend themselves 
 to teachers. Among these are the script illustrations as 
 models for trial balances, ledger accounts, time sheets, 
 accounts of sales, etc. These have been executed by the 
 skilful pen artist and illustrator, Mr. E. C. Mills. 
 
 Chapter IV on Fractional Parts presents a natural and 
 easy approach to fractions, and will be found of great practi- 
 cal value. The authors have had the courage to put deci- 
 
EDITOR'S INTRODUCTION ix 
 
 mals ahead of common fractions, treating them as they should 
 be treated, simply as a descending scale in our decimal nota- 
 tion, of which whole numbers are an ascending scale. 
 
 The book will be found to have little of the impractical 
 and troublesome G. C. D. and L. C. M. problems. Fractions 
 of large denominations are not introduced, as they present 
 difficulties and are almost never encountered except in the 
 arithmetics. Square root and the treatment of mensuration 
 are disposed of in connection with weights and measures. 
 The antiquated percentage problems to be solved by the use 
 of formulas are omitted, as are the conventional percentage 
 formulas themselves. Partial payments is relegated to an 
 unimportant place in the Chapter on Interest. The so-called 
 true discount is eliminated from the textbook as it is from 
 business; and partnership and proportion are so treated as 
 to bring them within the practice of the actual world. 
 
 In some particulars the book is not as revolutionary as 
 the authors would have desired, but it is believed to be as 
 revolutionary as it could be without breaking with the prac- 
 tice of the schools. This text, it is believed, will prove a 
 logical and easy completion of the average elementary course 
 in arithmetic. It should find a place in the last years of the 
 grammar school, as the finishing book of the ungraded school, 
 and for the first high school year. 
 
 Throughout, this book will be found to use the method 
 of concrete presentation. The pupil is constantly asked 
 to consider problems with the thought of determining the 
 particular solution which will get a result the most directly. 
 Thus there is an absence of any ".wooden " working by rule. 
 At every turn the pupil is required to select his solution, 
 and to use his head in applying the form selected. This 
 cannot fail to produce clear thinking and a facility in com- 
 putations which will give accuracy. The problems of easy 
 solution for mental arithmetic offer one valuable feature 
 
X EDITOR'S INTRODUCTION 
 
 of the book. These, largely used, will develop power of 
 accurate thought and power of expression. 
 
 A limited range of treatment, with freedom of explana- 
 tion and plenty of drill on the fundamentals ; not too much 
 arithmetic attempted, but what is attempted done well ; 
 not generalized and abstract number, but arithmetic related 
 to the life experiences of the child ; problems selected from 
 the world around about the child ; not a treatment which 
 shall be " milk for babes," but one which will afford such a 
 ruggedness of drill as will make arithmetic a means of disci- 
 plinary education and a book of accurate information, — 
 these are the standards which the "Elements of Business 
 Arithmetic " has sought to meet. Possibly it has failed in 
 some particular, but it is published in the confident belief 
 that there is a large place for such a presentation as is here 
 attempted. The authors and the editor have the satisfaction 
 of having worked long and faithfully. They invite correc- 
 tions and suggestions for the improvement of the book. 
 
 C. A. H. 
 
CONTENTS 
 
 CHAPTER PAGB 
 
 I. Addition and Subtraction 1 
 
 11. Multiplication and Division 16 
 
 III. Decimals 23 
 
 IV. Fractional Parts 32 
 
 V. Fractions 53 
 
 VI. Measures of Length 67 
 
 VII. Measures of Area . 72 
 
 VIII. Measures of Volume 109 
 
 IX. Measures of Time 126 
 
 X. Measures of Weight . . . . . . . 137 
 
 XL Measures of Value 141 
 
 XIL French Metrical System 149 
 
 XIII. Percentage 157 
 
 XIV. Trade Discount 172 
 
 XV. Commission 180 
 
 XVL Taxes and Duties 185 
 
 XVII. Interest 192 
 
 XVIII. Banking and Discount 205 
 
 XIX. Stocks and Bonds 221 
 
 XX. Insurance . . . 231 
 
 XXI. Proportion 240 
 
 XXII. Proportional Parts and Partnership . . ' . 250 
 
 Index 255 
 
 XI 
 
THE ELEMENTS OF BUSINESS AEITHMETIC 
 
 ADDITION AND SUBTRACTION 
 
 1. Combinations to 9. Pupils who use this book will 
 probably know the combinations of numbers up to 9, but it 
 may be presumed that they are not sufficiently quick and 
 accurate in the use of these combinations. Thorough drills 
 should be given and continued until pupils can think groups 
 as a whole and can instantly name sums without naming 
 the parts. 
 
 Note. — There are but forty-five possible combinations of numbers 
 up to 9. The common, device of having each one of the combinations 
 on a separate card and giving results at sight rapidly and in varied or- 
 der, forms a practical drill. Each card may have the same combina- 
 tion on its back with the order of the figures reversed. 
 
 DRILL TABLE 
 
 The forty-five two-figure combinations. Name sums at sight. 
 74241343314221189856455 
 
 7 
 
 6 
 
 5 
 
 3 
 
 7 
 
 6 
 
 2 
 
 3 
 
 2 
 
 5 
 
 1 
 
 2 
 
 1 
 
 3 
 
 1 
 
 9 
 
 9 
 
 8 
 
 5 
 
 1 
 
 4 
 
 3 4 
 
 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 
 
 2. Combinations to 19. The following combinations 
 should be drilled upon and learned in the same way as those 
 in Sec. 1. They are important for rapid addition (Sec. 3). 
 
 B 1 
 
2. ELEMENTS OF BUSINESS ARITHMETIC 
 
 The instant calling of results should be insisted upon. Drill 
 cards as above suggested will be useful. 
 
 Note. — In making these combinations, the second number should be 
 thought of as one ten and a given number of units and not as a given 
 number of units and one ten. Thus, in combining 14 and 13 we should 
 think of 24 and 3 ; in combining 17 and 18 of 27 and 8. 
 
 DRILL TABLE 
 
 Forty-five more combinations, 11 to 19. Name sums at sight. 
 
 12 13 11 17 11 17 12 15 12 16 19 17 11 14 15 12 19 
 15 18 11 17 16 14 12 16 13 13 14 13 14 15 17 11 16 
 
 15 
 
 18 
 
 17 
 
 12 
 
 16 
 
 15 
 
 18 
 
 17 
 
 13 
 
 16 
 
 11 
 
 19 
 
 14 
 
 19 
 
 16 
 
 19 
 
 13 
 
 15 
 
 18 
 
 16 
 
 17 
 
 12 
 
 13 
 
 16 
 
 18 
 
 11 
 
 16 
 
 19 
 
 18 
 
 13 
 
 15 
 
 14 
 
 19 
 
 JL3 
 
 14 14 18 18 19 12 18 17 17 12 11 
 1418121113141511191915 
 
 3. Adding by Groups. In adding columns of figures one 
 should think first of the sum of each group, then group the 
 
 ^ .^ sums and name the sum of the groups. In the 
 
 o problem given, think and name the sum of the 
 
 -. 07 fii^st group as 14 without naming the figures of 
 
 o the group, the sum of the second group as 13, 
 
 ^ 2_ then the sum of 13 and 14 is thought and named. 
 
 r Seeing the next group as 10, we think the sum of 
 
 Q w . 27 and 10, saying 37; then, 11, 48. In adding, 
 
 r name the results only, as : 14, 27, 37, 48. 
 
 ^ Note. — Double drill may be had from the same problems 
 
 by adding both up and down; e.g. 11, 21, 34, 48. This is 
 also a way of detecting errors. If the same result is obtained from 
 adding both ways, the sum is probably correct. 
 
 4. Adding Two Columns. By practice, two columns of 
 figures may be readily added at once. The tens should be 
 combined first and the units added to their sum (Sec. 3). 
 
ADDITION AND SUBTRACTION 3 
 
 The problems below may be used for practice in both 
 
 single and double column adding. Add both up and down. 
 Drill until the group sums can be named quickly. 
 
 Add. 
 
 37 95 67 68 56 45 27 35 46 24 99 48 71 89 76 32 33 45 
 
 40 36 76 66 75 72 88 87 72 98 76 79 88 15 90 54 38 98 
 
 63 36 76 97 57 83 33 77 32 68 71 90 13 31 16 58 84 17 
 
 84 66 54 78 86 96 38 55 33 19 10 34 42 34 86 92 10 57 
 
 67 88 38 86 84 48 47 94 65 56 25 56 75 85 83 17 42 56 
 
 22 71 95 84 32 -23 47 85 65 76 81 34 56 87 34 50 60 70 
 
 71 95 84 32 23 47 85 65 76 81 8 78 19 30 54 76 58 43 
 
 45 24 98 76 67 19 14 67 32 78 76 14 98 88 28 34 13 65 
 
 Practice adding by groups. 
 
 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 
 
 180 718 678 432 178 852 458 37 458 123 7689 23 1768 
 
 717 678 743 327 187 918 1079 1876 7 7453 4 717 324 
 
 324 187 509 834 778 415 2345 324 6789 1064 318 6324 5324 
 
 873 548 791 130 543 915 718 98 4317 8796 1745 6 1098 
 
 376 876 987 734 435 478 3458 1756 5692 187 678 1 367 
 
 324 578 713 105 791 698 1234 304 2934 67 5487 75 84 
 
 943 528 281 375 905 768 5432 32 701 5432 34 150 69 
 
 473 432 958 104 548 785 7981 7615 45 478 91 235 75 
 
 432 958 104 548 785 442 2701 89 6194 91 3605 768 150 
 
 573 473 109 423 478 871 __71 _432 5061 2398 _148 _^ 1785 
 
 Add across. Add' down. Pfoblems like these are useful in training 
 to accuracy, and frequently occur in offices. If the final results are not 
 alike, the student should find and correct mistakes without awaiting 
 correction by the the teacher. 
 
 1. 324+625 + 463 + 526 = ?/- 2. 103 + 184 + 216 + 135 + 320 = ? 
 462 + 375 + 836 + 472 = ? ' 413 + 204 + 311 + 401 + 405 = ? 
 378 + 643 + 728 + 645 = ? ■ 764 + 835 + 735 + 631 + 987 = ? 
 
 643 + 372 + 654 + 528 = ? 367 + 543 + 263 + 413 + 187 = ? 
 
 584 + 653 + 936 + 364 = ? v_^^_ 185 + 371 + 213 + 715 + 478 = ? 
 ?+? + ? + ?=? \vi\ \ ?+?+? + ? + ?=? 
 
4 ELEMENTS OF BUSINESS ARITHMETIC 
 
 3. 265 + 167 + 324 + 734 + 178 = ? ' ' 
 
 562 + 713 + 432 + 817 + 384 = ? . 
 276 + 810 + 305 + 978 + 308 = ? , 
 
 417 + 187 + 523 + 734 + 198 = ? .icSl 
 
 672 + 432 + 278 + 598 + 471 = ? T gj^ — ■ 
 
 ?+? + ? + ? + ?=? ' -'^^ 
 
 "\1*^ . : - - ' '^'' 
 
 5, Adding Long Columns. Civil Service Method. In add- 
 ing long columns of figures it is often desirable to retain 
 $791.52 *^® exact sum of each column. Errors are more 
 604.83 easily located and unnecessary re-adding of col- 
 879.26 umns which are correct is often avoided. Gen- 
 243.79 erally speaking, this practice should always be 
 732.46 used for columns of more than ten numbers. 
 47.95 This is sometimes known as the " Civil Service 
 
 856.43 Method," presumably from its large use in gov- 
 497.65 ernment offices. It may be frequently used to 
 541.26 advantage in many offices. Practice using this 
 616.72 iiiethod in the problems given after Sec. 6. 
 
 857.94 Note. — Drill in urriting, from dictation, long columns 
 
 ^ of numbers of varied size, so that neat vertical columns 
 
 Q g will be secured. Verify answers by adding downward, if 
 
 ro the first addition was upward, or vice versa. 
 
 ^^ 6. Addition Proof by Check. In counting- 
 
 rooms or elsewhere where long columns of fig- 
 qt) 0009.81 ^j,gg ^j.Q common, some system for checking the 
 correctness of results of addition is often resorted to. Such 
 devices are of various kinds, but all consist of some variation 
 of a casting out process, as of the 9's or ll's. 
 
 The Unitate method is here presented. It consists of re- 
 moving all the nines from the numbers added and from the 
 sum. The remainder would necessarily be less than nine 
 and being a single figure is termed a unitate. When the 
 unitate of the numbers added is the same as the unitate of the 
 8um^ the addition is usually correct. 
 
 60 
 
ADDITION AND SUBTRACTION 5 
 
 The process of finding the unitate of a given number or, 
 as it is often called, " casting out the nines," is based on the 
 facts that our system of notation is a decimal one and that 
 there is but a single unit of difference between 9 and the 
 basis of our notation, 10. 
 
 Every digit in our notation stands for the number of tens 
 (or powers of ten) represented by the digit. Now there are 
 as many nines in each of these as there are tens and there 
 would be a remainder of a single unit from each ten. But 
 as each digit represents the number of tens, there would be 
 a number of units equal to the digit itself remaining, after 
 all the nines had been taken out. The sum of the digits of 
 any number would represent, then, the number of units 
 remaining after all the nines had been "cast out." If this 
 sum is a number with more than one digit, it still contains a 
 nine and the sum of its digits would be the true remainder. 
 When this remainder consists of but one digit, it is the 
 desired unitate. 
 
 To find the unitate of 87,564,892, add the digits and their 
 sum is 49 ; adding these gives 13, and adding these gives 4, 
 the unitate. 
 
 Unitates. 
 
 Thus, 32,543 8 
 
 16,789 3 h'^i 
 
 11,942 2 : \ 
 
 27,683 1 
 
 38,578 5 
 
 96,541 3 
 
 73,847 5 
 
 297,923 5 
 
 In theory, the unitates of every addend are themselves 
 reduced to a unitate, which should equal the unitate of the 
 correct answer. 
 
6 
 
 ELEMENTS OF BUSINESS ARITHMETIC 
 
 In practice, the "unitate of the first addend (8) is written 
 and then added to the digits of the second addend and the 
 whole reduced to its unitate (3). This is again added to 
 the digits of the third addend and the third unitate is found 
 (2). This process is continued until the unitate of the last 
 addend is found, which should be the same as the unitate of 
 the answer. 
 
 Checking by the unitate is not an absolute proof. It will 
 not detect an error in transposition of digits, or where the 
 wrong digits total the same as the right ones. It is a satis- 
 factory check, however, in most cases. 
 
 PROBLEMS 
 
 Solve and prove the following : 
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ADDITION AND SUBTRACTION 
 
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8 ELEMENTS OF BUSINESS ARITHMETIC 
 
 4. List of appropriations by Congress, for bienniuni. 
 
 
 19_ 
 
 19_ 
 
 Deficiencies 
 
 $ 19,651,968.25 
 
 ^ 25,083,395.78 
 
 Legislative, Executive, and Judicial 
 
 27,598,653.66 
 
 28,558,258.22 
 
 Sundry Civil 
 
 61,763,709.11 
 
 49,968,011.34 
 
 Support of the Army 
 
 
 . 
 
 77,888,752.83 
 
 77,070,300.88 
 
 Naval Service . 
 
 
 
 81,876,791.43 
 
 97,505,140.94 
 
 Indian Service . 
 
 
 
 8,540,406.77 
 
 9,447,961.40 
 
 Rivers and Harbors . 
 
 
 
 20,228,150.99 
 
 10,872,200.00 
 
 Forts and Fortifications 
 
 
 
 7,188,416.22 
 
 7,518,192.00 
 
 Military Academy . 
 
 
 
 652,748.67 
 
 973,947.26 
 
 Pensions . 
 
 
 
 139,847,600.00 
 
 138,360,700.00 
 
 Consular and Diplomatic 
 
 
 
 1,968,250.69 
 
 2,020,100.69 
 
 Agricultural Department 
 
 
 
 5,978,160.00 
 
 5,902,040.00 
 
 District of Columbia 
 
 
 
 8,638,097,00 
 
 11,018,540.00 
 
 Miscellaneous . 
 
 
 
 3,025,064.95 
 
 2,860,828.52 
 
 Total 
 
 
 
 
 
 5. The cotton crop of the United States by states for five years. 
 
 States 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 North Carolina 
 
 480,000 
 
 400,000 
 
 426,000 
 
 504,000 
 
 400,000 
 
 South Carolina 
 
 960,000 
 
 874,000 
 
 948,000 
 
 955,000 
 
 845,000 
 
 Georgia 
 
 1,448,000 
 
 1,226,000 
 
 1,493,000 
 
 1,498,000 
 
 1,405,000 
 
 Florida 
 
 54,000 
 
 57,000 
 
 56,000 
 
 60,000 
 
 55,000 
 
 Alabama 
 
 1,161,000 
 
 1,136,000 
 
 1,287,000 
 
 1,065,000 
 
 1,040,000 
 
 Mississippi 
 
 1,776,000 
 
 1,349,000 
 
 1,460,000 
 
 1,418,000 
 
 1,385,000 
 
 Louisiana 
 
 577,000 
 
 651,000 
 
 851,000 
 
 864,000 
 
 832,000 
 
 Texas 
 
 3,143,000 
 
 2,575,000 
 
 2,682,000 
 
 2,575,000 
 
 2,446,000 
 
 Arkansas 
 
 921,000 
 
 665,000 
 
 771,000 
 
 938,000 
 
 855,000 
 
 Tennessee 
 
 381,000 
 
 240,000 
 
 229,000 
 
 303,000 
 
 255,000 
 
 All others 
 
 334,000 
 
 267,000 
 
 498,000 
 
 578,000 
 
 516,000 
 
 Total crop 
 
 r ■ . •; 
 
 v^Jioi.c 
 
 'V ' 
 
 ""' ■. ' 'i 
 
 '■ 
 
ADDITION AND SUBTRACTION 
 6. Add down. Add across. 
 
 7654 
 
 4753 
 
 1954 
 
 1763 
 
 8197 
 
 6548 
 
 ? 
 
 1458 
 
 3674 
 
 1756 
 
 3263 
 
 3954 
 
 3245 
 
 ? 
 
 5786 
 
 9876 
 
 1753 
 
 5132 
 
 6587 
 
 3642 
 
 ? 
 
 4327 
 
 9876 
 
 587 
 
 3674 
 
 5743 
 
 6798 
 
 ? 
 
 56 
 
 7815 
 
 2301 
 
 1567 
 
 4326 
 
 189 
 
 ? 
 
 1587 
 
 9817 
 
 3246 
 
 1685 
 
 8542 
 
 7432 
 
 ? 
 
 1563 
 
 6743 
 
 9816 
 
 1076 
 
 875 
 
 1904 
 
 ? 
 
 8543 
 
 3425 
 
 178 
 
 25 
 
 3607 
 
 98 
 
 ? 
 
 5674 
 
 8795 
 
 5432 
 
 5768 
 
 5843 
 
 2345 
 
 ? 
 
 + 
 
 7. Departmental sales for the week ending June 15, 19 
 
 Days 
 
 Clothing 
 
 Dry Goods 
 
 Furnish- 
 ings 
 
 Millinery 
 
 Groceries 
 
 Total 
 
 Monday 
 
 695.50 
 
 894.30 
 
 175.65 
 
 325.45 
 
 678.10 
 
 ? 
 
 Tuesday 
 
 546.15 
 
 716.98 
 
 243.25 
 
 817.42 
 
 313.48 
 
 ? 
 
 Wednesday 
 
 981.76 
 
 654.32 
 
 145.60 
 
 567.89 
 
 543.26 
 
 ? 
 
 Thursday 
 
 578.90 
 
 765.10 
 
 324.65 
 
 687.58 
 
 987.60 
 
 ? 
 
 Friday 
 
 842.45 
 
 918.75 
 
 216.40 
 
 561.46 
 
 674.15 
 
 ? 
 
 Saturday 
 
 985.50 
 
 818.40 
 
 456.12 
 
 764.55 
 
 925.48 
 
 ? 
 
 Total 
 
 ? 
 
 9 
 
 ? 
 
 ? 
 
 ? 
 
 ? 
 
 8. The records of a post-office show the following mail for six days : 
 Monday, registered letters, 625; ordinary letters, 14,570; postal cards, 
 2134; book packets, 957; parcels, 184; newspapers, 25,514. Tuesday, 
 registered letters, 541 ; ordinary letters, 13,576 ; postal cards, 2134 ; book 
 packets, 587; parcels, 146; newspapers, 26,156. Wednesday, registered 
 letters, 750; ordinary letters, 14,569; postal cards, 3456 ; book packets, 
 1056; parcels, 178; newspapers, 24,356. Thursday, registered letters, 
 587 ; ordinary letters, 13,452 ; postal cards, 2451 ; parcels, 143 ; news- 
 papers, 23,781. Friday, registered letters, 547; ordinary letters, 13,567; 
 postal cards, 1346 ; book packets, 890 ; parcels, 157 ; newspapers, 26,543. 
 
10 ELEMENTS OF BUSINESS ARITHMETIC 
 
 Saturday, registered letters, 857 ; ordinary letters, 15,472 ; postal cards, 
 3145 ; book packets, 789 ; parcels, 245 ; newspapers, 23,100. 
 
 Arrange these facts in tabular form, in six columns, under proper 
 headings. Find the total number of pieces of mail for each day, the 
 total number of pieces of each class, and the total number for six days. 
 
 7. Method of Subtraction. Whether the pupil subtracts 
 by the method of borrowing from the minuend or by 
 adding to the subtrahend, is immaterial. Ordinarily he 
 should be permitted to use the method previously taught 
 him. The effort should be toward facility, which may only 
 be acquired by practice. The examples here given are sug- 
 gestive rather than sufficient. If when these are completed 
 there is not the power of quick and accurate subtraction, 
 much additional work should be given. 
 
 If another method is desired, that known as the addition 
 method is suggested. It has the advantage of using the 
 addition combinations previously taught and involves no 
 new process. 
 
 In the accompanying problem, we say 4 and 5 are 9, writing 
 . the 5. Then, 8 and 8 are 16, writing the 8 and carrying the 1 
 ■ as in addition. Then, 10 (9 + 1) and are 10, writing ; and 1 
 
 (carried from 10) and 1 are 2, writing 1. 
 
 8. Making Change. The addition method of making 
 change is the one most generally used by tellers, and is un- 
 questionably the most accurate. Thus, a ten-dollar bill is 
 tendered in payment of a bill of $2.85. Counting out suc- 
 cessively a five-cent piece, a dime, two dollars, and a five- 
 dollar bill, the sums are named as follows: $2.85, $2.90, 
 $3, $4, $5, $10. 
 
 Name the coins and bills, and the amount of change to be 
 given in each of the following : 
 
 1. From $1, for a bill of: 17^, 43^, 65/-, 72;^, 28 J^, 10 ^ 15^, 40^, 
 35j^, 87/^. 
 
ADDITION AND SUBTRACTION 11 
 
 2. From |2, for a bill of: |1.26, $ 1.40, $1.47, $1.61, $1.75, $1.55. 
 $1.69, $1.83, $1.19, $1.33. 
 
 3. From $.5, for a bill of: $3.50, $1.12, $2.32, $3.37, $2.87, $0.79, 
 $4.11, $1.78, $3.56, $2.75, $4.15. 
 
 4. From $10, for a bill of: $3.50, $4.75, $6.32, $7.28, $4.87, $9.15, 
 $8.5.5, $6.70, $1.95, $4.28. 
 
 5. From $20, for a bill of: $17.35, $14.32, $13.45, $10.75, $9.15, 
 $ 3.85, $ 12.10, $ 14.30, $ 8.24. 
 
 9. Horizontal Subtraction. Subtract the following with- 
 out rearranging. Find' the sum of the minuends, the sum 
 of the subtrahends, and the sum of the remainders. 
 
 1. 3,264,873- 286,729 = ? 3. $867.50- $742.25 = ? 
 
 328,629- 124,962=? 329.87- 124.68 = ? 
 
 729,687 - 638,469 = ? 1768.42 - 938.89 = ? 
 
 382,962- 146,702 = ? 2762.48- 1262.34-? 
 
 2,678,212 - 1,476,388 = ? 3786.32 - 439.37 = ? 
 
 729,326- 384,578 = ? 9623.29- 3674.28 = ? 
 
 368,742- 176,386 = ? 9627.42- 7672.91=? 
 
 504,726 - 386,275 = ? 1076.16 - 729.78 = ? 
 
 ? - ? =? 
 
 2. 496,827 - 268,794 = ? 
 
 1,986,702 - 1,346,825 = ? 
 
 2,787,543 - 1,968,729 = ? 
 
 79,843 - 67,983 = ? 
 
 4,869,625 - 2,278,631 = ? 
 
 967,198 - 798,631 = ? 
 
 472,398 - 126,535 = ? 
 
 848,716- 432,567 = ? 
 
 V - ? r? ? _ ? =? 
 
 Find the new balances in the following banking individual 
 ledger accounts by adding the deposits to the balances and 
 subtracting the checks. Find the total of balances, checks, 
 and deposits. Prove. 
 
 ? - 
 
 ? 
 
 = ? 
 
 $6487.50- 
 
 $3228.25: 
 
 = ? 
 
 329.75 - 
 
 128.95 : 
 
 = ? 
 
 6756.50 - 
 
 2278.36 : 
 
 = ? 
 
 4798.60 - 
 
 3128.75 : 
 
 = ? 
 
 12986.72 - 
 
 9647.22 : 
 
 = ? 
 
 3678.45 - 
 
 2968.42 : 
 
 = ? 
 
 728.32 - 
 
 648.25 : 
 
 = ? 
 
 2198.65 - 
 
 1297.70 : 
 
 = ? 
 
12 
 
 ELEMENTS OF BUSINESS ARITHMETIC 
 
 1. 
 
 Names 
 
 Balances 
 
 Checks 
 
 Deposits 
 
 Balances 
 
 Ames Wm E 
 
 865.52 
 584.32 
 954.60 
 523.40 
 976.35 
 126.65 
 925.43 
 1214.34 
 752.30 
 178.95 
 986.57 
 
 321.56 
 127.55 
 532.58 
 305.45 
 532.78 
 
 38.72 
 413.86 
 615.47 
 456.87 
 
 34.56 
 485.74 
 
 675.80 
 220.00 
 276.50 
 560.00 
 125.40 
 423.80 
 575.94 
 213.44 
 435.87 
 698.00 
 325.34 
 
 
 Bentley, C. A. 
 
 
 
 Dayton, F. R. 
 
 
 
 Frank G A 
 
 
 G-ramrn D C 
 
 
 Hughes, C. M. 
 Innes U. P 
 
 
 
 Justin, John 
 King, A. S. 
 
 
 
 
 
 ? 
 
 ? 
 
 ? 
 
 ?. 
 
 Names 
 
 Balances 
 
 Checks 
 IN Detail 
 
 Total Checks 
 
 Deposits 
 
 Balances 
 
 Anson, E. M. 
 
 5671.80 
 1544.42 
 2345.60 
 5467.80 
 
 967.85 
 1267.98 
 
 845.34 
 3289.07 
 
 876.35 
 
 125.50 
 
 232.20 
 
 678.90 
 750.00 
 142.76 
 525.00 
 453.50 
 
 134.75 
 
 1100.00 
 190.00 
 253.78 
 
 1234.56 
 26.78 
 
 325.40 
 
 
 546.70 
 
 546.70 
 756.05 
 854.32 
 768.90 
 987.45 
 1200.00 
 584.32 
 542.08 
 
 
 Barnes, T. D. 
 
 
 
 Cowles, E. M. 
 
 
 
 Doyle, F. E. 
 
 
 
 Farish Bros. 
 
 
 
 Grim, G. L. 
 
 
 
 Haines, F. R. 
 
 
 
 Johns, B. I. 
 
 
 
 Love, P. V. 
 
 
 
 
 ? 
 
 V 
 
 ? 
 
 ? 
 
 ? 
 
ADDITION AND SUBTRACTION li' 
 
 10. Problems for Explanation. In problems where the pro- 
 cesses themselves are very simple, the more difficult work of 
 accurate expression in equation form and of explanation 
 should be carefully taught. Exactness in language is the 
 only proper expression for an exact science. 
 
 A suggestive statement and explanation are given. Set 
 forms of explanation are not desirable, but a correct use of 
 mathematical symbols and of clear and accurate English in 
 the explanation of a problem should be insisted upon. Use 
 the fewest words possible to express the thought clearly. 
 
 Problem. — During one season a jobbing carpenter built 
 five dwellings which cost him respectively 13176, $5194, 
 $1342, 16950, and $788. He received for building them 
 $3875, $6820, $1280, $7896, and $875. What were his 
 season's profits ? 
 
 $3176 + $5194 + $1342 + $6950 + $788 = $17,450. 
 $3875 + $6820 + $1280 + $7896 + $875 = $20,746. 
 $20,746 - $17,450 = $3296, season's profits. 
 
 Explanation. — The total cost of building the five build- 
 ings is the sum of $3176, $5194, $1342, $6950, and $788, or 
 $17,450. The total amount received is the sum of $3875, 
 
 $6820, $1280, $7896, and $875, or $20,746. His profits, 
 therefore, are the difference between $20,746 and $17,450, or 
 
 $3296. 
 
 PROBLEMS 
 
 1. A grain dealer bought 15,640 bushels of wheat, and sold at one time 
 3465 bushels, at another time 4205 bushels, and at another time 1080 
 bushels. How many bushels remained? 
 
 2. A man deposited in a bank .|9672. He drew out at one time 
 $4234, at another $1700, at another $762, and at another $49. How 
 much remained? 
 
14 ELEMENTS OF BUSINESS ARITHMETIC 
 
 3. A teacher's salary was $1200. His living expenses were $760. 
 He paid $314 for a lot and $95 for a horse. How much of his salary- 
 remained ? 
 
 4. A merchant in a year bought goods to the amount of $ 8750. He 
 paid for clerk hire $6735, and for rent $318. For how much must he 
 sell his goods in order to clear $ 1250 ? 
 
 5. I sold a farm for $ 9625 and a house for $ 3275. Lost $ 475 on the 
 farm and gained $ 360 on the house. What did each cost me ? What 
 was the total gain or loss ? 
 
 6. The population of Indiana in 1900 was 2,516,462. The population 
 of the principal cities of the state was as follows : Indianapolis, 169,164 ; 
 Evansville, 59,007; Fort Wayne, 45,115; Terre Haute, 36,673; South 
 Bend, 35,999. How much did the population of the state exceed that of 
 these cities ? 
 
 7. The distance from Chicago to Buffalo is 523 miles, and from 
 Chicago to New York 980 miles. How far is Buffalo from New York? 
 
 8. At a sawmill 120,000 feet of pine lumber were sawed in a month. 
 47,250 feet of it were sold to one man and 32,575 to another. How much 
 of the month's output remained ? 
 
 9. The imports of sugar and molasses into the U.S. in one year 
 amounted to $ 108,387,388, and ten years later to $ 101,100,000. What 
 was the amount of decrease ? 
 
 10. Three persons bought a hotel valued at $45,675. The first agreed 
 to pay $ 8575, the second twice as much as the first, and the third the 
 remainder. How much was the third to pay? 
 
 11. Borrowed of a bank at one time $ 875, at another $ 385, and at an- 
 other $ 528. Having paid $ 1275, how much do I owe? 
 
 12. A minister had his life insured for $5000. At the time of his 
 death, $375 of his salary was unpaid; he owned a farm worth $4675, 
 but upon it was a mortgage of $ 2385, and his small debts amounted to 
 $ 879. What was the value of his estate ? 
 
 13. A stock dealer bought 789 cattle from A, and 1249 from B. He 
 then sold 228 to C, 468 to D, and the remainder to E. How many did E 
 buy? 
 
 14. A merchant commenced business with $7500. The first year he 
 gained $1275, the second year he lost $2475, the third year he gained 
 $978, and the fourth year lost $674. How much had he left at the end 
 of the fourth year ? 
 
ADDITION AND SUBTRACTION 
 
 15 
 
 15. A bank had $422,785 on hand. During the day they received on 
 deposit .| 14,657, and paid out by check f 24,570. How much remained 
 on hand at the close of the day ? 
 
 16. The cost of my house and lot was $ 12,860. I expended 1 1367 
 for carpenter work, $ 567 for bricklaying, 1 6850 for plumbing, $ 587 for 
 painting, and $369 for sodding and fencing the grounds. I then sold 
 the property at a loss of f 135, receiving $ 7850 in cash and a note for 
 the balance. What was the face of the note? 
 
 17. During five years a firm gained $36,750. The first year they 
 gained $7565; the second $4125; the third as much as both the first 
 and second years ; and the fourth year the difference between the gains 
 of the first and second years. How much did they gain the fifth year? 
 
 18. The exports of cattle from the United States during a period of 
 nine years were as follows : $159,179; $439,987; $1,103,095; $13,344,195; 
 $14,304,103; $12,906,693; $31,161,131; $30,445,249; and $39,099,095. 
 During a later period of seven years; ' $23,032,428; $33,461,022; 
 $30,603,796; $34,560,672; $36,357,451; $37,827,500; and $30,516,833. 
 For which period were the exports greater, and how much ? 
 
 19. The expenditures for schools during one year in Alabama were 
 $1,583,250; in Arizona, $377,253; in Arkansas, $1,396,594; in Cali- 
 fornia, $6,401,439; and in Illinois, $18,167,219. How much more was 
 expended in Illinois than in the other states mentioned ? 
 
 20. Find the balance of the following : — 
 
 Svcit ^tional JSanfe, 
 
 Sa}>toti. 0. 
 
 .^>^^ 
 
 J ' ^ ^ 
 
 /- 
 
 .Jyy^^T^n^^^ 
 
 ■^^ 
 
 yLyJx^fy^^J- 
 
 Ul. 
 
 -^^-r>-/t 
 
 / ^ rA 
 
 Ul 
 
 sC/>./7^rr-H^^ 
 
 A-2^.,-?^ 
 
 ^^~i 
 
 ,J'>ird Ink) 
 
II 
 
 MULTIPLICATION AND DIVISION 
 11. Reference and Drill Table. 
 
 12 3 4 
 
 5 
 
 6 
 
 7 8 9 10 
 
 11 12 13 14 16 
 
 16 17 18 19 20 
 
 2 4 6 8 
 
 10 
 
 12 
 
 14 16 18 20 
 
 22 24 26 28 30 
 
 32 34 36 38 40 
 
 3 6 912 
 
 15 
 
 18 
 
 21 24 27 30 
 
 33 36 39 42 45 
 
 48 51 54 57 60 
 
 4 8 12 16 
 
 20 
 
 24 
 
 28 32 36 40 
 
 44 48 52 56 60 
 
 64 68 72 76 80 
 
 5 10 15 20 
 
 25 
 
 30 
 
 35 40 45 50 
 
 55 60 65 70 76 
 
 80 85 90 95100 
 
 6 12 18 24 
 
 30 
 
 36 
 
 42 48 54 60 
 
 66 72 78 84 90 
 
 96 102 108 114 120 
 
 7 14 21 28 
 
 35 
 
 42 
 
 49 56 63 70 
 
 77 84 91 98 105 
 
 112 119 126 133 140 
 
 8 16 24 32 
 
 40 
 
 48 
 
 56 64 72 80 
 
 88 96 104 112 120 
 
 128 136 144 152 160 
 
 9 18 27 36 
 
 45 
 
 54 
 
 63 72 81 90 
 
 99 108 117 126 135 
 
 144 153 162 171 180 
 
 10 20 30 40 
 
 50 
 
 60 
 
 70 80 90 100 
 
 110 120 130 140 150 
 
 160 170 180 190 200 
 176 187 198 209 220 
 
 11 22 33 44 
 
 ■^ 
 
 ■ST 
 
 77 88 99 110 
 
 121 132 143 154 165 
 
 12 24 36 48 
 
 60 
 
 72 
 
 84 96 108 120 132 144 166 168 180 
 
 192 204 216 228 240 
 
 13 26 39 52 
 
 65 
 
 78 
 
 91 104 117 130 143 156 169 182 195 
 
 208 221 234 247 260 
 
 14 28 42 56 
 
 70 
 
 84 
 
 98 112 126 140 154 168 182 196 210 
 
 224 238 252 266 280 
 
 15 30 45 60 
 
 75 
 
 90 105 120 135 150 165 180 195 210 225| 
 
 240 255 270 285 300 
 266 272 288 304 320 
 
 16 32 48 64 
 
 'W 
 
 ■w 
 
 112 128 144 160 176 192 208 224 240 
 
 17 34 51 68 
 
 85 102 119 136 152 168 187 204 221 238 255 272 289 306 323 340 
 
 18 36 54 72 
 
 90 108 126 144 162 180 198 216 234 252 270 288 306 324 342 360 
 
 19 38 57 76 
 
 95 114 133 152 171 190 209 228 247 266 285 304 323 342 361 380 
 
 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 
 
 12 3 4 
 
 5 
 
 6 
 
 7 8 9 10 11 12 13 14 15 16 17 18 19 20 
 
 12. Suggestions for Study. It is presumed that students 
 pursuing this course already know the multiplication tables 
 through the 12's. It will prove of great practical value to 
 know them well through the 20's. If undertaken, the pupil 
 should build his own tables and learn them. Drill on them 
 
 16 
 
MULTIPLICATION AND DIVISION 17 
 
 should not stop short of absolute mastery. A half-known 
 table of 15's will seldom be used. 
 
 Note. — Drill cards of convenient size, with the factors on one side 
 and the product on the other, are valuable aids in securing readiness and 
 accuracy. They are equally usable in multiplication and factoring (or 
 division), by varying the sides shown. 
 
 In learning a table, it should be kept in mind that the 
 product is the same regardless of the order of the factors. 
 Thus, 136 is the product of both 8x17 and 17 x 8. If this 
 is kept in mind, the new combinations to be learned in the 
 higher tables become constantly less ; e.g. if all the tables 
 through the 19's are known, the only new combination in the 
 20's to be learned is, 20 x 20 = 400. 
 
 After completing the 12's to 20, it will be found helpful 
 to pursue the following order : 
 
 Review the 5's and lO's and study the 15's and 20's. 
 Review the 3's, 6's, 9's, 12's, and take the 18's. 
 Review the 4's, 8's, 12's, and take the 16's. 
 Then take the 13's, 14's, 17's, and 19's. 
 
 Problems for practice in short multiplication and division 
 with abstract numbers may be drawn from the above table 
 or dictated to students at will. 
 
 13. Suggestions and Problems for Explanation. 
 
 Problem. — A horse worth |130 and 3 cows worth $36 
 each, were exchanged for sheep at $ 6 per head and $ 82 in 
 money. How many sheep were received ? 
 
 136 X 3 = 1108. 
 1108 + f 130 = $238, value of horse and cows. 
 1238 -$82 = 1156. 
 $ 156 -^ 1 6 = 26, the number of sheep received. 
 
 Explanation. — If 1 cow was worth 1 36, 3 cows were 
 worth three times $36, or $108. If the cows were worth 
 c 
 
18 ELEMENTS OF BUSINESS ARITHMETIC 
 
 $108 and the horse 1130, together, they would be worth 
 the sum, or $238. If $82 was received in money, the differ- 
 ence between $238 and $82, or $156, was received in sheep. 
 If each sheep was worth $ 6, as many would be received as 
 $6 is contained in $156, or 26. (See Sec. 10.) 
 
 Two kinds of division problems. Division, as commonly 
 used, includes two classes of problems, e.g. : 
 
 (1) When it is required to find how many times one quan- 
 tity is contained in another, and 
 
 (2) When it is desired to find the size of one of the equal 
 parts of the quantity. As examples, the following may be 
 given : 
 
 1. How many bags will be required to hold 42 bushels of 
 wheat, if each bag will hold 3 bushels ? 
 
 2. How many bushels in each bag, if 42 bushels are put 
 into 14 bags ? 
 
 In (1) the dividend and divisor are the same in kind. 
 
 In (2) the dividend and the result are the same in kind. 
 
 For the purposes of clear thinking and rigid explanation, 
 it is thought best to treat problems like (2) as problems in 
 fractions. 
 
 To assist in keeping the reasoning clear, it is suggested 
 that problems like (1) be written in equation form in this 
 way : 
 
 42 bu. -r- 3 bu. = 14, the number of bags required. 
 
 Problems like (2) should be written : 
 
 ^^ of 42 bu. = 3 bu., number in each bag. 
 
 It should be noted, however, that the results in both classes 
 of problems are obtained by the same process. 
 
 In the written solution of a problem, great care should be 
 taken to use the equation correctly. 
 
 An equation should be true in kind as well as in amount. 
 
MULTIPLICATION AND DIVISION 19 
 
 Concrete numbers should invariably be named, and should 
 be written first in the equation. 
 
 The sign of multiplication is the St. Andrew's Cross. It 
 should be read "multiplied by." If the word times is used, 
 the second number is read first. Thus, the first equation 
 above should read, % 36 multiplied by 3, or 3 times 1 36. 
 
 Reading. — When the reasons for the steps of the prob- 
 lem are well known by the pupil, much time may be gained 
 by a mere reading of the equations instead of requiring an 
 explanation. 
 
 The above statement would be read as follows : $ 36 mul- 
 tiplied by 3 is flOS; 1108 plus 1130 is $238; 1238 less 
 1 82 is $ 156 ; and 1 156 divided by 1 6 is 26, or the number 
 of sheep received. 
 
 14. Multiplication by Numbers Greater than 20. Multi- 
 plication, when the multiplier is greater than one's known 
 multiplication table, involves the use of partial cnof^ >. 
 
 products or the product of each figure of the ^^^ 
 
 multiplier and the whole of the multiplicand. i. rvj^r-r^ 
 The essential thing to be observed in the -ic^oR 
 process is that the right-hand figure of each Arj^c,^ 
 partial product should be directly under each ' ^ ^.-^ , 
 multiplier. The reason for this is easily seen 
 when we remember that the units figure is multiplied first 
 in each case. Thus, units times units give units, and units 
 multiplied by hundreds are hundreds, etc., the result each 
 time being of the same denomination as that of the figure 
 multiplied by. The fact that these partial products are to 
 be added to find the complete product explains the necessity 
 of arranging units under units and tens under tens. 
 
 When the multiplier contains a cipher, it is passed over and 
 the next figure to the left becomes the multiplier. Care should 
 be taken to begin the partial product under the new multiplier. 
 
20 ELEMENTS OF BUSINESS ARITHMETIC 
 
 When ciphers are to the right of either the multiplier or 
 multiplicand, or both, the last figures to the right of both 
 (other than the ciphers) are placed under each other. The 
 ciphers are disregarded in the partial products, but as many 
 are placed to the right of the product as are to the right of 
 both numbers being multiplied. 
 
 15. Long Division. If the quotient is placed above the 
 dividend, the reason for the place of each figure will be more 
 readily seen. Thus, there are 9 hundred thirty- ^rn 
 
 fours in 325 hundreds, and accordingly the 9 is 04.^09^79 
 placed directly above the hundreds of the divi- oa/> 
 dend ; likewise 34 into 197 tens gives tens^ and ~Tq7 
 should be placed directly over the tens of the -j,^^ 
 
 dividend, etc. "979 
 
 The ease with which one gets, to know the de- c)ncf 
 
 nomination of any partial quotient will often pre- 
 
 vent getting absurd answers, and will greatly simplify the 
 " pointing off " process in decimals to be met later. 
 
 Problems in which the divisor contains ciphers to the 
 right may be much shortened by cutting off as many figures 
 to the right of the dividend as there are ciphers to the right 
 of the divisor. The figures so cut off are always the whole 
 or part of the remainder, and if written fractionally should 
 be written over the entire divisor. 
 
 19^15. 
 
 00 
 
 15(5)9) 285)2^. 
 
 The process of long division may be ^ ,^^ 
 
 much shortened by subtracting the coyr^Aoo 
 
 partial products as we find them, writ- 91 a 
 
 ing only the remainder each time. 090 
 
 Drill on this will not only be found ^^^^ 
 
 valuable as a mental training, but of ^^ . , 
 
 . , . , ^ 12, remainder, 
 real value m lessening work. 
 
MULTIPLICATION AND DIVISION 21 
 
 PROBLEMS 
 
 1. 7668 - 36. 11. 1,674,918 - 1980. 21. 372,104 -4- 386. 
 
 2. 48,967 - 52. 12. 324,217 - 268. 22. 385,200 -^ 1370. 
 
 3. 20,982-78. 13. 356,686-682. 23. 466,830 -^ 2730. 
 
 4. 24,476 -^ 58. 14. 1,769,824 - 3800. 24. 8,326,900 - 1029. 
 
 5. 10,908 - 236. 15. 367,240 - 461. 25. 9,230,021 ^ 3120. 
 
 6. 98,340-^63. 16. 378,625-325. 26. 780,164-119. 
 
 7. 76,055-53. 17. 148,050-315. 27. 27,552 -f- 328. 
 
 8. 42,400 -i- 98. 18. 986,172 - 186. 28. 172,096 - 3600. 
 
 9. 39,628-76. 19. 34,572-129. 29. 34,216-^9203. 
 10. 97,266 -f- 78. 20. 3,467,000 ^ 360. 30. 1,647,756 - 198. 
 
 Solve, expressing in equation form, and explain : 
 
 1. Bought 568 bushels of corn at 48^ a bushel and 675 bushels of 
 wheat at 76 J^ a bushel. What did both cost? 
 
 Note. — While 48^ is the real multiplicand, the problem is shortened 
 by treating it (being the smaller number) as the multiplier. In 
 expressing the work in an equation, however, 48^ must appear as the 
 multiplicand. 
 
 2. Mr. McFarland has property valued at $6700. He buys land 
 and sells it at a gain of 1 5 per acre. He is then rated at $8075. How 
 many acres did he buy? 
 
 3. A stock buyer having $3540 buys 16 horses at $75 each, and 
 invests the remainder of his money in cattle at $36 per head. How 
 many head of cattle does he buy ? 
 
 4. At the rate of 45 miles an hour, how long will it take a train to 
 run 325 miles ? 
 
 5. If a book contains 255 pages, and there are 1864 ems to the page, 
 how many ems in the book ? 
 
 6. An electric railway company has 435 miles of track which was 
 built at a cost of $ 83,672 per mile. What was the total cost of con- 
 structing the road ? 
 
 7. A coal dealer bought 428 tons of coal at $ 7.50 per ton. He sold 
 200 tons of it at $8 a ton and the remainder at $ 8.75. Find his gain. 
 
 8. A man sold mining stock for $ 9600, a mill for $ 12,600, and three 
 houses for $ 12,530, $ 6780, and $ 9870 respectively. He invested t21-,^^r^===^ 
 
22 ELEMENTS OF BUSINESS ARITHMETIC 
 
 of the proceeds in the stock of a manufacturing company at $75 a 
 share, and the balance in railroad stock at $80 a share. How many 
 shares of each did he buy? 
 
 9. If a clerk receives $1850 a year, and pays $38 a month for 
 board and room, $2.40 a month for laundry, $3.80 a month for life in- 
 surance, $ 12 a month to a building and loan association, $ 376 a year for 
 clothing, and $ 325 a year for incidentals, how much does he have to 
 invest in business after a term of ten years ? 
 
 10. Into how many states as large as Texas (265,780 sq. mi.) could 
 the United States (3,616,484 sq. mi.) be divided? 
 
 11. A manufacturer bought 165 tons of steel billets at $32 a ton, 
 45,000 pounds of steel bars at $ 1.80 a hundred, and 75 tons pig iron at 
 $ 19 a ton. How much did he pay for all? 
 
 12. A young man takes out a life insurance policy for $2000 and 
 agrees to pay $3.78 a month for 20 years. How much money will he 
 have paid to the company at the end of that time ? 
 
 13. How many cords of 128 cu. ft. in a pile of wood containing 
 122,880 cu. ft. ? AVhat is it worth at $ 3.50 per cord ? 
 
 14. A dealer bought 370 tons of coal by the long ton (2240 lb.) at 
 $5 a ton. He retailed it at $7 a short ton (2000 lb.). What was his 
 total gain ? What would he have gained if he had sold it at the same 
 price per long ton? 
 
 15. A stock dealer sold 26 carloads of cattle, 26 head of cattle in each 
 car, at 3;^^ a lb. If the cattle averaged 935 lb., how much did he receive 
 for them? 
 
 16. I paid $ 12,400 for apples at $ 2.50 per barrel. The total loss in 
 storage was 38 barrels, and I paid 5^ a barrel for storage and 2^ a 
 barrel for drayage. How much did I gain, if I sold them for $ 3.80 per 
 barrel ? 
 
 17. A certain oil company produced 23,000,000 barrels of refined oil 
 in one year. How many tanks each holding 35,000 barrels would it take 
 to store this oil? On one of the company's farms there was stored in 
 tanks of the above capacity 2,450,000 barrels of crude oil. How many 
 tanks were there ? 
 
 18. If the area of all the continental divisions of the earth is 51,238,800 
 sq. mi. and the population is 1,487,900,000, how many people are there to 
 the sq. mi. ? 
 
Ill 
 
 DECIMALS 
 
 16. Decimal Notation. By decimal^ is meant a tenth. Our 
 notation (Arabic) is said to be a decimal notation because 
 each denomination is one tenth of the next larger denomina- 
 tion. Thus, units are tenths of tens, tens are tenths of hun- 
 dreds, and hundreds are tenths of thousands. 
 
 Likewise tenths of units are written to the right of units, 
 but separated from them by a decimal point (.). Tenths of 
 tenths, or hundredths, are written in the next place to the 
 right, and tenths of hundredths, or thousandths, in the next 
 or third place from the decimal point. Thus, 324.516 con- 
 sists of 3 hundreds, 2 tens, 4 units, 5 tenths^ 1 hundredth^ and 
 6 thousandths. 
 
 While the system is thus properly termed a system of 
 decimal notation, the meaning of the word decimal is made 
 more specific by restricting its use to the part of a number 
 which is to the right of the decimal point ; the whole num- 
 bers to the left being termed integers. 
 
 It will therefore be seen that the denominations to the 
 right of units are the same as those to the left, except that 
 they decrease by tens while those to the left increase by tens. 
 These decreasing denominations are distinguished by adding 
 the suffix ths to each denomination. 
 
 17. Reading and Writing Decimals. Beginning at the 
 decimal point, decimals are read exactly as whole numbers are 
 read, with the denomination of the last place to the right 
 named. The name of the denomination of the right-hand 
 
24 ELEMENTS OF BUSINESS ARITHMETIC 
 
 place is the same as if it were a whole number, with one 
 additional place, and with the suffix ths added. The word 
 and is always used between the whole number and the deci- 
 mal. Thus, 3,204.6103 is read : three thousand two hundred 
 four and six thousand one hundred three ten-thousandths. 
 
 Write decimals exactly as whole numbers are written, and 
 then set off to the right of the decimal point the num- 
 ber of places indicated by the name of the right-hand de- 
 nomination. The number of places to be set off is always 
 one less than the number of places required to write the corre- 
 sponding denomination as a whole number. This is because 
 the units place is always to the left, and never to the right, 
 of the decimal point. Thus, 506 ten-thousandths is written 
 w\t\i four places (.0506) because ten thousand as a whole 
 number requires five places. Ciphers are used to the left of 
 the number to be written as a decimal when necessary to set 
 it away from the decimal point. In whole numbers the deci- 
 mal point is presumed to be at the right of units figure, 
 whether it is written or not. 
 
 Write : 
 
 1. Sixty-eight thousandths. 
 
 2. Five tenths. Seven hundredths. Eighteen hundredths. 
 
 3. Six thousandths. Five hundred three thousandths. Eighteen 
 thousandths. Seven hundred thirty-five ten-thousandths. Eighty-four 
 hundred-thousandths. 
 
 4. Eighteen hundredths. Thirteen hundred-thousandths. Sixty- 
 four tenths. 
 
 5. Eighty-five ten-thousandths. Three hundred twenty-three and 
 fifty-six hundredths. Seventy-eight thousandths. Three hundred 
 twenty-three thousandths. 
 
 Read: 
 
 1. .5, .25, .025, .0625, .0005. 
 
 2. .00001, .3762536, .00875, .000025. 
 
 3. 2.5, 25.25, 25.025, 700.07, .303. 
 
 4. .193, 4.2, 6.3028, .03275, 176.4. 
 
DECIMALS 25 
 
 18. Addition and Subtraction of Decimals. The processes 
 and principles applied in the addition and subtraction of 
 decimals are the same as of integers. The same care should 
 be exercised in keeping like denominations in the same 
 column. The decimal point in the sum or difference is 
 placed directly under the decimal points in the numbers 
 added or subtracted. 
 
 All decimals smaller than thousandths are usually dropped, 
 one being usually added to the thousandths column when 
 the discarded decimal is five tenths or more. 
 
 
 
 PROBLEMS 
 
 
 
 1. 7.17 
 
 2. 6.789 
 
 3. 
 
 23.768 
 
 4. 6.34 
 
 5. 37.68 
 
 .9 
 
 .075 
 
 
 .45 
 
 .0074 
 
 1.045 
 
 .0006 
 
 74.8 
 
 
 6.985 
 
 43.325 
 
 174.632 
 
 4.76 
 
 .4 
 
 
 17.005 
 
 24.18 
 
 35.986 
 
 5.0017 
 
 24.986 
 
 
 3.178 
 
 3.457 
 
 1.706 
 
 6. .024 + 1.54 + 74.6 + 27.878 = ? 
 
 7. 234.96 + 756 + 40.5 + 6.03 + 1.005 = ? 
 
 8. 2.054 + 35.78 + .067 + .65 + .268 = ? 
 
 9. .9 + 13.564 + 234.96 + 8.5 + .306 + 41.87 = ? 
 
 10. .335 + 23.75 + 601.76 + .007 + 35.86 = ? 
 
 11. .91 + 13.564 + 234.96 + 8.5 + .306 + 41.87 = ? 
 
 12. l-.063 = ? 13.435 - .106 = ? 
 
 13. 3.5872 + 1.2834 = ? 
 
 14. 6-2.763 = ? 
 
 15. 8-3.234 = ? 
 
 16. 4.1 + 67.5 + 42.007 + 17.14 + .0009 = ? 
 
 17. 34.006-15.556 = ? 
 
 18. 68.215-36.5 = ? 
 
 19. 94.35 - 36.7 = ? 
 
 20. 46.235 - 22.065 = ? 
 
26 ELEMENTS OF BUSINESS ARITHMETIC 
 
 21. 216.745 - 176.89 = ? - 
 
 22. 681.34 - 95.275 = ? 
 
 23. 14.367 + 743.65 + .8 + .306 + 9.845 + 834 + 7.63 = ? 
 
 24. 3.8 + 1.576 + 3.42 + 4 + 2.372 + .8 + 354 = ? 
 
 25. 43.382 - 17.06785 = ? 
 
 19. Multiplication and Division by 1 with Ciphers Annexed. 
 
 Since the value of each place in both whole numbers and 
 decimals is one tenth of that to the left of it, any way by 
 which each figure can be shifted one place to the righU or 
 one place nearer units, will decrease its value to one tenth 
 of its former value. Shifting two places would likewise 
 decrease it to one hundredth, three places to one thousandth, 
 etc. In the same way shifting one place to the left would 
 multiply it by ten, two places by one hundred, etc. This 
 may be accomplished by moving the decimal point to the 
 left or right. Thus, 25 -j- 10 = 2.5 or 25 x 10 = 250, and 
 25 -^ 1000 = .025 or .25 x 1000 = 250. 
 
 In multiplying hy 1 with ciphers annexed^ the decimal point 
 is moved to the right as many places as there are ciphers in the 
 multiplier. Thus, 
 
 326.02x100 =32602, 
 and 100.2437 x 1000 = 100243.T. 
 
 In dividing hy 1 with ciphers annexed^ the decimal place is 
 moved as many places to the left as there are ciphers in the 
 divisor. Thus, 
 
 326.02-^100 = 3.2602, 
 and 3.42 -^100 = .0342. 
 
 These simple facts are of great convenience in performing 
 operations with decimals and in shortening multiplication 
 and division of whole numbers. 
 
 Note. — Problems like those above should be given until the pupil 
 thinks of no other way of multiplying or dividing by 10, 100, etc., than by 
 
DECIMALS 27 
 
 removing the decimal point the proper number of places to the right or 
 left. 
 
 20. Multiplication of Decimals. 
 
 (a) 9 times .25 = 2.25. 
 
 When a decimal is multiplied by a whole number, the 
 denomination of the decimal of the product is the same as 
 that of the multiplicand. 
 
 (h) .1 times 9 means one tenth of 9 or .9 (Sec. 19). 
 
 .25 x.l=.025. 
 
 Multiplying by .1 is the same as dividing by 10; by .01 
 the same as dividing by 100, etc. (Sec. 19). 
 
 (c) .25 X .7 = ? (t^) .25 X 6.25= ? 
 
 .25x.l = .025 .25 X .01 = .0025 (see 5) 
 
 .025 X 7 = .175 .0025x 625 = 1.5625 (see a) 
 
 In multiplying decimals by tenths, hundredths, etc., one 
 tenth, one hundredth, etc., is found, and the result is then 
 multiplied by the number of tenths, hundredths, etc., in the 
 multiplier. Or, 
 
 Remove the decimal place in the multiplicand as many places 
 to the left as there are decimal places in the multiplier^ then 
 multiply by the multiplier as a whole number. 
 
 PROBLEMS 
 
 1. 214.76 X 89.104 6. 24.075 x 16 
 
 2. 3.0046 X 43.25 7. 45.009 x 78 
 
 3. .8756 X .173 8. 50.13 x 4.321 
 
 4. .045 X 18 9. 176.84 x 4.321 
 
 5. 64 X .032 10. 95.817 x 1000 
 
 Find the cost of : 
 
 11. 24,800 bricks @ $7.35 per M. 
 
 12. 875 lb. hay @ 11.25 per hundredweight. 
 
 13. 186 bu. wheat @ 67^^ per bushel. 
 
28 ELEMENTS OF BUSINESS ARITHMETIC 
 
 14. 15,680 lemons at 65^ per hundredweight. 
 
 15. 357 bu. oats @ 30^ ^ per bushel. 
 
 16. 75 bbl. flour @ % 4.15 per barrel. 
 
 17. 70 bbl. mess pork @ $ 10.50 per barrel. 
 
 18. 14 bbl. beef @ $14.40 per barrel. 
 
 19. 5 cases shredded codfish @ % 4.90 a case. 
 
 20. 3 cases canned pineapples, 6 doz. each, @ % 2.87| per dozen. 
 
 21. 30 bbl. mess beef @ $ 14.85 per barrel. 
 
 21. Division when the Divisor is an Integer. 
 
 Solve : 
 
 1. .6 -4- 2 = ? Just as 6 bu. -^ 2 = 3 bu. (| of 6 bu.), so 
 6 tenths -2 = 3 tenths or .6 h- 2 = .3. 
 
 2. 2.6-^-2=? 
 
 2.6-2 = 1.3. 
 
 2)2.6 
 1.3 
 
 
 
 
 3. .396-3 
 
 5. 6.6 - 4 
 
 7. 
 
 .7236 - 9 
 
 9. 
 
 .16-5 
 
 4. 12.15-3 
 
 6. 3.68-4 
 
 8. 
 
 .084 - 4 
 
 10. 
 
 .54-6 
 
 In division the decimal point is written in the quotient when 
 it is reached in the dividend. In short division it should be 
 placed directly below the decimal point of the dividend ; in 
 long division directly above. 
 
 Illustrations. 
 
 .06 1.24 .072 
 
 {A) 5 ).18 (5) 15 ) .09 (C) 125 )7.50 {D) 26 )32.24 {E) 24 )1.728 
 
 .036 .006 7.50 62 48 
 
 104 
 
 From inspection of problems Ay B^ and j^, just above, it 
 will be seen that each figure in the dividend^ to the right of 
 the decimal pointy requires a figure (^or cipher^ in the quotient. 
 
 Ciphers may be added at will to the right of a decimal 
 without altering the value. They are, therefore, annexed 
 where necessary to permit further division, but in practice 
 they are carried in the mind only and not written. 
 
DECIMALS 29 
 
 In problems D and E^ the partial products are omitted, 
 the subtraction being performed mentally (Sec. 15). 
 
 So 
 1. 
 
 ilve: 
 48.24 - 3 
 
 10. 
 
 1.5-4 
 
 19. 
 
 174.9 -4- 75 
 
 2. 
 
 8.64^4 
 
 11. 
 
 .06 - 15 
 
 20. 
 
 43.58 -4- 671 
 
 3. 
 
 .465 -^ 3 
 
 12. 
 
 8.06 - 5 
 
 21. 
 
 345.9 -4- 329 
 
 4. 
 
 8.4-4-5 
 
 13. 
 
 .2^8 
 
 22. 
 
 56.89 - 137 
 
 5. 
 
 .648 - 6 
 
 14. 
 
 34.75 -4- 25 
 
 23. 
 
 .0789 -f- 703 
 
 6. 
 
 .114 - 7 
 
 15. 
 
 .0543 -4- 15 
 
 24. 
 
 .6789 -- 212 
 
 7. 
 
 .81-^9 
 
 16. 
 
 .0255 -4- 11 
 
 25. 
 
 17.68 ^ 245 
 
 8. 
 
 .63-9 
 
 17. 
 
 3.184-4-482 
 
 26. 
 
 334.4^76 
 
 9. 
 
 .12-4-9 
 
 18. 
 
 37.86 - 541 
 
 
 
 22. When the Divisor contains a Decimal. 
 
 Example: 1.728-^.12 = ? 
 
 If we could change the divisor .12 to a whole number and 
 solve as in Sec. 21, the problem of the location of the 
 decimal point in the quotient (the only way in which divi- 
 sion of decimals differs from division of whole numbers), 
 would be an easy one. 
 
 But dividing by one tenth is equivalent to multiplying by 
 ten, dividing by one hundredth to multiplying by one hun- 
 dred, etc. (Sec. 20). Thus dividing 1.728 by .01 = 1.728 x 100 
 or 172.8 (Sec. 19), and dividing by .12 would give -^ of 
 the result obtained through dividing by .01. Thus : 
 
 1.728 -f-. 01 = 172.8, 
 and 172.8-1- 12 = 14.4. 
 
 In division by a decimal, therefore, the decimal point of the 
 divisor should he moved to the right of the last digits and that of 
 the dividend an equal number of places to the right of its origi- 
 nal position (Sec. 19). Division should then be performed 
 as in Sec. 21, marking the decimal point in the quotient 
 when the decimal point is reached in the dividend. 
 
30 ELEMENTS OF BUSINESS ARITHMETIC 
 
 To avoid losing the identity of the original divisor and to 
 be able to fix the exact remainder, should there be one, the 
 position of the new decimal point should be indicated only. 
 A small St. Andrew's cross ( X ) forms a convenient mark 
 for that purpose. 
 
 Solve : 
 
 
 
 
 
 1. 6.336^1.44. 
 
 6. 
 
 784 -^ 4.235. 
 
 17. 
 
 1885 -- 28.47. 
 
 
 7. 
 
 .3416 -f- .0189. 
 
 18. 
 
 363.71 -^ 1.126. 
 
 4.4 
 
 8. 
 
 12.347 -f- .0074. 
 
 19. 
 
 83.078 -^ 3.57. 
 
 1.44x)6.33x6 
 
 9. 
 
 .8765 -^ 3.422. 
 
 20. 
 
 137.854 -- .425. 
 
 5 76 
 
 10. 
 
 9 -- 102. 
 
 21. 
 
 38.9007 -- .425. 
 
 57 6 
 57 6 
 
 11. 
 
 14.3768 -r- .9817. 
 
 22. 
 
 568.148 -r- 201.03. 
 
 12. 
 
 84.45 - .089. 
 
 23. 
 
 .81769 -- .0008175. 
 
 2. 7.345 -.29. 
 
 13. 
 
 3894.78 -i- 4287. 
 
 24. 
 
 $135 ^$.37^. 
 
 3. 250.754-^6.17. 
 
 14. 
 
 346.543 - 634.08. 
 
 25. 
 
 74 -- .0136. 
 
 4. 6.0534-^-19.23. 
 
 15. 
 
 34.25 -^ 84.6. 
 
 26. 
 
 .33614 -f- 13.45. 
 
 5. 132.5 -=- 734. 
 
 16. 
 
 9.1342 -f- 208.4. 
 
 27. 
 
 18.3467 -- 1.233. 
 
 PROBLEMS 
 
 1. Add eight and nine tenths ; seven hundred twenty-six and twenty- 
 five hundredths ; one hundred sixty-eight and ninety-seven hundredths ; 
 one thousand three and seven tenths; seven hundred sixty-eight and 
 seventy-four hundredths. 
 
 2. 967.45 -f 8.674 -f 23.997 + 864.325 + 37286 -f 42.1 + 6.5 = ? 
 
 3. From six thousand seven take one thousand two hundred twenty- 
 eight and seven hundredths. 
 
 4. What is the sum of 3.25, 1.8, 67.89, .0032, 879.435, and 23.067? 
 
 5. What is the sum of .125, 1.25, 12.5, 125, .0125? 
 
 6. From 675. take 67.893. 9. Multiply 543.002 by 18.6. 
 
 7. From 345.6703 take 43.52. 10. Divide 6423.38 by 28.87. 
 
 8. Multiply 54.054 by .0678. 11. Divide .00684 by .25. 
 
 12. At $ 9.25 per ton, how much coal can be bought for $ 67.53 ? 
 
 13. If a barrel of apples cost $5.15, how many can be bought for 
 $258.75? 
 
DECIMALS 31 
 
 14. At $ .26 per dozen, how many eggs can be bought for $ 185.32 ? 
 
 15. At 1 6.45 per ton, how many tons of soft coal can be bought for 
 $175? 
 
 16. A farmer sold 65 bu. wheat at $.62|, 34 bu. rye at $ .58^, 78 bbl. 
 of apples at $ 6.40. He bought 35 lb. sugar at $ .06, 25 gal. molasses at 
 $ .85, and a set of harness at $ 16.75. How much money had he left ? 
 
 17. A carpenter earned $ 15.60 a week for 8 weeks. The first week 
 he spent $8.75, the second week $11.45, and the remaining weeks he 
 spent $ 8.50 per week on an average. How much money had he at the 
 end of the time ? 
 
 18. A man who has an income of $ 6785 per year spends $ 1385.75. 
 How much does he save? 
 
 19. A hardware merchant had, at the beginning of the year, $ 9800. 
 During the year he bought goods to the amount of $ 7845, and sold to 
 the amount of $ 7856. If the goods he had on hand at the end of the 
 year were worth $ 8340.65, what did he gain or lose during the year ? 
 
 20. A man bought a mower for $ 67, a wagon for $ 56.50, a plow for 
 $6, and a rake for $1.75. If he gave the merchant two one-hundred- 
 dollar bills, how much change did he receive ? 
 
 21. Mr. A. W. Springer bought of C. E. Dunlap, 75 bbl. flour at $5.14, 
 and 34 bbl. buckwheat flour at $3.95. What was the amount of the 
 biU? 
 
 22. Mr. S. H. Detmore bought of Wm. King, 124 boxes oranges at 
 $ 1.12|, 12 boxes figs at $ 1.62^, 33 boxes apricots at $ 2.15, 15 boxes 
 citrons at $ 1.33|, and 12 boxes layer raisins at $2.95. Find the amount 
 of the bill. 
 
 23. How many cases tomatoes, 70 doz. to the case, at 92 ^ per dozen, 
 can be bought for $ 277.55 ? 
 
IV 
 FRACTIONAL PARTS 
 
 23. Use of Fractional Parts. The necessities of everyday 
 business require the use of few processes in arithmetic more 
 frequently than that of finding some part of a given quantity. 
 Not only is this true in the simpler problems of business, but 
 the processes of percentage, discount, partnership, interest, 
 taxes, etc., are, fundamentally, finding fractional parts. 
 
 Finding the fractional parts requires the use of the proc- 
 esses of multiplication and division, together with the simpler 
 principles of decimals. Under the usual arrangement, it 
 forms a case in fractions ; viz., multiplication by a fraction. 
 Its common use and wide application is thereby obscured, 
 and the simplicity of the operation seldom realized. Then, 
 too, such problems are commonly included in division and 
 cause a great deal of haziness in reasoning. It is believed 
 that the claims of clear thinking, and the importance of the 
 power sought, warrant treatment in a separate chapter. 
 
 24. Finding a Single Fractional Part. 
 Find J^ of 291.9. (See Sec. 13.) 
 
 20.85 Ana. 
 14)291.9 
 
 28 
 
 119 
 112 
 
 70 
 
 70 
 Find: 
 
 1. ^ff of 5763 (Sec. 19), 320, 2000, 17,722, 1023, 1879, 98,450. 
 
 2. ^ of 150, 337, 20.0400, 5061, 17,002, 279.63, 56,840, 96.789. 
 
 32 
 
FRACTIONAL PARTS 33 
 
 3. i^ of 740 (i of yij), 648, 10.25, 2047, 50.011, 3452, 6.073, 3004. 
 
 4. i of 450, 645, 705, 64.2, 687, 1018, 50,670, 58,605, 1600.115. 
 
 5. ^ of 7218, 6345, 117,927, 14.4639, 5017, 72668.54, 576,943. 
 
 6. I of 588, 1672, 976, 847, 734,68^p^r33, 20,053, 1,129,734. 
 
 7. \ of 3876, 327,450, 389.90, 263,87^2^, 32,876, 42,680. 
 
 8. \ of 42,744, 62,728, 88,102. \ of 67,833, 300.0321, 67431.76318. 
 ^^ of 27,924,12, 102,603. ^V of 367,485, 790,532. J^ of 362,922, 758.492. 
 yig of 786,496, 710.43, 876,014. f^ of 72,632, 43.2004, 432,132. 
 
 9. ^ of 7364, 5467, 18,998, and 6555.88. ij of 4763, 9878, 190.63, and 
 121,264.396. ^V of 28,652. 10,431,278 and 624.0156. 
 
 25. Divisibility. The following facts will be found help- 
 ful in finding fractional parts and in short division. 
 
 1. When the right-hand digit of a number is divisible by 
 2, the whole number is divisible by 2. When exactly 
 divisible by 2, a number is said to be even; when not so 
 divisible, it is odd. 
 
 2. When the right-hand digit of a number is 5 or 0, the 
 whole number is divisible by 5; when it is 0, the whole 
 number is divisible by 10. 
 
 3. When the number expressed by the two right-hand 
 digits is divisible by 4, the whole number is divisible by 4. 
 When the number expressed by the three right-hand digits 
 is divisible by 8, the whole number is divisible by 8. 
 
 4. When the sum of all its digits is divisible by 3 or 9, 
 the number is divisible by 3 or 9. 
 
 5. When an even number is divisible by 3, it is also 
 divisible by 6. 
 
 26. Factors and Multiples. A number exactly divisible by 
 another number, other than 1, is said to be composite^ ^.e., 
 composed of other numbers called factors^ which multiplied 
 together will produce the number. 
 
 A number which is not exactly divisible by another (ex- 
 cept by unity), is said to be prime. 
 
34 ELEMENTS OF BUSINESS ARITHMETIC 
 
 Factors may, themselves, be either prime or composite. 
 
 A number which will exactly divide two or more numbers 
 is said to be a common divisor or a com,mon factor. The 
 largest number which will exactly divide two or more num- 
 bers is their greatest common factor or greatest common divisor. 
 If two or more numbers have no common divisor, they are 
 said to be relatively prime. 
 
 A number which is two or more times another number is 
 its multiple. When a number is a multiple of two or more 
 numbers, it is their common multiple. When it is the smallest 
 number which is a multiple of two or more numbers, it is 
 their least common multiple. 
 
 27. Averaging. By average is meant the size of a part if 
 the sum of a given series of numbers is distributed into 
 equal parts. Thus, we say the average cost, average monthly 
 expenses, average daily attendance, the assessed valuation 
 per capita., etc. 
 
 In business, we may more clearly judge given conditions by 
 the use of an average cost, output, etc., than if we were com- 
 pelled to keep in mind the several amounts. In administra- 
 tive statistics a more intelligible summary of facts may often 
 be made by using averages. In many similar ways averaging 
 enters into business, and it should be early understood. 
 
 Problem. — A merchant's receipts for one week were : 
 $140.45, 1217.20, $200, $209.80, $432.75, and $630.40. 
 Find his average daily receipts. 
 Solution. —$140.45 
 217.20 
 200.00 
 209.80 
 432.75 
 630.40 
 \ of $1830.60 = $305.10, average daily receipts. 
 
FRA.CTIONAL PARTS 
 
 35 
 
 The general method of averaging seen in the above is to 
 find the sum of the quantities to be averaged, and divide 
 that sum into as many equal parts as there are quantities 
 added. 
 
 PROBLEMS 
 
 1. If a man's gains for the year were as follows: $628, $75, $220, 
 $865, $2205, $3600, $1780, $1500, $1240, $3275, $825, $775, what was 
 his average monthly gain? 
 
 2. A grain dealer bought during the week: 2600, 3850, 4506, 7870, 
 9675, and 5490 bushels of wheat. What was the average daily purchase? 
 He paid the market prices each day, which were quoted at 85^, 85^ )2^, 
 88)25, 90)*, S4:^, and 80 (2^. What was the average cost per bushel? How 
 would he know what price per bushel to ask that he may not lose in the 
 transaction ? 
 
 3. An agent's expenses were as follows : January, $ 125, February, 
 $75, March, $80, April, $95, May, $105, and June, $225. What were 
 his average monthly expenses? 
 
 4. Find the total time, the amount due, and the average daily wages 
 for each laborer in the following time sheet, counting 8 hrs. to the day ; 
 also find the average wage for all. 
 
 Time Sheet for Week Ending April 30, 19—. 
 
 PAY ROLL. 
 
 
 
 
 
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 5. A student's grades for his high school course were as follows: 
 Algebra 92, Geometry 86, Beginning Latin 89, Caesar 94, Cicero 94, 
 
36 ELEMENTS OF BUSINESS ARITHMETIC 
 
 Virgil 92, German 90, Literature 83, Essays 85, Chemistry 92, Physics 
 90, Physiology 95, Greek History 87, Roman History 88, U. S. History 
 87, Arithmetic 85, Bookkeeping 90. What was his average grade for the 
 course ? 
 
 28. To find the Value of more than One Part. Tenths. 
 (Sec. 19.) To find -^-^ take \. To find any other number of 
 tenths, find J^ as in section 19, and multiply by the number 
 of tenths desired. 
 
 Solve : 
 
 i\ of 323,496, T-^ of 769,284, ^ of 6789.40, xV of 81,145, ^ of 72,910, ^ 
 of 432,561, i-^ of .63245, t% of 237,684. 
 
 Twentieths. (|- of 1 tenth.) J^ = i, -f^ = -J^, etc. The 
 simplest form of the fraction should be taken and that frac- 
 tion of the number then found. Practice until student uses 
 the simplest form at first sight. 
 
 Solve : 
 
 1. 3j%of360 5. ^^ of 3240 
 
 2. la of 8.460 6. if of 7.380 
 
 3. 5^ of 3750 7. 1^ of 960 
 
 4. ^5^ of 7260 8. f|of490 
 
 Fifths. | = A,| = TVand-| = ^V 
 Fourths, f = J. For |, subtract \. 
 Sixths -^ = -1 ^ = 1 4 _ 2 
 
 toiJ^LUS. 62' 63' 63* 
 
 Eighths, f = i 2 ^ i^ | = |^ etc. 
 
 Twelfths, -f^ = ^> iV = h -\2 — h ®^^* ■^^^' 11 subtract ^2 • 
 In general, use simpler equivalent fractions wherever 
 
 possible. If there are none, find the value of one part and 
 
 multiply by the number of parts desired. 
 
 Solve : 
 
 1. T^Yof726 4. I of 911 7. t'^j of 728 
 
 2. ^^ of 543 5. ^5^ of 614 8. f of 329 
 
 3. I of 789 6. i3T0f729 9. f of 628 
 
 9. 
 
 H of 4260 
 
 10. 
 
 ^ of 428 
 
 11. 
 
 M of 5.061 
 
 12. 
 
 A of 576 
 
FRACTIONAL PARTS 37 
 
 10. 
 
 j\ of 7028 
 
 17. 
 
 j\ of 48,689 
 
 24. 
 
 1 of 36,428 
 
 11. 
 
 j\ of 7028 
 
 18. 
 
 iV of 45,186 
 
 25. 
 
 T% of 7829 
 
 12. 
 
 jS of 328.65 
 
 19. 
 
 j\ of .3876 
 
 26. 
 
 r% of 1.7206 
 
 13. 
 
 35V of 16,924 
 
 20. 
 
 j% of 4263 
 
 27. 
 
 f of 72,963 
 
 14. 
 
 j% of 6724 
 
 21. 
 
 1 of 67.908 
 
 28. 
 
 j\ of 34,102 
 
 15. 
 
 j% of 72,365 
 
 22. 
 
 f of 7252 
 
 29. 
 
 j\ of 4291 
 
 16. 
 
 j\ of 98,642 
 
 23. 
 
 f of 81.720 
 
 30. 
 
 j\ of 28,367 
 
 Solve : 
 
 
 
 
 
 1. 
 
 a- of 68,734 
 
 11. 
 
 j\\ of 19,643 
 
 21. 
 
 j\\ of 47.6297 
 
 2. 
 
 If of 57,836 
 
 12. 
 
 U of 42,782 
 
 22. 
 
 fW of 62,587 
 
 3. 
 
 H of 196,732 
 
 13. 
 
 If of 32,785 
 
 23. 
 
 /^«j of 29,410 
 
 4. 
 
 j%\ of 67.941 
 
 14. 
 
 H of 42,363 
 
 24. 
 
 7^^ of 25,000 
 
 5. 
 
 If of 94,362 
 
 15. 
 
 ^ of 89.642 
 
 25. 
 
 if'j of 67,894 
 
 6. 
 
 II of 6946 
 
 16. 
 
 ^Vi7 of 2632 
 
 26. 
 
 iH of .456781 
 
 7. 
 
 j^^j of 198,674 
 
 17. 
 
 if of 15,922 
 
 27. 
 
 n of 86,543 
 
 8. 
 
 H of .4575 
 
 18. 
 
 rV5 of 58,596 
 
 28. 
 
 f 1 of 219,876 
 
 9. 
 
 tV^ of 9.7176 
 
 19. 
 
 tV^ of 155,476 
 
 29. 
 
 ^ of 98,634 
 
 10. 
 
 /A of 32,642 
 
 20. 
 
 T«^ of 167.1918 
 
 30. 
 
 xVx of 186,792 
 
 29. Simplified Processes. It often happens that the num- 
 bers to be multiplied or divided are such that the operations 
 of multiplication and division may be much shortened. The 
 simplicity of the shortened method, too, often lends itself to 
 clearer thinking and lessens the liability of error. The fol- 
 lowing suggestions are not given to be learned as "short 
 methods," but as examples of what may be done to simplify 
 problems if a little thinking is done before the pupil plunges 
 into the mechanics of a solution. They should form a part 
 of the methods habitually employed and not merely referred 
 to when longer processes have been drilled into habit by 
 much practice. A keen appreciation of the possibilities for 
 shortening and simplifying many of the arithmetical pro- 
 cesses will lead to quick work and accurate results. 
 
38 ELEMENTS OF BUSINESS ARITHMETIC 
 
 Illustrations : 
 
 1. 426x100=? By 1000 = ? By 10 = ? (Sec. 19.) 
 
 Suggestion. In multiplying by a number consisting of 1 with 
 ciphers annexed, the product is found to consist of the multiplicand with 
 the ciphers annexed. Thus, 728 x 1000 = 728,000. 
 
 2. 53,648 ^ 1000 = ? (Sec. 19.) 
 
 Since the divisor will take out even thousands, the hun- 
 dreds, tens, and units figures will form the remainder. Thus, 
 53,648^1000=53 and -j^*^, or 53.648. Likewise, 8269 
 -i- 100 = 82 and {-^% or 82.69. 
 
 3. 7268 X 25 = ? 
 
 Multiplying by 25 would give ^ as much as multiply- 
 ing by 100. Thus, the above may be written at sight as 
 182,300 (1 of 726,800). Likewise 7268x50 = 1 of 726,800 
 or 363,400. 
 
 4. 742-25 = ? 
 
 The quotient is manifestly 4 times what it would be if 
 100 were the divisor. In practice, multiply first and divide 
 by 100. Thus, 742 x 4 = 2968, which divided by 100 = 29 
 and ^%8_ or 29.68. Likewise 742 ^ 50 = 7.42 x 2 or 14.84. 
 
 5. 246 X 121 = ? (1 of 24,600.) 
 
 6. 368^121=? (368x8-^100.) 
 
 7. 1284x15 = ? 
 
 If the table of 15's has not been learned, the usual process 
 may be shortened by annexing a cipher and adding J of the 
 number thus formed to itself. Thus, 
 
 12,840 
 
 6420 
 
 19,260 
 
 8. Divide 5286 by 20. By 30. 
 The quotient, dividing by 20, is clearly -J as much as 
 when divided by 10. Thus, J of 528.6 = 264.3. 
 
FEACTIONAL PARTS 39 
 
 9. 8246 X 98 = ? 
 It is evident that the product would be 100 times the mul- 
 tiplicand less twice the latter. Thus, 
 
 
 
 
 824,600 
 
 16,492 
 
 808,108 
 
 
 To 
 
 multiply by 
 
 11 
 
 or any multiple 
 
 of 11 
 
 
 328 X 
 
 11 
 
 = ? 476x66 = 
 
 :? 
 
 10. 
 
 To multiply by 11 is to multiply by 10 + 1. Therefore, 
 multiply 328 by 10 and add 328 ; or add the digits as fol- 
 lows : 8; 8 + 2 = 10; 2 + 3 + 1 (carried) = 6 ; bring down 3. 
 The result is 3608. 
 
 66 is 11 times 6. Multiply 476 by 11, and that by 6. It 
 may be done as follows : 6 x 6 = 36 ; write 6 and carry 3. 
 
 6 + 7 = 13; 6x13 + 3 (carried) = 81 ; write 1 and carry 8. 
 
 7 + 4 = 11; 6x11 + 8 (carried) = 74 ; write 4 and carry 7. 
 6x4 + 7 (carried) = 31 ; write 31. The result is 31,416. 
 
 11. When one part of the multiplier is contained in an- 
 other, the multiplication may be shortened as shown in the 
 following ; 
 
 342 X 248 = ? 167 x 412 = ? 
 
 Since 24 is 3 times 8, it is evident that if 342 be multi- 
 plied by 8 and that result by 3, the final result will be the 
 same as though we multiplied 342 by 248 in the usual man- 
 ner (a). 
 
 167 multiplied by 4 gives 668. 668 mul- 
 tiplied by 3 (12-^4) gives 2004. The 
 
 final result is the sum of 668 written in 2,736 668 
 hundreds' place and 2004 in units' place, or 82,08 2004 
 68,804 (5). 84,816 68,804 
 
 12. 52x225 = ? 
 
 (a) 
 
 (^) 
 
 342 
 
 167 
 
 248 
 
 412 
 
40 ELEMENTS OF BUSINESS ARITHMETIC 
 
 In the multiplier 225, we see twice and ^ of 100 times 52, 
 or 10,400 
 
 1,300 
 11,700 
 
 13. 7558x125 = ?^ | of (7558 x 1000). 
 
 14. 57,632 -f- 370 = 5763.2 -V- 37. 
 
 Solve : 
 
 1. 37,685 X 10 11. 32,764 x 12^ 21. 16,324 x 95 
 
 2. 49,652 X 100 12. 25,670 x 742 22. 17,264 x 320 
 
 3. 29,627 X 1000 13. 62,482 x 250 23. 98,643 - 20 
 
 4. 72,864x25 14. 41,628x75 24. 76,523x300 
 
 5. 22,345 X 15 15. 29,863 x 99 25. 8629 h- 12i 
 
 6. 28,364 X 125 16. 36,486 x 175 26. 37,520 - 25 
 
 7. 48,627 X 55 17. 28,634 x 325 27. 4286 x 88 
 
 8. 26,327 X 98 18. 67,450 x 217 28. 36,742 x 97 
 
 9. 4652 X 225 19. 6284 x 1500 29. 75,267 x 420 
 10. 63,245x30 20. 729x40 30. 2672-50 
 
 30. Decimal Equivalents of the More Common Fractional 
 Parts. 
 
 Solve : 
 
 1. ^ = how many tenths ? 
 
 i = ^ of 10 tenths = 2 tenths or .2. 
 
 2. J = how many hundredths? 
 
 ^ = i of 100 hundredths or .25. 
 
 3. i = how many thousandths ? 
 
 ^ = i of 1000 thousandths or .125. 
 
 4. ^ = how many tenths? How many hundredths? 
 
 5. ^ = how many tenths ? How many hundredths ? 
 
 6. ^ = how many tenths? How many hundredths? How many 
 thousandths ? 
 
 ^ = ^ of ten tenths or .3|. 
 
 7. J = how many tenths? How many hundredths? How many 
 thousandths ? 
 
FRACTIONAL PARTS 41 
 
 8. ^ = how many tenths? How many hundredths? How many 
 thousandths ? 
 
 9. ^ =: how many tenths? How many hundredths? How many 
 thousandths ? 
 
 10. j*^ = how many tenths? How many hundredths? How many 
 thousandths? 
 
 11. 2V = I'ow many tenths? How many hundredths? How many 
 thousandths ? 
 
 12. ^i^ = how many tenths? How many hundredths? How many 
 thousandths ? 
 
 Table of decimal equivalents. Memorize. 
 \ = .2 i = .125 | = .14f 
 
 Name Decimal Equivalents at Sight : 
 
 1. I 5. } 9. x% 
 
 2. f 6. I 10. I 
 
 3. f 7. f 11. 11 
 
 4. f 8. I 12. f 
 
 31. Fractional Parts of One Dollar. 
 
 The more common fractional parts of $1 are : 
 
 50^= fi io^=iJ^ ^^^=^ 61^=1^5 
 
 The more usable multiples of these parts are : 
 
 371^ = $! 871^ = ||- 75^ = || 60^ = If 
 
 621^ = If 66|^=lf 40^ = $1 80^ = || 
 
 32. Finding the Cost of Articles. A large part of the 
 multiplication in business consists in finding the cost of 
 articles at a given price each or per dozen, As prices are 
 in a large number of cases a simple fractional part of one 
 
 
 
 ^ = 
 
 •08J 
 
 
 
 ■h = 
 
 .05 
 
 
 
 ^ = 
 
 .04 
 
 13. 
 
 f 
 
 
 17. 1 
 
 14. 
 
 f 
 
 
 18. f 
 
 15. 
 
 ^ 
 
 
 19- iV 
 
 16. 
 
 f 
 
 
 20. 1 
 
42 ELEMENTS OF BUSINESS ARITHMETIC 
 
 dollar, this fact may be used to shorten the work, and what 
 is of greater importance, to secure greater accuracy in result. 
 Thus, 
 
 (a) Find cost of 344 yd. of calico at 12|^^ per yard. At 
 12 J ^ it would cost I as much as it would at 81. I of $344 
 = 143. 
 
 Note. — In problems like this where the multiplicand is an easier 
 number to multiply by than the real multiplier, it is so used, keeping in 
 mind the denomination of the product. 
 
 (h) Paid 1403 for corn at 20^ per bushel. How many 
 bushels did I buy ? 
 
 403 bu. X 5 = 2015 bu. 
 
 The number of bushels bought is clearly 5 times greater 
 than if it were worth $ 1 per bushel. 
 
 Find Cost of: 
 
 1. 7286 bu. wheat @ 50)*. 
 
 2. 1456 yd. prints @ 12^ ^. 
 
 3. 764 yd. cloth @ 33^^. 
 
 4. 324 bbl. mess beef @ $ 15. 
 
 5. 750 bbl. pork @$ 15.12^. 
 
 6. 24 doz. cans tomatoes at 1 1.12J. 
 
 7. 90 doz. cans peas @ $ 1.37^. 
 
 8. 35 boxes figs @ $ 1.62|. 
 
 9. 14911b. tea @33|^. 
 
 10. 450 sugar-cured hams, 5400 lb., @ 12J^. 
 
 11. 348 boxes oranges @ $ 1.75. 
 
 12. 117 bbl. flour® $6.75. 
 
 13. 16 hams, 195 lb., @ 16f ^. 
 
 14. 48 doz. cans of tomatoes @ 75 ^. 
 
 15. 108 bbl. N.Y. buckwheat flour @ $4.25. 
 
 16. 125 doz. cans tomatoes @ $2.66|. 
 
 17. 96 doz. cans peas @ $ 1.37^. 
 
FRACTIONAL PARTS 43 
 
 18. 6 tierces refined lard, 2096 lb., @S^^. 
 
 19. 330 doz. jars of mustard @ 87^)?. 
 
 20. 368 1b. coif ee@ 20 j«. 
 
 21. 178 bbl. beef @ 120. 
 
 33. Cost of Goods Sold by the Hundred. Freight tariffs ^ 
 are usually based on one hundred pounds. Live stock is 
 quoted by the hundredweight, and many other articles are 
 sold in lots of one hundred. Since cents are hundredths 
 
 of a dollar, goods sold hy the hundred will cost as many cents 
 per unit as dollars per hundred. Thus $5 per cwt. is 5^ per 
 pound, and f 3.75 per C is f .0375 per pound. 
 
 Find Cost of: 
 
 1. Freight charges on 4230 lb. of merchandise at $1.40 per cwt. 
 1.014 X 4230 = 159.22. (|4.23 x 14 = $59.22.) 
 
 2. 1280 lb. nails @ 32^ per cwt. 
 
 3. 2963 lb. rock salt @ $1.20 per cwt. 
 
 4. 2974 lb. fence wire @ $4.50 per cwt. 
 
 5. 1280 posts, split, @ $18 per C. 
 
 6. 375 lb. lead @ $4.75 per cwt. 
 
 7. 5 cattle, averaging 925 lb., @ $4.50 per cwt. 
 
 8. 11,580 posts, round, @ $25 per C. 
 
 9. 9850 lb. pork @ $7.50 per cwt. 
 
 10. Freight on a carload of grain, 46,500 lb., @ 38)^ per cwt. 
 
 11. 1378 1b. poultry @ $6.50 per cwt. 
 
 12. 1865 lb. beef @ $14 per cwt. 
 
 13. 975 lb. bran @ 90^ per cwt. 
 
 34. Goods Sold by the Thousand. Brick, lumber, shingles, 
 and many other articles are sold by the thousand. Gas is 
 sold by the thousand cubic feet. The amount consumed is 
 shown by a meter, upon the face of which are usually three 
 dials. The dial to the right shows hundreds of cubic feet, 
 that in the middle shows thousands, and that to the left 
 
44 ELEMENTS OF BUSINESS ARITHMETIC 
 
 shows tens of thousands. Reading the last figure passed by 
 the pointers, and going from left dial to right, the number of 
 hundreds of cubic feet consumed is shown. Thus, the ac- 
 companying illustration reads 323 hundreds or 32,300 cubic 
 
 A Gas Meter 
 
 feet. The reading for the preceding period is shown by the 
 dotted lines. The reading for the preceding period is sub- 
 tracted from the present reading to find amount of gas 
 consumed. 
 
 A mill being j-^q-q of a dollar, articles sold hy the thousand 
 are as many mills per article as dollars per thousand. Thus, 
 137 per M is 37 mills per pound or foot, or 126.50 per M is 
 26.5 mills (f .0265) each. 
 
 Find Cost of: 
 
 1. 1625 ft. oak lumber @ $36 per M. 
 
 One foot would cost 36 mills (1.036), and 1625 ft. would cost $.036 x 
 1625 ($1,625 X 36) or $58.50. 
 
 2. 20,408 ft. flooring @ $17.50 per M. 
 
 3. 14,450 ft. 2 X 6-15 @ $15.50 per M. 
 
 4. 10,458 ft. 6 X 6-18 @ $22.50 per M. 
 
 5. 10,448 ft. sheathing© $12.75 per M. 
 
 6. 612 ft. pine lumber @ $15 per M. 
 
 7. 456 ft. spruce @ $12.50 per M. 
 
 8. 7750 shingles @ $5.25 per M. 
 
 9. The meter readings for gas consumed in a residence for six 
 months were as follows: Sept. 1, 28,400; Oct. 1, 31,000; Nov. 1, 
 34,400; Dec. 1, 37,600 ; Jan. 1, 40,700 ; Feb. 1, 43,200; Mar. 1, 45,100. 
 
FRACTIONAL PARTS 45 
 
 At 90 ^ a thousand, what was the total of the monthly gas bills for the 
 six months? 
 
 10. A contractor bought material for a building as follows : 567,800 
 brick @ $6.75 ; 35,657 ft. matched pine @ $ 20 ; 14,720 ft. hemlock @ $12 ; 
 4680 ft. walnut @ $45 ; 75,250 shingles @ |4.75. What was the total cost ? 
 
 11. If there is a gas meter in the school building, read it from month 
 to month and compute the cost of gas. Also read the meters in your 
 home and compute the cost of gas. 
 
 35. Goods Sold by the Ton. Coal, hay, and other articles 
 are sold by the ton. If coal is quoted at $ 6.50 per ton, it is 
 $3.25 per 1000 pounds, and 3.25 mills (1.00325) per pound. 
 In other words, goods sold hy the ton are one-half as many 
 mills per pound as dollars per ton. Thus, $3.80 per ton is 
 1.9 mill (1.0019) per pound. 
 
 PROBLEMS 
 
 1. 3640 lb. hay @ $8.50 per T. 
 
 1 of $8.50 = $4.25 per M., or 4.25 mills per lb., $.00425 x 3640=$15.47. 
 
 2. 42,300 lb. hay @ $9 per T. 
 
 3. 2860 lb. soft coal @ $3.75 per T. 
 
 4. 84,375 lb. steel rails @ $20 per T. 
 
 5. 225,780 lb. ore @ $25 per T. 
 
 6. 3500 lb. fertilizer @ $22 per T. 
 
 7. 4525 lb. anthracite coal @ $8.75 per T. 
 
 8. 38,960 lb. salt @ $3.35 per T. 
 
 9. 265,700 lb. clover hay @ $ 6.20 per T. 
 
 10. What will be the cost of the freight on 5 cars of coal at $ 2.75 per 
 ton, the cars weighing respectively : 87,560, 75,605, 54,780, 85,670, and 
 70,840 pounds net? 
 
 11. Find the Value of Each: 
 
 Article 
 
 Gross Weight 
 
 Tabb 
 
 Priob 
 
 A load of hay 
 
 3450 lb. 
 
 1256 lb. 
 
 $8.75 per T. 
 
 A load of straw 
 
 2975 lb. 
 
 856 lb. 
 
 $2.60 per T. 
 
 A load of coal 
 
 5475 lb. 
 
 1680 lb. 
 
 $7.50 per T. 
 
 A load of beets 
 
 3475 lb. 
 
 1240 lb. 
 
 $12..50 per T. 
 
 A load of stone 
 
 4250 lb. 
 
 1250 lb. 
 
 $14.50 per T. 
 
46 
 
 ELEMENTS OF BUSINESS ARITHMETIC 
 
 36. Invoicing, Bills, and Accounts. When goods are 
 shipped, or sold on account, an invoice or bill is mailed, or 
 sent with the goods. A detailed statement of the amount, 
 kind, and prices of the goods, together with the names of the 
 parties to the transaction, term of credit, condition of sale, 
 discount allowed, etc., is called either a bill or invoice. The 
 term bill is applied particularly to a statement of goods 
 bought, of services rendered, or of work performed. Prompt- 
 ness in billing goods shipped, and accuracy in computing the 
 amount of the bills, are business essentials. 
 
 Formerly, the term " invoice " was used when goods were 
 shipped on consignment only, but it is now often used inter- 
 changeably with "bill." A clause on the invoice, stating that 
 the goods remain the property of the consignor until paid for, 
 makes the sending them out a consignment rather than a sale. 
 
 At stated periods, usually the first of each month, a state- 
 ment of account is sent to debtors. A statement merely 
 gives the amount of bills previously rendered, and credits, 
 with their dates. Its chief purpose is to remind the debtor 
 of the debt, but it is also of assistance in checking errors in 
 accounts. 
 
 PROBLEMS 
 
 Rule forms and copy the following invoices, filling in the missing 
 amounts. 
 
 Philadelphia, Pa., Jan. 19, 19— 
 1. The Amos King Co., 
 Baltimore, Md. 
 
 
 
 Bought of JOHN 
 
 WANAMAKER 
 
 
 
 
 Terms : "Net 30 days 
 
 
 1310 
 
 10 
 
 Roll Top Office Desks, 
 
 143.75 
 
 
 
 
 
 X1338 
 
 15 
 
 Typewriter Desks, 
 
 4.50 
 
 
 
 
 
 1317 
 
 20 
 
 Office Tables, 
 
 7.00 
 
 
 
 
 
 1238 
 
 10 
 
 Office Chairs, 
 
 6.75 
 
 
 
 
 
 Note. — The figures to the left of the ruling show the catalogue numbers. 
 
FRACTIONAL PARTS 
 
 47 
 
 2. 
 
 PitUburg, Pa^ /i^r^yy 
 
 /. 
 
 _19= 
 
 - 
 
 
 Xh^-iT-i^^^^T'T^^y/^-^^/^ Uy. - 
 
 
 in.ccmitwhh A. D. HIRSCH & COMPANY 
 
 
 
 
 ^^. 
 
 
 
 
 
 /^Tjiyf^. 
 
 / 
 
 -^.^rr^ a^^.^<h-fy^yy7^yA:y^^.^^^^^ ^^,^ 
 
 .^r 
 
 ^-r^ 
 
 
 
 
 /^ 
 
 .. %^^.^ 
 
 / 
 
 
 
 
 i.^ 
 
 
 ,^,6-r. 
 
 7.^; 
 
 
 
 
 ~^^y 
 
 
 X— ' 
 
 
 
 
 /^ 
 
 ■"■?W<?/ .^^^r^r^A^^^ 
 
 / /O o 
 
 or? 
 
 
 
 
 ?-^/ 
 
 
 ,?/^/^ 
 
 (L^ 
 
 
 
 
 
 
 
 
 
 ' 
 
 
 
 
 
 
 
 
 Chicago, 111., May 6, 19— 
 
 3. Mr. S. a. Daubb, 
 
 Evanston, 111. 
 
 Bought of 
 
 Terms : 30 da. 2 % 10 da. 
 
 1690 ft. N.C. Ceiling $18.50 
 
 20,165 ft. Flooring 28.50 
 
 3520 ft. Studding 10.80 
 
 12,500 Shingles 5.75 
 
 4. Make out bill and find the amount of the following invoice of 
 goods, shipped to Messrs. Chase and Witherspoon, Richmond, Ind., by 
 The Sprague Grocery Co., Chicago, 111., May 1, 19 — ; terms: 60 da. net, 
 3% 10 da. : 20 hf. chests Japan Tea, 1200 #, @ 23j2!; 20 hf. chests Oolong 
 Tea, 1000 #, @ 48^; 16 cases Ceylon Tea, 800 #, @ 35^; 10 bags Mocha 
 Coffee, 1500 #, @ 24^; 20 bags Java Coifee, 1500 #, @ 25^. 
 
 Find the amount of the following invoices : 
 
 5. N. E. Nash & Co., Chicago, sold Young & Hooper, Ft. Wayne, Ind., 
 135 bbl. Mess Pork @ $11.50; 40 bbl. Mess Beef @ $11.75; SOtigtcfiS;,^^^^ 
 R'fd Lard, 10,500 #, @^,f\ 450 Sugar-cured Hams, 1500 #, (^,J:^^RY^ 
 
48 
 
 ELEMENTS OF BUSINESS ARITHMETIC 
 
 6. A. D. Cunningham bought of C. E. Kinsley on account 30 da., 
 24 pr. Congress Shoes @ .|1.60; 25 pr. Men's Heavy Shoes @ $1.75; 
 22 pr. Ladies' French Kid Shoes @ $3.25 ; 12 pr. Boys' Shoes @ $.80; 14 
 pr. Infants' Shoes @ $.90; 22 pr. Calf Boots @ $2.25. Make out the 
 bill, supplying dates and places, and take off | for cash. 
 
 7. 5 Brass Bedsteads No. 200 @ $30; 6 rolls Wilton Carpet, 280 yd., 
 @ 80^; 4 Hair Mattresses, Style BB, @ $12.50; 16 rolls Ingrain Carpet, 
 1256 yd., @ 40 ^ ; 8 rolls Moquette, 432 yd., @ 85 )^ ; 12 pr. Portieres @ $ 8 ; 
 5 Antique Oak Bedsteads @ $15.20; discount ^V 
 
 New York, N. Y., ~^6/l^ 
 
 7-f7, 
 
 _19^= 
 
 — 
 
 
 /t^^f^^^^^--p-?^.-^ /^^.^ 
 
 
 
 ^'^ R H. MACY ca. COMPANY 
 
 Terms; ^.^.d^-A— 255 Broadway 
 
 
 
 S 
 
 ^2l^^^<P^N-^.^7^^d^ 7Z.J/^ Z3 4 
 
 /^ 
 
 „r^ 
 
 
 
 
 Z 
 
 >^v^^^^^y^-^>^ /r. .. ^r. 
 
 ,3 7 
 
 .To 
 
 
 
 
 7- 
 
 r.^^^^ — '/.^f;-?^^f:7jt?h-^^^~ tc^^ /. ^' 
 
 7 
 
 ? 
 
 
 
 
 ,? 
 
 JJJ. .J^^^/?^.w^J^^Z^.^ "" ^.^ 
 
 /a 
 
 ,r^ 
 
 
 
 
 
 
 
 
 
 
 
 
 ^^^^^^^A-^..,:^Sr/^^^i^^^^.^^ 
 
 
 
 
 
 
 
 -^^<A^^^^. ^^^ 
 
 
 
 
 
 
 
 ^y^^-i^-/y/7?7 
 
 
 
 
 
 
 
 
 
 
 
 
 Render the following statements: 
 
 9. Jan. 31, 19 — , the debits and credits of John C. Scott in account 
 with the Simmons Hardware Co., of St. Louis, Mo., were as follows : 
 
 Debits : Jan. 1, To account rendered, $ 157.25 
 Jan. 7, To Mdse. $32 
 Jan. 12, ToMdse. $345 
 
 Credits : Jan. 5, By cash, $ 150 
 
 Jan. 10, By 20-day note, $60 
 
 10. Oct. 31, 19—, the debits and credits of W. C. McClure with Lord 
 & Taylor, New York City, were : 
 
FRACTIONAL PARTS 
 
 49 
 
 Debits : Oct. 1, To account rendered, $86.25 
 
 Oct. 10, To Mdse., $ 165 
 
 Oct. 19, ToMdse., $425 
 
 Oct. 29, To Mdse., $ 136.50 
 Credits : Oct. 5, By cash, $ 80 
 
 Oct. 20, By note for 60 days for balance due 
 
 11. On Nov. 30, 19 — , the account of Wm. E. Bacon with American 
 Prism Co., Cincinnati, O., was as follows : 
 
 Debits : Nov. 2, To Mdse., $ 109.60 
 Nov. 8, To Mdse., $67.80 
 Nov. 21, To Mdse., $240 
 Credits : Nov. 15, By cash, $ 80 
 
 Nov. 20, By 10-day note, $75 
 Nov. 29, By cash, $ 200 
 
 37. Pay Rolls. 
 
 PAY ROLLi 
 
 
 
 
 
 For the week ending _ 
 
 
 
 19 
 
 N. 
 
 Number of H<M>r. Work 
 E.ch D., 
 
 Told 
 No. oi 
 Hours 
 
 per 
 Hour 
 
 Toial 
 W^ei 
 
 Rco-rk. 
 
 M 
 
 T 
 
 w. 
 
 T. 
 
 F 
 
 s. 
 
 . ^-^-^2^. ^ 
 
 r 
 
 f 
 
 r 
 
 ^ 
 
 
 ^ 
 
 ^f 
 
 X^l 
 
 /? 
 
 
 
 2 ^^-^...7>.y 
 
 <^'/^ 
 
 
 T'A 
 
 
 
 r'/, 
 
 ur'A 
 
 .^09 
 
 /// 
 
 <:< 
 
 
 3 iCP'OfOh.^J^ 
 
 ^ 
 
 ■r 
 
 r 
 
 ^'/„ 
 
 
 r 
 
 
 7//-^ 
 
 
 
 
 4 ^(;^.-^i&.-r>nf.A^ 
 
 r 
 
 r 
 
 r 
 
 )(• 
 
 
 f 
 
 
 x.r<^ 
 
 
 
 
 s -^y/Jh/h.^^^ 
 
 ,P 
 
 f 
 
 -r 
 
 f 
 
 -r'/, 
 
 r 
 
 
 3S<^ 
 
 
 
 
 6 ^^^^^.A,-^^ 
 
 ^ 
 
 . 
 
 e% 
 
 ^ 
 
 f 
 
 r 
 
 
 'C'A 
 
 
 
 
 r 'Ofy^%'^.y.^^ 
 
 ^ 
 
 r 
 
 r'A 
 
 ^ 
 
 r'A 
 
 rf 
 
 
 yp/. 
 
 
 
 
 8 0-^::^:^y.^ 
 
 ^ 
 
 r 
 
 /f 
 
 f 
 
 ^'/, 
 
 
 
 ZJ-</' 
 
 
 
 
 9 ""0^:^^^^^... 
 
 a- 
 
 
 a 
 
 rf 
 
 ^ 
 
 n"/. 
 
 
 ^•7^M 
 
 
 
 
 10 (2^^-\d^^.y^^y 
 
 jp 
 
 ^ 
 
 
 ^'A 
 
 
 
 
 ?f7|/. 
 
 
 
 
 ^ J J^ - 
 
 
 
 
 
 
 
 
 
 
 
 
 
 _ 
 
 
 
 
 
 
 
 
 
 
 
 
 Copy the above pay roll and fill out the blank spaces as 
 shown in Nos. 1 and 2. 
 
 Some firms use checks in paying off employes, but many 
 find it more convenient to pay in cash by the envelope 
 system. To pay off men by the envelope system, it is neces- 
 sary to find first the amount of money needed, then the bills 
 and fractional currency needed. To do this, a blank called 
 
50 
 
 ELEMENTS OF BUSINESS ARITHMETIC 
 
 a change memorandum^ similar to the one shown below, is 
 generally used. 
 
 Fill out the accompanying change memorandum for the 
 pay roll given above. Make totals and check results. 
 
 No. 
 
 Bills 
 
 Coins 
 
 
 $20 
 
 $10 
 
 $5 
 
 $2 
 
 $1 
 
 50 j^ 
 
 25^ 
 
 nf 
 
 5^ 
 
 1^ 
 
 1 
 
 2 
 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 10 
 
 
 1 
 1 
 
 
 1 
 
 2 
 
 
 1 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 FIRST NATIONAL BANK 
 
 Union City, Mo. 
 
 Pay-roll Memorandum 
 
 Union City Mfg. Co. require the 
 
 following : 
 
 Pennies ... 8 .08 
 
 Nickles 
 
 
 
 6 .30 
 
 Dimes . 
 
 
 
 7 .70 
 
 Quarters 
 
 
 
 8 2.00 
 
 Halves . 
 
 
 
 6 3.00 
 
 Dollars 
 
 
 
 4 4.00 
 
 2'8 . . 
 
 
 
 7 14.00 
 
 5's . . 
 
 
 
 3 15.00 
 
 lO's . . 
 
 
 
 9 90.00 
 
 20'8 . . 
 
 
 
 1 20.00 
 
 149.08 
 
 When the amount of the pay 
 roll and the necessary bills and 
 fractional currency have been 
 determined, a pay-roll memo- 
 randum^ similar to the accom- 
 panying form, is taken to the 
 bank and the necessary amount 
 of each denomination of money 
 secured. The pay-roll memo- 
 randum should foot the same as 
 the pay-roll book. 
 
 Make out a pay-roll memo- 
 randum for the preceding pay 
 roll, using the ajCcompanying 
 form as a model. 
 
FRACTIONAL PARTS 51 
 
 PROBLEMS 
 
 1. A man worked 9 months, 26 days per month, and received $ 912.60. 
 What was his daily wage ? 
 
 2. I hold $ 2000 worth of stock in a company capitalized at $ 36,000. 
 A dividend of $ 4200 is declared. What is my share ? 
 
 3. I have $1200 on interest at 5%. What does it yield annually? 
 (5%=^.) 
 
 4. Load deliveries of coal at a school building for the month of Feb- 
 ruary were as follows: 4500, 4650, 3850, 4000, 3675, 3800, 4200, 4350, 
 4500, 4100, 4350, 4400, 4050, 3900, 3600, 3550, and 3850 lb. net. Find 
 the total delivery, the average weight per load, the total cost at $ 4.20 
 per ton, and the average cost per load. 
 
 5. A society, after raising $ 17,675 by subscription, erected a building 
 that cost $ 22,500. How much did each of its 25 members have to pay 
 on a per capita assessment ? 
 
 6. In a woolen factory there are 48 looms. If in 208 days they wove 
 269,568 yd. of cloth, how much was woven on each loom ? How much 
 on all in one day ? On each in one day ? 
 
 7. A shoe dealer bouoht 35 cases of boots, containing 12 pairs each. 
 For the lot he paid |2100. How much did they cost per case? Per 
 pair? 
 
 8. The entire cost of constructing a railroad 25 miles long was 
 $ 1,736,075. What was the cost per mile ? 
 
 9. The population of Rhode Island in 1900 was 428,556. The area is 
 1250 sq. mi. What was the number of inhabitants to the square mile ? 
 
 10. A schoolroom containing 72,500 cu. ft. of air is occupied by 125 
 pupils. How much air is that to each pupil? If the air is completely 
 changed every 10 minutes, how much fresh air is provided for each J)upil 
 per hour? 
 
 11. A county containing 488,000 acres is divided into 27 townships 
 of equal area. How many acres in each township ? 
 
 12. The population of the United States in 1900 was 74,627,907. The 
 number of congressmen was 357. What was the average number of 
 persons represented by each congressman ? 
 
 13. An agent earns monthly commission as follows : $ 75, $ 92, $ 83, 
 $ 66, $ 55, $ 28, ^ 82, 1 160, $ 32, $ 280, $ 175, $ 196, and $ 215. What is 
 his average commission per month ? 
 
52 
 
 ELEMENTS OF BUSINESS ARITHMETIC 
 
 14. In one year the national banks of New York City loaned 
 $ 236,327,598. What was the average weekly loan ? 
 
 15. The total cost of building the Union Pacific Railway was 
 $ 18,941,400. The road is 1177 miles long. What was the cost per mile ? 
 
 16. A man bought a house and lot for $6107. He paid $1392 down 
 and agreed to pay the balance in 115 monthly payments. What was the 
 amount of each payment ? 
 
 17. If a gas burner consumes 1100 cu. ft. of gas in 160 hours at a 
 cost of $1.42 per thousand, what is the cost per hour? How many 
 cubic feet does it burn in one hour ? 
 
 18. How much will it cost to ship a carload of corn containing 
 45,000 lb. from Crete, Neb., to Chicago, if the freight rate is 6^ per hun- 
 dred pounds? What would be the cost per bushel of 56 lb. ? 
 
 19. The average wages of a steel mill employing 4500 men is $ 3.50 
 per day. If a reduction of ^ is made in wages, how much is the com- 
 pany's pay roll reduced? 
 
 20. Find the amount of the following invoice of goods : 
 
 10 pc. sateen, 55|, 51, 50|, 52, 57, 54^, 50, 56-1, 5.31, 51 @ 6| ^ ; 11 pc. 
 flannel, 62^, 60, 65, 67|-, 65, 64, 63^, 67, 78, 62|, 63 @ 6| ^ ; 15 pc. ging- 
 ham, 50, 51, 52, 54^, 56, 57, 58|, 54|, 59, 60, 61, 56i, 52^, 53|, 54 @ 12^;^. 
 
 21. Find the amount of the following pay roll (a) on the basis of an 
 8-hour day ; (6) on the basis of a 10-hour day : 
 
 Name 
 
 MON. 
 
 TuES. 
 
 Wed. 
 
 Tm-RS. 
 
 Fri. 
 
 Sat. 
 
 Eate 
 
 Lawrence Dodd 
 
 7 
 
 8 
 
 10 
 
 9 
 
 8 
 
 11 
 
 $3.00 
 
 Henry Wahl 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 10 
 
 2.50 
 
 James Ringo 
 
 6 
 
 7 
 
 5 
 
 8 
 
 9 
 
 1 
 
 3.50 
 
 Leroy Hines 
 
 9 
 
 6 
 
 8 
 
 10 
 
 12 
 
 12 
 
 3.00 
 
 Thomas Cox 
 
 8 
 
 8 
 
 8 
 
 9 
 
 9 
 
 10 
 
 1.75 
 
 Amos Zeering 
 
 9 
 
 9 
 
 9 
 
 9 
 
 9 
 
 8 
 
 2.50 
 
 Adam Frick 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 3.00 
 
 Warren Hastings 
 
 9 
 
 10 
 
 10 
 
 10 
 
 11 
 
 12 
 
 2.25 
 
 Gordon Bang 
 
 7 
 
 6 
 
 5 
 
 6 
 
 8 
 
 8 
 
 3.00 
 
 Emil Ehman 
 
 8 
 
 8 
 
 9 
 
 5 
 
 6 
 
 5 
 
 4.00 
 
 22. Make pay-roll memorandum and change memorandum for above 
 problem. 
 
FRACTIONS 
 
 38. Definitions. By fraction is meant one or more of the 
 equal parts of some quantity. Tlie size of a part is indicated 
 by the denominator in showing the number of parts into 
 which the wliole is divided. Inasmuch as fourths and halves 
 are as truly different things as bushels and pecks, it follows 
 that the use of the denominator is to denominate or name 
 the kind of parts. The number of parts used is indicated by 
 the numerator (numberer). Thus, in | and |, and in 5 bu. 
 and 3 pk., the denominators 8 and 7 and the words bushels 
 and pecks indicate the kind of parts or units, while 5 and 3, 
 in each case, show the number of those units. The numerator 
 and denominator are called the terms of the fraction. 
 
 By common or vulgar fractions are meant fractions of 
 ordinary kind, i.e. fractions expressed by a numerator written 
 above a line and a denominator written below the line. A 
 common fraction whose denominator is 1, with a number of 
 ciphers annexed, is called a decimal fraction, because its 
 denominator expresses a decimal denomination (Sec. 16). 
 Such a fraction may be written at pleasure as a decimal 
 fraction or as a decimal. (Sec. 17.) 
 
 A fraction which represents a part of a quantity less 
 than unity is called a proper fraction; if it is equal to or. 
 greater than unity, it is improper. The numerator of a 
 proper fraction is numerically less than the denominator; 
 the numerator of an improper fraction is equal to or greater 
 than the denominator. A mixed number contains a whole 
 number and a fraction. 
 
 53 
 
54 ELEMENTS OF BUSINESS ARITHMETIC 
 
 Business computations require constant use of fractions. 
 A practical knowledge of fractions, however, necessitates 
 only a thorough mastery of operations with the simpler 
 fractions, and skill in their use. Little attention need be 
 given to fractions having larger terms. 
 
 39. Reduction of Fractions. The same fraction may be 
 expressed in different denominations. Thus, ^ may be 
 written as | or |. 
 
 A change in denomination requires a change in the de- 
 nominator. If the value of the fraction is not to be changed, 
 then the numerator must also be changed and in exactly the 
 same ratio. In other words, the terms of a fraction may he 
 multiplied or divided hy the same number without changing the 
 value of the fraction. 
 
 When the change results in terms of less numerical value, 
 the fraction has been simplified or reduced to lower terms. 
 
 It should be remembered that one may add, subtract, or 
 divide like quantities only. Three quarts may not be added 
 to five pints until they are changed to the same denomina- 
 tion. Neither is it possible to add | and | or to find how 
 many |'s there are in | without reducing to like denomina- 
 tion. In division of fractions this step is obscured by the 
 commonly used process of inversion, but it is nevertheless 
 performed. The recognition of these facts and their con- 
 stant application will do much to simplify the whole subject 
 of fractions and give facility in computations involving them. 
 
 Simplify : 
 
 • 1. m 7. Itl 13. ^\% 19. ^^ 25. ^^, 
 
 2. tVt 8. ^%% 14. IM 20. lU 26. i|f 
 
 3. H 9- n 15- .Vr . 21. m 27. f|« 
 
 4. iU 10. U 16. m 22. ^^\ 28. lit 
 
 5. M 11. U 17. i^, 23. ^^^ 29. Hf 
 
 6. j^^ 12. 11} 18. ,^ 24. ^f 30. ill 
 
FRACTIONS 55 
 
 40. Reduction to Like Denomination by Inspection. Chang- 
 ing fractions to fractions of like denomination is called 
 reducing to a common denominator. Fractions of like de- 
 nominators are similar fractions. The practice problems 
 here given deal with the more commonly used fractions and 
 are intended to develop the power of readily thinking one 
 fraction in terms of another; e.g. |, l, ^, as -f^, -f-^^ and -f^. 
 This is one of the operations in fractions most frequently 
 needed. 
 
 DRILLS 
 
 1. I = how many 4ths, Gths, lOths, 16ths, 18ths, 24tlis, 28ths, 36ths, 
 72ds, 84ths, 60ths, 56ths, 42ds, oOths, 44ths, 70ths? 
 
 2. i = how many 9ths, 15ths, 21sts, 24ths, 27ths, 36ths, 45ths, 48ths, 
 51sts, 54ths, 63ds, 72ds? 
 
 3. ^ = how many 8ths, 12ths, 16ths, 24ths, 28ths, 32ds, 36ths, 48ths, 
 52ds, 56ths, 60ths ? 
 
 4. ^ = how many 15ths, 25ths, 30ths, 35ths, 45ths, 50ths, 55ths, 65ths ? 
 
 5. ^ = how many 12ths, 18ths, 24ths, 30ths, 36ths, 42ds, 48ths, 
 54ths, 60ths, 66ths, 72ds? 
 
 6. i == how many 16ths, 24ths, 32ds, 40ths, 48ths, 56ths, 64ths, 
 72ds? 
 
 7. f = how many 14ths, 21sts, 28ths, 35ths, 42ds, 49ths, 56ths, 
 63ds, 70ths? 
 
 8. ^-^ = how many 20ths, 30ths, 40ths, 50ths, 60ths, 70ths? 
 
 9. ^ = how many ISths, 27ths, 36ths, 45ths, 54ths, 63ds? 
 
 10. fr = how many 22ds, 33ds, 44ths, 55ths, 66ths, 77ths? 
 
 11. j\ = how many 24ths, 36ths, 48ths, 60ths, 72ds ? 
 
 12. fj = how many 26ths, 39ths, 52ds, 65ths, 78ths? 
 
 13. j\ = how many 28ths, 42ds, 56ths, 70ths? 
 
 14. ^3 = how many 30ths, 45ths, 60ths, 75ths? 
 
 15. 2V = tow many 40ths, 60ths, 80ths? 
 
 16. From comparison with their equivalents, as required in the above 
 problems, to what could one change or reduce halves, fourths, and eighths 
 so that they would be of the same units or kinds of parts ? What halves, 
 
56 ELEMENTS OF BUSINESS ARITHMETIC 
 
 fourths, eighths, and sixths ? What halves, fourths, eighths, sixths, and 
 twelfths? 
 
 17. Reduce thirds, sixths, and ninths to common denominators. 3ds, 
 6ths, 12ths, and 15ths. 3ds, 6ths, 9ths, and 12ths. 
 
 41. Operations with Easily Reducible Fractions. 
 
 1. $5 + S3 +12 = $10. 23bu. + 15bu.4-12bu. = 50bu. 
 
 Likewise ^ + ^ + 1 + ^= 2^3.^ or 5|. 
 
 2. $56-112 = 144. 24 pk. - 19 pk. = 5 pk. 
 
 Likewise | — -| = |, or -|. 
 
 3. 181^19 = 9. 72bu. -f-6bu. = 12. 
 
 Likewise §J -5- ^\ = 24 -- 3, or 8. 
 Take note that in the division of fractions having the same 
 denominator^ the denominators are disregarded^ and the quotient 
 of the fractions is the quotient of the numerators. 
 
 Reduce to Similar Fractions by Inspection and Add: 
 
 1. i 1, h h ^ 
 
 
 
 
 5. 
 
 h h t\, t\ 
 
 
 
 2. h h A, 
 
 It\ 
 
 
 
 
 6. 
 
 4J A) TSf ^i 
 
 
 
 3. 1,1, A, 
 
 hj\ 
 
 
 
 
 7. 
 
 h ^% h If 
 
 
 
 4. f,>,T\, 
 
 A 
 
 
 
 
 8. 
 
 2) 6J 3, 9 
 
 
 
 Subtract 
 
 : 
 
 
 
 
 
 
 
 
 1. A-^ 
 
 
 5. 
 
 f- 
 
 ■f 
 
 • 
 
 9. 
 
 15^ 
 
 -n 
 
 2. f-f 
 
 
 6. 
 
 tV- 
 
 -1% 
 
 
 10. 
 
 16|. 
 
 -5| 
 
 3. l-i 
 
 
 7. 
 
 j% 
 
 -z\ 
 
 
 11. 
 
 163- 
 
 -98f 
 
 4. i-T^B 
 
 
 8. 
 
 i- 
 
 -ih 
 
 
 12. 
 
 .85- 
 
 -14| 
 
 Divide : 
 
 Note. — All easily reducible fractions are more quickly and accurately 
 divided by reducing to similar fractions. In practice, write the numera- 
 tors only. 
 
 
 
 i-^T%=^%-^ 
 
 •if = 
 
 = T^ir 
 
 
 
 1. 
 
 f^f 
 
 5. U-^ii^ 
 
 
 
 9. 
 
 16^ -- 4 J 
 
 2. 
 
 ^^i 
 
 6. !|-^t\ 
 
 
 
 10. 
 
 f-f-41 
 
 3. 
 
 tV-A 
 
 7. 1^1 
 
 
 
 11. 
 
 61-^ H 
 
 4. 
 
 f-^^ 
 
 8. A^f 
 
 
 
 12. 
 
 t¥t-^t'j 
 
2)^-9-^-j[^-7-14-^-12 
 3 ) 9 7 6 
 
 3 7 2 
 
 FRACTIONS 57 
 
 42. Addition and Subtraction through finding the Least 
 Common Multiple. For fractions not easily reducible to a 
 common denominator by inspection, the method of finding 
 the Least Common Multiple of all the denominators is used 
 in order to know the least denomination to which all of the 
 fractions may be reduced. 
 
 Problem. —Add f, ^, 8f , f^, 5f, ^^, 7|, f|-. 
 In order to find the Least Common Multiple, it is usual to 
 write all the denominators, then strike out any one that is 
 
 repeated or that is a factor 
 of any other; as, 3, 6, 14, 
 7, and 2. The remaining 
 numbers are then divided 
 2x3x3x7 X 2 = 252, L.C.M. y^^ ^„y divisor of two or 
 
 more of them, bringing down any number not divisible. 
 This is repeated, if necessary, until no two of the numbers 
 remaining have a common divisor. The product of the 
 divisors and all of the numbers remaining will be the Least 
 Common Multiple of the given numbers. 
 
 43. To reduce to a Common Denominator. Since 252 is a 
 multiple of each of the denominators of the fractions given 
 in the preceding section, each fraction may now be reduced 
 
 168 to an equivalent fraction having 252 for its 
 
 112 denominator. To do this, it is well to arrange 
 
 210 the fractions vertically, placing the desired 
 
 90 denominator below a line and directly beneath 
 
 216 the fractions. To the right of a line drawn 
 
 162 vertically, and opposite each fraction, is written 
 
 126 the numerator of the equivalent fraction. The 
 
 231 method used in finding this equivalent fraction 
 
 3 
 
 of 
 
 71 
 11 
 
 12 
 
 252jl816(5 is the same as by inspection. (Sec. 40.) 
 
 1260 i = ^AO'«-*of |f|),andf = i|f. J=AV 
 
 ii^ and| = Ji|. H^jandf=e|,etc. Add- 
 
58 ELEMENTS OF BUSINESS ARITHMETIC 
 
 20 
 
 ing the numerators, the sum of the fractions 
 is seen to be ^gV^ ^^ ^ ^^^ /A* Added 
 to 20, the sum of the whole numbers, the 
 sum required is found to be 25 and ^-^. 
 
 PROBLEMS 
 
 1. Add 7, 8f, 9i^, 6j\, 5t\, and 3^. 
 
 2. Add i I, 5|, 12^23, and xV- 
 
 3. Subtract |f from 7||^. 
 
 4. Subtract yff ^ from 256ff. 
 
 5. From If + llf take 12\ - 9|. 
 
 6. From the sum of 75| and 94f take the sum of 36^^ and 24|. 
 
 44. Multiplication of Fractions and Whole Numbers. 
 
 (a) Five times 3 bu. = ? Five times 3 fourths = ? Five 
 times f=? |x5=? How many wholes ? (Use multi- 
 plication sign correctly. Sec. 13.) 
 
 5 times | = J^ or 3|. 
 
 (6) 5 X f = ? This may also be written, | of 5 = ? In 
 the latter form it is recognized as finding the value of frac- 
 tional parts. See Chapter IV. 
 
 I of 5 = f X 3 = -I45. or 3f . 
 
 A fraction may be multiplied hy multiplying its numerator. 
 Dividing the denominator also multiplies the fraction by 
 increasing the size of the denomination. A fraction may be 
 used as a multiplier hy dividing hy its denominator and multi- 
 plying the quotient hy the numerator. This process may some- 
 times be shortened by first multiplying by the numerator 
 and then dividing by the denominator. 
 
 In multiplying mixed numbers, the whole numbers and 
 fractions should be multiplied separately and the partial 
 products added. Solve each problem in the shortest possible 
 way. 
 
FRACTIONS 59 
 
 PROBLEMS 
 
 1. 348 X f = ? 5. 284 x 3f = ? 
 
 2. tV X 8 = ? 6. 71f X 3 = ? 
 
 3. f of 721 = ? 7. 256 X I = ? 
 
 4. 12f X 8 = ? 8. 5xV X 9. 
 
 45. Multiplication of Fractions. 
 Solve : 
 
 1. f of,^. 
 
 Since -J of -^^ is -^^ f is f ^ or f . 
 
 2. Find I of -|. 
 
 Since J of |- is -j%, f is j-^^. 
 
 Note. — i x f = ^\, and § x | = y^^ ; that is, the product 
 of the numerators is divided by the product of the de- 
 nominators. 
 
 3. Find I off. 
 
 Since i of f (I X i) = l | of f = f or i 
 
 These steps may be performed together and shortened by 
 writing the fractions, canceling the factors common to the 
 numerators and denominators, and multiplying the remain- 
 ing factors in the numerators for the numerator of the prod- 
 uct, and the remaining factors in the denominators for the 
 denominator of the product. Thus, 
 
 2x^-1 and 1 of ?^-_22ll-12^ 
 
 2 5 13 
 
 4. Multiply 5| by 6f . 
 
 Mixed numbers should ordinarily be reduced to fractions, 
 then multiplied as in 3. Thus, 
 
 9 
 
 .2 ^ns 17 ^ ;27 17x9 153 _ oo. 
 5|x6| = — x^ = -^- = — or 38^. 
 
60 ELEMENTS OF BUSINESS ARITHMETIC 
 
 5. Multiply 144| by 9f . 
 
 Reducing a large number, like 144|, would give too large 
 a numerator to be handled readily. In such cases, the whole 
 number and the fraction of the multiplicand may be multi- 
 plied separately by the whole number and the fraction of 
 the multiplier, and the partial products added. Thus, 
 
 
 
 144| 
 
 
 
 n 
 
 
 
 A = (fx|) 
 
 
 
 108 = (f X 144) 
 
 
 
 3i^o=(fx9) 
 
 
 
 1296 =(144x9) 
 
 
 
 1407^ 
 
 
 
 PROBLEMS 
 
 1. 
 
 |xf xf=? 
 
 6. 2i X 3| = ? 
 
 2. 
 
 |x|xf=? 
 
 7. f X If = ? 
 
 3. 
 
 f X A X ^\ = ? 
 
 8. ^ xi\ = i 
 
 4. 
 
 T% X f X f X 1 =: 
 
 = ? 9. T5 of 2^ X i of 7} 
 
 5. I of f of I of I = ? 10. 9f X 6| = ? 
 
 11. A man owns f of a store and sells | of his share for ^5000. 
 What was the store worth, at this rate ? 
 
 Find cost of : 
 
 12. 127 bu. wheat at 62^^ per bushel. 
 
 13. 125 cords of wood at $ 6| per cord. 
 
 14. 17f tons coal at $ 3| per ton. 
 
 15. 26 bu. clover seed at $ 7^ per bushel. 
 
 16. 22f thousand feet lumber at $17^ per M. 
 
 17. A merchant sold 25 yd. satin at $ If per yard, 26| yd. silk at $2f, 
 26 yd. carpet at $|, 56 yd. calico at 3|/'. What was the amount of the 
 sale? 
 
 18. Having bought f of a farm of 180 acres, I sell f of my share at 
 $40 per acre. How much do I receive for it? 
 
 19. A quantity of provisions will last 25 men 12| days. How long 
 will it last one man ? 
 
FRACTIONS 61 
 
 46. Division of Fractions. Division of fractions which are 
 not easily solved by the method in Sec. 41 is most readily 
 performed by the method of inversion. In the latter, the 
 terms of the divisor are inverted and the fractions are mul- 
 tiplied. Thus, 
 
 l^i = ixf = f (Sec.45.) 
 
 As stated in Sec. 39, we cannot divide fractions without 
 first reducing them to like denominations. In the division, 
 then, of dissimilar fractions the two steps are : first, reduc- 
 tion to like denomination, and second, the division of the 
 numerators. These two steps are accomplished simul- 
 taneously by the inversion of the terms of the divisor and 
 multiplying, thereby shortening the written process. Thus, 
 placing the denominator, 6, above the line, forms, with the 
 denominator of the dividend, | or 2. This is the number 
 necessary by which to multiply the numerator and denomi- 
 nator of the fraction (|) to change it to sixths, that it may 
 be of like units with the divisor (|). If this multiplication 
 were actually made, the J would now be |. The placing of 
 the numerator of the dividend, 5, below the line indicates 
 the division of 4 (the numerator of the dividend) by it, or 
 performs the second step in the operation. This gives, as 
 the result, -I, the required result, since the numerators only 
 are to be divided when the fractions are similar. (Sec. 41.) 
 The ease with which inversion and simple multiplication ac- 
 complishes these steps as | x f = f , explains the economy in 
 its use for all problems not mentally reducible to a common 
 denominator. 
 
 Note. — The example used in the above illustration could be more 
 readily solved by mentally reducing both fractions to sixths, when the 
 quotient would be readily seen to be f. The method of inversion is only 
 shorter when the fractions have much larger terms, or are not readily 
 reducible. 
 
 A fraction may be divided by a whole number by dividing the numera- 
 
62 ELEMENTS OF BUSINESS ARITHMETIC 
 
 tor, or by multiplying the denominator. The latter divides by decreas- 
 ing the size of the denominator. A whole number may be divided by 
 a fraction by reducing it to a similar fraction and dividing the 
 numerators. 
 
 Divide: 
 
 
 
 
 1. T^byf. 3. ^by^^^. 5. 
 
 fibyxir. 
 
 7. tfbyf. 
 
 9. HbyH. 
 
 2. Ifbyf. 4. ilbyf. 6. 
 
 H by f ?. 
 
 8. fby^. 
 
 10. ^byl6. 
 
 11. tfby^f. 13. 
 
 If by 14. 
 
 15. 
 
 8bys. 
 
 12. xfirbyM. 14. 
 
 1 by 13. 
 
 16. 
 
 213 by H. 
 
 47. Decimal Fractions. By definition (Sec. 38), a deci- 
 mal fraction is a common fraction whose denominator is 1 
 with ciphers annexed. Thus, ^y^^, loVW ^^^ -^^ are 
 decimal fractions. 
 
 These decimal fractions may be written without the de- 
 nominator (Sec. 18), by pointing off from the right of the 
 numerator one less than the number of digits in the denomi- 
 nator. This would always be equal to the number of ciphers 
 in the denominator. The above decimal fractions would 
 thus be written as decimals : .105, .0017, and 210.5. 
 
 The position of the decimal point in a decimal always indi- 
 cates, then, the number of ciphers at the right of the 1 in 
 the denominator, if written as a decimal fraction. Decimals 
 may thus be changed to common fractions by writing as 
 decimal fractions, and then reducing, if desired, to lower 
 terms. 
 
 1. Write as decimals : ^^, jiUh^ j^U^ f |^, and m%. 
 
 2. Write as decimal fractions : 5.625, .00052, 1.1, and 32.0225. 
 
 3. Change to decimal fractions and reduce to lower terms : .125, .00875, 
 .0625, 75.015, and 300.05 
 
 48. Changing Fractions to Decimals. 
 
 (a) When the denominator of the required fraction is 
 known. 
 
FRACTIONS 63 
 
 Example. -^ = how many thousandths ? 
 
 A 
 
 = A of \m = -M^ thousandths or .ISTJ. 
 
 Ordinary business problems do not require the use of deci- 
 mals beyond the thousandths. Practice in reduction from 
 fractions to decimals should largely be, then, to acquire facil- 
 ity in changing to tenths, hundredths, and thousandths, and 
 in writing at sight the decimal form of the simpler fractions. 
 After the method of reduction is understood, these decimal 
 equivalents should be drilled upon until they can be written 
 at sight. 
 
 (5) When merely a change of form is desired. 
 
 Example. Reduce -f^ to a decimal. 
 16 )3.0000 
 .1875 
 This is the method of reduction usually given. It consists 
 of annexing decimal ciphers to the numerator, and dividing 
 by the denominator. The business necessities for the use 
 of this method are not frequent. It should, however, be 
 thoroughly understood. 
 
 Change to Decimals: 
 
 
 
 To hundredths. 
 
 To thousandths. 
 
 1. /^ 6. 1 
 
 1. jh 
 
 6. H 
 
 2. /^ 7. 1 
 
 2. zh 
 
 7. ^ 
 
 3. M 8. f 
 
 3. M 
 
 8. M 
 
 4- M 9- A 
 
 4. fa 
 
 9. H 
 
 5. tW 10. ^ 
 
 5. It 
 
 10. If 
 
 Reduce to Decimals: 
 
 Do not carry beyond five places. 
 
 1. A 6. i 
 
 2. li 7. ^^ 
 
 3. ^ 8. il 
 
 4. Yh 9. H ■ 
 
 5. ^2 10. 3f 
 
64 ELEMENTS OF BUSINESS ARITHMETIC 
 
 MISCELLANEOUS PROBLEMS IN FRACTIONS 
 
 1. An estate is divided among three sons so that the first gets ^, the 
 second ^j^, and the third the remainder, $3600. What is the amount of 
 the estate? 
 
 2. If a clerk spends | of his weekly salary for board, ^ for clothing, 
 and i for books and papers, and has left $ 4, what is his salary ? 
 
 3. If I sell a house for $3600, thereby gaining |, what was the cost 
 of the house ? 
 
 4. A real estate agent rents a house for $850, which is f of its cost. 
 What is its cost ? 
 
 5. A book dealer paid $21 for a set of books and sold them for $24. 
 The gain was what part of the cost? 
 
 6. A farmer sold two horses for $48 each. On one he lost f of the 
 cost, on the other he gained j of the cost. How much did he gain or 
 lose by the transaction ? 
 
 7. If 3 yd. of cloth cost 37^ ^, how much change would you receive 
 from a $5 bill if you buy 15 yd. ? 
 
 8. A grain dealer had $14,000. He spent f of it for wheat at 75 ^ a 
 bushel, and f of the remainder for oats at 30 j^^ a bushel. How many 
 bushels of each did he buy, and how much money had he left ? 
 
 9. A man spent i of his money for a suit, | of the remainder for a 
 shotgun, and has left $40. How much had he at first? 
 
 10. A locomotive runs | of a mile in f of a minute. At what rate 
 per hour does it run ? 
 
 11. The silver dollar weighs 412.5 grains ; ^ of its weight is alloy. 
 How many grains of pure silver are there in one dollar ? 
 
 12. Gold is 19.36 times as heavy as water; copper, 8.97; lead, 11.36. 
 Find the weight of a cubic foot of each, if a cubic foot of water weighs 
 62^ lb. 
 
 13. The total crop of cotton in the United States in a certain year 
 was 10,758,000 bales. Of this amount 6,482,849 bales were exported to 
 Europe. What fraction (decimal) of the crop was exported ? Retained 
 at home? 
 
 14. If a coal dealer gains ^ by selling coal for $8 a ton, how much 
 would he gain on a sale of 8.8 tons ? 
 
FRACTIONS 65 
 
 15. After paying $ 74.85 for mileage, ^ 37.50 for hotel bills, and $ 13.65 
 for sundry expenses, a traveling agent finds that he has expended | of 
 his money. How much had he at first ? 
 
 1^. pind-the value of a sheep which dressed as follows : 
 
 Leg 22.51b., 121;? 
 
 Loin 17.5 lb., O^f 
 
 Rib 14.8 lb., 9f^ 
 
 Chuck 19.6 1b., 2f J? 
 
 17«- The price of corn as quoted at the close of the market each day 
 was as follows: Monday, 58^^; Tuesday, 58|^; Wednesday, 56^;''; 
 Thursday, J^7f^ ; Friday, 57f ^ ; Saturday, 56|^. Find the average price 
 for the week. 
 
 18. A can do a piece of work in 4 days, B can do it in 5 days. How 
 long will it take them both to do it ? 
 
 19. A general store's sales of dry goods for a month amounted to 
 $ 6300, and f of the sales of dry goods was f the sales of groceries. What 
 was the sales of groceries ? 
 
 20. A bank teller received during the day $ 60,000 in silver and paper 
 money. There was f as much silver as paper money. How much of 
 each did he receive ? 
 
 21. An automobile cost $ 1600. If f the cost of the automobile is 4 
 times the cost of a carriage, what is the cost of a carriage? 
 
 22. A grocer bought eggs at the rate of 4 for 5 f and 6old them at the 
 rate of 5 for 9 ^. How much did he gain on each dozen ? 
 
 23. A piece of cloth is 20 yd. long and | yd. wide. How wide is 
 another piece which is 12 yd. long and contains as many square yards 
 as the first ? 
 
 24. If I pay $48 for a buggy after receiving a discount of |, and a 
 further discount of f of the latter price for cash, what was the asking 
 price? 
 
 25. A broker sold stocks at $ 82 and gained :^o- What would he have 
 gained or lost had he sold them a few days later when they were quoted 
 at $64? 
 
 26. If an ordinary gas burner consumes ^q cu. ft. of gas per second, 
 what would be the cost per hour to light a room with 50 burners at $1.25 
 per thousand cubic feet? If a Welsbach burner consumes ^ as much gas, 
 how much would be saved in a day of 6 hours by installing Welsbach 
 burners ? How much in a month of 30 days V 
 
 i 
 
66 ELEMENTS OF BUSINESS ARITHMETIC 
 
 27. A owned f of a store and sold | of his share to C. C sold 
 I of what he bought to B for $4000. At this rate, what was the store 
 worth ? 
 
 28. James and John hire a pasture for $ 35. James puts in 4 cows 
 and John puts in 3. What must each pay? 
 
 29. A merchant sold a quantity of coffee for $1280, and thereby 
 gained ^ of the cost. If he had sold it for $ 1000, would he have gained 
 or lost, and how much ? 
 
 30. The net profits of a business for two years were $6400. The 
 second year's profits were | greater than the first year's. What were the 
 profits each year ? 
 
 31. I paid $22,500 for two farms. If f of the cost of one is equal to 
 f of the cost of the other, what did I pay for each one ? 
 
 32. A merchant bought 300 crates of peaches at 87^^ a crate. He 
 sold I of them at an advance of 10^ a crate, | of the remainder at 80 j^ a 
 crate, and the remainder at a loss of 3^ ^ a crate. What did the mer- 
 chant gain or lose? 
 
 33. A man invested ^ of his money in bonds, ^ of it in real estate, 
 I in mining stock, and the balance, $3900, in bank stock. How much 
 did he have in all ? How much in each investment ? 
 
 34. A man spent f of his money for a house, invested ^ of the re- 
 mainder in stocks, and had $3200 left. How much had lie at first? 
 
 35. A tree fell, breaking in three pieces. The first wa^ f as long as 
 the second, and the third was I as long as the other two pieces. What 
 was the length of each piece, if the total length was 180 ft. ? 
 
 36. A speculator invested ^ of his money and $600 in land, ^ of his 
 money and $250 in bank stock, I of his money and $144 in bonds, and 
 the remainder, which was $1400, in a house and lot. What did he in- 
 vest in each kind of property ? 
 
 37. George's money is^'f/of James'. James' money is | of Clara's. 
 Clara's is 1^ times Daniel's. How much had each, if ^ of George's 
 money is $60? 
 
VI 
 
 MEASURES OF LENGTH 
 
 49. Measurement. Measurement of quantity enters so 
 largely into life that to make arithmetic really practical, our 
 concepts of the units of measurement should be very accu- 
 rate. 
 
 The first efforts of the learner should be toward the 
 building of accurate concepts of the various units, rather 
 than toward proficiency in repeating tables or in changing 
 to higher or lower units. 
 
 Note. — Pupils should have the actual measurement units present to 
 their senses. Do not make the mistake of talking of rods, miles, acres, 
 etc., without bringing these quantities actually before you. Use a tape 
 or string a rod long ; view and walk a mile, or view and walk around 
 an acre, etc. Practice judging length, extent, or weight, and test your 
 accuracy by measurement. 
 
 50. The Unit of Length. The unit for measuring length 
 is the yard. Formerly the unit for the United States was 
 the same as the English yard, but the law of 1893 made the 
 yard fffy of the international unit, the meter. A standard 
 yard is kept in the Bureau of Standards at Washington 
 (Sec. 127). 
 
 TABLE 
 
 There are: 12 inches ('^) in 1 foot ('). 
 3 feet in 1 yard (°). 
 5J yards in 1 rod. 
 320 rods in 1 mile. 
 
 1 mile = 5280 feet or 63,360 inches. 
 67 
 
68 ELEMENTS OF BUSINESS ARITHMETIC 
 
 51. Surveyor's Measure. The unit of land measure has 
 long been the Gunter's chain, 4 rods, or 6Q feet long. This 
 was divided into 100 parts or links, each link being 7.92 
 inches long. Eighty of these chains make a mile. 
 
 In civil engineering and at the customhouse, the inch and 
 foot are divided into tenths, hundredths, and thousandths, in 
 lieu of the usual subdivisions of halves, quarters, eighths, etc. 
 This is indicative of a tendency in all measurements toward 
 a larger use of decimal divisions. Some of the advantages 
 of a decimal system of measurement, the strongest argument 
 for the metric system, are thereby secured. 
 
 52. Nautical Units. Numerous special units have become 
 established by usage in particular vocations. Many of these 
 are of local use, others have varying values in different 
 localities, and still others have become or are becoming ob- 
 solete. A few nautical units of length or distance are here 
 given. 
 
 There are : 6 feet in 1 fathom. 
 
 1.15 statute miles in 1 geographic (some- 
 times called nautical) mile or knot. 
 
 3 geographic miles in 1 league. 
 
 60 geographic miles in 1 degree. 
 
 •69.16 statute miles in 1 degree of the earth's 
 equator. 
 
 360 degrees in 1 circumference of the earth. 
 
 CLASS EXERCISES 
 
 1. Mark off 1 foot in length on the board. Scan it carefully. 
 
 2. Mark off 1 foot without using the ruler. Test accuracy. 
 
 3. Draw a horizontal line i foot long ; a vertical line ; an oblique 
 line. Test them. 
 
 4. Estimate the number of feet in length of your desk top ; its width; 
 the length and width of the door; of the window. Verify your es- 
 timates by measuring. 
 
MEASURES OF LENGTH 69 
 
 5. Estimate the height of the room ; its width ; its length. Test. 
 
 6. Practice judging length of articles in the room, and of distances, 
 until you can estimate a foot with considerable accuracy. An inch. 
 A yard. 
 
 7. Carefully measure off a rod. Estimate length in rods until you 
 can do so accurately. 
 
 8. Make a chain, either Gunter's or 100-foot chain. In company with 
 another pui^il, if it can be arranged after school hours, measure off a 
 mile. Mark its limit by some signal, so that the eye may judge the dis- 
 tance. Walk the mile. Estimate distances in miles and, if possible, 
 verify estimates. 
 
 PROBLEMS 
 
 1. How many inches in 7 ft.? In 3 yd. ? In a rod? 
 
 2. How many feet in 90 in. ? In 4 rd. ? In a half mile ? 
 
 3. A ship travels 18 knots per hour. How many miles does she travel 
 in 6 hours ? 
 
 4. A garden is 90 ft. square. How many yards of fence will it take 
 to inclose it? 
 
 5. A well is 25 yd. deep. What will be the cost of a pump stock, 
 that reaches to the bottom, at 8/^ per foot? 
 
 6. If potato rows are 3 ft. apart, how many rows are there in a lot 
 4 rd. wide ? 
 
 7. A gardener has a bed 16 ft. long and 6 ft. wide. He wishes to 
 have 4 rows of plants, 6 in. apart, in the row. How many plants will it 
 take ? How far apart are the rows ? 
 
 8. How many miles does a boy ride in a month of 26 days, if he 
 rides 342 rd. to and from work each day ? 
 
 9. If a man digs a ditch 3 rd. long in a day, how long will it take 
 him at the same rate to dig a ditch I mi. long? 
 
 10. If railroad ties are laid 18 in. apart (from center to center), how 
 many ties will it take to lay a mile of track? What will they cost at 
 75 f apiece ? 
 
 11. What will it cost to place a hedge around a lot, the distance around 
 which is 72 rd., if the plants are placed 6 in. apart and cost |4.25 per 
 hundred ? 
 
70 ELEMENTS OF BUSINESS ARITHMETIC 
 
 12. The distance between two cities is 12 mi. What will the wire 
 to build a telephone line cost at «|2.90 per hundredweight, if it weighs 
 l^lb. to the rod? 
 
 13. A barn roof is 84 ft. long. What will it cost to place an eave- 
 trough along two sides, if the trough costs 6d^ per 10 ft. length? 
 
 14. The length of a rectangular lot is 100 yd., its width 40 yd. 
 What will it cost to fence it with wire netting at $3.60 per 150 ft., and 
 posts, set 10 ft. apart, at 35 j^ each ? 
 
 15. A water company wishes to lay a line of pipe along a mile of 
 street. If the pipe is worth f 1.25 per foot, what will it cost ? 
 
 16. A gentleman has a field, the perimeter of which (i.e. the distance 
 around it) is 320 rd. He wishes to build a fence of eight wires, with 
 posts set 8 ft. apart around it. If the wire weighs 1 lb. to the rod and 
 costs f 2.50 per hundredweight, and the posts cost 17^ each, what will it 
 cost for material to build the fence? 
 
 17. What will it cost to fence a lot, with boards 16 ft. long and 4 in. 
 wide, at 10)^ per board, if the perimeter of the lot is 63 rd. and the fence 
 is to be 6 boards high? 
 
 18. My lot is 32 rd. long and 20 rd. wide. I built a picket fence 
 around it, using pickets 4 in. wide, and placing them 2 in. apart. How 
 many pickets were required? What did they cost at $3.25 per M? 
 
 19. A plot of ground is 4 rd. long and 2 rd. wide. How many straw- 
 berry plants will it take to set the plot in rows 2 ft. apart, running 
 lengthwise, if the plants are set 10 in. apart in the row? 
 
 20. What will the material to build 125 mi. of railroad cost, if the 
 rails are 30 ft. long, weigh 25 lb. to the foot, and cost $28 per ton of 
 2000 lb., and the ties, costing 75^ apiece, are laid 3050 to the mile? 
 
 21. What will it cost to build an electric line between two cities, 15 
 mi. apart; the rails weighing 25 lb. to the foot, and costing $18 per ton; 
 the ties costing 70 ^ apiece, laid 2 ft. apart ; the wire (one strand) weigh- 
 ing 5 lb. to the rod and costing $10 per hundredweight; the posts being 
 set 19 rd. apart, and costing 80 j* apiece? 
 
 22. A tennis court has four lines 78 ft. long, two lines 36 ft. long, 
 two lines 27 ft. long, and one line 31 ft. long. What will the tape to mark 
 these lines cost at i^ per foot? 
 
 23. How many yards of carpet will be required for a flight of 15 steps 
 1 ft. wide and 6 in. high, and a landing 6 ft. 8 in. wide? What will 
 it cost at 70 ^ per yard ? 
 
MEASURES OF LENGTH 71 
 
 24. The two aisles of a church are each 6 ft. wide and 85 ft. long. 
 At $ 1.85 per linear yard, what will the carpet cost to cover them? 
 
 25. In surveying the route for a proposed railroad the surveyors ap- 
 plied the Gunter's chain 6450 times. How many miles of road ? How 
 many rails, 30 ft. long, would it require ? How many ties, counting 
 3050 to the mile? 
 
 26. A boat is rowed at the rate of 10 knots an hour. The current 
 runs at the rate of 4 knots an hour. If the boat is rowed with the cur- 
 rent, how many miles will it go in 10 hr. ? 
 
 27. The elevator in the Washington monument goes 500 ft. above 
 the base. How many rods of cable in the eight strands extending from 
 bottom to top ? 
 
VII 
 
 MEASURES OF AREA 
 
 53. Area Units. Area is the extent of surface. The units 
 for its measurement are rectangular in form and correspond 
 to the units of length. They are the square inch, square 
 foot, square yard, square rod, and square mile. The stand- 
 ard unit is the square yard. For ordinary surfaces, the 
 square inch and square foot are most frequently used. The 
 square mile is used only in land measure, together with a 
 special unit, the acre. 
 
 Table 
 There are : 
 
 144 square inches (sq. in.) in 1 square foot (sq. ft.) 
 
 9 square feet in 1 square yard (sq. yd.) 
 
 30^ square yards in 1 square rod (sq. rd.) 
 
 160 square rods in 1 acre (^0 
 
 640 acres in 1 square mile (sq. mi.) 
 
 36 square miles in 1 township (Tp.) 
 
 CLASS EXERCISES 
 
 1. Study the accompanying figure of a square inch. Cut from paper 
 a square inch without using a ruler. Test accuracy. Try it until nearly 
 correct. 
 
 2. Draw a square inch. Verify. Practice. 
 
 3. Estimate the number of square inches in one side of your book 
 cover. Test accuracy. Estimate the area in square inches of pages in 
 different books, of desk top, of window pane, of pictures, and of fresh 
 air or foul air register. Test the accuracy of each estimate. 
 
 72 
 
MEASURES OF AREA 
 
 73 
 
 4. Practice estimating areas in square inches until you acquire 
 considerable accuracy. Use the ruler only after the area has been 
 judged. 
 
 5. Develop power of estimating area in square feet in the same way 
 as above. Measure, cut, draw, and practice estimating. 
 
 6. Measure off a square yard. Practice estimating in square yards 
 until a fairly clear concept is in your mind. Less time and practice are 
 necessary for this than for the square inch and foot, because less fre- 
 quently needed. 
 
 7. In the same way develop concept of a square rod. Practice for 
 accuracy. 
 
 8. The pupils of a class should measure off an acre, using chains of 
 their own making. Let them mark the corners with flags and walk 
 over the ground in an effort to fix the size of an acre. Then have pupils 
 estimate in acres and verify their estimates by measurement. 
 
 9. If possible, view a square mile or section of land. Use what oppor- 
 tunities you have for fixing its size in your mind. 
 
 54. Area of Rectangles. Finding the area of anything is 
 merely finding the number of square units on its surface. 
 In square-cornered surfaces, or rectangles, these units are 
 always in rows. In the accompanying figure, 2 by 4 inches, 
 
 A Rectangle 
 
74 ELEMENTS OF BUSINESS ARITHMETIC 
 
 there are four square inches along one edge or in one row, 
 and two such rows. There are, therefore, 2 times 4 square 
 inches or 8 square inches in the figure. 
 
 In all rectangles there are as many square units in one 
 row or along one edge as linear units along that edge. Like- 
 wise, there are as many rows of square units as linear units 
 along the other edge. To find the area, then, the number of 
 square units in one row should he multiplied hy the number of 
 rows. 
 
 In a room 10 by 12 feet, there are 10 rows of 12 square 
 feet or 120 square feet in the floor. The multiplicand is 
 thus to be always a number of square units, and the product, 
 therefore, square units. 
 
 PROBLEMS 
 Find the area of the following rectangles : 
 
 1. 6 by 9 inches. 6. 35 by 46 feet. 
 9 sq. in. X 6 = 54 sq. in. 7. 15 by 36 rods. 
 
 2. 13 by 24 rods. 8. 45 by 86 yards. 
 
 3. 16 by 87 yards. 9. 32 by 54 inches. 
 
 4. 41 by 65 feet. 10. 40 by 60 yards. 
 
 5. 11 by 47 inches. 11. 24 by 48 feet. 
 
 12. A barn floor is 40 by 60 ft. What will it cost to cement it at 12/ 
 per square foot ? 
 
 13. How many rolls of weather paper each containing 100 sq. ft. 
 will it take to cover a roof 42 by 50 ft. ? 
 
 14. A board fence 7 ft. high surrounds a lot 32 by 120 ft. How 
 many square feet of boards in this fence? What would it cost to paint 
 it at 4 / a square foot ? 
 
 15. A box is 6 by 8 in. and 4 in. high. How many square inches in 
 its 4 sides? 
 
 Note. — Think of the sides as a rectangle having 4 rows of 28 sq. in. 
 
 16. A room is 12 by 18 ft. and 10 ft. high. How many square feet in 
 the four walls? lu the floor and ceiling? 
 
MEASURES OF AREA 75 
 
 17. What will it cost to tint the four walls of a room 15 by 18 ft. and 
 10 ft. high, at 40 ;* a square yard ? How many square yards in the ceil- 
 ing? What will be the cost of tinting the ceiling at the same rate? 
 What was the total cost of plastering the room at 25^ per square yard? 
 
 18. What will it cost to build a walk of stone, 6 by 360 ft., at 60 ^ 
 per square foot ? 
 
 19. A barn is 40 by 80 ft. and 25 ft. high. What will it cost to paint 
 the four sides at 25 f a square yard? 
 
 20. A hall floor is 15 by 30 ft. How many tiles 6 by 6 in. will it 
 take to lay the floor ? 
 
 21. There are 48 sq. ft. of boards on the side of a building. If each 
 board is 6 in. wide, and there are 12 of them, how long are they? 
 
 22. What will it cost to calcimine the walls and ceiling of a room 
 18 by 15 ft., and 9 ft. high, at 20^ per square yard? 
 
 23. A schoolroom is 30 by 40 ft. What will it cost to put a slate 
 board along one side and end, the slate being 4 ft. wide and costing 18;^ 
 per square foot? 
 
 24. How many bricks 4 by 8 in. will be needed for a walk 36 yd. 
 long and 4 yd. wide? What will they cost at $6.50 per M? 
 
 25. How many square feet of sidewalk 6 ft. wide will be required to 
 surround a lot 250 by 360 ft.? What will it cost to lay it at 12;^ a 
 square foot? 
 
 26. The owner of a lot of ground 1080 ft. long and 600 ft. wide cuts 
 two streets, each 75 ft. wide, through the middle, one running east and 
 west, and the other running north and south. How many acres has he 
 left? What will brick 6 by 9 in. cost to pave these streets at $ 7 per M? 
 
 27. A farmer owned a rectangular piece of land 40 by 80 rd. He 
 sold four lots, 7 by 10 rd., 8 by 25 rd., 10 by 15 rd., and 15 by 20 rd. How 
 many acres did he sell? How many remained? How many rods of 
 fence will it take to fence the remaining land, if the lots are taken, one 
 from each corner? 
 
 55. Squaring a Number. A rectangle which contains the 
 same number of rows of square units as there are square 
 units in each row is called a square. The number of units in 
 each row of a square is called the square root. 
 
76 ELEMENTS OF BUSINESS ARITHMETIC 
 
 CLASS EXERCISES 
 
 1. The square on 3 ft. has how many rows and how many square 
 units in each row? How many square units in the square, or its area? 
 
 2. A square on 9 in. has how many square inches in each row? 
 
 3. When a square contains 2.5 sq. in., it is how many inches long and 
 wide? 
 
 4. What is the square root of 9 sq. ft. ? Of 16 sq. ft. ? Of 25 sq. ft. ? 
 
 5. What is the square root of : 81, 49, 100, 225, 2500, 625, 169? 
 
 6. Drill upon the following table : 
 
 12 = 1 62 = 36 112 = 121 162 ^ 256 
 
 22 = 4 72 = 49 122 ^ 144 202 = 400 
 
 32 = 9 82 = 64 132 = 169 252 = 625 
 
 42 = 16 92 = 81 142 = 196 502 ^ 2500 
 
 52 = 25 102 = 100 152 = 225 1002 = 10000 
 
 FINDING THE SQUARE ROOT 
 
 66. Root Periods in the Square. In squaring ][2 _ ^ 
 numbers from 1 to 10, it may be noted that the 2^ = 4 
 square of a number composed of one figure never 32 — 9 
 contains less than one or more than two figures. ^2 _ iq 
 Squaring 10 and 99, it will be seen that the ^2 __ 25 
 square of a number composed of two figures has ^2 _, 3^ 
 not less than three nor more than four figures. ^2 = 49 
 In the same way it may be seen that the square g2 __ 54 
 of a number composed of three figures contains 92 _ g;[ 
 not less than five nor more than six figures. It ■^q2 _ iqO 
 may be concluded, then, that for every two fig- 992 _ g^Qi 
 ures in a square there is one figure in the root. 
 If wie mark off the number whose square root we are seek- 
 ing, into periods of two figures each, beginning at the right, 
 we will get one figure in the root for each period. 
 
 57. Square of a Number composed of Two Figures. Take 
 a paper 24 in. square, and carefully mark it off into square 
 inches. From a lower corner cut out the square on 20 in.. 
 
MEASURES OF AREA 
 
 77 
 
 and from the opposite upper corner cut out the square on 
 4 in. What are the dimensions of the two oblongs remain- 
 ing after the squares have been removed ? Observe that the 
 square on 20 in. is the square of the tens of the number 
 squared (24), and that the other square is the square of the 
 units (4). The dimensions of the two oblongs are also, 
 respectively, the same as the tens and units of the original 
 number. 
 
 The area of the square of a number composed of two figures 
 is, then, equal to the square of its tens plus two rectangles with 
 the tens and U7iits as dimensions, plus the square of its units. 
 
 A Square 
 
 f 
 
78 
 
 ELEMENTS OF BUSINESS ARITHMETIC 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r f 
 
 f^ - 
 
 
 
 
 
 
 
 
 
 ^, 
 
 L^ 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ~ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 8. 
 
 F 
 
 inc 
 
 lin 
 
 g 
 
 the 
 
 Th 
 
 jS 
 
 E i 
 
 qu 
 
 ar 
 
 CE 
 P 
 
 3 I 
 
 Sqi 
 RO 
 
 JAB 
 
 BL] 
 
 
 3ei 
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 AB 
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 AT] 
 
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 L S 
 
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 lar 
 
 e 
 
 whose area is 576 sq. ft., or extract the square root of 576. 
 
 5'76 sq. ft. (24 
 The square on 2 tens, or 
 20 sq. ft. X 2 = 40 sq. ft. 
 
 4 00 
 
 _4 sq. ft. 
 44 sq. ft. 
 Sq. ft. in one row of 2 rec- 
 tangles and smaller square. 
 
 1 76 sq. ft., the area of 2 
 rectangles and smaller 
 square. 
 
 1 76 sq. ft. the square feet 
 in 4 rows. 
 
MEASURES OF AREA 79 
 
 Marking off 576 according to Sec. 56, we find that its 
 root will contain two figures, tens and units. From Sec. 57, 
 also, we see that the second period, or 6 hundreds, contains 
 the square of the second or tens figure. 
 
 Taking out the square of the largest tens (2 tens) con- 
 tained in 6 hundreds, or 4 hundreds, we have left 176 sq. ft. 
 as the area of the remaining two rectangles and the smaller 
 square. We know (Sec. 57) that the rectangles have each 
 two tens, or 20 sq. ft., along one edge or in one row. We do 
 not as yet know the dimensions of the smaller square, but 
 we know they are the same as the other dimension of the 
 rectangles. 
 
 If the area of the rectangles is nearly 176 sq. ft., and the 
 number of square feet along one edge of both is 40 ; then 
 there must be nearly as many rows of 40 sq. ft. as 40 sq. ft. 
 is contained in 176 sq. ft., or 4 rows. If this is correct, the 
 smaller square is 4 ft. square, and along one edge there 
 would be 4 sq. ft. 
 
 Along one edge of both the rectangles and the smaller 
 square, there would be, then, 44 sq. ft., and if 4 rows wide, 
 the area of the rectangles and square would be 176 sq. ft., 
 or the same as the area remaining after removing the larger 
 square. 
 
 The other dimension of the rectangles, therefore, must be 
 4 ft., and the units figure 4, making the square root 24. 
 
 STATEMENT OF PROCESS 
 
 The principles applied in finding the square root of larger 
 numbers are the same as when the root contains two figures. 
 The two periods to the left are treated as containing the 
 square of the number composed of two figures, and when 
 these are found, they are considered as the known tens' figure, 
 and the units' figure is sought in the next period to the right, 
 and so on, until the complete root has been found. 
 
80 ELEMENTS OF BUSINESS ARITHMETIC 
 
 The following is. a summary of the steps in the method : 
 
 (1) Beginning at the right, mark off periods of two figures 
 each. 
 
 (2) By inspection use the square root of the largest square 
 contained in the left-hand period as the first figure of the 
 root, and subtract its square from that period, bringing 
 down the next period to the right. 
 
 (3) Treating the first figure thus obtained as the tens and 
 adding a cipher, we have the number of squares along one 
 edge of one rectangle. Multiplying this by 2 and dividing 
 the product into the number brought down (allowing some 
 for the area of the smaller square), the probable width of 
 the rectangles or second root is obtained. 
 
 (4) Add the root to the number of squares along one 
 edge of both rectangles, and multiply the sum by the same 
 figure (as indicating the number of rows) to complete the 
 square. 
 
 (5) Subtract this from the number brought down, and 
 bring down the next period, proceeding as in (3) and (4). 
 
 59. Square Root of Decimals and Fractions. When the 
 square consists of a decimal or a whole number and decimal, 
 mark off periods right and left from the decimal point, and 
 proceed as in whole numbers. The root figure for the first 
 decimal period would be tenths; the second, hundredths, 
 etc. 
 
 The square root of a fraction may be found by extracting 
 the square root of the numerator for a new numerator and 
 of the denominator for a new denominator. 
 
 60. Applications of Square Root to the Right Triangle. 
 
 Draw a right triangle (Sec. 68) having a base of 3 inches 
 and a perpendicular of 4 inches. What is the length of 
 the hypotenuse ? 
 
MEASURES OF AREA 
 
 81 
 
 Erect a square on the perpendicular. Erect a square on 
 the hypotenuse, and on the base. How many square inches 
 in the squares on the base and 
 on the perpendicular? In the 
 square on the hypotenuse ? 
 
 Draw a right triangle with base 
 and perpendicular, respectively, 
 6 and 8 inches. Draw squares 
 on the three sides. Compare the 
 area of the square on the hy- 
 potenuse of the triangle with 
 those on the other two sides. 
 
 These illustrate the geometrical truth, that the square on the 
 hypotenuse of a right-angled triangle is equal to the sum of 
 the squares on the other two sides. 
 
 ?I> 
 
 \D THE 
 
 Square Root of: 
 
 
 
 
 
 1. 
 
 4225 
 
 4. 53,361 
 
 7. 
 
 72,984 
 
 10. 
 
 ^m 
 
 2. 
 
 5625 
 
 5. 17,424 
 
 8. 
 
 1900.96 
 
 11. 
 
 \m 
 
 3. 
 
 1225 
 
 6. 97,344 
 
 9. 
 
 97.8121 
 
 12. 
 
 ■i^ 
 
 13. The base of a triangle is 8 ft., the altitude 6 ft. What is the 
 hypotenuse ? 
 
 14. Find the base of a triangle whose altitude is 12 ft., hypotenuse 
 16 ft. 
 
 15. The hypotenuse of a triangle is 45 ft., the base 27 ft. Find the 
 
 altitude. 
 
 16. A square farm contains 160 A. What is the length of one side ? 
 
 17. A ladder is 25 ft. long and, when the foot is placed 15 ft. from 
 the foot of a wall, just reaches the top. How high is the wall ? 
 
 18. A lot is in the form of a right-angled triangle, whose base is 16 
 chains and altitude 12 chains. How many rods of fence will be needed 
 to inclose it ? 
 
82 ELEMENTS OF BUSINESS ARITHMETIC 
 
 19. Find in rods the diagonal of a square field that contains 12 acres. 
 
 20. What is the distance from one lower corner to the opposite upper 
 corner of a room, that is 36 by 72 ft. and 12 ft. high ? 
 
 PRACTICAL APPLICATIONS OF AREA 
 
 61. Pitch of Roofs and Roofing. The degree of slant given 
 to the sides of a roof is called its pitch. 
 
 When the height of the gable is one fourth the width of 
 the building, the roof is said to have one-fourth pitch. When 
 the height is one half the width, the roof has one-half pitch; 
 when five eighths, it has a five-eighths or Gothic pitchy etc. 
 One half is the pitch commonly used. 
 
 In estimating the cost of roofing in accordance with plans 
 of a building under consideration, it is sometimes necessary, 
 from the size of the building and the roof pitch, to find the 
 dimensions of the two sides of a roof. 
 
 Knowing the width of the building, the degree of pitch 
 will give the height of the gable or the perpendicular of 
 the right triangle, one half the width of building forming 
 the base. Applying the principle of Sec. 60, the length of the 
 hypotenuse or the width of one side of the roof is then found. 
 The distance the eaves extend over the sides of the building 
 should be added to the roof width, and the extension of the 
 gables over the ends should be added to the length of the 
 building for the roof length. 
 
 The standard width of a shingle is 4 inches. This may 
 vary, however, according to grade or style. In ordinary 
 shingles there is no uniformity in width, but the average 
 width in a bundle is supposed to be 4 inches. 
 
 The standard exposure to the weather is, likewise, 4 inches. 
 That is, in laying, all but 4 inches of the length of a shingle 
 is covered. Thus, the average area of the exposed surface 
 of a shingle is 16 sq. in. 
 
 In selecting the proper widths of shingles so that all joints 
 
MEASURES OF AREA 
 
 PITCH OF ROOFS 
 
 83 
 
 40' 
 One-fourth Pitch 
 
 40' 
 One-half Pitch 
 
 40' 
 Five-eighths or Gothic Pitch 
 
84 ELEMENTS OF BUSINESS ARITHMETIC 
 
 will be covered, and discarding defective shingles, there 
 is more or less waste, varying with the grade of shingle 
 used. One ninth is the amount usually allowed for such 
 waste. 
 
 Thus, 9 shingles, at 16 sq. in. each, would cover a surface 
 of 144 sq. in. or 1 sq. ft. With i allowance of waste, 10 
 shingles would be required for 1 sq. ft., and 1000 shingles 
 for 100 sq. ft., the surface unit. It is called the square (100 
 sq. ft.), and is used in roofing, flooring, slating, etc. As 
 bundles are usually made up to contain 250 shingles, it 
 would take/oi^r to cover a square^ or one bundle would cover 
 a surface of 25 sq. ft. Thus, a roof 24' by 30' would be 
 720 sq. ft. on each side, or 1440 sq. ft. on both sides. This 
 would make 14.40 squares and would require 14 J thousand 
 shingles or 58 bundles. 
 
 The outer edge of the roof is usually laid double. This is 
 offset, however, by the width of the ridge board. 
 
 Bundles of shingles are not usually broken, so if the sur- 
 face shows that a fraction of a bundle is needed, a whole 
 bundle must be purchased. 
 
 When, for any purpose, an exposure other than the stand- 
 ard 4-inch is allowed, the number of shingles or bundles are 
 found in the usual way, and the result is modified to meet 
 the conditions. Thus, if the exposure is to be 3'' instead of 
 4'', \ more shingles will be required ; if 2'\ twice as many ; if 
 5", \ less; and if 6'', ^ less. If the width of the shingle 
 varies from the standard 4-inch, a correction may be made 
 in the same way. 
 
 Roofing, other than shingling, is also figured by the square. 
 In slating, the size of the slates varies from &' x 12'' to 16" 
 X 24". The number of slates needed per square would, 
 therefore, vary from 533 to 86. Contractors usually figure 
 from prepared tables, showing the number of slate at a given 
 size per square. 
 
MEASURES OF AREA 
 
 85 
 
 62. Flooring in Wood. Boards for flooring are tongued 
 and grooved. This entails a loss in width of | of an inch for 
 each board. That is, a board bought as a 3-inch board will 
 cover but 2|- inches in width. Three-eighth inch loss on 
 3 inches is |-inch loss on 1 inch. In other words, each 1 inch 
 of board width purchased will cover but J inches of floor 
 space. It will require | more lumber, therefore, than it 
 would if there were no waste. 
 
 Tongued and Grooved 
 
 How much flooring would be required for a room 24^ x 32' 
 if 3" flooring were used ? 
 
 24 sq. ft. X 32 = 768 sq. ft. Surface area. 
 
 768 sq. ft. + 1 of itself = 878 sq. ft. Flooring required. 
 
 Spruce or pine flooring is made 3 in., 4 in., or sometimes 
 51 in. wide. Hardwood flooring is but 2 in. or 2| in. wide, 
 or even less. In the same way, allowances for loss from 
 grooving, for the usual widths of flooring, i^ as follows : for 
 2 in. flooring, add -^^ ; for 2^ in., add -^y ; for 4 in., add ^; 
 
86 ELEMENTS OF BUSINESS ARITHMETIC 
 
 and for 5 J in., add -^^ of the actual floor space. Prove the 
 correctness of these fractions, as above. Carpenters charge 
 by the square for laying floors. 
 
 PROBLEMS 
 
 1. The roof of a shed is 12 by 20 feet. How many slates 6 by 12 
 inches will be required to cover it? How many shingles? How many 
 bundles of shingles ? 
 
 2. What will it cost to shingle the two sides of a roof, if each side is 
 20 by 80 feet, with shingles costing $3.20 per M? 
 
 3. A barn is 40 by 80 feet, and the roof is J pitch. How many 
 shingles must I buy to roof it, if the roof extends 18 inches over each 
 side and end? 
 
 4. The roof of a church is Gothic pitch. The building is 40 by 72 
 feet, and the roof extends 2 feet over each side and end. If shingles are 
 laid 3 inches to the weather, what will be their cost at $4.50 per M? 
 
 5. The roof of a house is 56 feet long and each side is 18 feet wide. 
 There are two verandas, each having a roof 10 by 20 feet. What will 
 the shingles cost at $4.75 per M, if they are laid 5 inches to the 
 weather ? 
 
 6. A mill is 120 feet long. If each side of the roof is 32 feet wide, 
 and it extends 2| feet over each side and end, what will the shingles cost 
 at $ 3.80 per M, if they are laid 4 inches to the weather ? 
 
 7. Figure the flooring lumber bill for house shown on pages 90-91, as 
 follows : Hard pine at $48 per M on veranda, living room, dining room, 
 and entire upper floor, except bathroom, which is to be tiled ; two-inch 
 maple at $72 on kitchen; two-inch oak at $80 on reception hall, library, 
 and parlor; four-inch soft pine on porch, at $60; spruce, 5^ inches, at 
 $34 on laundry and cellar in basement. 
 
 63. Carpeting. The standard width of Kidderminster and 
 Ingrain carpets is 36 inches. That of Brussels, Wiltons, 
 Axminsters, etc., is 27 inches. Borders are usually 221 
 inches wide. 
 
 Rugs, whether pattern rugs or made up, are sold by the 
 piece. All other carpeting is sold from the roll and by 
 linear measure. 
 
MEASURES OF AREA 87 
 
 The direction the carpet is to be laid will often determine 
 the cost, as there might be more waste if laid one way than 
 if laid the other. 
 
 The length of each strip would be the length of the room 
 plus the allowance for matching. The number of strips 
 would be determined by dividing the width of the room by 
 the width of the carpet, counting a full strip for all fractions, 
 as any extra width must be cut off or turned under. Thus, 
 for a room 15 by 24 feet, the strips would be 8 yards long, 
 and with an allowance of 9 inches for matching, 8^ yards 
 long. If the carpet were Brussels, there would be 6 strips 
 and 18 inches additional, requiring 7 strips. 7 strips of 8J 
 yards each would be 57J yards. 
 
 PROBLEMS 
 
 1. How many yards of carpet 1 yard wide will it take to cover a floor 
 15 by 18 feet? 
 
 2. A room is 18 by 20 feet. How many yards of carpet f yard wide 
 will it take to cover the floor ? 
 
 3. How many yards of carpet 27 inches wide will be required to cover 
 the dining room in the floor plan on p. 90, if the strips run length- 
 wise ? What will it cost at ^ 1.20 per yard V 
 
 4. If the library shown on p. 90 is carpeted in the most economical 
 way, with carpet 30 inches wide, what will be the cost at $ 1.25 per yard? 
 
 5. What will it cost to cover the kitchen of the house shown on p. 
 90 with linoleum costing $1.35 per square yard? 
 
 6. The three bedrooms in the plan of the second floor on page 91 are 
 to be covered with carpet 1 yard wide, 8 inches waste in matching. 
 What will be the cost at 95 ^ per yard, if the strips run in the direction 
 to leave the least waste ? 
 
 7. The stairway in the floor plan (pp. 90-91) is 11 feet 4 inches high, 
 the tread of each stair is 12 inches, and the riser 8 inches. What will be 
 the cost of carpet at $1.35 per yard? 
 
 8. A rug 8' X 13' is placed in the living room at a cost of $42.50. 
 The remaining space is painted at a cost of 36 ^^^ a square yard. What 
 is the total cost for covering the floor ? 
 
88 ELEMENTS OF BUSINESS ARITHMETIC 
 
 64. Lathing and Plastering. The square yard is the unit 
 by which estimates are made for the cost of lathing and plas- 
 tering. Contracts are made at a given price per square yard 
 of plastering, or lathing, or both. 
 
 In estimating the cost of lathing and plastering for scratch 
 (first) and brown (second) coat on wood lath, the follow- 
 ing quantities are generally allowed for 100 square yards : 
 1400 to 1500 laths (laths are put up in bundles of 50, and 
 sold by the bundle or thousand); 10 pounds of 3-penny 
 lathing nails ; 2^ barrels, or 500 pounds, of lime ; 45 cubic 
 feet, If loads, or 15 barrels, of sand ; and 4 bushels of hair 
 (hair is put up in bushel bundles, containing 5 packages). 
 For the best quality of white coat, the estimate is 90 pounds 
 of lime to 50 pounds of plaster of Paris, and 50 pounds of 
 marble dust. Stucco is put up in bags containing 100 
 pounds each, and is sold by the ton ; 900 to 1000 pounds are 
 required for 100 square yards of surface. 
 
 Custom varies as to an allowance for doors and windows. 
 Some contractors make no allowance, holding that the extra 
 care and time needed in lathing and plastering around an 
 opening make up for saving of material. When an allow- 
 ance is made, it is usually 20 sq. ft. for each opening, and 
 not the exact measurement. In estimating material alone, 
 allowance is usually made for all openings. A fraction of a 
 square yard is counted as a square yard in the final result. 
 
 PROBLEMS 
 
 1. What will it cost to plaster a room 20 x 24 ft. and 10 ft. high, 
 at 20 ^ per square yard ? How much lime, hair, and sand will it take 
 for two coats ? 
 
 2. How many square yards in the walls and ceiling of a room 18 by 
 20 ft. and 9 ft. high ? How many laths, and how much stucco will it 
 take to cover it ? 
 
 3. A hall is 5 X 15 ft. and 10 ft. high. What will it cost to lath and 
 plaster it at 25^ a square yard ? How many laths will it take ? 
 
MEASURES OF AREA 
 
 89 
 
 FLOOR PLANS OF HOUSE 
 
 a □ 
 
 1 
 
 LAUNDRY 
 
 
 COAL BIN 
 
 1 I4'6"x 17'6^ 
 
 8'xl4'6" 
 
 8' X 14' 6" 
 
 1 1 1 
 
 
 
 
 n 
 
 1 
 
 r CFI 1 AR 
 
 
 1 HEATER J 
 
 
 L hS 22' e" 
 
 
 16' X 22' 6" 
 
 . 
 
 1 1 1 
 
 a 
 
 □ a 
 
 Basembnt 
 
 D 
 
 D 
 
90 ELEMENTS OF BUSINESS ARITHMETIC 
 
 PORCH 
 6' 6" X I { 
 
 \^ 
 
 V 
 
 KITCHEN 
 
 <■ 
 
 LIBRARY 
 
 13 X 14 
 
 \ 
 
 LIVING ROOM 
 
 DINING ROOM 
 
 \ 
 
 <f 
 
 HALL 
 9'6"x 11 
 
 PARLOR 
 
 VERANDA 6 6 X 32 
 
 First Floor 
 
MEASURES OF AREA 
 
 91 
 
 
 
 \ PORCH ROOF / 
 
 
 
 
 
 1 BED ROOM \ 
 1 14' 6" X 15' ^ 
 
 Ir-il 
 
 \ 
 
 L HALL 
 
 / 
 
 CLC 
 
 4' 
 
 BED ROOM 
 lo'x 13' 
 
 1 
 
 SET 
 
 x6' 
 
 4' X ( 
 
 CLOSE 
 
 1 
 
 
 6' 6" X 8' 
 BATH ROOM \ 
 
 9 
 
 
 \ 
 
 NTURSLR/ 
 
 lO' X lo'e" 
 
 
 7' X 8' 6" 
 
 
 
 
 
 
 
 
 
 
 
 r 
 
 
 
 
 
 
 
 
 ALCOVE 
 
 
 A 
 
 BED ROOM 
 13' X 13' 
 
 1 1 
 
 
 
 
 < 
 
 I 
 
 _DOWN 
 
 
 
 
 k. 
 
 ■• 1 II , \ 
 Q fi V Q \ 
 
 
 
 
 
 
 
 
 
 
 P 
 
 
 
 
 
 
 / 
 
 / VERANDA ROOF \ 
 / 7'6"x 34' \ 
 
 Second Floor 
 
92 ELEMENTS OF BUSINESS ARITHMETIC 
 
 4. A dining room is 18 by 20 ft. and 11 ft. high. At 45^ per square 
 yard what will it cost to lath and plaster the walls and ceiling ? 
 
 5. A bedroom is 12 ft. square and 10 ft. high. What will it cost to 
 plaster it at 30 f per square yard ? 
 
 6. The rooms in the floor plan on pp. 90-91 are 11 ft. high. Making 
 allowance for doors and windows as suggested in the discussion, what 
 will it cost to plaster the house, if sand costs 75 f per cubic yard, lime 
 % 1.25 per barrel, and hair 25 ^ per bushel, and the cost of putting it on 
 is 20 i^ per square yard ? 
 
 7. What would it cost to lath, and plaster the house with stucco 
 (900 pounds to the 100 square yards), if laths are worth 50 ^ a bundle, 
 stucco $ 10.50 per ton, and the cost of putting on the laths and stucco is 
 25^ per square yard? 
 
 65. Papering. Wall paper is put up in single rolls 8 
 yards long, and double rolls 16 yards long. English rolls 
 are single rolls only, and are 12 yards long. The width of 
 ordinary or figured paper is 18 inches, that of ingrain is 30 
 inches. Higher grades vary somewhat in width, from 18 to 
 22 inches, and from 30 to 36 inches. Border and friezes are 
 sold by the linear yard. They vary in width. 
 
 In a roll of 18-inch paper, there would be IJ rows of 24 
 sq. ft., or 36 sq. ft. In 30-inch paper, there would be 2 J 
 rows of 24 sq. ft., or 60 sq. ft. In 22-inch paper, there 
 would be 1| rows, or 44 sq. ft., etc. 
 
 The surface to be papered in a room, 18 x 22 ft., 12 ft. 
 high, would be as follows : In the four walls there would be 
 12 rows of 80 (the perimeter of the room) sq. ft., or 960 
 sq. ft., and in the ceiling 18 rows of 22 sq. ft., or 396 sq. ft. 
 In finding the number of rolls necessary to cover the surface, 
 the waste due to matching must be considered, and allowance 
 made for windows and doors. The latter is usually 20 sq. ft. 
 for each opening, counting archways and double doors as 
 double openings. As whole rolls must be bought, it often 
 happens that there is a part of a roll remaining, sufficient, 
 perhaps, to cover waste due to matching, after the exact 
 
MEASURES OF AREA 93 
 
 surface has been provided for. If a border were used, its 
 width would be deducted from the height of the room. 
 
 If there were two doors and three windows in the above 
 room, 100 sq. ft. would be deducted, leaving 860 sq. ft. in 
 the walls. There being 36 sq. ft. in a roll, 23| rolls would 
 cover the surface; 24 rolls, at least, would be required 
 Q roll only being allowed for matching). Whether this 
 would exactly cover the surface would depend upon the 
 pattern. For the ceiling, 11 rolls would exactly cover the 
 surface, and 12 would probably be necessary to provide for 
 waste in matching. The border would be 80 ft. or 27 yd. 
 long. 
 
 PROBLEMS 
 
 1. A room is 20 by 24 ft. and 10 ft. high. What will it cost to paper 
 it with 18-inch paper (always use single rolls unless otherwise mentioned) 
 at 50 ^ a roll, if there are two doors and three windows ? 
 
 2. A dining room is 18 by 24 ft. and 11 ft. high. What will it cost 
 to paper it with paper 22 in. wide, worth 90)^ a double roll, if there are 
 two doors and four windows ? 
 
 3. How many rolls of paper 30 in. wide will it take to paper the 
 parlor in the plan on page 90, if a border 22 in. wide is used ? 
 
 4. How many rolls, 36 in. wide, to cover the dining room (p. 90) if 
 a border 18 in. wide is used ? Cost at $ 1 a roll for body, 60 ;^ a yard for 
 border, and 65 j2^ a roll for ceiling ? 
 
 5. Find cost of papering the library (p. 90) with paper 22 in. 
 wide, at 95^ a roll, and a border 18 in. wide at 95^. Deduct for a 
 wainscoting 3 ft. high all around the room. 
 
 6. Find cost of papering the kitchen (p. 90) with 18-inch paper at 
 45 ;* a roll. 
 
 7. Find cost of papering the halls of the house (pp. 90-91), deducting 
 for the stairway and doors, with 22-inch paper costing $1.10 a double 
 roll. Use a border of the same width costing 30 )* a yard, and 18-inch 
 paper for ceiling at 40 ^ a roll. 
 
 8. Find cost of papering the bedrooms (p. 91) with 18-inch paper 
 at 35 j^ a roll, using a border of the same width and costing 20 ^ a yard 
 
94 ELEMENTS OF BUSINESS ARITHMETIC 
 
 LAND MEASURE AND SURVEY 
 
 66. The Township. For purposes of public record, so 
 that land ownership may be secure and transfers of title 
 conveniently and safely made, a system of land survey is 
 necessary. 
 
 In the United States prior to the year 1785, each tract of 
 land was surveyed and its boundary line described as start- 
 ing from some natural object, thence a certain distance in 
 some direction, thence in another direction to another object, 
 etc. The results of such surveys were irregular shapes of 
 land tracts and often endless litigation. This method is still 
 in use in the Eastern states. 
 
 By an ordinance of the Continental Congress, a system of 
 rectangular survey was adopted in 1785 for the Northwest 
 Territory, and later this was applied to all the Central and 
 Western states. 
 
 By this system, the land is surveyed into rectangular 
 townships^ six miles square, with boundary lines conforming 
 to the points of the compass. To locate a particular township 
 certain meridians are designated as principal meridians and 
 certain parallels as base lines or standard parallels. Twenty- 
 four principal meridians have been established, the first one 
 being in Ohio and the last one in Oregon. Parallels most 
 convenient in the survey of a state are selected for base lines. 
 
 The townships are numbered, in order, north and south 
 of the base line, and the particular row of townships is 
 shown by the number of rows or ranges east and west of the 
 principal meridian. Thus, T 7 N, R 4 W, would mean the 
 seventh township north of the given hase line, in the fourth 
 range of townships west of the given principal meridian. 
 
 Starting from the principal meridian, the surveyors mark 
 off points every six miles on the base line. From these 
 points, lines are run to the north by compass. Because the 
 
MEASURES OF AREA 
 
 95 
 
 meridians approach as they near the pole, it is clear that 
 these north lines would soon be less than six miles apart as 
 they extend northward. To preserve the townships as nearly 
 uniform in size as possible, new parallels are run at frequent 
 intervals as correction lines^ upon which the north lines are 
 erected again, starting six miles apart. 
 
 The shaded township in the accompanying illustration 
 
 Government Survey 
 
96 
 
 ELEMENTS OF BUSINESS ARITHMETIC 
 
 would be described as Township 6 North, Range 2 East of 
 the Principal Meridian. 
 
 67. Subdivisions of the Township. The surveying unit, 
 the township, is six miles square, and contains 36 sq. mi. 
 
 6 
 
 5 
 
 4 
 
 3 
 
 2 
 
 •1 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 18 
 
 17 
 
 16 
 
 15 
 
 14 
 
 13 
 
 1.9 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 30 
 
 29 
 
 28 
 
 27 
 
 26 
 
 25 
 
 31 
 
 32 
 
 33 
 
 34 
 
 35 
 
 36 
 
 A TOWNSHIF 
 
 These square miles are called sections^ and are numbered 
 from 1 to 36, beginning at the northeast corner, going west 
 on the northern row of sections, east on the second row, west 
 on the third, etc. 
 
 Since (Sec. 53) there are 640 acres in a square mile or 
 
MEASURES OF AREA 
 
 97 
 
 section, in half a section there are 320 acres, and 160 acres 
 in a quarter section. One half of the latter is called an 
 "eighty." A tract of land within a section is located by 
 the use of fractions with descriptive directions ; thus, N W ^ 
 is the northwest quarter of the section, and the S J of NW 
 ^ is the south half of the northwest quarter section. 
 
 LOCATION OF SECTIONS 
 
 
 N.E. 1/4 of 
 N.W. 1/i 
 
 N.E.1/; 
 
 
 
 8.1/4 
 
 A Section 
 
 A complete description of a tract of land might be: 
 the NE ^ of the NW I sec. 24, T 12 N, R 9 E of the 
 5th Principal Meridian, which would be read as the north- 
 east one fourth of the northwest quarter of section 24, 
 township 12 North, range 9 East of the 5th Principal 
 Meridian. 
 
98 ELEMENTS OF BUSINESS ARITHMETIC 
 
 PROBLEMS 
 
 SUBDIVISION OF SECTIONS 
 
 1. Make a diagram of a township. Locate Sec. 20, 12, 9, and 13. 
 
 2. Make a diagram of a section. Locate the NW ^, and S ^. 
 
 3. Mr. Brown owns 40 acres in the SE corner of section 13, township 
 5 N, range 3 E. Locate his land. 
 
 4. A man bought the E i of the NW i, the W J of the NE \, and the 
 NE i of the SE ^. How many acres did he buy? Locate. 
 
 5. I sold the E ^ of a section of land for $24.50 per acre, the NW ^ 
 for $ 28, the E i of SW I for $ 23.75, and the remainder of the section 
 for $ 20. How much did I get for the entire section ? 
 
 6. A man bought the following property : the SE I, the NE J, the 
 N I of NW i, the NW \ of the SW i, and the E i of the SW \ of Sec. 
 16, T 13 N", R 3 E. How many acres did he buy ? 
 
 MEASUREMENT OF NONRECTANGULAR SURFACES 
 
 68. Surface Forms. A surface is the outside or face of a 
 solid. 
 
 A plane surface is one that is flat. A line connecting any 
 two points in such a surface will lie wholly within the sur- 
 face. Being the outside or face merely, a surface is not con- 
 sidered as having thickness. It is said, therefore, to have 
 but two dimensions, length and breadth. 
 
 The edges of surfaces form lines. These lines may be 
 straight, when their direction is never changed ; or curved, 
 when continually changing, so that no three adjacent points 
 are in the same straight line. 
 
 An angle is the difference in direction between two straight 
 lines that meet. 
 
 When two meeting lines go in exactly opposite directions, 
 the angle formed is a straight angle (a). The angles formed 
 
 b a c on either side of (a), by lines ab 
 
 « and ac, leaving the point a in 
 
 exactly opposite directions, are straight angles. 
 
MEASURES OF AREA 
 
 99 
 
 A right angle is one half a straight angle. 
 When one line meets another so as to form a 
 two equal angles, as ade and adc^ the 
 angles formed are right angles (5). 
 
 An acute angle is less than a right angle, and an ohtu%e 
 angle is greater than a right angle (c? and d^. 
 
 c d 
 
 Lines are parallel when they extend in the same direction 
 and are at all points equally distant from each other (e), 
 
 . Lines are perpendicular when the 
 
 angles formed by their meeting are 
 
 right angles (5) ; horizontal when 
 parallel to any line connecting two points on 
 the horizon of vision (/) ; vertical when per- 
 pendicular to a horizontal plane (5). 
 
 A triangle is a plane surface bounded by 
 three straight lines. 
 
 A triangle is equilateral when its three sides 
 are equal (/) ; isosceles when two of its sides 
 are equal (^), and scalene when all three sides are unequal (^). 
 A right triangle has 
 one right angle (A), an 
 obtuse triangle has one 
 obtuse angle (^), and an 
 acute triangle has three 
 acute angles (/). 
 
 A polygon is a plane 
 9 surface of more than 
 
 four sides and bounded by straight lines. 
 
 The aggregate length of all the lines bounding a polygon 
 is called its perimeter. 
 
100 
 
 ELEMENTS OF BUSINESS ARITHMETIC 
 
 A quadrilateral is a plane surface 
 bounded by four straight lines. 
 
 A parallelogram is a quadrila- 
 teral the opposite sides of which 
 are parallel (/). 
 
 A trapezoid is a quadrilateral two 
 sides of which are parallel (Ic). 
 A trapezium is a quadrilateral with no two sides parallel (Z). 
 
 h 
 
 A rectangle is a parallelogram the angles of w^hich are 
 right angles (m), 
 
 A square is a rectangle, the sides of which are equal (n). 
 
 ni 
 
 Other polygons are named likewise from the number of 
 sides, i.e. pentagon (o), hexa- 
 gon (jo), octagon, decagon 
 (five, six, eight, ten), etc. 
 
MEASURES OF AREA 
 
 101 
 
 A circle is a plane sur- 
 face, every point in the 
 boundary of which is 
 equidistant from a point 
 within called the center. 
 The curved line forming 
 the boundary of a circle is 
 called the circumference. 
 The distance from a given 
 point on the circumfer- 
 ence through the center to 
 the opposite point is the 
 diameter. One half this 
 diameter is the radius Qq) . 
 
 QUADRILATERALS 
 
 69. Oblique Parallelogram. Take a piece of paper and 
 make a rectangle 3 by 5 in. Mark off one inch along the 
 upper edge from the right upper corner. Draw a line from 
 this mark to the right lower corner. Fold along this line, 
 crease, and tear off. 
 
 Along the lower edge or base, mark off 1 in. from the left 
 lower corner, and draw a line to the left upper corner, crease, 
 
 and tear. The resultant figure 
 is an oblique parallelogram. 
 Why a parallelogram? Why 
 oblique ? 
 
 Again, on the lower base, 
 mark off 1 in. from the right 
 lower corner, and draw a line 
 to the right upper corner. Crease and tear. Place the 
 triangle to the left edge of the parallelogram so that the 
 oblique edges meet. The oblique parallelogram has been 
 changed into a rectangle. Has the base been changed? 
 
102 
 
 ELEMENTS OF BUSINESS ARITHMETIC 
 
 The altitude ? The perpendicular height of a figure is its 
 altitude. 
 
 An oblique parallelogram is equivalent to a rectangle with the 
 same base and altitude. 
 
 70. Trapezoid. Cut out a 5-inch paper square. Two 
 inches to the left of its right upper corner draw a line to the 
 
 right lower corner. Fold along this 
 line, crease, and tear. The four-sided 
 piece is a trapezoid. Why ? 
 
 Fold the trapezoid so that the parallel 
 sides will meet. Crease and tear. Turn 
 the upper piece upside down and place 
 it to the right of the lower one, so that 
 the oblique edges will meet. What 
 have you formed? Compare the base 
 of the rectangle with the combined 
 length of the two parallel sides (called bases) of the trape- 
 zoid. How does its altitude compare with that of the original 
 trapezoid ? 
 
 A trapezoid is equivalent to a rectangle with a base equal to 
 the sum of the two bases of the trapezoid^ and one half its 
 altitude. 
 
 PROBLEMS 
 
 1. One side of a walk is 30 ft. long and the other 27^ ft. and the 
 width 5 ft. How much will it cost to build it at 30/ per square yard ? 
 
 2. One side of a field in the shape of a trapezoid measures 180 yd., 
 the other 200 yd. How many acres in the field if the perpendicular dis- 
 tance between the sides is 75 yd. ? 
 
 3. What is the area of an oblique parallelogram, the perpendicular 
 height being 24 ft. and the length 56 ft. ? 
 
 TRIANGLES 
 
 71. Right Triangles. Take a piece of paper 4 inches square. 
 Fold and cut along the diagonal. You now have two right 
 
MEASURES OF AREA 
 
 103 
 
 angles. Take one of them and fold so that the vertex of the 
 upper angle will meet the vertex of the right angle. Crease 
 and tear. Place the upper part 
 upside down and with its oblique 
 side next to the oblique side of the 
 larger piece. What have you made ? 
 How does the base of the rectangle 
 compare in length with that of the 
 triangle ? The height of the rec- 
 tangle with the altitude of the 
 triangle ? 
 
 A right triangle is equivalent to a rectangle on the same base 
 with half the altitude. 
 
 72. Isosceles Triangle. Take a 4-inch paper square. 
 From the middle point of the upper edge draw lines to the 
 
 lower corners. Fold along the lines, 
 crease, and tear. The large triangle 
 remaining is an isosceles triangle. 
 Why? 
 
 Fold the upper vertex down so it 
 will just touch the middle point of 
 the base. Crease and tear. Con- 
 nect the middle points of the two 
 bases by a straight line. Fold along 
 this line, crease, and tear. Turn the two pieces upside down 
 and place one on each side of the triangular piece, with 
 oblique edges together. What have you formed ? Compare 
 base of the triangle with that of the original triangle. Com- 
 pare their altitudes. 
 
 An isosceles triangle is equivalent to a rectangle on the same 
 base and with half the altitude. As an equilateral triangle is 
 isosceles, the same would hold true. 
 
104 ELEMENTS OF BUSINESS ARITHMETIC 
 
 73. Scalene Triangle. Cut a paper triangle, no two sides of 
 which are equal, and not a right triangle. Fold over the 
 vertex to meet the base, and so that the line of the fold will 
 
 be parallel with the base. Crease 
 and open. Fold again along a line 
 drawn perpendicular from the ver- 
 tex to the base. Crease and tear. 
 Tear both new triangles at first fold. 
 Place the oblique sides of the two 
 parts of the new triangles together. 
 Then place the two resultant rec- 
 tangles side by side. Compare the entire base with that of 
 the original triangle. Compare their altitudes. A scalene 
 triangle is equivalent to a rectangle on the same base and with 
 one half the altitude. 
 
 As the same truth is seen, in the above, to hold good for 
 equilateral, isosceles, and scalene triangles, whether right, 
 obtuse, or acute-angled, the general statement may be made 
 that anT/ triangle is equivalent to a rectangle on the same base 
 and with half the altitude, 
 
 74. Other Polygons. All other polygons may be resolved 
 into rectangles or triangles. Finding the area of the parts, 
 the whole may be found. For example, among the more 
 common geometrical figures, a trapezium may be resolved 
 into two triangles ; a hexagon and octagon may be resolved 
 into six and eight triangles focusing at the center. 
 
 PROBLEMS 
 
 1. How many square inches in a right triangle, if the base is 18 in. and 
 the altitude 24 in. ? 
 
 2. The base of an isosceles triangle is 12 ft. and the altitude is 16 ft. 
 What is the area? 
 
 3. How many square feet in a scalene triangle, with the base 15 ft. 
 and altitude 20 ft. ? 
 
MEASURES OF AREA 105 
 
 4. A triangular field is 40 rd. along its base and 26 rd. along the alti- 
 tude. What is it worth at $72 per acre? 
 
 5. What is the area of a hexagon composed of equilateral triangles, 
 each side of which is 8 in. ? (Use the nearest inch as altitude.) 
 
 6. What is the cost of painting a church steeple, the base of which is 
 an octagon 8 ft. on each side, and whose slant height (altitude) is 90 ft., 
 at 25^ per square yard? 
 
 7. I have a triangular lawn that I wish to sod. The base is 25 yd. 
 and the altitude is 24 yd. What will it cost at $1.25 per square rod? 
 
 8. A hexagonal silo is 12 ft. on a side and the distance from the 
 middle point of a side to the center is 10.4 ft. What is the area of 
 the bottom ? 
 
 9. I have a plot of ground in the form of a trapezium. One of the 
 diagonals measures 42 yd., and the altitudes of the two triangles measure 
 20 yd. and 18 yd. respectively. What is its area ? 
 
 75. Ratio of Circumference of Circle to its Diameter. Pro- 
 cure a well-cut circular disk, or a wheel. Measure carefully 
 its diameter. Mark distinctly some point on the circumfer- 
 ence. Starting with that point at the end of a line drawn on 
 some flat surface, roll the disk or the wheel along that line 
 until the marked point on the circumference again touches 
 the line. Measure the length of the line thus marked off. 
 This will be the length of the circumference. Divide this 
 length by the length of the diameter. The quotient or ratio 
 will be found to be nearly 3^. In other words, the circum- 
 ference of a circle is 3^ times its diameter. For greater 
 accuracy, the ratio of 3.1416 is used. In mathematics, its 
 symbol is the Greek letter tt (pronounced pi). For ordinary 
 purposes 3^ is sufficiently exact. 
 
 76. Area of a Circle. Fold a rectangular piece of paper, 
 5 by 6 inches, so that the long edges meet ; crease but do not 
 open. Fold so that the short edges meet ; crease but do not 
 open. Fold so that the folded edges will meet, and crease. 
 
106 ELEMENTS OE BUSINESS ARITHMETIC 
 
 Mark off two inches from the point on both the edges, and 
 midway between the edges. Connect the three points thus 
 made by a uniformly curved line. Cut through the paper 
 on this line. Open the paper, and a circle is shown. Cut 
 through all the creased lines, forming eight equal sectors. 
 Arrange seven of these sectors in a row, with alternate points 
 
 and arcs along 
 
 C "7^^ ^TC 7\ ^ one side. Fold 
 
 the oblique 
 
 V edges of the 
 
 \ eighth sector 
 
 _ 3 together; crease 
 
 The Sectors of a Circle ^^^ ^^^* Flace 
 
 a half on each 
 end of the row. What have you approximately formed 
 (see figure)? What part of the circumference forms the 
 base ? What is the altitude ? If the sectors were sixteenths, 
 the resultant would be more nearly a perfect rectangle ; if 
 thirty-seconds, still more nearly, etc. We may conclude, 
 then, that : 
 
 A circle is equivalent to a rectangle with one half the circum- 
 ference as a base, and the radius as its altitude. 
 
 Note. — The geometrical formula for the area of a circle is ttt^, or the 
 square of the radius multiplied by 3^. 
 
 PROBLEMS 
 
 1. The area of a circle 14 in. in diameter with a circumference of 
 44 in. is equivalent to a rectangle with a base of inches Q the cir- 
 cumference), and an altitude of inches (radius). 
 
 2. What are the dimensions of a rectangle that may be formed from a 
 circle 7 in. in diameter and a circumference of 22 in. ? 
 
 3. The diameter of a circle is 21 in., the circumference 66 in. Find 
 the area. 
 
 4. Find the area of a circle with a diameter of 8 in. 
 
MEASURES OF AREA 107 
 
 OTHER SURFACES 
 
 If a paper 6 by 8 inches were rolled into a cylinder with 
 the long edges meeting, what would be the circumference of 
 the cylinder ? What the length ? 
 
 The surface of a cylinder is equivalent in area to that of a 
 rectangle having the length of the cylinder as a hase^ and its cir- 
 cumference as the altitude. 
 
 The surface of a pyramid consists of the rectangular base 
 and triangular sides. The triangles have a common altitude, or 
 slant height, hence the lateral area of a pyramid is equivalent to 
 that of a rectangle having the perimeter of the pyramid as a base, 
 and one half the slant height as the altitude. 
 
 The lateral area., likewise, of a cone is equivalent to that of a 
 rectangle having the circumference of the cone as a hase^ and 
 one half the slant height as the altitude. 
 
 The surface of a sphere is equivalent to a rectangle having the 
 circumference of the sphere for a hase^ and its diameter for the 
 altitude. 
 
 GENERAL PROBLEMS 
 
 1. What is the area of the surface of a globe 6 in. in diameter ? 
 
 2. What is the entire surface of a cylinder whose diameter is 6 in., 
 height 2 f t. ? 
 
 3. A cylindrical cistern is 10 ft. deep and 7 ft. in diameter. What 
 will it cost to cement it, at 35 /' a square yard ? 
 
 4. In a warehouse there are six hoppers of pyramidal shape, 6 ft. 
 high and 8 ft. square. How many square feet on their surfaces? 
 
 5. What will it cost, at 24- ^ per square yard, to paint a spire of conical 
 shape, if the slant height is 75 ft. and the diameter of the base is 16 ft. ? 
 
 6. A rectangular park is 80 by 160 ft. In the left- and right-hand 
 corners at one end are flower beds which are 5 ft. wide, and extend 15 ft. 
 both ways from the corners. In the left-hand corner at the other end is 
 a triangular bed, the altitude of which is 12 ft. and the base 16 ft. In 
 the right-hand corner is a bed in the form of a trapezoid 7 ft. broad, 
 whose sides are respectively 24 and 18 ft. In the center is a pond 42 ft. 
 in diameter, with a walk 3 ft. wide extending from one side of the park 
 
108 ELEMENTS OF BUSINESS ARITHMETIC 
 
 to and around the pond. The remaining space is a lawn. How many 
 square feet in the lawn? In each bed? In the pond? In the walk? 
 
 7. A street in a city is 100 ft. wide and 1 mi. long. What will it 
 cost to have it paved with brick, 4 by 8 in., at $ 6.20 per M. ? 
 
 8. A horseman has a field 20 rd. square in which he puts a circular 
 track 50 ft. wide. What is the area of the track ? Its length ? 
 
 9. A man owned a lot 80 rd. square. A corporation bought the right 
 of way for an electric line at $195 per acre. One fence inclosing the 
 track was to commence at the northwest corner of the lot and run to a 
 point 20 rd. east of the southwest corner ; the second fence to commence 
 at a point 15 rd. south of the northeast corner and run to a point 12 rd. 
 east of the first. What did the right of way cost? How many posts, 
 set 8 ft. apart, will it take for the fences? Find cost of the fencing at 
 65 ^ a rod. 
 
 10. A room is 15 by 18 ft. How many yards of carpet, 27 in. wide, 
 will it take to cover it, allowing 9 in. for matching? 
 
 11. A building is 40 by 80 ft. and 15 ft. high. The altitude of the 
 gable ends is 14 ft., and the roof projects 2 ft. over each side and end. 
 What will it cost to cover this roof with shingles, laid 4^ in. to the 
 weather, at |4.50 per M? What will it cost to paint the roof and 
 the body of the building, at 25 f per square yard ? 
 
 12. A buys a piece of land 20 rd. square, paying $250 an acre. He 
 divides it into lots 46 by 100 ft., after deducting for a street 50 ft. wide 
 around the piece and an alley 30 ft. wide through the center. The ex- 
 pense of setting out trees, grading, etc., was $ 754.75. If he sold the 
 corner lots for % 250 apiece, and the others for % 200 apiece, what did he 
 gain by the transaction ? 
 
 13. A house has 8 rooms, with dimensions as follows : 3 rooms 15 by 
 18 ft., 2 rooms 10 by 14 ft., and 3 rooms 12 by 16 ft., and 8, 9, and 10 ft. 
 high respectively. There are 24 windows and 8 doors. Find the cost of 
 lath to cover them at 30 ^ a bundle. How much lime, sand, and hair will 
 it take to plaster the house ? How much stucco would it take, counting 
 900 lb. to 100 square yards ? 
 
 14. The first three rooms in Problem 13 are to be covered with 18-inch 
 paper, the next three with 22-inch paper, and the last two with 30-inch 
 paper. Allowing 20 sq. ft. for openings, how many rolls of each kind of 
 paper will it take to cover the rooms, if the doors and windows are 
 divided equally among the rooms ? 
 
VIII 
 
 MEASURES OF VOLUME 
 
 77. Units of Volume. Volume is the quantity of space oc- 
 cupied. Any given space has length, breadth, and thick- 
 ness. Therefore volume has to do with these three 
 dimensions. Volume is measured by cubic units, i.e. units 
 whose length, breadth, and thickness are the same. 
 
 A rectangular body whose dimensions are equal is called 
 a cube. These cubic units correspond to units of length, viz. 
 a cubic inch, a cubic foot, and a cubic yard. 
 
 The standard unit is the cubic yard. From the fact that 
 volume is measured by cubic units, measures of volume are 
 called cubic measure. 
 
 Table of Cubic Units 
 There are : 
 
 1728 cubic inches (cu. in.) in 1 cubic foot (cu. ft.). 
 27 cubic feet in 1 cubic yard (cu. yd.). 
 
 EXERCISES 
 
 1. Procure or make a number "of inch cubes. Through touch and 
 sight, seek to develop an accurate concept of a cubic inch. Cut, without 
 measurement, an inch cube from any available material. Test its accuracy. 
 Practice. 
 
 2. Practice estimating the volume in cubic inches of a box, a book, 
 the desk top, a piece of wood, etc. Use a ruler for testing the accuracy 
 of your judgment. 
 
 3. By pasting the edges of six one-foot squares of pasteboard together, 
 a cubic foot may be made. 
 
 109 
 
110 
 
 ELEMENTS OF BUSINESS ARITHMETIC 
 
 
 s. \. 
 
 4. Develop a concept of the one-foot cube in the same manner as 
 above. 
 
 5. Estimate the volume in cubic feet of large boxes, rooms, halls; 
 foundations of buildings, etc. Test correctness with a ruler. Practice for 
 accuracy in judgment of the number of cubic feet in different volumes. 
 
 6. Measure oif a cubic yard from the corner of a room. Fix as accu- 
 rate a concept of it as possible. 
 
 7. Estimate volume of various solids in cubic yards. Test in practice. 
 
 78. Finding Volume. Finding the volume of a solid con- 
 sists in finding the cubic units of space occupied by it. In 
 finding the volume of rectangular solids, we first think or 
 
 image tlie cubic 
 unit, then the 
 number of these 
 units along one 
 edge or in one 
 row, then the 
 number of such 
 rows forming a 
 layer, and finally 
 the number of 
 layers of cubic 
 units in the solid. The number of cubic units in one row is 
 the same as the number of linear units along an edge or one 
 dimension; the number of such rows is the same as the 
 number of linear units along an edge forming another 
 dimension ; and the number of layers is equal to the linear 
 units along the edge forming the third dimension. We say, 
 then, that the length indicates the number of units in one 
 row, the breadth the number of rows, and the thickness the 
 number of layers. 
 
 To find the volume of any rectangular solid, multiply the 
 number of cubic units in one row by the number of rows form- 
 ing one layer^ and that product by the number of such layers. 
 
 (a) A cubic unit. (6) A row of four cubic units, (c) A 
 layer of 8 cubic units, (d) A solid of 16 cubic units. 
 
MEASURES OF VOLUME 111 
 
 Thus, a solid 6 in. long, 5 in. wide, and 4 in. high has 6 cu. 
 in. in each row, 30 cu. in. in five rows, or one layer, and 120 cu. 
 in. in the four layers of the solid. This may be expressed as 
 follows : 
 
 6 cu. in. X 5 X 4 = 120 cu. in. 
 
 PROBLEMS 
 
 1. A box is 10 by 20 inches and 4 inches deep. How many cubic 
 inches in each layer ? How many in the box ? 
 
 2. A box is 6 feet long, 4 feet wide, and 4 feet deep. How many 
 layers of cubic feet? How many cubic feet in the box? 
 
 3. How many cubic feet of water will a tank hold if it is 12 by 6 feet 
 and 3 feet deep ? 
 
 4. What will it cost to dig a cellar 22 by 16 feet and 10 feet deep, at 
 $1.35 a load (cubic yard)? 
 
 5. How many cubic feet in a piece of timber 6" x 8" x 40' ? 
 
 6. How many cubic feet of wheat in a bin 6 by 12 by 8 feet high ? 
 
 7. A schoolroom is 60 feet long, 40 feet wide, and 12 feet high. If 
 there are 120 pupils, how many cubic feet of air to each pupil? How 
 often must the air be changed to give fresh air, if a pupil requires 2400 
 cubic feet of air per hour? 
 
 8. How many cubic yards of earth must be removed in digging a 
 trench for the walls of a building, if the building is 40 by 100 feet, the 
 walls 2 feet wide, and the trench 3 feet deep? 
 
 9. I wish to build a road. The grade would cost $ 80 a mile. The 
 gravel is to be spread 9 inches deep and 16 feet wide. What will the 
 road cost per mile, if the gravel is worth $ 2 per cubic yard? 
 
 10. My lawn is 70 by 120 feet. What will be the cost for dirt to raise 
 it 18 inches at 50^ a load? 
 
 11. The abutments of a bridge are 30 feet long and 6 feet wide and 
 15 feet high. Find the pressure exerted, counting 160 pounds to the 
 cubic foot. 
 
 12. A man has a pond 5 by 20 rods. The ice is 9 inches deep. Count- 
 ing 1 of the space for sawdust, and ^^r loss of ice in handling, what will be 
 the height of a building 80 feet long and 40 feet wide that will hold it ? 
 
112 ELEMENTS OF BUSINESS ARITHMETIC 
 
 79. Cubes and Roots. A cube as defined in Sec. 77 is a 
 rectangular body whose three dimensions are equal. That 
 is to say, the ' number of cubic units along one edge is the 
 same as the number of rows and the number of layers of 
 such units. In finding the volume of such a body, the same 
 number will, therefore, be used as a factor three times, and 
 the resultant volume is said to be expressed by the cube of 
 the number, or the number is said to be raised to the third 
 power. Thus, the cube of three may be expressed : 3 cubic 
 units X 3 X 3 = 27 cubic units, or, without reference to defi- 
 nite units ; as, 3 x 3 x 3 = 27. This may also be expressed 
 as 33= 27. 
 
 Finding the volume of a cube when the number of cubic 
 units along one edge only is given, is called cubing a number. 
 When the volume is known, and the problem is to find the 
 number of cubic units along one edge, the process is known 
 as finding the euhe root. Thus, the cube of 2- is 8, or of 2 
 cu. ft. is 8 cu. ft. The cube root of 8 is 2, or the cube root 
 of 8 cu. ft. is 2 cu. ft. 
 
 EXERCISES 
 
 1. The cube of three feet has how many cubes in each row? How 
 many rows? How many layers? How many cubic feet in all? 
 
 2. The cube of 5 feet contains how many rows ? How many cubic 
 feet? 
 
 3. A cube of 9 inches has how many cubic inches ? 
 
 4. A cube containing 125 cu. ft. has how many cubic feet along one 
 edge? 
 
 5. A cube containing 27 cubic inches has how many cubic inches in 
 each row ? The cube root of 27 is ? 
 
 6. The cube of 4 is? Of 6 is? Of 7 is? Of 8 is? 
 
 7. The cube of 10 is ? Of 11 is ? Of 12 is ? Of 15 is ? 
 
 8. The cube of 20 is? Of 25 is? Of 30 is? Of 50 is? Of 40 is? 
 Of 75 is? 
 
 9. The cube root of 64 is? Of 27 is? Of 125 is? 
 
MEASURES OF VOLUME 113 
 
 10. The cube r9ot of 1000? Of 1728? Of 1331? Of 3375? 
 
 11. The cube root of 8000? Of 15,625? Of 27,000? Of 1,000,000? 
 
 Memorize the following table : 
 
 13=1 
 
 73 = 343 
 
 153=3375 
 
 23=8 
 
 83 = 512 
 
 203 = 8000 
 
 33=27 
 
 93 = 729 
 
 253=15,625 
 
 43=64 
 
 103 = 1000 
 
 503 = 125,000 
 
 53 = 125 
 
 113 3= 1331 
 
 1003 = 1,000,000 
 
 63=216 
 
 123 = 1728 
 
 
 APPLICATIONS OP VOLUME 
 
 80. Wood Measure. Volume measurement forms a con- 
 venient way of finding the amount of many things bought 
 and sold in bulk. Among the simplest of its applications 
 is wood measure. For this purpose a special unit is used. 
 Wood used for fuel, when first prepared for sale, is usually 
 cut 4 feet long. A pile of wood, 4 feet high and 8 feet long, 
 is called a cord. There would be, then, 4 layers of 32 cubic 
 feet, or 128 cubic feet in one cord. Rough timber cut in 
 4-foot lengths is called cordwood. A cord foot is 4 feet 
 wide, 4 feet high, and 1 foot long, and contains 16 cubic 
 feet. There are 8 cord feet in a cord. 
 
 81. Stone Measure. Stone is divided into two classes at 
 the quarry, viz. dimension stone and rubble. The first con- 
 sists of pieces which are quarried in regular shapes and to 
 a fixed size. This class of stone is generally sold by the 
 cubic foot. 
 
 Rubble is the waste from quarrying the larger stones, and 
 includes pieces of various sizes and shapes. It is usually 
 sold by the carload, or in small quantities by the perch or 
 cord, and in some places by the ton. 
 
 In the wall, dimension stone is generally measured by the 
 square foot. Rubble is measured by the perch, which varies 
 
114 ELEMENTS OF BUSINESS ARITHMETIC 
 
 from 16 to 25 cu. ft., usually 24f cu. ft.; or the cord, of 128 
 cu. ft., or the cubic yard. About I of the volume, is usually 
 allowed for mortar, which is commonly mixed in the propor- 
 tion of 3 parts of sand to 1 of lime. 
 
 Walls of the same thickness and height are thought of 
 as one long wall, using the outside measurements of the 
 building to determine its length. No allowance is made for 
 openings in estimating cost of labor, unless openings are 
 many or very large, when | the volume is sometimes 
 deducted. 
 
 82. Bricklaying. In estimating the number of bricks for 
 a given foundation or building, the walls are considered as 
 one wall, with the outside perimeter of the four walls as the 
 length. The height being known, the area of the side of 
 such a wall is then found. If the wall is to be a single brick 
 in thickness, 7J bricks will be required for each square foot ; 
 if 2 bricks thick, 15 ; and if 3 bricks thick, 22|- bricks. In 
 estimating the cost of labor, allowance for openings is not 
 usually made unless these are many and large. When 
 deduction is made for openings, it is for one half their 
 volume. 
 
 The above estimates are based on the size of the common 
 brick, which is about 2x4x8 inches. As the bricks are 
 laid flat, the exposed side of each brick would be 2 x 8 inches, 
 or 16 square inches. Nine bricks, if laid without mortar, 
 would be required for one square foot (16 sq. in. x 9 = 144 
 sq. in.). Allowing about -^ for mortar, 7 J bricks would be 
 required. 
 
 Bricks are now made in such a variety of sizes that it is 
 often necessary to figure the number to a square foot, from 
 the known size of the brick to be used in the particular 
 building. 
 
 From 1 to IJ barrels of lime, according to quality, and |- 
 
MEASURES OF VOLUME 115 
 
 of a cubic yard of sand are required for each 1000 bricks. 
 No allowance, as a rule, is made for openings in estimating 
 labor. Four bricklayers and one tender will lay from 7200 
 to 8000 bricks per day. 
 
 PROBLEMS 
 
 1. How many cords of wood in a pile 24 ft. long, 8 ft. high, and 4 ft. 
 wide? 
 
 2. How many cords of stone in a wall 8 by 60 ft. and 2 ft. thick? 
 
 3. What will a pile of wood 4 by 6 by 48 ft. cost, at $4 per cord? 
 
 4. What will it cost to build the stone foundations for a barn, 20 by 
 40 ft., if the wall is 6 ft. high and 2 ft. thick, at $ 15 per cord? 
 
 5. If a cellar is 18 by 36 ft. and the walls 10 ft. high, how many 
 bricks will it -take to build it two bricks thick? 
 
 6. How many bricks would be necessary for the foundations of a 
 building 25 by 30 ft., if it is 4 ft. high and 3 bricks thick? 
 
 7. The foundation of a house is 3 bricks thick and 3 ft. high. The 
 total length, if extended, is 175 ft. How many bricks will it take? How 
 much lime and sand ? What will be the cost of the bricks at $ 9.50 per 
 M ? How long will it take four bricklayers and one tender to lay the 
 foundation ? 
 
 8. Commencing at the northwest corner, a wall extends east 30 ft., 
 then south 10 ft., then west 10 ft., then south 10 ft., then east 10 ft., 
 then south 10 ft., then west 15 ft., then north 5 ft., then west 15 ft., then 
 north to the point of starting. The wall is 3 bricks thick and 4 ft. high. 
 How many bricks will it take to lay it? How much sand and lime? 
 
 9. In the floor plan on p. 88, how many cords of stone would it 
 take to lay the foundation 18 in. thick and 7 ft. high? How many cubic 
 yards? How many perch of 24| cu. ft.? How much lime and sand 
 would it take? How many bricks would it take for a wall 3 bricks 
 thick? What will be the cost of brick at $ 8.75 per M? What will be 
 the cost of lime and sand for either brick or stone wall, the lime at 
 $ 1.00 per barrel ; sand at 75^ a load of one cubic yard? 
 
 83. Board Measure. Lumber is measured by the hoard 
 foot. One square foot of surface on a board one inch thick 
 
116 ELEMENTS OF BUSINESS ARITHMETIC 
 
 or less is called a board foot. A board foot is added for 
 each added inch in thickness. 
 
 According to a method used by large dealers, the board 
 12' long and 12' ' wide is used as a basis for computation. 
 The total number of board feet in the given number of 
 boards is increased or decreased by such a fraction of the 
 result as the length is of 12' and the width is of 12". A 
 board 12' long and 12" wide would contain, manifestly, 12 
 board feet. A board 6" wide would contain ^ as much, or 
 6 board feet; 8" wide, f of 12 board feet, or 8 board feet, etc. 
 
 Twelve-foot boards, then, contain as many hoard feet as the 
 hoard is inches wide. Thus, 24 such boards 8" wide would 
 contain 24 times 8 board feet, or 1092 board feet ; if 14-foot 
 boards, they would contain \ more, or 1274 board feet ; if 
 16' long, \ more, or 1456 board feet. The width of boards 
 is counted as inches and half inches. If a board is not an 
 even size, it is considered as the next smaller half inch. 
 
 84. Log Measure. For estimating the amount of lumber 
 that may be cut from a given log, or for estimating the value 
 of logs to be sold, it is necessary first to find a mean diameter. 
 This is found by taking \ of the sum of the diameters of the 
 two ends of the log, inside the bark. 
 
 Slabs are taken off to get the log into rectangular form 
 and wide enough to be of use. 4" of the diameter (2" on 
 each of the four sides) is deducted for slabs, and \ of the 
 solid contents for waste in sawing (kerf) and in edging. 
 Thus, a log with an average diameter of 14", 12' long, would 
 form a rectangular log 10" by 10", 12' long, and would con- 
 tain 100 board feet. Allowing \ for kerf and other waste, 
 the board product would be 75 board feet. If the length of 
 the log was 16', there would be \ more, or 100 board feet. 
 Computations for other lengths are made in the same way. 
 (Sec. 83.) 
 
MEASURES OF VOLUME 
 
 117 
 
 The accompanying diagrams show two of the many ways 
 of sawing logs. 
 
 85. Lumberman's Reference Table. The following table, 
 similar in form to the Lumberman's Reference Table, may 
 be used for drill in finding mentally the number of board 
 feet in boards, joists, etc., of various dimensions, using the 
 table for verifying results. Notice that if the answer con- 
 tains a fraction less than |-, it is dropped, while if it is more 
 than J, another foot is added. 
 
 PROBLEMS 
 
 1. How many board feet in 50 boards 8 inches wide and 12 feet long? 
 
 2. How many board feet in 60 boards 10 inches wide and 12 feet long? 
 
 3. How many board feet in 10 boards 2 x 4-12 ? 
 
 4. How many board feet in 20 boards 3 x 6-16 ? 
 
 5. How many board feet in 60 boards 1 x 10-20 ? 
 
 6. How many board feet in 10 boards 6 x 6-24 ? 
 
 7. The Koontz Lumber Company sold the following bill of lumber : 
 20 pc. 2x6-10, 15 pc. 4x4-14, 25 pc. 4x4-16, 40 pc. 2x6-14, 75 pc. 
 2 x 10- 8 at $20 per M ; 100 po. 2 x 10-20, 50 pb. 2 x 6-22, 45 pc. 2 x 8-18, 
 10 pc. 2 X 12-16, 75 pc. 2 x 8-18, at $22 per M ; 2050 posts at $24 per C ; 
 25 pc. 6x8-24 at $20 per M; 125 bunches shingles at $4.50 per M. 
 Find the amount of the bill. 
 
118 
 
 ELEMENTS OF BUSINESS ARITHMETIC 
 
 Eeference Table 
 
 BOARD MEASURE, JOISTS, SCANTLING, AND TIMBER 
 
 Size 
 
 12 
 
 14 
 
 16 
 
 18 
 
 20 
 
 22 
 
 24 
 
 26 
 
 28 
 
 30 
 
 2x4 
 
 8 
 
 9 
 
 11 
 
 12 
 
 13 
 
 15 
 
 16 
 
 17 
 
 19 
 
 20 
 
 2x6 
 
 12 
 
 14 
 
 16 
 
 18 
 
 20 
 
 22 
 
 24 
 
 26 
 
 28 
 
 30 
 
 2x8 
 
 16 
 
 19 
 
 21 
 
 24 
 
 27 
 
 29 
 
 32 
 
 35 
 
 37 
 
 40 
 
 2x10 
 
 20 
 
 . 23 
 
 27 
 
 30 
 
 33 
 
 37 
 
 40 
 
 43 
 
 47 
 
 50 
 
 2x12 
 
 24 
 
 28 
 
 32 
 
 36 
 
 40 
 
 44 
 
 48 
 
 52 
 
 56 
 
 60 
 
 3x6 
 
 18 
 
 21 
 
 24 
 
 27 
 
 30 
 
 33 
 
 36 
 
 39 
 
 42 
 
 45 
 
 3x8 
 
 24 
 
 28 
 
 32 
 
 36 
 
 40 
 
 44 
 
 48 
 
 52 
 
 56 
 
 60 
 
 3x10 
 
 30 
 
 35 
 
 40 
 
 45 
 
 50 
 
 55 
 
 60 
 
 65 
 
 70 
 
 75 
 
 3x12 
 
 36 
 
 42 
 
 48 
 
 54 
 
 60 
 
 66 
 
 72 
 
 78 
 
 84 
 
 90 
 
 3x14 
 
 42 
 
 49 
 
 56 
 
 63 
 
 70 
 
 77 
 
 84 
 
 91 
 
 98 
 
 105 
 
 4x4 
 
 16 
 
 19 
 
 21 
 
 24 
 
 27 
 
 29 
 
 32 
 
 35 
 
 37 
 
 40 
 
 4x6 
 
 24 
 
 28 
 
 32 
 
 36 
 
 40 
 
 44 
 
 48 
 
 52 
 
 56 
 
 60 
 
 4x8 
 
 32 
 
 37 
 
 43 
 
 48 
 
 53 
 
 59 
 
 64 
 
 69 
 
 75 
 
 80 
 
 6x6 
 
 36 
 
 42 
 
 48 
 
 54 
 
 60 
 
 66 
 
 72 
 
 78 
 
 84 
 
 90 
 
 6x8 
 
 48 
 
 56 
 
 64 
 
 72 
 
 80 
 
 88 
 
 96 
 
 104 
 
 112 
 
 120 
 
 6x10 
 
 60 
 
 70 
 
 80 
 
 90 
 
 100 
 
 110 
 
 120 
 
 130 
 
 140 
 
 150 
 
 8x8 
 
 64 
 
 75 
 
 85 
 
 96 
 
 107 
 
 117 
 
 128 
 
 139 
 
 149 
 
 160 
 
 8x10 
 
 80 
 
 93 
 
 107 
 
 120 
 
 133 
 
 147 
 
 160 
 
 173 
 
 187 
 
 200 
 
 8x12 
 
 96 
 
 112 
 
 128 
 
 144 
 
 160 
 
 176 
 
 192 
 
 208 
 
 224 
 
 240 
 
 10x10 
 
 100 
 
 117 
 
 133 
 
 150 
 
 167 
 
 183 
 
 200 
 
 217 
 
 233 
 
 250 
 
 10x12 
 
 120 
 
 140 
 
 160 
 
 180 
 
 200 
 
 220 
 
 240 
 
 260 
 
 280 
 
 300 
 
 12x12 
 
 144 
 
 168 
 
 192 
 
 216 
 
 240 
 
 264 
 
 288 
 
 312 
 
 336 
 
 360 
 
 12x14 
 
 168 
 
 196 
 
 224 
 
 252 
 
 280 
 
 308 
 
 336 
 
 364 
 
 392 
 
 420 
 
 14x14 
 
 196 
 
 229 
 
 261 
 
 294 
 
 327 
 
 359 
 
 392 
 
 425 
 
 457 
 
 490 
 
 8. How many board feet in a log 2 ft. in diameter at one end and 
 2' 6" at the other, and 12 ft. long? 
 
 9. A tree is 60 in. in circumference at the base and 48 in. at the top, 
 and 42 ft. high. How many feet of lumber may be sawed from it? 
 
 10. A man built a close board fence 4 ft. high around a lot 120 yd. 
 square. The boards were nailed in upright position to two railings 2 by 
 
MEASURES OF VOLUME 119 
 
 4 in., and the posts were set 8 ft. apart. He paid |20 per M for the 
 railings, $30 per C for the posts, and |25 per M for the boards. What 
 did the material cost ? 
 
 11. How many feet of lumber will it take to board a barn 40 by 80 
 ft. and 25 ft. high, if the distance from the girders to the ridge of the 
 roof is 18 ft. ? Cost at $ 32 per M ? 
 
 12. A lot 50 by 160 ft. is inclosed by a picket fence. The pickets are 
 4 ft. long, 3 in. wide, and 1 in. thick, and are placed 3 in. apart. The 
 posts are placed 8 ft. apart, 2 x d's being used at top and bottom for railing, 
 and a baseboard 10 in. wide. What will the fence cost, if the posls are 
 $20 per C, the lumber $22 per M, and pickets $4.25 per C ? 
 
 13. Mr. A. H. Ward buys the following bill for a small house : 1 pc. 
 6x8-20, 2 pc. 2x8-26, 1 pc. 2 x 8-20, 4 pc. 2x8- 18 @$26; 45 pc. 
 2 X 8 - 14, 34 pc. 2 X 8 - 12, 44 pc. 2 x 6 - 12, 38 pc. 2x6- 12, 3 pc. 2 x 6 - 
 18@$23; 130pc. 2x4-18, 30 pc. 2x4-16, 30 pc. 2x4-14, 50 pc.2x 
 4 - 12 @ $28 ; 1700 ft. # 1 sheathing, 10,000 ft. # 1 fence flooring, 2000 ft. 
 # shiplap, 2000 ft. # 2 fence flooring, 700 ft. straight grain flooring, 1360 
 ft. cedar siding @ $24; 320 ft. polished brass ceiling, 20 pc. 1x6-1.5, 
 G. P. finishing @ $ 28; 5 rolls best paper, 2500 ft. @ $ 1.50 per C ; 7 win- 
 dows, 10x16 @ $1.80; 3 doors, 2 ft., 8 in. by 6 ft. 8 in. @ $5; 10 bbl. 
 lime @ $1.15; 5000 hard brick @ $9.25 per M; 200 ft. 3^ in. crown 
 molding @ $ 2..50 per hundred linear feet ; 200 ft. 2-in. crown molding @ 
 $ 2.50 per hundred linear feet. What is the amount of the bill ? 
 
 MEASURES OF BULK 
 
 86. Dry Measure. For measuring small solids, such as 
 grains, fruits, vegetables, etc., vessels are used whose volume 
 or capacity is known. The volume of liquids must be ob- 
 tained in the same way. Inasmuch as the solid form and the 
 varying shape of grains, fruits, etc., leave unoccupied spaces 
 between the solids, while liquids are compact, different unit 
 measures have been established. The units for measuring 
 solids form what is known as dry measure. 
 
 The established unit for dry measure is the bushel. It is 
 usually cylindrical in form, 18^ inches in diameter, and 8 
 inches deep, containing 2150.42 cubic inches. For grain, 
 shelled corn, etc., these measures are filled to the level, called 
 
120 ELEMENTS OF BUSINESS ARITHMETIC 
 
 "stricken measure." For apples, potatoes, ear corn, etc., 
 the measure is "heaped," and the bushel is supposed to 
 contain 2747.71 cubic inches. 
 
 For finding the capacity, in bushels, of a grain bin, granary, 
 or elevator, the bushel expressed in cubic feet is more con- 
 venient. A cubic foot (1728 cu. in.) is slightly more than 
 .8 of a bushel (2150.42 cu. in.). To find the capacity of a 
 bin, then, multiply .8 bushel by the number of cubic feet in 
 the bin. For greater accuracy add |- of a bushel for each 
 100 cubic feet. If capacity in heaped bushels is desired, 
 multiply .63 bushels by the cubic feet and correct as above 
 for greater accuracy. 
 
 Thus, a bin 8 ft. wide, 10 ft. high, and 20 ft. long would 
 contain 1600 cu. ft., and .8 bu. x 1600 = 1280 bu. Adding 
 ■^ of a bushel, the contents would be 1285 bu. 
 
 To estimate the number of bushels of shelled corn in a 
 crib of ear corn, if high grade and dry, take J as many 
 bushels as there are cubic feet. 
 
 Table 
 
 There are: 2 pints (pt.) in 1 quart (qt.) 
 8 quarts in 1 peck (pk.) 
 
 4 pecks in I bushel (bu.) 
 
 Note. — The practical worth of a knowledge of dry measure consists 
 not alone in knowledge of its units, but in power to estimate bulk in 
 terms of units. Actual measures should be made in the classroom, and 
 practice given in estimating bulk, verifying estimates. 
 
 For determining the number of bushels of different grains by weights 
 see p. 139. 
 
 87. Liquid Measure. For measuring liquids, the standard 
 unit is the gallon. It contains 231 cu. in. A cylindrical 
 vessel 7 in. in diameter and 6 in. high contains a standard 
 gallon. 
 
MEASURES OF VOLUME 121 
 
 For finding the capacity of cisterns, tanks, etc., the num- 
 ber of cubic inches may be divided by 231 cu. in. Since there 
 are 1728 cu. in. in a cubic foot, there would be 7.4805"^ gallons 
 in a cubic foot. A shorter method for finding the number 
 of gallons in a tank or cistern would be, then, to multiply 1^ 
 gallons by the number of cubic feet, and subtract -^^^ (^ of 
 Y^) of itself. This will give the correct result to a fraction 
 of a gallon. Thus, a cistern 6 x 8 x 15 ft. would contain 
 720 cu. ft. 7J gal. x 720 = 5400 gal. ^ of j^-^ of 5400 
 gal. = J of 54 gal., or 131 gal. 5400 gal. - 131 gal. = 53861 
 gal. 
 
 Casks, hogsheads, pipes, tuns, butts, tierces, carboys, etc., 
 are indefinite standards, and their capacity is determined by 
 measurement, and it is usually stamped on them. Barrels of 
 oil and. liquors are also indefinite. The barrel of 31 J gal. 
 was formerly a standard, and is still so used for some purposes. 
 
 Table 
 
 There are: 4 gills (gi) in 1 pint (pt.) 
 2 pints in 1 quart (qt.) 
 
 4 quarts in 1 gallon (gal.) 
 
 88. Apothecaries' Liquid Measure. For measuring drugs, 
 still smaller units are used. The gill is divided into fourths, 
 known as fluid ounces. The size of small bottles is usually 
 designated by the number of fluid ounces they contain. The 
 further subdivisions are shown in the following table. 
 
 Table 
 
 There are: 60 minims (M) in 1 fluid dram 
 8 fluid drams in 1 fluid ounce 
 16 fluid ounces in 1 pint 
 8 pints in 1 gallon 
 
122 ELEMENTS OF BUSINESS ARITHMETIC 
 
 PROBLEMS 
 
 1. How many pinfcs in 6 quarts? In 3 pecks? In 4 bushels? 
 
 2. How many pints in 8 quarts ? In 5 gallons ? In 6 gallons and 3 
 quarts ? 
 
 3. How many quarts in 5 pecks? In 4 bushels? In 3 bushels 3 pecks ? 
 In 14 pecks ? 
 
 4. What will 8 bu. 10 qt. cranberries cost at 12^ a quart? 
 
 5. What will 3 pk. 5 qt. beans costs at 5)^^ a quart? 
 
 6. Reduce 25 gal. 3 qt. 1 pt. to pints. 
 
 7. Change 196 pints to gallons. 
 
 8. Change 680 quarts to bushels. 
 
 9. Reduce f bu. to pints. 
 
 10. How many fluid drams in 3 fluid ounces? How many in 3 pints? 
 How many in 1 gallon ? 
 
 11. What is the difference in the size in cubic inches of a quart dry 
 measure and a quart liquid measure ? 
 
 12. A bin for corn is 8 x 9 x 24 ft. How many bushels of shelled corn 
 can be obtained from the ear corn it would hold ? 
 
 13. A swimming pool in a gymnasium is 15 x 10 x 40 ft. How many 
 gallons of water will it hold ? 
 
 89. Forms of Solids. A solid is a body of matter or a 
 defined portion of space. 
 
 A solid, whose surfaces are planes is called a polyhedron. 
 Polyhedrons are named tetra, hexa, octa, dodeca, or icosahe- 
 drons, as they have four, six, eight, twelve, or twenty faces. 
 
 A prism is a solid whose bases are equal, similar, and 
 parallel, and whose sides are parallelograms. Prisms are de- 
 scribed as triangular, square, oblong, etc., according to the 
 form of their bases. They are right or oblique, as their 
 sides are perpendicular or oblique to their bases. 
 
 A solid with two similar, parallel, and plane surfaces as 
 bases, and curved surfaces as sides, is called a cylinder. If 
 the bases are circles, it is said to be a circular cylinder. In 
 speaking of a cylinder, a circular cylinder is usually meant. 
 
MEASURES OF VOLUME 123 
 
 A cone is a solid having a circle for a base, and tapering 
 to a point or vertex. 
 
 A sphere is a solid or space contained within a given sur- 
 face, every point of which is equidistant from a central 
 point within. • 
 
 MEASUREMENT OF NON-RECTANGULAR SOLIDS 
 
 90. Non-rectangular Prisms. In studying the measure- 
 ments of area, all non-rectangular forms were reduced to 
 equivalent rectangles. So in volume all solids not rectan- 
 gular prisms should be considered as equivalent rectangular 
 prisms, and then the volume computed, as in Sec. 78. 
 
 In all non-rectangular prisms, first determine the number 
 of cubic units in one layer covering the base. If this base 
 were rectangular, there could be placed upon it, in one layer, 
 as many cubic units as there are square units in the area. 
 By finding the area, then, of a rectangle equivalent to the 
 base, we have the volume of one layer. The altitude or 
 height of the prism is the number of layers. 
 
 The volume of one layer of cubic units on the base of a tri- 
 angular^ trapezoidal^ or any other non-rectangular prism, is 
 equal to the volume of one layer on a rectangle equivalent to such 
 hase^ and there are as many such layers as the prism is corre- 
 sponding units high. 
 
 91. Volume of Pyramids. Cut from- cardboard four tri- 
 angles with a 3-inch base and.5|-ihch altitude. Sew the 
 edges together to form a pyramid, having an altitude of 5 
 inches. 
 
 Also cut four rectangles 3 by 5 in., and one 3 in. square ; 
 sew the edges of the rectangles together and to the edges of 
 the square, to form a rectangular prism 5 in. long. 
 
 The prism and the pyramid thus formed have the same 
 base, 3 in. square, and their common altitude is 5 in. 
 
124 ELEMENTS OF BUSINESS ARITHMETIC 
 
 Fill the pyramid with dry sand, and empty it into the 
 prism. It will require three times the contents of the pyra- 
 mid to fill the prism. Its volume is, then, one third that of 
 the prism. 
 
 The volume of a pyramid is equivalent to that of a rectangular 
 prism with the same base arid one third the altitude. 
 
 92. Cylinders. As in the prisms (Sec. 90), the number of 
 cubic units on the base of a cylinder would be the same as on 
 a rectangle equivalent, to the area of the base. The volume 
 or capacity of a cylinder is equivalent to the volume of a rec- 
 tangular prism whose base is one half the circumference in length- 
 and the radius in widths and whose height is the same as the 
 length of the cylinder. 
 
 93. Cones. Make from pasteboard, or procure a hollow 
 cone, and a cylinder having the same base and altitude. Fill 
 tlie cone with dry sand and empty it into the cylinder. 
 Three times the contents of the cone will be found necessary 
 to fill the cylinder. 
 
 The volume of a cone is equivalent to one third that of a 
 cylinder with the same base and altitude. 
 
 94. The Sphere. A sphere may be considered as made up 
 of many pyramids having their bases on the surface and their 
 apexes at the center. Applying the principle of the pyramid 
 (Sec. 91), the following statement is derived : 
 
 The volume of a sphere is equivalent to that of a rectangular 
 prisjn with a base equal to the surface area of the sphere (Sec. 
 76) and height equal to one third its radius. 
 
 PROBLEMS 
 
 1. How many cubic inches in a prism 2 ft. high, the sides of the rec- 
 tangular base being 10 and 16 in. respectively ? 
 
 2. An ash hopper is in the form of a pyramid. If the base is 6 ft. 
 square and the height 5 ft., how many bushels of ashes will it hold? 
 
MEASURES OF VOLUME 125 
 
 3. A cylindrical cistern is 12 ft. in diameter and 15 ft. deep. How 
 many gallons of water will it hold? 
 
 4. A water filter in the form of a cone has a depth of 10 ft. The 
 diameter of the base is 4 ft. How many bushels of charcoal will it hold? 
 
 5. The diameter of a cylindrical tank is 14 ft., the length is 18 ft. 
 How many gallons of oil will it hold ? 
 
 6. A man dug a well 4 ft. in diameter and 30 ft. deep. He got 4^ a 
 cubic foot and the dirt for digging. He sold the dirt for 50^ a cubic 
 yard. What did he get for his work ? 
 
 7. A man has a bin 20 ft. long and 10 ft. high. If it is 4 ft. wide 
 at the bottom and 6 ft. at the top, how many bushels of oats will it hold ? 
 
 8. How many cubic feet of stone in a pyramid 24 ft. square and 54 
 ft. high? 
 
 9. The gas tank of a company is 60 ft. in diameter, and when full of 
 gas stands 50 ft. high. How many cubic feet of gas has been used 
 when the tank stands 5 ft. high ? 
 
 10. Each side of an octagonal room is 12 ft., the distance from the 
 center to each corner is 16 ft., and the room is 15 ft. high. How many 
 cubic feet of air does it contain ? 
 
 11. A well is 34 ft. deep and 5 ft. in diameter. The w^ater stands 10 
 ft. from the top. How many standard barrels does it contain? 
 
 12. A farmer in burying his potatoes makes 3 cone-shaped piles 3 ft. 
 high and 12 ft. in circumference at the base. How many bushels has he ? 
 
IX 
 
 MEASURES OF TIME 
 
 95. Unit. The unit by which time is measured is the 
 solar day. It is the amount of time required for the earth 
 to make one complete revolution upon its axis. It is com- 
 puted from the instant the sun is at the highest point on any 
 stated meridian until it again shines at the highest point on 
 that meridian. When the sun is at the highest point, it is 
 noon, and the time is known as 12 M., or meridian. If the 
 sun has passed the highest point, the time is afternoon, and 
 it is marked p.m., or post-meridian (after meridian); if this 
 time has not yet arrived, the time is forenoon, and it is 
 marked a.m., or ante-meridian (before meridian). 
 
 The day as used in practice begins when the sun is on the 
 opposite side of the earth, or to midnight, and lasts until 
 the succeeding midnight. 
 
 The solar year is the time required for the earth to make 
 one complete revolution around the sun. It is 365 da. 5 hr. 
 48 min. 49.7 sec. long. This being nearly 365^ da., the 
 length of the year is 365 da., with 1 da. added in each 4 yr., 
 which day is counted as the 29th day of February. Since 
 in 4 yr. the 5 hr. 48 min. and 49.7 sec. each year will not 
 amount to one full day, the extra day is omitted from each 
 centennial year not divisible by 400. Thus, the year 1900 
 was not a leap year, but the year 2000 will be. 
 
 Note. — In most business transactions 30 days are considered 1 
 month, and 12 months a year. The calendar length of the months varies. 
 January, March, May, July, August, October, and December have 31 
 days ; April, June, September, and November have 30 days ; while Feb- 
 ruary has 28 days' in all years except leap years, when it has 29 days. 
 
 X26 
 
MEASURES OF TIME 127 
 
 Table 
 There are : 60 seconds (sec.) in 1 minute (min.) 
 
 60 minutes 
 
 in 1 hour (hr.) 
 
 24 hours 
 
 in 1 day (da.) 
 
 7 days 
 
 in 1 week (wk.) 
 
 30 days 
 
 in 1 month (mo.) 
 
 12 months 
 
 in 1 year (yr.) 
 
 365 days 
 
 in 1 year 
 
 366 days 
 
 in 1 leap year 
 
 96. Difference in Time. There are two methods of finding 
 the difference in time between certain dates. The exact 
 method takes into account the calendar months intervening. 
 It is used in determining the date of maturity of notes or 
 other legal paper when the time period is given in days, and 
 also by many banks in calculating discount. The eorhmercial 
 method counts the year as made up of 12 months of 30 days 
 each. In this method the difference in time is usually 
 found by what is termed compound subtraction, 
 
 97. Compound Subtraction in Time. If we desire to find 
 
 the difference in time between May 24, 1918, and March 16, 
 
 1922, we write the dates as above, ^ ^^^ « -m j 
 
 , - , , 1922 3 mo. 16 da. 
 
 puttinsr months under months, days ^rv^D c ^a j 
 
 -.1 . _ . _ -^ 1918 5 mo. 24 da. 
 
 under days, etc. As 24 days can- 5 7: ^7^7-3 — 
 
 1 ,. -. /» 1 1 o yr. 9 mo. 22 da. 
 not be taken from 16 days, 1 
 
 month must be reduced to days, making 46 days in all. 
 There would, then, be a difference of 22 days. One month 
 being used, there are but 2 months remaining in the minu- 
 end, and 1 year or 12 months must be added, making 14 
 months. Subtracting 5 months, the remainder is 9 months. 
 One year of 1922 being used, the date becomes 1921; and 
 subtracting 1918, we have 3 years. The whole difference 
 in time, then, is 3 years, 9 months, 22 days. 
 
128 ELEMENTS OF BUSINESS ARITHMETIC 
 
 LONGITUDE AND TIME 
 
 98. Measurement of Circles. For measuring angles, a unit 
 called a degree is used. It is -^^ of a right angle. The en- 
 tire angular measure around the center of a circle is equiva- 
 lent to four right angles, or 360 degrees. 
 
 The part of a circumference opposite each degree angle, 
 or g^Q of a circumference, is also called a degree (written °). 
 In the same or equal circles, one degree of arc measures one 
 degree of angle. If the circle is larger, the length of the 
 arc increases, while the angle remains the same. The degree 
 of arc means, simply, ^l^ of a circumference. The derived 
 units are shown in the table given below. 
 
 Table 
 
 There are : 60 seconds ('^) in 1 minute (') 
 60 minutes in 1 degree (°) 
 360 degrees in 1 circle 
 
 99. Latitude and Longitude. When the units of circular 
 measure are applied to the measurement of the earth's circum- 
 ference, they are of use for locating places on the globe and 
 for measuring distances. 
 
 For fixing locations north and south, great imaginary cir- 
 cles termed parallels are drawn one degree apart and parallel 
 with the equator. The number of degrees north or south 
 of the equator determines the latitude. These circles being 
 parallel, the distance between them, or the length of a degree 
 of latitude, is always nearly the same. Owing, however, to 
 the curvature of the earth, a degree of latitude varies slightly, 
 it being 68.72 miles at the equator, and 69.34 miles at the 
 poles. Sixty-nine and sixteen hundredths miles has been 
 adopted as the standard length of a degree of latitude. -^^ of 
 a degree of latitude is sometimes known as a geographical 
 
MEASURES OF TIME 129 
 
 mile, to distinguish it from the ordinary or statute mile. 
 The geographical mile is, therefore, equal to l.lo2| statute 
 miles. 
 
 For measuring east and west, great circles, called me- 
 ridians, pass through the poles. The distance between them 
 varies from nothing at the poles, where they meet, to ^1^ of 
 the circumference of the earth at the equator. A degree of 
 longitude varies in length, then, from 69.16 miles at the 
 equator to nothing at the poles. 
 
 Distance east or west is indicated by the number of 
 degrees or meridians east or west of a given meridian desig- 
 nated as a principal meridian for such purpose. The me- 
 ridians of Greenwich, England, and Washington, D.C., are 
 both used as principal meridians. 
 
 100. Difference in Longitude. If two places are both east 
 or both west of the principal meridian, the difference in their 
 longitude is found by subtracting the 
 longitude of one from that of the otjier. /-^n 
 
 Thus, the longitude of New York is 
 74° 0' 3'' W. (Greenwich), and that of ^' "^^ ^"^ ^* 
 Chicago is 87° 36^ ^2" W. To find ^4 3 W. 
 the difference, write the less number 
 under the greater, putting degrees 
 under degrees, minutes under minutes, 
 and seconds under seconds (<x). When 
 the minuend numbers are larger, ordi- 
 nary subtraction will give the result. 
 In this case it is 13° 36' 39'^ Should 
 some of the minuend numbers be 
 smaller, the same method as that used 
 in finding the difference of time (Sec. 
 97) should be used. Thus, in finding 
 the difference in longitude between New 76° 20' 18'' 
 
 13° 
 
 36' 39" 
 
 
 (J) 
 
 84° 
 
 26' 0" 
 
 74° 
 
 0' 3" 
 
 10° 
 
 25' 57" 
 
 
 (0 
 
 74° 
 
 0' 3" W. 
 
 2° 
 
 20' 15" E. 
 

 Cd) , 
 
 87° 
 
 36' 42'' W. 
 
 13° 
 
 23' 43" E. 
 
 130 ELEMENTS OF BUSINESS ARITHMETIC 
 
 York and Cincinnati (84° 26' 0' W.) 
 (6), it is necessary to change one minute 
 to seconds before the subtraction of 3" 
 can be made. 101° 0^ 25" 
 
 If the two places are one east and one 
 west of the principal meridian, the difference in their longi- 
 tude is found by adding them. Thus, the difference in 
 longitude between New York and Paris (2° 20' 15" E.) is 
 found, by adding, to be 76° 20' 18" ((?), and between Chicago 
 and Berlin (13° 23' 43" E.) is found to be 101° 0' 25" {d). 
 
 101. Relative Time. Difference in longitude is often 
 expressed by the difference in the time of two given places. 
 When the time of two places is given at the same instant, 
 their difference in time may be found by compound subtrac- 
 tion. If the times of two places are both A.M. or both P.M., 
 the difference in their time is found by subtracting (a). 
 
 
 C«) 
 
 
 New York 4 hr. 
 
 55 min. 
 
 53 sec. P.M. 
 
 Boston 4 hr. 
 
 44 min. 
 
 15 sec. P.M. 
 
 11 min. 38 sec. 
 
 When the times given are one a.m. and one p.m., the dif- 
 ference in their time is found by adding 12 hr. to the p.m. 
 time and subtracting the a.m. time from the sum (5). 
 
 
 
 (^) 
 
 
 Berlin 
 
 Ihr. 
 
 20 min. 
 
 42 sec. P.M. 
 
 Chicago 
 
 6hr. 
 
 36 min. 
 
 40J sec. A.M. 
 
 6 hr. 44 min. 1| sec. 
 
 102. Comparative Tables. As the earth revolves through 
 360° of longitude in 24 hours, it revolves through 1° in ^J-j^ 
 of 24 hours, or 4 minutes. In this way the following tables 
 are derived. 
 
MEASURES OF TIME 
 
 131 
 
 Table 1 
 There are : 360° longitude in 24 hours of time 
 
 1° longitude in 4 minutes of time 
 1' longitude in 4 seconds of time 
 1" longitude in -f^ seconds of time 
 
 Table 2 
 There are : 24 hours for 360° 
 
 1 hour for 15° 
 1 minute for 15' 
 1 second for 15" 
 
 103. Reduction. Inasmuch as the time of day depends 
 upon the position of the sun in the heavens, a place west 
 of another will have an earlier time, and a place east will 
 have a later time. A difference of longitude, then, will 
 make a difference in time, and if we know the difference in 
 longitude between two places, we may calculate their differ- 
 ence in time. Thus, the difference in longitude between 
 New York and Paris is 76° 20' 18''. Since 1° in longitude 
 makes a difference of 4 minutes in time, 76° would make a 
 difference of 304 minutes (4 min. x 76), or 5 hr. 4 min. 1' 
 making 4 sec. difference, 20.3' (20' 18") would make 81.2 
 sec, or 1 min. 21.2 sec. All together, there would be a dif- 
 ference in time of 5 hr. 5 min. 21.2 sec. The problem may 
 be written in the manner shown, 
 and solved by the method of 
 compound multiplication, which 
 combines multiplication of each 
 part and a reduction to higher 
 denominations. 
 
 The first product (a) is that from simple multiplication, 
 and the second (5) is the result of the reduction. The 
 seconds of longitude should be reduced to decimals of a 
 minute before multiplication. 
 
 76 
 
 20.3 
 4 
 
 (a) 304 81.2 
 
 (5) 5 hr. 5 min. 21.2 sec. 
 
132 ELEMENTS OF BUSINESS ARITHMETIC 
 
 On the other hand, when the difference in time between 
 
 two places is known, the difference ,^c^r^r • r»-i n 
 
 . , ., T 1 n 11 T .1 4)305 mm. 21.2 sec. 
 
 in longitude may be lound by divid- r-^o orT^r 
 
 ing such time by 4, using the method 
 
 of compound division. Thus, if the difference in time be- 
 tween New York and Paris were known to be 5 hr. 5 min. 
 21.2 sec. (305 min. 21.2 sec), the longitude difference 
 would be found by taking | of the time (expressed in min- 
 utes and seconds).^ This would be 76° 20.3' (T6° 20' 18''). 
 
 The minute remaining in the first division is changed to 
 seconds and added to the second dividend before again 
 dividing by four. 
 
 104. Standard Time. If 15° of longitude make a differ- 
 ence of one hour in time (Sec. 102), it is plain that as one 
 traveled west the time, as shown by his watch, would grow 
 faster than the local (meridian) time along the route. This 
 varying time led to many inconveniences, particularly in 
 railroad management. 
 
 In 1883, a system of standard time was adopted, and it 
 is now in general use. By this plan, the United States is 
 divided into four time belts, each about 15° in width. 
 
 Over the whole of each belt exactly the same time is used. 
 It is the local or sun time at each standard meridian. These 
 meridians are the 75th, which passes near Philadelphia ; the 
 90th, near St. Louis ; the 105th, near Virginia City, Mont. ; 
 and the 120th, near Carson City, Nev. The belts are known 
 respectively as Eastern, Central, Mountain, and Pacific, and 
 extend about 7J° on either side of the meridian. The stand- 
 ard meridians, being 15° apart, the difference in time between 
 them and their belts is exactly one hour. Hence, in travel- 
 ing westward, watches should be set back one hour on cross- 
 ing the dividing line between these belts, and in traveling 
 eastward set forward one hour. 
 
MEASURES OF TIME 
 
 133 
 
 The dividing line between the belts is quite irregular, 
 owing to the necessity of railroads avoiding a change of 
 time at any but division stations. Similar devices for using 
 
 BXiMDABD XIM£ IS ICHE IFM l.TF.n aiiXES 
 
 exactly the same time over given areas are now in use in 
 the countries of Europe and in the European colonies of 
 Asiatic and East Indian countries. 
 
 105. International Date Line. In traveling around the 
 earth east or west, the fact that the earth is rotating from 
 west to east has an effect upon the apparent time of day. 
 In traveling east, the earth travels with one, but the sun 
 seems to travel in an opposite direction, and one therefore 
 shortens his day. In traveling around the earth to the east, 
 one's time would be so shortened that it would take a day 
 more for the journey than the actual time consumed would 
 make if divided into days of twenty-four hours each. 
 
 On the other hand, if one traveled west around the world, 
 it would seem to take a day less than the actual time con- 
 sumed. 
 
 In navigation, this phenomenon caused annoyance, for ships 
 going in opposite directions would often be a day apart in 
 
134 
 
 ELEMENTS OF BUSINESS ARITHMETIC 
 
 their dates. To avoid this, a line called " The International 
 Date Line " has been established. This line follows, approxi- 
 mately, the 180th meridian from Greenwich, diverging from 
 such meridian whenever necessary to avoid land, because of 
 the confusion of dates that would result through the use of 
 different dates on the same body of land. The fact that 
 the 180th meridian east or west of Greenwich passes between 
 
 105° 136° 
 
 e)l0lGlQlQlel©l0lOlQ 
 
 000 
 
 3 P.M. 5 P.M. 7 pIm. 
 
 V\\' 
 
 5 
 
 EB, 
 
 8 PIM. 
 
 EaBt 180° 160° 120° 90° 60° Long. 30° West 0° Long. 30° Eaat 60° 90° 120' 
 
 iNTERNATIONAIi DATE LiNE 
 
 North America and Asia, and traverses the ocean save for 
 a few small islands, makes it a convenient longitude for such 
 a purpose. Whenever ships pass this line going eastward, 
 they subtract a day from their reckoning, and when going 
 westward, they add a day. 
 
 106. Longitude of Leading Cities. The longitude of the 
 cities given below may be determined from their difference 
 in time from the time of Greenwich, which is given with the 
 direction indicated. 
 
 H. M. S. H. M. S. 
 
 Albany 4 55 6.8 W. Annapolis, Md. 5 5 56.5 W. 
 
 Ann Arbor, Mich. 5 34 55.2 W. Baltimore, Md. 5 6 26.0 W. 
 
DIFFERENCE IN TIME FROM GREENWICH 135 
 
 
 H. 
 
 M. 
 
 8. 
 
 
 
 H. 
 
 M. 
 
 s. 
 
 
 Berlin 
 
 
 
 53 
 
 34.9 
 
 E. 
 
 Hong Kong 
 
 7 
 
 36 
 
 41.9 
 
 E. 
 
 Bombay 
 
 4 
 
 51 
 
 15.7 
 
 E. 
 
 Liverpool 
 
 
 
 12 
 
 17.3 
 
 W. 
 
 Bordeaux, France 
 
 
 
 2 
 
 5.4 
 
 W. 
 
 Madrid, Spain 
 
 
 
 14 
 
 45.4 
 
 W. 
 
 Brussels, Bel. 
 
 
 
 17 
 
 28.6 
 
 E. 
 
 Melbourne, Vic. 
 
 9 
 
 39 
 
 54.1 
 
 E. 
 
 Calcutta 
 
 5 
 
 53 
 
 20.7 
 
 E. 
 
 Moscow 
 
 2 
 
 30 
 
 17.2 
 
 E. 
 
 Cambridge, Eng. 
 
 
 
 
 
 22.7 
 
 E. 
 
 New Orleans 
 
 6 
 
 
 
 13.9 
 
 W, 
 
 Canton, China 
 
 7 
 
 33 
 
 46.3 
 
 E. 
 
 New York 
 
 4 
 
 55 
 
 54.6 
 
 W. 
 
 Cape Good Hope 
 
 1 
 
 13 
 
 58.0 
 
 E. 
 
 Paris 
 
 
 
 9 
 
 20.9 
 
 E. 
 
 Chicago 
 
 5 
 
 50 
 
 26.7 
 
 W. 
 
 Philadelphia 
 
 5 
 
 
 
 38.5 
 
 W. 
 
 Cincinnati 
 
 5 
 
 37 
 
 41.3 
 
 W. 
 
 Rome 
 
 
 
 49 
 
 55.6 
 
 E. 
 
 Denver, Col. 
 
 6 
 
 59 
 
 47.6 
 
 W. 
 
 San Francisco 
 
 9 
 
 9 
 
 42.8 
 
 W. 
 
 Dublin, Ireland 
 
 
 
 25 
 
 21.1 
 
 W. 
 
 Seattle 
 
 8 
 
 9 
 
 19.9 
 
 W. 
 
 Edinburgh 
 
 
 
 12 
 
 43.1 
 
 W. 
 
 St. Louis 
 
 6 
 
 
 
 49.1 
 
 W. 
 
 Glasgow 
 
 
 
 17 
 
 10.6 
 
 W. 
 
 Providence, R.I. 
 
 4 
 
 45 
 
 37.5 
 
 W. 
 
 PROBLEMS 
 
 1. A note dated May 15 was paid July 6. How many days did it 
 run? ^ 
 
 Solution : 16 days in May. 
 
 30 days in June. 
 _6 days in July. 
 52 days from May 16 to July 6. 
 
 Note.' — The day on which the note is given is not counted, but the 
 day on which it is due is counted. 
 
 Find exact number of days from : 
 
 2. Aug. 7 to Oct. 28. 
 
 3. Dec. 14 to Feb. 12. 
 
 4. May 20 to July 25. 
 
 5. Aug. 18 to Nov. 14. 
 
 6. Sept. 28 to Jan. 31. 
 
 By compound subtraction find difference in time between : 
 
 1. Nov. 18, 1903, July 7, 1906. 4. Oct. 17, 1903, Dec. 6, 1907. 
 
 2. May 17, 1899, Sept. 16, 1903. 5. Aug. 26, 1902, Feb. 15, 1908. 
 
 3. Feb. 11, 1910, Jan. 17, 1903. 6. May 25, 1901, Aug. 3, 1904. 
 
 7. March 13 to Nov. 26. 
 
 8. May 3 to Dec. 5. 
 
 9. Jan. 1 to March 13. 
 10. Sept. 15 to Dec. 31. 
 
136 ELEMENTS OF BUSINESS ARITHMETIC 
 
 Reduction : 
 
 1. The difference in time between two places is 2 hr. 15 min. What 
 is the difference in longitude ? 
 
 2. When it is noon at Chicago, what time is it 15° 15' west of Chi- 
 cago? 32° 15' 30" east? 
 
 3. A man travels from Cincinnati until his watch is 45 minutes fast. 
 In what direction and through how many degrees has he traveled? 
 
 4. What is the difference in time between two places whose longi- 
 tudes are 80° W. and 65° W. ? Two places 16° 18' E. and 90° 0' 12" W. ? 
 
 5. By use of the preceding table, find the longitude of Albany, Balti- 
 more, Berlin, Calcutta, Dublin, Madrid, Paris, Rome, and Providence. 
 
 6. When it is 11 a.m. at Washington, it is 10 hr. 7 min. 4 sec. at St. 
 Louis. What longitude is St. Louis west of Washington? 
 
 7. Determine the time and date at Washington, D.C., Hongkong, 
 Cape Good Hope, and Moscow when it is midnight, Jan. 1, at Green- 
 wich. 
 
 8. When it is noon, Feb. 15, at Glasgow, what time and date is it at 
 Denver ? At Melbourne ? At New York ? At Brussels ? 
 
 9. When it is 8 p.m. at Philadelphia, what is the time at Liverpool? 
 At Paris? At Berlin? At Rome? 
 
MEASURES OF WEIGHT 
 
 107. Units of Weight. Weight is the downward pressure 
 of bodies toward the earth, or the measure of the attraction 
 of gravitation. The English Ttoj pound has been adopted 
 as the unit. As fixed by the English law, this is the weight 
 of 22.7944 cu. in. of pure water at its greatest density. 
 The use of this unit and its Troy subdivisions is restricted 
 to weighing gold, silver, jewels, etc. The Troy pound is 
 divided into twelfths, known as ounces, which in turn are 
 divided into 20 pennyweights, each composed of 24 grains. 
 There are, then, 5760 grains in a pound Troy. 
 
 Troy Weight Table 
 There are : 
 
 24 grains (gr.) in 1 pennyweight (pwt.) 
 20 pennyweights in 1 ounce (oz.) 
 12 ounces in 1 pound (lb.) 
 
 Comparisons 
 
 1 lb. = 12 oz. = 240 pwt. = 5760 gr. 
 
 1 oz. = 20 pwt. = 480 gr. 
 
 1 pwt. = 24 gr. 
 
 108. Apothecaries' Weight. For weighing drugs, in com- 
 pounding prescriptions, etc., the Troy ounce is differently 
 divided. The table of Apothecaries' weight follows. 
 
 137 
 
138 ELEMENTS OF BUSINESS ARITHMETIC 
 
 Table 
 
 There are : 
 
 TABLE 
 
 20 grains (gr.) in 1 scruple (sc.) 
 3 scruples in 1 dram (dr.) 
 8 drams in 1 ounce (oz.) 
 
 12 ounces in 1 pound (lb.) 
 
 109. Avoirdupois Weight. For weighing ordinary bulky- 
 articles, such as coal, grain, groceries, etc., the Avoirdupois 
 pound is used. It consists of 7000 Troy grains, and has 
 been fixed by Congress (1901) as the weight of 27.7015 cu. 
 in. of distilled water, at 62° Fahrenheit, weighed with brass 
 weights in air, with the barometer at 30 in. 
 
 Table 
 
 There are : 16 ounces (oz.) in 1 pound (lb.) 
 
 100 pounds in 1 hundredweight (cwt.) 
 
 20 hundredweight in 1 ton (T.) 
 
 Comparisons 
 
 1 ton = 2000 lb. 
 1 lb. = 7000 grains 
 1 oz. = 437.5 grains 
 
 Note. — At the Custom House, and to some extent in mining, the 
 long ton of 2240 pounds is still used. As a subdivision, 112 pounds are 
 called a hundredweight. 
 
 110. Bushel Weights. In handling grains and many 
 other products for shipment, the dry-measure units are sel- 
 dom used. It is much more convenient to weigh in bulk 
 and allow a certain weight per bushel. Many states have, 
 by statute, fixed the number of pounds per bushel of various 
 products. The result of this practice has been a lack of 
 uniformity among the states. In 1901 Congress fixed t^ 
 minimum weight per bushel of certain articles of produce 
 
MEASURES OF WEIGHT- 139 
 
 Pounds Pounds 
 
 Wheat 60 Dried apples 26 
 
 Corn, ear 70 Dried peaches 33 
 
 Corn, shelled 56 Clover vseed 60 
 
 Buckwheat 48 Flax seed 56 
 
 Barley 48 Millet seed 50 
 
 Oats 32 Hungarian 50 
 
 Peas 60 Timothy seed 45 
 
 White beans 60 Blue grass 44 
 
 Castor beans 46 Hemp seed 44 
 
 Rye 56 Corn meal 48 
 
 White potatoes 60 Ground peas 24 
 
 Sweet potatoes 55 Bran .20 
 
 Onions 57 Malt 34 
 
 Turnips 55 
 
 Note. — In many places the weight per bushel has been discarded, 
 grain and other materials being bought by hundredweight. This plan 
 has many advantages. 
 
 Salt. The weight per bushel of salt as adopted by dif- 
 ferent states ranges from 50 to 80 pounds. Coarse salt in 
 Pennsylvania is reckoned at TO pounds, and in Illinois at 
 50 pounds per bushel. Fine salt in Pennsylvania is reck- 
 oned at 62 pounds, in Kentucky and Illinois at 55 pounds 
 per bushel. 
 
 PROBLEMS 
 
 1. How many pounds in 1 T. 6 cwt. ? In 4 cwt. ? In 6 T. 3 cwt. ? 
 
 2. How many ounces in 5 lb.? In 8 lb.? In 10 lb.? In 16 lb.? 
 (Troy.) 
 
 3. How many grains in 3 pwt.? In 5 pwt.? In 8 pwt.? In 
 12 pwt.? In 16 pwt.? 
 
 4. How many tons in 40 cwt.? In 50 cwt.? In 60 cwt.? In 80 
 cwt.? In 100 cwt.? 
 
 5. How many ounces Troy in 4 lb. ? In 8 lb. ? In 20 lb. ? In 15 lb. ? 
 
 6. How many grains in 3 oz. 16 pwt. 12 gr. ? In 4 oz. 6 pwt. 20 gr. ? 
 
140 ELEMENTS OF BUSINESS ARITHMETIC 
 
 7. How many bushels of ear corn in a load weighing 2650 lb.? 
 
 8. What will 2680 lb. shelled corn cost at 60^ per bushel? 
 
 .15 1430 
 Solution : $.^0 x %^^^ - % 30.64: 
 
 u 
 
 7 
 
 9. What will 3670 lb. wheat cost at 78 f per bushel ? 
 
 10. What will 2750 lb. potatoes cost at 45 ^ per bushel ? 
 
 11. If a man weighs 156 lb. Avoirdupois, what will he weigh in Troy- 
 weight ? 
 
 12. A farmer raises 7600 lb. rye, 1856 lb. clover seed, 1375 lb. flax, 
 4532 lb. onions, 458 lb. timothy seed. How many bushels of each did he 
 raise ? 
 
 13. A cubic foot of water weighs 62| lb. At that rate, what will be 
 the weight of the water in a tank 4 x 6 x 15 ft. ? 
 
 14. A grocer's boy sold 10 lb. coffee by mistake, weighing it on a 
 druggist's scales. How much did the grocer lose or gain, if the coffee 
 was worth 20 ^ a pound ? 
 
 15. A farmer sold 27,650 lb. corn at 65 ^ per bushel, 48,500 lb. wheat 
 at 82/ per bushel, 15,810 lb. oats at 35/ per bushel. Find the total 
 selling price. 
 
 16. A horse requires 12 lb. of oats and 20 lb. of hay per day. What 
 will it cost to feed him a year, if oats are worth 26/ a bushel, and hay 
 $ 9.75 a ton ? 
 
 17. A dealer purchased in a day 26,040 lb. oats at 28/ a bushel, 
 19,654 lb. wheat at 76 / per bushel, 28,642 lb. corn at 49 / per bushel, and 
 15,060 lb. barley at 98 / per bushel. The oats are sold at an advance of 
 4 /, the wheat at 5 /, the corn at 7 /, and the barley at 6 /. Find the gain. 
 
 18. A farmer raised on his farm 28,540 lb. wheat, 27,486 lb. rye, 
 8752 lb. potatoes, 54,660 lb. hay, 40,500 lb. oats, and 600 lb. beans. He 
 sold the wheat for 65 /, the rye for 57 /, the potatoes for $ 1.25, the beans 
 for $ 3.50, the oats for 32 / per bushel, and the hay for $ 8.75 per ton. 
 Find total returns. Find his gain, if he paid % 375 for labor, $ 245 for 
 seed, $175 for machinery, and $60 for hauling the crops to market. 
 
 19. A coal dealer bought 326 tons of coal by the long ton at $ 3.75 
 per ton, and sold it by the short ton at $6.50. How much did he gain ? 
 
XI 
 
 MEASURES OF VALUE 
 
 111. The Unit of Value. Congress has designated the dol- 
 lar as the unit of value for the United States, to be com- 
 posed of 23.22 grains of pure gold. For greater hardness, 
 the pure metal is mixed with copper or an alloy of copper 
 and silver, nine parts gold and one of alloy, making the full 
 weight of a gold dollar 25.8 grains. The silver dollar con- 
 tains 371.25 grains of pure silver; 41.25 grains of copper 
 are added for hardness. 
 
 From the standard unit, f 1, other units are derived, using 
 decimal divisions and multiples; thus, 810 is called an eagle, 
 ■^^ of f 1 is called a dime (French disme or ten) or ten cents ; 
 ^^^ of a dollar is 1 cent (Latin centum or a hundred), and 
 l-Q^^ of a dollar is 1 mill (Latin mille or a thousand). While 
 theoretically all these divisions are made, practically we use but 
 dollars and cents in writing or reading expressions of value. 
 
 112. Coins. For the convenience of business, Congress 
 has caused to be minted the following coins : 
 
 Grold. Twenty dollars, sometimes called the double eagle, 
 ten dollars or eagle, and five dollars or half eagle. 
 
 Silver, One dollar, half dollar or fifty cents, quarter or 
 twenty-five cents, and dime. All except the first are subsid- 
 iary. 
 
 Minor Coins, Nickel or five cents, and one-cent piece. 
 
 Note. — The coinage of the gold dollar was discontinued pursuant to 
 an act of Congress, Sept. 26, 1890. 
 
 113. Paper Money. For larger denominations, and for 
 convenience in handling larger amounts of money. Congress 
 
 141 
 
142 ELEMENTS OF BUSINESS ARITHMETIC 
 
 has authorized the issue of five different kinds of paper 
 money, printed in various denominations. These are United 
 States Notes or Greenbacks, Silver Certificates, Gold Certif- 
 icates, Treasury Notes of 1890, and National Bank Notes. 
 
 114. Legal Tender. To facilitate trade and render it 
 more secure, governments declare certain kinds of money 
 to be "legal tender," i.e. money that must be accepted by a 
 creditor in payment of a debt. 
 
 The following kinds of money are full legal tender : all 
 gold coins, silver dollars. Treasury notes of 1890, and United 
 States notes (except for duties on imports and interest on the 
 public debt). National Bank notes and gold and silver certif- 
 icates are not legal tender, but, because they are redeemable 
 at the United States Treasury in legal tender money, they pass 
 current without question. Subsidiary silver coins are legal 
 tender for ten dollars or less, and minor coins for 25 ^ or less. 
 
 115. Sending Money and Valuables. For a fee of 10 cents 
 a letter or parcel containing money or other articles of value 
 may be sent by registered mail. Registered mail is delivered 
 only to the addressee, or on his written order, and, if not 
 known to the authorities, such addressee must be identified. 
 A receipt is taken upon delivery, and is returned to the 
 sender of the letter or parcel. This receipt is, under the law, 
 prima facie evidence of delivery. In case of loss, the gov- 
 ernment indemnifies the sender for a value up to $50. 
 
 B^ Express. Express companies will transport money under 
 guarantee of its safe delivery, and will indemnify losses in 
 full. The cost of such service, however, and its inconvenience, 
 render it little used for ordinary business transactions. 
 
 Exchange. By exchange is meant a system of making 
 money payments without the actual transportation of money. 
 It is accomplished through the exchange of credits. The 
 principal systems of exchange are given below. 
 
MEASURES OF VALUE 143 
 
 116. Post OflBice Money Orders. The government provides 
 a simple and safe means of exchange through the postal sys- 
 tem. Offices designated as "money-order offices" are em- 
 powered to issue orders on other similarly designated offices 
 for an amount not exceeding f 100 to any one order. For 
 sums above $100 more than one order must be purchased. 
 
 To obtain a money order, a prescribed form of application 
 blank must be filled out. The rates charged, in addition to 
 the face value of the order, are printed on the back of the 
 order blank, and the cost of an order for a given amount 
 may be easily computed. The government is liable for pay- 
 ment to the wrong person, provided the wrong payment was 
 not brought about through the fault of the remitter, payee, 
 or indorsee. Money orders may be obtained or cashed at 
 over 35,000 offices in the United States. 
 
 117. Express Money Orders. Express companies issue 
 money orders in much the same form and at practically the 
 same rates as the Post Office Department. A receipt is 
 given the sender, and, in case the order is lost, a refund may 
 be had upon filing an indemnity bond guaranteeing the com- 
 pany against loss should the lost order be cashed. 
 
 The limit of each order is $ 50, but any number may be pur- 
 chased. When indorsed, these orders are usually accepted 
 by banks at their face value. 
 
 118. Bank Exchange. Banking houses maintain deposits 
 or accounts with other banks, called "correspondents." 
 They sell orders, called drafts, on these correspondents, 
 charging a small fee therefor. In all ordinary business and 
 for relatively small sums, the fee is a fixed charge. Ten 
 cents for small amounts, and fifteen cents per hundred dollars, 
 are the rates charged in many banks. For larger amounts, 
 and between large business houses, a certain fraction (per 
 cent) of the amount of the draft issued is sometimes charged. 
 
144 ELEMENTS OF BUSINESS ARITHMETIC 
 
 Note. — As this phase of exchange deals with per cents and banking 
 procedure, it is reserved for further discussion in the chapter on " Bank- 
 ing and Discount." 
 
 Some banks issue drafts payable at any one of a number of 
 other banks. Such drafts are known as hank money orders^ 
 and are usually charged for at the same rate as post office 
 or express money orders. 
 
 119. Exchange by Wire. Telegraph companies, for a fixed 
 fee in addition to the cost of sending the necessary tele- 
 grams, instruct payees by wire to call at a designated tele- 
 graph office to receive a certain sum of money, and, in the 
 same way, direct the manager of such office to pay the amount 
 to a particular person on demand. 
 
 Express companies also order money paid by telegram, in- 
 structing their agent at a certain place to deliver to the 
 payee a given amount. 
 
 Telephone companies direct their managers at other places 
 to pay given amounts. Banks, too, will order the payment 
 of money by telegraph. 
 
 Exchange by wire is necessarily more expensive, as the 
 fees are quite large. Such a method is intended for use 
 only in emergencies. 
 
 120. Foreign Exchange. 
 
 Foreign Coins 
 
 English Money. There are : 4 farthings in 1 penny (<f.) 
 
 12 pence in 1 shilling (s.) 
 20 shilling in 1 pound (X) 
 
 French Money. There are : 10 centimes in 1 decime 
 
 10 decimes in 1 franc 
 
 German Money. There are : 100 pfennigs in 1 mark 
 
 Spanish Money. There are : 100 centesimos in 1 peseta 
 
 Note. — France, Belgium, Greece, Italy, and Switzerland constitute 
 what is known as the " Latin Monetary Union," and their coins are 
 
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 OOOOOOOOOOOOOOOcc 
 
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 OOOO OOOOOOO 
 
 lis 
 
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 ^ C C eS I) 
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 Ira 
 
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 •aa 
 
 ^ rt o 'x ■^ > 2 
 
 s *> ? N 
 
 J2^ bto 
 
 S .S 2 S S S S-o-^ 5-5 «^ o £ o 5 « S2o3 
 
 •a 
 
 •E ;^o ^ 
 
 a ^o 'g 
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 "" M e c a 
 t.r' <o 01 oj « 
 
 ^■^ § 2 ® 
 
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 Sa^ -2 
 
 •S g-s 2 '3 
 
 liil i 
 
 s -S^ 2 >S -^^ 
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MEASURES OF VALUE 145 
 
 alike in weight and fineness, occasionally differing, however, in name. 
 Most of the Central and South American states possess a standard coin. 
 
 121. Bills of Foreign Exchange. Drafts payable in foreign 
 countries are either banker's or commercial bills of exchange. 
 They are based upon credits in the foreign cities in the same 
 way as are bank drafts upon cities in our own country, the 
 latter being known, by way of distinction, as domestic ex- 
 change. Banker's bills were formerly drawn in duplicate or 
 triplicate, each copy being sent by a different route for safety. 
 Rapid and safe mail service renders this no longer necessary, 
 particularly between Europe and America. Accordingly, 
 foreign bills are now usually drawn singly, the same as 
 domestic drafts. 
 
 All foreign bills are payable in the money of the country 
 upon which they are drawn, and quotations of prices of 
 foreign exchange give the cost in our money of one mone- 
 tary unit of such country. Thus, an English pound sterling 
 may be quoted as $4.86 or 14.88, etc. 
 
 The ebb and flow of commerce, with the balance of trade 
 shifting from one country to another, causes changes in the 
 value of credits in one country which may be owned in 
 another, and these fluctuations are shown in daily quotations 
 issued by banks and clearing houses. When there is a larger 
 foreign credit than is needed for the demands of business, 
 then foreign exchange is at a discount ; when less than busi- 
 ness requires, it is at a premium. 
 
 Certain banks in larger cities maintain reciprocal relations 
 as correspondents with banks in certain foreign centers. 
 Banks in smaller places often arrange to draw on such foreign 
 correspondents through these larger banks, paying therefor 
 as per quotations furnished. 
 
 122. Letters of Credit. Ba-nks also issue letters of credit 
 to persons who wish to travel abroad. They are addressed 
 
146 ELEMENTS OF BUSINESS ARITHMETIC 
 
 to any of a list of correspondent banks in various cities in 
 Europe, requesting them to furnish funds to the order of the 
 holder up to a certain limit. Upon reaching one of the men- 
 tioned cities, a draft drawn by the holder of the letter for an 
 amount within the limit will be cashed by the correspondent 
 bank. The amounts drawn are indorsed on the letter by the 
 banks cashing the drafts, and the letter itself accompanies 
 the last draft which exhausts the limit of the credit. The 
 advantage of these letters is in one being able to draw varying 
 amounts, according to his needs, in a large number of places. 
 
 123. Foreign Money Orders. Foreign money orders^ pay- 
 able in the currency of the principal countries of the world, 
 may be purchased of the United States government at any 
 money-order office. 
 
 Some express companies issue money orders or drafts on 
 important cities in foreign countries, and these drafts are 
 payable in the money of such country at the company's 
 agencies, or at certain banks with which they have arrange- 
 ments as correspondents. 
 
 Travelers'' checks^ issued in certain denominations, with 
 the foreign-money values in various countries printed on 
 the face thereof, may be purchased in any number. These 
 checks are payable at various banks and agencies, and pass 
 current in almost any country for their face value in pay- 
 ment of ordinary expenses. A supply of these checks, in 
 different denominations, forms a very convenient kind of ex- 
 change for travelers. 
 
 Some express companies also issue letters of credit^ en- 
 titling the holder to draw money or travelers' checks thereon 
 up to a certain limit. For such a letter of credit either cash 
 must be deposited, or it must be secured either by a deposit 
 of securities or the guarantee of a responsible bank, trust 
 company, or banker. 
 
MEASURES OF VALUE 147 
 
 Oahle Transfers. Money may be paid by cable in the same 
 way as by telegraph. 
 
 Shipments of gold coin or bullion are frequently made to 
 restore a balance of credits in foreign exchange. 
 
 Miscellaneous Measures 
 
 124. Tradesman's Table. 
 
 There are : 12 units in 1 dozen (doz.) 
 
 12 dozen in 1 gross (gr.) 
 
 12 gross in 1 great gross 
 Twenty units are often called a score. 
 
 125. Stationer's Table. 
 
 Paper, particularly of the finer kinds, is commonly put up 
 and sold according to the following units : 
 
 There are : 24 sheets in 1 quire 
 
 20 quires in 1 ream (480 sheets) 
 2 reams in 1 bundle 
 5 bundles in 1 bale 
 
 Paper houses are gradually adopting the commercial ream 
 of 500 sheets, particularly for book and print papers. The 
 larger units are disappearing from use. 
 
 PROBLEMS 
 
 1. The United States paid Spain $20,000,000 for the Philippines. 
 What was the face of the bill, if the rate of exchange was 19.38 cents? 
 
 2. A merchant owes a bill of £300 85. 10c?. in London. What will it 
 cost him for a bill, if the rate of exchange is $4,865? 
 
 3. A Boston importer owed a Dresden manufacturer 26,450 marks, and 
 paid him by a bill of exchange. If the rate was 23.85 cents, what did it 
 cost him ? 
 
 4. I owe 4250 francs in Paris. If the rate is 19.38 cents, what wiU it 
 cost me to pay the debt ? 
 
148 ELEMENTS OF BUSINESS ARITHMETIC 
 
 5. A merchant bought 4 cases musical instruments in Berlin, amount- 
 ing to 3598.6 marks, and received a discount of i^. If the exchange is 
 worth 23.87 cents, what will the bill of exchange cost? 
 
 6. What will be the value of a sight draft on Berlin, amounting to 
 3598.6 marks, if the exchange is worth 23.88 cents ? 
 
 7. What must be paid for a bill on Rome for 6750 lire, at 19 ^ ? 
 
XII 
 
 FRENCH METRICAL SYSTEM 
 
 127. The Metric System. The rapidly expanding use of 
 the measures of the P'rench Metrical System renders a 
 knowledge of its units more and more a necessity in a busi- 
 ness education. Its use is required in some of the depart- 
 ments of the government and is authorized in others. 
 Government comparisons of standards are made by reference 
 to the international metric units. The metric system is the 
 legal standard in the Philippines and Porto Rico. So ex- 
 tensive is its use in foreign countries that manufacturers, 
 particularly of scientific instruments and machinery, are put- 
 ting out their products in metric sizes for the export trade. 
 
 The metric system is founded primarily upon the meter^ 
 which is the unit of length. All other units are so related 
 to this as greatly to simplify calculations in which weights 
 and measures are involved. It is this simple relation to one 
 fundamental unit which makes it a system. Add to this the 
 fact that it is a decimal system, and its extreme simplicity 
 both in plan and ease of calculation is explained. 
 
 With a desire to found the unit upon something fixed, the French 
 mathematicians took one ten-thousandth part of the earth's quadrant for 
 a Kilometer, of which the more usable meter is a thousandth part. Under 
 the direction of an International Bureau of Weights and Measures, es- 
 tablished near Paris by the principal nations of the world, standard 
 meters and kilograms were cast from platinum-iridium, and distributed 
 to the various nations. By the law of 1893, all our common units are 
 derived from these international standards, which are kept in the office 
 of the National Bureau of Standards at Washington. 
 
 149 
 
150 ELEMENTS OF BUSINESS ARITHMETIC 
 
 The plan of naming the derived units is equally simple. 
 Derivatives from the Greek, viz. deka for ten, hehta for 
 hundred, hilo for thousand, and myria for million, are pre- 
 fixed to the name of the units for all multiples, while the 
 fractional units are indicated by the Latin prefixes, deei for 
 tenth, centi for hundredth, and milli for thousandth. 
 
 A practical knowledge of the metric system should be 
 acquired by a study and use of the measures themselves. 
 Each unit studied should be fixed by actual use, without 
 regard to the nearest English unit, and the power to esti- 
 mate accurately in its terms developed through drill. After 
 such knowledge of the various units has been acquired, and 
 one is able to think in these units, a study of comparative 
 values between them and the English units has its value. 
 Comparative tables are appended for reference only. 
 
 MEASURES OF LENGTH 
 
 128. The Unit of Length. As before stated, the meter is 
 one thousandth part of a kilometer, which is one ten-thou- 
 sandth part of the earth's quadrant. It is, therefore, one ten- 
 millionth of the distance from the equator to the pole. From 
 it all the other units of the French Metrical System are 
 derived. The surveyor's chain is a dekameter or half deka- 
 meter in length. 
 
 Table 
 There are : 
 
 10 millimeters (mm.) in 1 centimeter (cm.) 
 iO centimeters in 1 decimeter (dm.) 
 
 10 decimeters in 1 meter (i^O 
 
 10 meters in 1 dekameter (Dm.) 
 
 10 dekameters in 1 hektameter (Hm.) 
 
 10 hektameters in 1 kilometer (Km.) 
 
FRENCH METRICAL SYSTEM 151 
 
 MEASURES OF AREA 
 
 129. Unit of Area. The unit of area is the square meter. 
 In a like manner each denomination of linear measure is 
 squared, forming the units of area. A square meter being 
 10 decimeters wide and 10 decimeters long, has 100 square 
 decimeters. The same principle is applied to the other 
 units. 
 
 There are: ^^^^^ 
 100 sq. millimeters (sq. mm.) in 1 sq. centimeter (sq. cm.) 
 
 100 sq. centimeters in 1 sq. decimeter (sq. dm.) 
 
 100 sq. decimeters in 1 sq. meter (sq. m.) 
 
 100 sq. meters in 1 sq. dekameter (sq. Dm.) 
 
 100 sq. dekameters in 1 sq. hektameter (sq. Hm.) 
 
 100 sq. hektameters in 1 sq. kilometer (sq. Km.) 
 
 The chief use of the larger area units is in the measure- 
 ment of land surfaces. For that purpose, the square deka- 
 meter is given a special name, and is known as the are., and 
 the square hektameter as the hectare. The latter is the 
 more commonly used land -area measure, while the square 
 kilometer has no practical application. 
 
 MEASURES OF VOLUME 
 
 130. Units of Volume. The units of volume are derived 
 in the same way, each linear unit being cubed. A cubic 
 meter being 10 decimeters along each edge, contains 1000 
 cubic decimeters. Likewise, there are 1000 of each cubic 
 denomination in the next higher one. 
 
 There are: ^^^^^ 
 
 1000 cu. millimeters (cu. mm.) in 1 cu. centimeter (cu. cm.) 
 1000 cu. centimeters in 1 cu. decimeter (cu. dm.) 
 
 1000 cu. decimeters in 1 cu. meter (cu. m.) 
 
 1000 cu. meters in 1 cu. dekameter (cu. Dm.) 
 
152 ELEMENTS OF BUSINESS ARITHMETIC 
 
 131. Capacity Units. For measuring capacity, the cubic 
 decimeter is used. The special name of liter is given to it, 
 from which are derived the denominations of capacity with 
 the constant ratio of ten between successive units, and desig- 
 nated by the usual prefixes. 
 
 There are : Table 
 
 10 milliliters (ml.) in 1 centiliter (cl.) 
 10 centiliters in 1 deciliter (dl.) 
 
 10 deciliters in 1 liter (1.) 
 
 10 liters in 1 dekaliter (Dl.) 
 
 10 dekaliters in 1 hektaliter (HI.) 
 
 Wood Measure. For the measure of wood, the cubic meter 
 is given a special name, the stere. This, likewise, may be 
 treated decimally and a series of denominations formed. 
 
 MEASURES OF WEIGHT 
 
 132. Unit of Weight. For a unit of weight, one cubic 
 centimeter of distilled water is taken, at its greatest density, 
 in the latitude of Paris and at sea level. It is called a gram. 
 Decimal denominations are derived from this in the same 
 way as from the liter. A cubic decimeter or liter of pure 
 water contains 1000 cubic centimeters, and hence weighs 
 1000 grams, or one kilogram. For very heavy articles the 
 weight of a cubic meter of water is used, or 1000 kilograms, 
 and this is called a tonneau. 
 
 There are : Table 
 
 10 milligrams (mg.) in 1 centigram (eg.) 
 
 10 centigrams in 1 decigram (dg.) 
 
 10 decigrams in 1 gram (g.) 
 
 10 grams in 1 dekagram (Dg.) 
 
 10 dekagrams in 1 hektagram (Hg.) 
 
 10 hektagrams in 1 kilogram (Kg.) 
 
 100 kilograms in 1 tonneau (T.) 
 
FRENCH METRICAL SYSTEM 153 
 
 133. Decimal Methods. As the successive units of the 
 metric system usually bear the ratio of ten to each other, 
 various denominations are very conveniently written as a 
 decimal of one denomination. Thus, 3 Dm. 5 m. 6 dm. 7 cm. 
 
 5 mm. would usually be written as 35.675 m. 
 
 For the same reason a change from one denomination to 
 another may be accomplished by merely moving the decimal 
 point. Thus, the above 35.675 m. may also be written as 
 3.5675 Dm., or 356.75 dm., etc. Obviously this ease of 
 reduction is a decided advantage in the use of the metric 
 system. 
 
 PROBLEMS 
 
 1. Write 5463 cm. as kilometers; as decimeters; as dekameters; 
 as meters. 
 
 2. Write 5360 sq. m. as ares ; as hektares. 
 
 3. Write 5200 cl. as liters; as hektaliters. 
 
 4. At 6 ^ a meter, what will it cost to build a fence 54 Dm. 6 m. 
 long? 
 
 5. How many rails 9 m. 4 dm. long will it take to build a railway 
 20 Km. 4 Hm. 2 Dm. long? 
 
 6. What is the weight of a liter of water in grams ? What is the 
 weight of a cubic meter of water ? How many liters in it ? 
 
 7. A circular lot is 27 meters in diameter. What is the area in 
 ares? 
 
 8. How many ares in a rectangular field 8 Dm. long and 5 Dm. 4 m. 
 
 6 dm. wide? 
 
 9. How many cubic meters of water in a tank 10 m. long, 6 m. 7 dm. 
 high, and 8 m. wide ? 
 
 10. A fence is 5 boards high and 12 Hm. 2 Dm. long. How many 
 boards 3 m. long are there in it ? 
 
 11. How much carpet 1 m. wide will be needed to cover a room 6.4 m. 
 long and 5.5 m. wide ? 
 
 12. What will 50 1. of mercury weigh, if it is 13.5 times heavier than 
 water? 
 
154 ELEMENTS OF BUSINESS ARITHMETIC 
 
 13. If a pile of wood is 32 m. long, 5 m. 2 dm. wide, and 3 m. 6 dm. 
 high, what is it worth at $2 a stere? 
 
 14. How much wheat will a bin 4 m. long, 3 m. 4 dm. wide, and 2 m. 
 high hold? What is the value at 60;* a liter? What is the weight ? 
 
 15. How many tiles 40 cm. x 20 cm. will be used in tiling a floor 9 m. 
 6 dm. long and 5 m. 4 dm. wide? 
 
 16. If copper is 8.8 times as heavy as water, what is the weight of 
 8 cu. dm. of the metal? 
 
 17. What will it cost to plaster the walls and ceiling of a room that 
 is 6 m. 5 dm. by 5 m. 8 dm. by 4 m. 2 dm., at 32;* a square meter? 
 
 18. What will it cost to carpet a room 5 m. 4 dm. by 3 m. 2 dm. with 
 carpet 8 dm. wide at 80;* a meter? 
 
 19. What will it cost to paint the walls of a barn 15.5 m. long, 10 m. 
 wide, and 9.5 m. high, at 35;* per square meter? 
 
 20. Find the capacity of a tank 5 m. long, 3 m. 6 dm. wide, and 2 m. 
 6 dm. high. 
 
 21. What will be the cost of 4608 Kg. of hay at $ 12 a ton ? 
 
 22. How many steres of wood in a pile 10 Dk. long, 3 m. 4 dm. wide, 
 and 1.5 m. high? 
 
 23. Gold is 19.5 times as heavy as water. What is the weight of a 
 cubic centimeter of gold ? 
 
 24. How many jars, each containing 2.5 liters, can be filled from a 
 cask containing 145.5 DL? 
 
 25. A room 6.3 m. long and 4 m. wide will require how many meters 
 of carpet 8 dm. wide to cover it? 
 
 26. Find the area of the four walls of a room, 10.5 m. long, 6.5 m. 
 wide, and 5.4 m. high. Area of ceiling? Cost to plaster the room at 
 35 ;* a square meter ? 
 
 27. What will it cost to paint the walls and ceiling of a room 8.5 m. 
 X 5.4 m. X 4 m., deducting for four windows each 2 m. x 1 m., at 15^ per 
 
 square meter ? 
 
 28. A room is 8 m. x 5 m. x 3.2 m. Deducting for five windows, 
 each 2.1 m. x 1 m., 2 doors each 2.8 m. x 1.4 m., and a baseboard 2 dm. 
 high, what will be the cost of plastering, at 45^ a square meter? The 
 cost of paper 4 dm. wide at $4.20 per roll of 10 meters? What is the 
 cost of carpet 5.2 dm. wide, at $1.50 per meter? 
 
FRENCH METRICAL SYSTEM 
 
 155 
 
 134. 
 
 lin. 
 1ft. 
 1yd. 
 1 mi. 
 
 1 cu. in. 
 1 cu. ft. 
 1 cu. yd. 
 1 bushel 
 
 COMPARATIVE TABLES 
 
 1. Customary Units to Metric Units. 
 
 Length 
 
 25.4001 mm. 
 .304801 m. 
 .914402 m. 
 1.60935 Km. 
 
 Volume 
 
 = 16.387 cu. cm. 
 = .02832 cu. m. 
 = .765 cu. m. 
 = .35239 HL 
 
 Area 
 
 1 sq. in. = 6.452 sq. cm. 
 Isq. ft. =9.290 sq. dm. 
 1 sq. yd. = .836 sq. m. 
 1 acre = .4047 Hm. 
 
 1 fl. dr. 
 1 fl. oz. 
 1 qt. 
 IgaL 
 
 Capacity 
 = 3.70 cu. cm. 
 = 29.57 mm. 
 = .94636 1. 
 = 3.78543 L 
 
 Weight 
 
 1 gr. = 64.7989 mg. 
 
 1 av. oz. = 28.3495 g. 
 1 av. IK = 45359 Kg* ^ 
 1 troy oz. = 31.10348 g. 
 
 2. Metric Units to Customary Units. 
 
 Length Square 
 
 1 sq. m. = 1550 sq. in. 
 sq. m. = 10.764 sq. ft. 
 sq. m. = 1.196 sq. yd. 
 
 1 m. = 39.3700 in. 
 1 m. = 3.28083 ft. 
 1 m. = 1.093611 yd 
 1 Km. = o62137 mi. 
 
 1 
 1 
 IHa. 
 
 = 2.471 A. 
 
 1 cu. cm. 
 1 cu. dm. 
 1 cu. m. 
 1 cu. m. 
 
 Cubic 
 
 = .0610 cu. in. 
 = 61.023 cu. in. 
 = 35.314 cu. ft. 
 = 1.308 cu. yd. 
 
 Capacity 
 1 mm. = .27 fl. dr. 
 1 cl. = .338 fl. oz. 
 11. = 1.0557 qt. 
 1 Dl. = 2.6417 gaL 
 IHL = 2.8337 bu. 
 
156 ELEMENTS OF BUSINESS ARITHMETIC 
 
 Weight 
 
 1 mg. = .01543 gr. 
 1 Kg. = 15432.36 gr. 
 1 Hg. = 3.5274 av. oz. 
 1 Kg. = 2.20462 av. lb. 
 1 T. = 2204.6 av. lb. 
 
XTII 
 PERCENTAGE 
 
 135. Meaning and Use. By per cent is meant hundredths. 
 The term is an abbreviation of the Latin per centum^ or " by 
 the hundred." As a further abbreviation, % is used. 
 
 Just as decimals are a development of certain forms of 
 fractions having advantages in ease of operation, so percent- 
 age is the development of a certain kind of decimal fraction, 
 having advantages for comparison in business transactions. 
 A wide range of variation in the size of different parts of a 
 whole may be indicated by hundredths, while operations are 
 often made easy by frequent opportunities for using simple 
 fractions. Since hundredths may be written as decimals, 
 operations in percentage also possess all the advantages of 
 decimals in ease and quickness of computation. 
 
 In Sees. 19 and 20 the methods of finding any number of 
 hundredths were developed. In the treatment of simplified 
 processes under Fractional Parts (Chapter IV), suggestions 
 were made for shortening those methods. While not enter- 
 ing now upon the study of anything essentially new, a 
 further and more systematic study of hundredths must be 
 made that we may better understand percentage in its vari- 
 ous applications to business. 
 
 136. Profit and Loss. Business enterprises are carried on 
 in the expectation of making a profit for those who invest 
 money or other wealth in them. At regular intervals, usu- 
 ally every year, a careful inventory is made of the results of 
 the year's business, to find out how much profit has been made. 
 
 167 
 
158 ELEMENTS OF BUSINESS ARITHMETIC 
 
 It sometimes happens that such an inventory shows that the 
 business has been carried on at a loss. Whether a loss or a 
 profit, the amount is first ascertained, and then, for purposes 
 of comparison, the per cent that the loss or profit is of the 
 capital invested is found. 
 
 Profit or loss is expressed, then, in terms of per cent. 
 Because of the simplicity of the ideas involved, problems in 
 profit and loss are used in the study of percentage processes. 
 
 137. Finding 50%, 25 %, or 20 %. Since there are one hun- 
 dred hundredths in the whole of anything, 100 % of anything 
 is equal to the whole of it. If 100 % of anything is all of it, 
 50 % is ^ of it. Likewise, since J of 100 % equals 25 %, then 
 25 % of anything is ^ of it ; 20 % of anything is ^ of it, etc. 
 
 What per cent of anything is | of it ? What f ? What 
 I? Whatf? 
 
 Summary 
 
 100% =1^8- 25% =1 75% =1 60% =1 
 
 50%=i 20%=i 40% = | 80% = | 
 
 ILLUSTRATIVE PROBLEMS 
 
 1. Mr. King invested $ 400 in grain, and lost 50 %. How much did 
 he lose ? 
 
 Since 50% of anything is ^ of it, then his loss is i of $400, or 
 $200. 
 
 Stated thus : 
 
 50%= h 
 i of $400 = $200 
 
 2. A clerk spends $ 30, which is 25 % of his salary. What is his 
 salary ? 
 
 Since 25 % of anything is i, then $ 30 is J of his salary, and f are 4 
 times $ 30, or $ 120, his salary. 
 
 Statement : 
 
 25%= i 
 J of salary = $ 30 
 
 $ 30 X 4 = $ 120, salary. 
 
PERCENTAGE 159 
 
 3. A man having $ 75, spends $ 15 for a coat. What per cent of his 
 money does he spend ? 
 
 Since he has $ 75, and spends $ 15, he spends f|, or | of his money. 
 But ^ of anything is 20 % of it ; therefore he spends 20 % of his money 
 for the coat. 
 
 Statement : 
 
 H = i 
 
 ^ = 20 % spent for coat. 
 
 4. A man owns 300 A. of land, which is 50% more than he owned 
 the year before. How much did he own the year before ? 
 
 Since 50 % of anything is | of it, he would have ^ more, or 1^ times 
 what he had the year before. But 1^ times anything is | of it ; there- 
 fore 300 A. is f of what he had, and ^ is ^ of 300 A., or 100 A. The 
 whole of what he had, then, was twice 100 A., or 200 A. 
 Statement : 
 
 50% = ^. 
 
 I of what he had = 300 A. 
 
 ^ = ^ of 300 A., or 100 A. 
 
 f = 100 A. X 2, or 200 A., what he had the year before. 
 
 5. I sell a house for $ 600 and lose 20 % by doing so. What did the 
 house cost ? 
 
 Since 20% of anything is^ of it, I have lost ^, and have f remaining. 
 Then, $ 600 is f of the cost, and ^ is i of $ 600, or $ 150. The whole cost, 
 then, would be 5 times $150, or $750. 
 
 Statement : 
 
 20% = i 
 4 of cost = $600 
 
 i = iof $600, or$150 
 
 I = $ 150 X 5, or $ 750, cost. 
 
 PROBLEMS 
 
 ^1. A farmer invested $600 in hogs and lost 50 7o- How much did 
 
 he lose ? 
 
 V 2. A bookkeeper spends $28, which is 25% of his monthly salary. 
 
 What is his salary ? 
 
 ^ 3. A man owns 600 sheep, which is 50% more than he owned the 
 
 year before. How many did he own the year before ? 
 
 ^ 4. A horseman bought a horse for $160 and sold it for $200. What 
 
 per cent did he gain ? 
 
160 ELEMENTS OF BUSINESS ARITHMETIC 
 
 5. A farmer had 150 hogs and lost 20 % of them. How many did 
 he have left ? 
 
 6. An agent buys books for $3.25 and sells them for $ 6.50. What 
 per cent did he gain ? 
 
 7. A firm buys lots for $520 and sells them for $650. What per 
 cent is the gain ? 
 
 8. A man paid $ 80 for a horse and sold it for $ 120. What per 
 cent did he gain ? 
 
 * 9. A poultry man sold 240 chickens, which were | of his flock. 
 How many were there in the flock? What per cent of the flock did he 
 sell? 
 
 it 10. Mr. Frank paid $ 80 for a horse. What per cent of his money 
 did he pay if he had $200? 
 
 11. I sell a farm for $ 6000 and gain 25 % by doing so. What did the 
 farm cost me ? 
 
 12. I sell a horse for $ 600 and lose 20 % by doing so. What did the 
 horse cost me ? 
 
 ^13. A farmer lost 60 hogs by disease, which was 40% of his herd. 
 How many had he at first? 
 
 14. A land agent had 1600 acres of land and sold 60% of it. How 
 many acres did he sell ? 
 
 15. A capitalist gave $ 5400, which was 75 % of the amount he gained, 
 to a public library. What did he gain? 
 
 16. A broker sold 30 shares of stock, which was 20 % of all he had. 
 How many shares did he have ? 
 
 17. If A's money, $42, is 20% more than my money, how much 
 have I? 
 
 18. If 84 sheep is 20 % less than the number of sheep I have, how 
 many have I? 
 
 19. A farmer buys goods for $ 60 and sells them for $48. What per 
 cent does he lose ? 
 
 138. Finding 33^%, 16|%, 12|%, and 14f %. 
 
 l-of 100% = 33i% 
 ^ofl00%=16f% 
 I of 100% =121% 
 |ofl00% = 14f% 
 
PERCENTAGE 161 
 
 Then, 
 
 331% = i 14|% = i 371% = 3 
 
 16|%=| 66|% = J 621% = ! 
 
 28^% =f 
 
 It is well to remember that | = -J, or 33|^ % ; | = ^, or 50 % ; 
 f = f, or 66|%; | = i, or 25%; f = i or 50%; and f = f, 
 or 75%. 
 
 PROBLEMS 
 
 1. If $15 is 33^% of my money, how much have I? 
 
 2. A has 172. B has 33^% less. How much has B? 
 
 3. If A has $48, which is 33^% less than B*s money, how much 
 hasB? 
 
 4. Mr. Hartman buys a buggy for $63, which is 12|% less than he 
 paid for a horse. What did the horse cost? 
 
 5. C has 70 acres of land, which is 16f % as much as B's. How much 
 hasB? 
 
 6. A drove 21 miles in a day, which was 16|% farther than B drove. 
 How far did B drive ? 
 
 7. A merchant buys goods for $480 and sells them for $540. What 
 is his per cent of gain ? 
 
 t^e. A broker buys stock for $4800 and sells it for $5600. What per 
 cent did he gain ? 
 
 9. Mr. Jacobs buys goods for $39 and sells them for $52. What 
 per cent did he gain ? 
 
 10. Mr. Gould buys a lot for $1200 and sells it for $2400. What per 
 cent did he gain ? 
 
 >^11. A raised 80 bushels of potatoes, which was 16|% less than B 
 raised. How many bushels did B raise? 
 
 12. If I buy a horse for $120 and sell it for $140, what per cent do 
 I make ? 
 
 13. A received a salary of $720 and spends 66|% of it. How much 
 did he spend? 
 
 14. If $49 is 874% of my wages, what are my wages? y^^ 
 
 
162 ELEMENTS OF BUSINESS ARITHMETIC 
 
 15. If 37^% of my money is |33, how much have I? 
 
 16. A merchant makes a profit of $4200 from his business and spends 
 33 1 % of it for family expenses. How much does he save? 
 
 17. A merchant sold 24 dozen eggs, which was 37^% of all he had. 
 How many had he ? 
 
 18. A farmer had 24 hogs in one pen and 12 in another. He took 4 
 hogs from the first and put them in the second. What per cent decrease 
 in the first pen ? What per cent increase in the second ? 
 
 19. He again took from the first 4 hogs and put them in the second. 
 What per cent increase in the second and what decrease in the first? 
 
 20. Books that cost $21 were sold so as to gain 28f%. Find the 
 selling price? 
 
 f^ 21. Goods that cost $88 were sold so as to gain $77. What is the 
 per cent of gain? 
 
 139. Finding 10 %, 1 %, 5 %, and \ %. 
 
 Then, 
 
 Jj of 100% = 10% 
 
 10%=tV 
 
 30%=T% 
 
 ^^oflOO%=l% 
 
 1% =Tb 
 
 70%=Jj 
 
 2V of 100% = 5% 
 
 5% =2V 
 
 90% = ^^, etc, 
 
 ^ofl00%=i% 
 
 \1o =jk 
 
 
 By Sec. 19, we learned that to find J^, we should point 
 off one decimal place or remove the decimal point one place 
 to the left ; to find ^-J^j, we remove the decimal point two 
 places to the left. Then to find 10 % or 1 % of any number, 
 we remove its decimal point respectively one or two places 
 to the left. Thus, 10% of 33 is 3.3, and 10% of §54.20 
 is 15.42. Also, 1% of 256 is 2.56, and 1% of 15425 is 
 154.25. 
 
 Since ^^ is \ of J^, we may find ^, or 5%, by taking \ of 
 Jjjofit. Thus, 5% of $976 is 1 of §97.60, or $48.80. Like- 
 wise, -|-% or glo is 1 of 1%, or \ of -j^^. Thus, \% of 
 1976.25 is 1 of §9.7625, or §4.88. 
 
PERCENTAGE 163 
 
 PROBLEMS 
 
 1. A miller makes 56 bushels of wheat into flour in one day. If 
 this is 10 % of the wheat he has, how many days will it take him to grind 
 all of it at the same rate ? 
 
 2. James has $76, which is 5% less than I have. How much have I? 
 
 3. A man owed $6700 when he died. His property was worth 
 90 % of this amount. What was the value of his property ? 
 
 4. Mr. Berry sold a carriage for $195, which was 30% more than it 
 cost him. What did it cost him ? 
 
 5. A merchant sold goods that cost him $34 at a loss of 1%. Find 
 the selling price. 
 
 6. A house worth $6000 is rented for 5% of its valuation. What is 
 the rent? 
 
 7. A bedroom suite was sold for $77, which was 10% above cost. 
 Find the cost. 
 
 8. A farmer sold two cows for $ 38 each, gaining 5 % on one and losing 
 5% on the other. Find the cost of each. Find the loss or gain in the 
 transaction. 
 
 9. An implement dealer buys a wheat drill for $54, and sells it at 
 a profit of 10%. What is the selling price ? 
 
 10. A young man had $600 and received $4200 by will. What per 
 cent was the increase in his wealth? 
 ^ 11. A ton of coal was sold for $9, which was a gain of 12|%. Find 
 the cost. 
 
 12. A father had $1400. He gave 25% to his son, 40% of the re- 
 mainder to his elder daughter, and the remainder to the younger daughter. 
 How much did each receive ? 
 
 13. A merchant sells $56 worth of goods to-day, which is 12^% less 
 than he sold the day before. Find amount of his sales the day before. 
 ^14. A grocer buys a hogshead (56 gal.) of molasses and sells 30% of 
 it the first day. How many gallons had he left? 
 
 15. Mr. Burkett sold a buggy for $85, which was 70% more than it 
 cost him. Find cost. 
 
 16. I loan $600 and receive $30 interest. What per cent is this of 
 the amount loaned ? 
 
 17. A commission merchant charges $20 for selling $4000 worth of 
 wheat. What per cent is that of the selling price of the wheat ? 
 
164 ELEMENTS OF BUSINESS ARITHMETIC 
 
 140. Finding Other Per Cents. Much the larger part of 
 all business problems involving percentage has to do with 
 per cents given in Sections 137-139. Most of the prob- 
 lems in interest, bank discount, and trade discount may be 
 solved with them or with slight modifications of them, and 
 these are the most commonly used of all percentage applica- 
 tions. A thorough mastery of the per cents thus far devel- 
 oped, then, is of first importance, and they should be 
 practiced upon until problems involving any phase of these 
 per cents can be solved quickly and with accuracyo 
 
 Finding any per cent, other than those mentioned, is based 
 on the method of finding 1 %. Thus, 7,% is -^^ or 7 times 
 1 % ; 13 % is JqS. or 13 times 1% ; 34 % is -^^-^, or 34 times 
 1 % ; etc. To find any given per cent first find 1 % by 
 removing the decimal point two places to the left, and then 
 multiply by the number of per cent required. 
 
 Thus, 4 % of 13296 is 132.96 x 4 =1157.04. 
 
 PROBLEMS 
 
 1. A farmer raised 450 bu. of potatoes and sold 6 % of them. How 
 many bushels had he left ? 
 
 2. I spent $ 28, which was 14 % of my money. How much had I at 
 first? 
 
 3. I have ^65 and spend 12 7o of it for trousers. What do they cost? 
 
 4. A buys goods for $36 and sells them for $39. What is his per 
 cent of gain ? 
 
 5. Mr. Bunger sells his horse for $65 and by doing so gains 8^%. 
 What was the cost? 
 
 6. If Street & Co. pay $45 for dishes and sell them for $48, what 
 per cent gain have they? 
 
 7. If a coal dealer bought coal for $7.50 and sold it for 6|% advance, 
 what did he gain ? 
 
 8. Sold goods so as to gain $ 86, which was a gain of 12 %. What 
 was the cost? What the selling price? 
 
 9. I sold goods for $ 376 and lost 6%. Find the cost. 
 
PERCENTAGE 165 
 
 10. 1 have 320 sheep and buy 8 more. What per cent do I add to my 
 flock? 
 
 11. I have $39 and spend 1 21. What per cent do I spend ? 
 
 12. A merchant paid |17 for goods and sold them so as to gain |3, 
 What per cent did he make ? 
 
 13. A man spent 28% of his money for a coat that cost $56. How 
 much money did he have ? 
 
 14. A miller lost 36% of his wheat by fire and had 1280 bushels 
 remaining. How many bushels did he lose ? 
 
 15. A had 240 chickens and sold 84 of them. What per cent of his 
 flock had he left? 
 
 16. A dealer sold grain at a profit of 16% and received $696. What 
 did it cost ? 
 
 17. A speculator sold land for $9200, which was 8% less than it cost 
 him. What did it cost him ? 
 
 18. A stockman bought horses for $84 and sold them for $ 96. What 
 per cent did he gain ? 
 
 19. Isellgoodsfor $184 and lose 8%. What did they cost ? 
 
 20. A man sold his horse for $ 147, which was $ 33 less than he paid 
 for it. What per cent did he lose ? 
 
 GENERAL PROBLEMS 
 
 1. If a grocer adds a pound of Java coffee to every four pounds of 
 Mocha, what per cent of the mixture is of each coffee ? 
 
 2. A man owning | of a mill, sells f of his share. What per cent of 
 the mill does he still own ? 
 
 3. A man's money invested at 10 % annual interest yields $ 125 a 
 month. How much has he invested ? 
 
 4. My agent sold $ 625 worth of goods and charged me \% commis- 
 sion for selling. How much money did I pay him ? 
 
 5. A mechanic has $42 a month left after paying 6f % of his wages 
 for car fare. What are his wages? How much does he pay for car fare? 
 
 6. Parker & Co. pay $60 for a buggy and sell it for $64. What per 
 cent do they make ? 
 
 7. A hardware man sold a stove for $84 and lost, by so doing, 6|%. 
 What was the cost of the stove ? 
 
166 ELEMENTS OF BUSINESS ARITHMETIC 
 
 8. A jeweler sold a watch so as to gain $2, which was 2J% of the 
 cost. What was the cost ? 
 
 9. The annual interest on my money loaned at 4% is $75. How 
 much have I loaned? 
 
 10. A man invested 60 % more money in a business than his partner, 
 and the difference between their investments was $3000. How much 
 did each invest ? 
 
 11. A man bought a horse and a cow for $300, the horse costing 50% 
 more than the cow. How much did each cost? 
 
 12. I of James's money is 75 % of Henry's, and | of Henry's is 25 % of 
 John's. John has 1 32. How much has James ? 
 
 13. A capitalist had a half interest in a ranch and sold | his interest 
 for 13600. The sale was made at a gain of 25%. What was the cost of 
 the ranch? 
 
 14. A bank building rents for |4200 a year, which is 12^% of its 
 value. What did it cost, if it had increased in value 40 %? 
 
 15. If a firm quits business with property worth $2600 and owes 
 $3900, what per cent of their debt can they pay? How many cents on 
 the dollar? 
 
 16. If I sell $ 6000 worth of goods for my principal and remit him 
 $5900, what per cent commission did I charge him? 
 
 17. A farmer lost 16|% of his hogs, sold 40% of the remainder, and 
 had left 120. How many had he at first ? 
 
 18. A man bought property which increased in value 16|% the first 
 year. He finally sold it at an increase of 2o% over this, and received 
 $ 8400. What was the cost ? 
 
 19. A lady bought a piano for $480, which was 25% of the money she 
 had in the bank. The money she had in the bank was 40% of the 
 value of real estate that she owned. What was the value of the real 
 estate ? 
 
 20. A confectioner sold candy for 60 j^ a pound. If this were 25% 
 more than it cost him, what was the cost? 
 
 21. A clothier sold a suit of clothes for $ 18 and lost 40 %. He then 
 sold another at a profit of 16|% and gJtined as much as he had lost on 
 the first. What was the cost of each suit? 
 
 22. I sold a house and lot for $ 1600, losing 20 %. For how much 
 should I have sold it to gain 20 % ? 
 
PERCENTAGE 167 
 
 23. A merchant engaged in business, investing cash $ 3500. At the 
 end of one year he found that he had paid for merchandise, f 3250; for 
 rent, $450; for clerk hire, $1200; for incidentals, $725. He had sold 
 merchandise to the value of $ 6758.40. What was his per cent of profit 
 on his investment? 
 
 24. A man's real estate is now worth $12,000. The first year it 
 increased 20 % in value, and the second year 33^%. What did it cost two 
 years ago? 
 
 25. A dry goods merchant marked cloth at 25% advance on the cost, 
 but was obliged to sell it at 20% less than the marked price. If it cost 
 him $1 a yard, what did he sell it for? 
 
 26. A man willed 50 7o of his property to his wife, 60% of the 
 remainder to his invalid daughter, and the remainder to his church. 
 The church received $650. What did the wife and daughter receive? 
 
 27. A hardware dealer engaged in business and lost 12 % of his money 
 the first year and gained 33|% the second year. If he started with 
 $ 4500, how much has he now ? 
 
 28. If a man loses $2400 by selling at a loss of 12^%, at what should 
 he sell to gain 12^%? 
 
 29. Mr. McFarland bought a square piece of land, containing 40 
 acres, paying $800 for it. He opened a street through the center of it 
 and divided the land on each side into lots 4 rods wide. If he sold the 
 corner lots for $540 apiece, and the other lots for $400 apiece, and the 
 expense of opening up the land was $1250, what was his per cent of 
 profit or loss? 
 
 30. A speculator bought a section of land for $ 3 per acre. He sold J 
 of it at $8 per acre, and the remainder at $9.50 per acre. If the cost of 
 making the sales was $ 135, what was his gain per cent ? 
 
 31. A merchant's profit the second year was 66|% greater than it was 
 the first. The profits of the two years amounted to $ 8120. What was 
 the profit of each year ? 
 
 32. I offered my house and lot for sale at 50 % above cost, but sold 
 them for 25% below the asking price and gained $600. What was the 
 cost ? What the gain per cent ? 
 
 33. If my wheat cost 90 j^ a bushel and I lose 10% by shrinkage, 
 for what must I sell it to make a net gain of 12 J % ? 
 
 34. If I sell J of my land for what | of it cost me, what per cent do I 
 gain? 
 
168 ELEMENTS OF BUSINESS ARITHMETIC 
 
 35. I paid $22,500 for two houses. If 75% of the cost of the one is 
 equal to 150 % of the cost of the other, what did each cost ? 
 
 36. A has 25% more money than B, B has 20% more than C, C has 
 12| less than D. How much has each, if together they have 1 33,900 ? 
 
 37. I paid $580 for a horse, wagon, and harness. The wagon cost 
 40 % less, and the harness 66|% less, than the horse. What was the cost 
 of each? 
 
 38. A miller in 3 years made gains amounting to $6336. The second 
 year's gain was 20% greater than that of the first, and the third 10% 
 greater than the second. What was each year's gains? 
 
 39. A grocer bought apples at 60^ a bushel, and marked them so as 
 to sell at a gain of 20%, but sold them at a reduction of 12-|% from the 
 marked price. If he gained $42.80, how many bushels had he? 
 
 40. A man sold goods that cost $425 at an advance of 40%. He lost 
 25 % in bad debts and paid 5 % for collecting. What was his gain or 
 loss? 
 
 41. Our stock of goods decreased in value 33p/o, and again 20%. It 
 then increased 20%, and again 33i%, and was sold at a loss of $66. 
 What was it worth at first ? 
 
 42. If the retail profit is 33^%, what do I make on goods that cost 
 $180, if I sell them at wholesale for 10% less than at retail? 
 
 43. Last year a merchant gained $2000. This year he gained 20% 
 more, which is 44|% of what he gained the year before last. What did 
 he gain each year ? 
 
 44. A barrel of cider had lost 20% by leakage and was sold for 50% 
 above cost. What per cent gain was that ? 
 
 45. I buy a barrel of vinegar, containing 52 gallons, at 20 J^ a gallon. 
 If four gallons leak out, for what must I sell the remainder per gallon to 
 gain 25%? 
 
 46. If I buy stocks for 80% of their value, and sell them for 110% of 
 their value, what per cent do I gain ? 
 
 47. Sold a lot of cotton at a gain of 33^%. With the money I bought 
 another lot and sold it for $480 at a loss of 20 %• What did the first lot 
 cost me? 
 
 48. A manufacturer sold at a profit of 33| % to a wholesale dealer, 
 who sold at a profit of 25% to the retail dealer. The retail dealer sold 
 
PERCENTAGE 169 
 
 at a profit of 20% and received $60 for the article. What was the 
 original cost? 
 
 49. A merchant increased his investment 50%. He then withdrew 
 66|% of his capital and invested it in bonds. He now has $4800 in the 
 business. How much did he invest at first ? 
 
 50. An implement man bought a mower for $ 42. How much must 
 he ask for it in order to make a discount of 25% and still gain 16f %? 
 
 51. I buy goods and sell at a loss of 10%. I reinvest the money and 
 gain 10%. What per cent do I gain or lose? 
 
 52. A dairyman sold two cows for $120, and gained 20% on the one 
 and lost 20 % on the other. What was the cost of each, if he sold the 
 first for 50 % more than the other ? 
 
 53. A real estate dealer sold a building for $ 15,000 and lost 40%. He 
 sold a house and lot at the same time and gained 16f 7o- He did not gain 
 or lose on the two transactions. What was the cost of each property? 
 
 54. A man bought a business for $9775, which was 15% more than 
 the former owner paid for it. He then sold it at a profit of 6%. What 
 was the selling price ? What the gain ? What the original cost ? 
 
 55. The imports of sugar and molasses into the United States in 
 1891 amounted to $108,387,388; in 1900 they amounted to $101,100,000. 
 What was the per cent of decrease ? 
 
 56. The exports of wheat and flour in 1891 and 1900 were as follows : 
 1891, $106,125,188; 1900, $140,997,966. What was the per cent of 
 increase ? 
 
 57. The world's supply of sugar in 1900 was 8,800,000 lb. The sup- 
 ply of cane sugar was 2,850,000 lb. What per cent of the total supply 
 was cane sugar ? 
 
 58. The exports of the United States in a given year were : Europe, 
 $697,614,106; Asia and Oceania, $43,813,519 ; British North American 
 Possessions, $37,345,515; West Indies, $33,416,178; South America, 
 $33,226,401; Mexico and Central America, $21,236,545; Africa, 
 $4,738,847; all other, $879,172. Find what per cent of the total was 
 exported to each country. 
 
 59. A wholesale dealer had sold $1500 worth of goods to a retailer. 
 The retail dealer failed and could pay only 75 f on the dollar. K the 
 wholesale dealer paid 5% for the collection of ,the debt, what was his per 
 cent of loss ? 
 
170 
 
 ELEMENTS OF BUSINESS ARITHMETIC 
 
 60. The estimated total cut of lumber in the United States from 
 1880 to 1906 was 706,712,000 board feet. Of this amount, Michigan 
 produced 93,436,000. Michigan's output was what per cent of the 
 total? 
 
 61. The freight moved over the several lines of railroad in the United 
 States in a given year was as follows : 
 
 Class of Commodity 
 
 Tonnage keported as 
 
 OKIGINATING ON LiNE 
 
 Pek Cent op 
 Aggkegatb 
 
 Products of agriculture 
 
 Products of animals 
 
 Products of mines 
 
 Products of forests 
 
 Manufactures 
 
 Merchandise 
 
 Miscellaneous 
 
 56,102,838 
 15,145,297 
 269,372,556 
 60,844,933 
 71,681,178 
 21,697,693 
 26,493,338 
 
 ? 
 ? 
 ? 
 ? 
 ? 
 ? 
 ? 
 
 Grand total 
 
 9 
 
 ? 
 
 Europe, $215,000,000 
 
 Find the aggregate tonnage. Find the per cent to two decimal places 
 that each item is of the aggregate. 
 
 62. If the exports of manu- 
 factured products from the 
 United States is as shown in 
 the accompanying square, find 
 what per cent of the aggregate 
 is exported to each division. 
 
 63. The total imports into 
 the United States for a certain 
 year were $903,320,948. Of 
 this amount, the United King- 
 dom sent 18%; Germany, 11%; 
 France, 9%; Brazil, 8.7%; Brit- 
 ish North America, 5.3%; all 
 other countries, 48%,. Find the 
 amount sent by each division 
 to approximate millions, and 
 Problem 62. 
 
 North America, $96,000,000 
 
 Asia, $34,000,000 
 
 Oceania, $29,000,000 
 
 South America, $27,000,000 
 
 Africa, $11,000,000 
 
 represent it in graphical form as in 
 
PERCENTAGE 171 
 
 64. From the accompanying graph, showing the per capita consump- 
 tion of coffee in pounds, find the 
 
 per cent of the whole amount United Kingdom 72 
 
 consumed by each of the five ~~ 
 
 countries, the population of the -^^^v ^° 
 
 United Kingdom being approxi- Austria Hungary 2.04 
 
 mately 43,650,000; of Italy, ^ ' 
 
 33,750,000 ; of Austria-Hungary, Germany 4.62 
 
 47,000,000; of Germany, 
 
 60,650,000; and of the United United States 10.79 
 
 States, 90,000,000. 
 
 65. If the per capita consumption of sugar in Russia is 13.9 ; in 
 Portugal, 14.2 ; in Austria, 17.6 ; in Belgium, 23.0 ; in Netherlands, 34.3 ; 
 in France, 36.9 ; in Norway and Sweden, 40.6 ; in Denmark, 48.7 ; in 
 Switzerland, 52.0; in the United States, 65.2; in the United Kingdom, 
 91.1 pounds, find total amount consumed by all, the population of each 
 being, in approximate millions, as follows: Russia, 107^; Portugal, 5| ; 
 Austria, 26; Belgium, 6|; Netherlands, 5|; France, 39; Norway and 
 Sweden, 7^; Denmark, 2.6; Switzerland, 3|; United States, 90; and 
 United Kingdom, 43.6. Find the per cent of this amount which each 
 consumes. Represent in graphical form as in Problem 64. 
 
XIV 
 
 TRADE DISCOUNT 
 
 141. Trade Discount is an allowance from the price of 
 goods made because of special conditions, or for settlement 
 of the account within a specified time. Such discounts are 
 usually made "to the trade" or to persons engaged in the 
 retail trade in a given line of goods, hence the term. The 
 amount of the allowance is expressed in per cent or fractions. 
 Discount plays a large part in the arithmetic of mercantile 
 business. 
 
 142. Marked and List Prices. Merchants generally mark 
 or list their goods, so they may gain a certain per cent of the 
 cost as profit. If strictly a " one price " house, the per cent 
 of profit is the only element to be considered by a retailer in 
 determining the " marked " price. If th'e goods are to be so 
 marked that a discount may be allowed and still a certain 
 per cent of profit be made, then both discount and profit 
 must be considered. 
 
 Wholesalers and manufacturers very generally publish a 
 " list price " from which a certain per cent is allowed as a 
 discount "to the trade." Fluctuations in the market prices 
 may in this way be provided for by varying the discounts 
 quoted instead of changing the printed or list prices. Goods 
 listed and not subject to discounts of any kind are usually 
 marked "net." 
 
 143. Term Discount. Even when goods are billed at such 
 
 172 
 
TRADE DISCOUNT 173 
 
 discounted prices, a further discount is often allowed if the 
 bttl is paid within a certain time, say 30 or 60 days ; while 
 that discounted price maybe still further discounted for <?a«A. 
 
 Thus, goods may be listed at $60, and quoted at dis- 
 counts of 20, 10, and 5. The goods would usually be billed 
 at "20 off" (160 less |), or 148. 10% off for payment 
 within a time period marked, say 30 days, would reduce the 
 bill ($48-14.80) to $43.20, and 5% further discount for 
 cash would make the cash value of the bill J of $4.32 
 ($2.16) less, or $41.04. 
 
 When discounts are quoted in series as above, it should be 
 .noted that the discount indicated by the first per cent given, 
 is taken from the amount of the bill, then the second dis- 
 count is computed upon the remainder and deducted, and 
 so for the third, each discount being a per cent of the 
 remainder after deducting the previous discount. 
 
 144. Fractional Discounts Used. Whenever the nature of 
 the business will allow, prices are listed so that the discounts 
 used are those per cents which are the equivalents of simple 
 fractions. Thus, instead of allowing a discount of 15%, the 
 list price is raised slightly so that a discount of either 16| % 
 or 20 % may be given. Finding | is not only purely a mental 
 operation, but the process is simpler, shorter, and less liable to 
 error than multiplying y^-g- of the amount by 15. For these 
 reasons, probably, the trend of business usage for years past 
 has been away from a purely decimal to a fractional discount. 
 
 145. Finding Net Price. 1. 200 arithmetics were billed to 
 a dealer for $ 80, on which a discount of 20 % was allowed. 
 What was the net cost ? 
 
 20% =i 
 I of $80 = $16 
 $80 -$16 = $64, net cost. 
 
174 ELEMENTS OF BUSINESS ARITHMETIC 
 
 2. A discount of $25 was allowed on a bill of goods for 
 $150. What was the per cent or rate of discount ? 
 
 $25 = ^2^^ of 1150. 
 
 i¥iT = 6 ^^ l^t %^ ^^t® of discount. 
 
 146. Net Price with a Discount Series. 1. Discounts of 25, 
 20, and 5 per cent were quoted on a bill of goods amounting 
 to $360. What was the net price? 
 
 $360. 
 
 = the list price. 
 
 90. 
 
 = 25% 
 
 or I of $360. 
 
 270. 
 
 = price 
 
 after first discount. 
 
 54. 
 
 = 20% 
 
 or 1 of $270. 
 
 216. 
 
 = price 
 
 after second discount. 
 
 10.80 
 
 = 5% or 2V of $216. 
 
 $205.20 
 
 = net price. 
 
 2. A bill of goods netted $ 216, after being discounted at 
 25 % and 20 %. What was the face of the bill? 
 
 $216 = 20 % or ^ less than the amount of the last discount. 
 
 I = $216. 
 
 | = f of $216, or $270. 
 $270 = 25 % or ^ less than face of bill. 
 
 f = $270. 
 
 I = f of $270 or $360, face of bill. 
 
 147. Single Equivalent of Discount Series. 1. Discounts 
 of 25% and 20% are equivalent to what single discount? 
 
 100 % = the whole of anything. 
 
 25 % = J of 100 %, or first discount. 
 
 75 % = price after first discount. 
 
 15% = 20%, or^of 75%. 
 
 60 % = net price after second discount. 
 100 % = 60 % = 40 %, equivalent single discount. 
 
TRADE DISCOUNT 175 
 
 2. A bill of goods netted $216, after being discounted 25 
 and 20 per cent. What was the face of the bill ? 
 
 Discounts of 25 and 20 = 40 % discount (Prob. 1). 
 100 % -40 % = 60 %, net price. 
 60%, or I = 1216. 
 
 f = f of 1216, or 1360. 
 
 PROBLEMS 
 
 1. A merchant bought a bill of goods amounting to $400. He re* 
 ceived a discount of 10 % and 8^ % off for cash. What was the cash 
 value of the bill ? 
 
 2. Fonts & Co. bought a bill of goods amounting to $600, on which 
 they received a discount of 20 % and 10 % off for cash. What was the 
 cash value of the bill ? 
 
 3. A carriage is marked at $100, but the dealer tells me I may have 
 it on 30 days' time at a discount of 20 %, and, if I pay cash, an additional 
 discount of 2| % will be given. What must I pay for it in cash? 
 
 4. The marked price of a mower is % 64. I sell it to a farmer, giving 
 him a discount of 12| % and 14| %. What did he pay ? 
 
 5. I buy a bill of goods amounting to % 500, on which I get a dis- 
 count of 12| % and 4 % off for cash. What did I pay ? 
 
 6. Smith & Co. buy a bill of goods amounting to $ 500, on which 
 they get a discount of 16|% and 10% off for cash. What did they 
 pay? 
 
 7. McKee & Co. sell goods for % 120. They had given a discount of 
 20 % and 5 % off for cash. What was the list price of the bill ? 
 
 8. What is the cash value of a bill of goods that lists $49, if a dis- 
 count of 14f % and 16|% is given? 
 
 9. Mr. Kerr sells a wagon on 60 days' lime for $72. If paid in 
 30 days, the buyer will get a discount of 16|%; if he pays in 10 
 days, he will get 10 % additional. What would the " cash in 10 days " 
 payment be? 
 
 10. A dealer bought goods at 20% and 25 % off, and sold them at 10% 
 and 10 % off the same marked price. What was his per cent of gain ? 
 
176 
 
 ELEMENTS OF BUSINESS ARITHMETIC 
 
 11. How much better is a series of 25%, 20%, and 10% than a single 
 discount of 60%? 
 
 12. What is the amount of the following bill if it is paid within 10 
 days? 
 
 M ^^^^.yU 
 
 Chicago, DL,. 
 
 fyP-^^^-<^.A^^^ 
 
 «' W. M. WELCH & COMPANY 
 
 Tenns: ^ , ./^^. ^ ^ J, .9 - /. / ^ dr,^ "^ "«"«^ Street 
 
 13. Sold Lew A. Wallace, terms 5/10-n/30 ; 12 bx. Apricots @ % 2.15 ; 
 12 bx. Layer Raisins @ $2.95; 5 bx. Dried Citron @ $1.33^; 3 tierces 
 Refined Lard, 1065 #, @ 9^; 3 cases Shredded Codfish, 6 doz., @ |1.12^; 
 4 cases Salmon, 8 doz., @ f 2.12|. Find cash value. 
 
 14. Sold A. O. Bennett, terms 3/10-n/60; 4 cases Rio Coffee, 400 #, 
 @Vl\f',2 bags Mocha Coffee, 185 #, @ 28^ ; 2 bags M. G. Coffee, 378 #, 
 @ 22 ^ ; 4 bbl. Herring @ % 6.25. What is the cash value of the bill? 
 
 15. Sold W. B. Lambert, terms 10-2/10-n/30; 5 doz. Baseballs 
 @$15; 3 doz. Bats @$9; 24 Infielders Gloves @ $2.50; 3 Catcher's 
 Gloves @ $6.25; 12 Baseball Uniforms @ $14; 12 pr. Shoes @ $3.50. 
 What is the cash value of the bill ? 
 
 16. Bought of J. A. Amsbaugh & Co., 225 bbl. Flour @ $4.45; 165 
 S. H. Hams, 1960 #, ©14^^; 5 tierces Refined Lard, 745#, @ 9^2^; 6 bbl. 
 Mess Pork @ $15.20; 8 bbl. G. Flour @ $6.10; 24 bx. Figs @ $1.85; 
 5' cases G. Coffee, 300 #, @ $0.12|. I am allowed a discount of 20% and 
 an additional discount for cash of 5 %. What is the cash value of the 
 bill? 
 
 17. A man desires to buy 48,000 ft. of pine lumber. One firm offers 
 the lumber at $60 per M, less discounts of 20% and 5%; another firm 
 
tra.de discount 177 
 
 offers the lumber at $75 per M, less discounts of 33^% and 6|%. The 
 terms offered by both are l/lO-n/30. Which is the better offer, and 
 how much does the lumber cost? 
 
 148. Marking Goods for the Shelf. In determining the 
 selling price of goods, it is usual to add to the cost a certain 
 per cent as profit. Thus, if a set of books, costing 83.60, is 
 to be sold at a profit of 25 %, they must be marked at ^ more 
 than they cost, or 14.50. 
 
 In marking goods for sale, merchants often place the cost 
 price as well as the selling price on each article. For this 
 cost price a " cipher " is often used. Any word or phrase of 
 ten letters, or any ten arbitrary characters, may be used as a 
 cipher. Thus, 
 
 Repeaters x and y. 
 
 Repeaters # -/- 
 
 When a figure occurs twice or more in succession, a re- 
 peater is used instead of repeating the character for the fig- 
 ure. This renders deciphering more difficult. 
 
 The cost and selling price may both be written on the 
 tag. If so, the selling price is written below a horizontal 
 line, and may be in cipher or plainly |_r|v 1.40 
 
 written. LHT 1-75 
 
 Sometimes the cost price is written in an 
 entirely different cipher, known only to the mon 2.48 
 
 buyer or proprietor. pmr 3.25 
 
 149. Listing Goods for Catalogues. Goods to be listed in a 
 catalogue are usually so priced as to permit a discount or a 
 discount series, and still to sell at a price which will afford 
 the desired per cent of profit. 
 
 1 2 
 
 3 
 
 4 5 6 7 8 9 
 
 
 
 i m 
 
 P 
 
 or t a n c 
 or 
 
 e 
 
 / -2 
 
 3 
 
 ^ ^ ^ y /^ 
 
 f ^ 
 
 L C 
 
 7 
 
 n T J. f- J- 
 
 J X 
 
178 ELEMENTS OF BUSINESS ARITHMETIC 
 
 Thus, if the set of books (Sec. 148) costing $3.60 is to be 
 so listed that a discount from the list of 16| % may be given 
 and still yield a profit of 25 %, the selling price of $4.50 
 (Sec. 186) is J less than the list price, or | of list. The list 
 price, then, would be 14.50 x |, or 15.40. 
 
 If it were desired so to list an article, costing $ 3.92, that a 
 discount of 25%, 16|%, and 2% might be given, and still 
 provide for a profit of 25%, the steps in the process would 
 be as follows : 
 
 I of 13.92 = 1 4.90, selling price. 
 
 $ 4.90 = 98 % or |f of price discounted 2 %. 
 $4.90 X |^ = $5.00. 
 
 $5.00 = f of price discounted 16| %. 
 $5.00 X f = $6.00. 
 
 $ 6.00 = 75 % or f of list price. . 
 $6.00 X I = $ 8.00, list price. 
 
 PROBLEMS 
 
 1. I buy goods for $24 and desire to sell them so I may make 33|%. 
 What must I ask for them that I may give a discount of 20% and still 
 make my desired gain ? 
 
 2. What must goods that cost 20 ^ a yard be marked so as to make 
 25 % after a discount of 16| % ? 
 
 3. What must I ask for goods for which I paid $ 16, if I desire to 
 give a discount of 20% and still gain 25%? 
 
 4. A book seller bought a set of encyclopedias for $40. Remarked 
 them so as to gain 10 % and give a discount of 20 %. What did he ask ? 
 What did he get? 
 
 5. A hardware merchant buys stoves for $24 each. He desires to 
 sell them so as to make 12|% after deducting 25 7o- What did he ask? 
 What did he get? 
 
 6. I wish to sell goods at cost that cost me $400. In order to do so, 
 I mark them up 25%. What per cent discount can I give ? 
 
 7. I buy plows at $ 9.60. If I wish to make a gain of 25 % and still 
 offer a discount of 25% and 10%, what must I ask for them? 
 
TRADE DISCOUNT 179 
 
 8. A dealer marked a carriage by mistake at $ 160, and sold it at a 
 discount of 25 %, thereby losing 16| %. What should he have marked it 
 in order to have gained 25%? 
 
 9. Wishing to sell goods at cost that cost me $5, 1 mark them at 
 20% advance. What discount must I give? 
 
 10. A merchant buys goods for $ 1200. What must he ask in order 
 to give a discount of 16|% and still gain 10%? 
 
 11. How must goods that cost 25 ^ be marked so as to make 20 % and 
 still give a discount of 16| %? 
 
 12. Find the list price, if goods that cost $ 600 are sold at a gain of 
 25 % after a discount of 20 %, 5 %, and 2 %. 
 
 Using the characters in Sec. 148, mark the cost and selling price of 
 the following articles : 
 
 13. Shoes costing $5 and selling for $ 6.50. 
 
 14. Gloves costing $ 22.50 per dozen and selling at 20% gain. 
 
 15. Caps costing $ 7.25 per dozen and selling at 33^ %, gain. 
 
 16. Shoes costing 1 1.97 and selling at 25 % gain. 
 
 17. Boots costing $2.68 and selling at $3.75. 
 
 18. Make a key from the letters contained in the words "blacksmith" 
 and " authorizes," and mark the articles given in the above examples. 
 
XV 
 
 COMMISSION 
 
 150. The Commission Business. In the development of 
 our industries, certain cities become the centers for han- 
 dling particular products. Thus, Chicago is a great grain 
 market; and Chicago, Kansas City, and Omaha are great live- 
 stock markets. 
 
 Where important products are handled so largely, there 
 will naturally be a large number of buyers, and sellers will 
 ship their products from the tributary territory. It would 
 be too expensive to accompany their shipments to market, 
 and yet they want their goods sold to the highest bidder. 
 They must, then, have some one to represent them. This 
 representative or agent of the owner of the product is 
 known as a commission merchant. 
 
 These merchants receive the goods, see that they are 
 properly cared for, effect a sale, and return the proceeds to 
 the owner, less expenses in handling and a fee for the serv- 
 ices rendered. This fee is usually computed as a per cent 
 of the amount of the sale, and is called a commission. 
 
 151. Terms Used. Goods sent to an agent for sale are 
 called shipments^ and are received by the agent or consignee 
 as a consignment^ the title to the goods remaining with the 
 consignor. The statement of the goods received, their sales 
 in detail, with the expenses and charges thereon, showing 
 the net proceeds, is called an account sales. 
 
 In a like manner commission merchants often receive 
 
 180 
 
COMMISSION 181 
 
 orders to buy, as agents, certain goods. For this service 
 they are paid a commission. A detailed statement of the 
 purchases, with expenses, charges, etc., is called an account 
 purchase, 
 
 152. Commission in General. In general any percentage 
 of money or value handled, which is received by an agent 
 for his services as such agent, is a commission. Thus, a real 
 estate agent receives a commission on property sold; an 
 insurance agent a commission on insurance written ; a col- 
 lector, a commission on money collected, etc. 
 
 153. Commission on Purchases. Sometimes, an amount of 
 money is sent an agent witli» instructions to expend it in the 
 purchase of goods. The present tendency in business is to 
 charge commission on the whole amount sent (gross). But 
 many commission houses still charge only on the purchase 
 (net). When the amount sent is to include the sum used 
 for the purchase, together with the commission, it is equal 
 to 100 cfo of the purchase to be made, plus the rate to be 
 paid as commission on such purchase. 
 
 Thus, if 1367.50 were sent to a commission merchant 
 with which to buy wheat, after deducting his commission 
 of 5%, what would he invest in wheat? 
 
 The amount of money he received would include 100 % of 
 the value of the wheat purchased and 5 % of such purchase as 
 commission. $367.50 would then equal 105 % of the amount 
 to be expended on wheat. 105 % = f^. Then $367.50 = |^ 
 of the wheat purchase money. gV would be 2^ of $367.50, 
 or $17.50, the commission, and |^ would be $350, or the money 
 for the purchase of wheat. 
 
182 
 
 ELEMENTS OF BUSINESS ARITHMETIC 
 
 PROBLEMS 
 
 1. How many pounds of coffee, at 27^ per pound, can be bought for 
 $8424, if the agent is allowed 4% of purchase price for buying? 
 
 2. I remitted $2612.90 to a New York agent for the purchase of 
 gloves. If the agent's commission is 4% net, and he makes an added 
 charge of 1 % for guaranteeing quality of goods, how many dozen pairs of 
 gloves, at $8.50 per dozen, should he send me? 
 
 3. I remitted $600 to an agent for the purchase of peaches. If the 
 agent's charges were 5% net for purchase and $12 for inspection, how 
 many baskets did he buy at 43/? 
 
 4. Rule a sheet of paper, copy the following account sales, making 
 
 all necessary extensions, etc. 
 
 diicago, UL, 
 
 u f^y^ 
 
 ^ ( ?^ ,.,.,19. 
 
 Sold for the account of 
 
 By HOWARD PAYNE & COMPANY 
 
 COMMISSION MERCHANTS 
 159 Randolph Street 
 
 ^ 
 
 Z2. 
 
 7^-^-»<%r7^ JP^^^^-^^lf^- 
 
 2^ 
 
 -i9-J/^J..^.,^ i(?.^jlM^. 
 
 
 t^l/^^ 
 
 L£. 
 
 2.f- . ,. . -f-/^^^. 
 
 z/:£e. 
 
 "-(^-^Co 
 
 ^•^-a^^fdr^ 
 
 ^ 
 
 ■^^^^^■i^y^ 
 
 >.^nr^J^^ 
 
 /y J 2-. - 
 
 ^d^ 
 
 ^zbZ£«-^== — //. 
 
 ^ 
 
 C<^7^t-f^^.^ .^^^^ 
 
 .x^^^^ri^.^^^^ 2- ye 
 
 7/ytrryt^^ ^d^^U.^^-^, 
 
 J^ 
 
 . Vfj'^-^^*>^j>jt^..i^ 
 
 5. Prepare an account sales under date of May 15, 1910 for 6000 bu. 
 Wheat, 3000 # Beef, sold Westerfield Bros., Greenville, O., for the 
 account of W. C. Pierse & Co., Union City, Ind., Sales: April 10, 3000 
 bu. Wheat @62^j^, 1500 # Beef @ 9^^; May 12, 3000 bu. Wheat @65^, 
 1500 # Beef @10j^. Charges: freight, $125; cartage, $15; storage, 
 $17.50; insurance, \%\ commission, 2%. 
 
COMMISSION 
 
 183 
 
 6. Rule a sheet of paper and copy the following purchase, making 
 all extensions, etc. 
 
 New York, N.Y., 
 
 6^-^-/ Zff 10 
 
 Purchased by 
 
 L. M. BARKER CS, CO. 
 
 For the account and risk of 
 
 {yf'?^,^^ ^^ JlY^^1^:<J^^ ^:Jr^^ 
 
 c nJ7^ ~C-).-^^ 
 
 JJ ^# 
 
 o-^ 
 
 ^-^-y 
 
 i^r o-- 
 
 .f Wf 
 
 L^ 
 
 -^ ^-ff?^ 
 
 70 a=^ 
 
 !St 
 
 Z^ 
 
 7^ 
 
 O'v^ yltyp^ 
 
 JL24 
 
 •^icr^^^y^ 
 
 L^ 
 
 ■y?.,^^^?^. 
 
 
 Amount charged to your account 
 
 7. Prepare an account purchase for coffee purchased by E. H. Reed 
 & Co., June 15, for the account and risk of Ames, Spencer & Co. Pur- 
 chases: 8 mats G. P. Coffee, 600 # at 2Zf\ 16 mats M. Coffee, 1300 # at 
 34^; 24 mats J. Coffee, 1800 # at 30^. Charges: cartage, |6.70; com- 
 mission, 2 % net. 
 
 8. A man places a claim of $2500 in the hands of an attorney for 
 collection. If the debtor is a bankrupt, having liabilities to the amount 
 of $20,000 and resources to the amount of $15,000, how much should 
 the creditor receive after a commission of 2^% is deducted for collection ? 
 
 9. A principal sends his agent $2500 with which to buy corn. After 
 deducting his commission of 2i% on the amount sent, he buys corn, pay- 
 ing 45^ a bushel. How many bushels does he buy? 
 
 10. Find the rate of commission and guaranty on a purchase if the 
 total cost is $4,143.10, the commission $174.60, the cartage $40, and 
 guaranty $48.50. 
 
 11. A land agent receives $51,000 with which to buy land. After 
 deducting his commission of 5% on the amount sent, how many acres 
 can he buy at $35 per acre? 
 
184 ELEMENTS OF BUSINESS ARITHMETIC 
 
 12. I sent my agent $4080 with which to buy corn. After deducting 
 his commission of 2% net, how many bushels can he buy at 54 j* a 
 bushel? 
 
 13. A dairyman received from his city agent $980 as the net proceeds 
 of a shipment of butter. If the agent's commission is 2^%, delivery 
 charges $12.55, how many pounds at 25 f^ a, pound must he have sold? 
 What was the agent's commission? 
 
 14. An agent remits to his principal $3656.25 after deducting a com- 
 mission of $93.75. What was the selling price of the goods? What 
 was the rate of commission ? 
 
 15. An agent sells cotton for $3600 and charges 2% commission. 
 After deducting his commission of 3% for buying, he buys corn at 30^ 
 a bushel. How many bushels does he buy ? 
 
 16. Find the proceeds of a sale, if the agent charges 2|% commission, 
 $17.50 for cartage, $10.45 for storage, and $4.25 for insurance. The 
 net proceeds are $1545.74. 
 
 17. I sell goods for $3600 and charge 3% commission. I then invest 
 the proceeds, less a commission of 3% on the proceeds, in grain. How 
 much do I invest in grain ? 
 
 18. An agent charged me 5% for selling wheat, and 2% for investing 
 the proceeds in cotton. His commission was $558.90; what was the 
 selling price of the wheat? 
 
 19. I sent my agent in Missouri $2000 with which to buy apples at 
 $1.75 per barrel and to pay commission and drayage. His charges were : 
 commission 4%, drayage 5^ per barrel. How many barrels did he buy, 
 and what was his unexpended balance? 
 
 20. Render in full the following account sales, supplying rates for 
 insurance and commission : April 16, Jacob Speer & Co., Portland, Ore., 
 sold Henry Williams & Co., Chicago, 12,000 Salmon at 11^; 18,000 
 Halibut at 16^. Charges: freight, $250; insurance, $32; commission, 
 $128. 
 
 21. Render an account sales for the following sales, supplying all 
 rates: The Henry St. Clair Co., Cleveland, O., sold for A. W. Ray- 
 mond, Sandusky, O., 250 bu. Beans at $1.06; 400 lb. Cheese at 10^; 40 
 bbl. Flour at $ 5.60 ; 1500 bu. Wheat at 65 ft. Charges : freight, $ 84.60 ; 
 storage, $24.30; inspection, $4; insurance, $3.76; commission, $75.20. 
 
XVI 
 
 TAXES AND DUTIES 
 TAXES 
 
 154. Property Tax. Money levied by the government for 
 defraying its expenses is called a tax. For local, county, and 
 state purposes this tax is usually levied upon property. 
 
 An assessor lists all the property in his district and places 
 a valuation upon it. This valuation is either an estimated 
 actual cash value or a certain per cent of it, as may be deter- 
 mined by law or custom in the locality. There is usually 
 a board of review or equalization in the local district, county, 
 and state, which compares valuations of different classes of 
 property made by the different assessors, and equalizes them. 
 The total of all the property valuations constitute the assessed 
 valuation of the township, village, school district, county, or 
 state. An estimate is then made of the amount needed for 
 the governmental expenses of each of such units, and the 
 per cent such an amount is of the assessed valuation is the 
 tax rate levied. This per cent is so often fractional that it is 
 usually expressed in mills ; meaning the number of mills 
 paid as tax on each dollar of property valuation. 
 
 155. Kinds of Property. Property is of two kinds — real 
 estate^ or lands with their improvements and buildings, and 
 personal property/, which includes all other property. Assess- 
 ments are made separately for each kind of property, and 
 often the personal and real taxes are due and become delin- 
 quent at different dates. Where taxes become delinquent, 
 a penalty is usually charged for their nonpayment on time. 
 
 185 
 
186 ELEMENTS OF BUSINESS ARITHMETIC 
 
 156. Other Forms of Taxation. Taxes levied solely upon 
 tangible property often lead to serious inequalities. Efforts 
 to distribute the burdens of government according to the 
 benefits derived therefrom and according to the ability 
 to pay, have led to many forms of special taxation, such as 
 taxes upon capital stock, gross earnings, or mileage of wires 
 or railroad bed, etc. Franchises^ granted by local legislative 
 bodies, giving to private corporations the right to use the 
 streets for pipes, rails, or wires, etc., are often taxed. Inheri- 
 tances and incomes are also taxed in some countries and states. 
 Revenue is derived, too, from the sale of licenses^ from fees 
 for official acts, and from fines imposed for breaches of law. 
 In some states a tax is laid upon each voter, regardless of 
 property owned. This is a per capita tax, and it is known as 
 a poll tax. 
 
 157. Finding the Rate of Taxation. Example. A township 
 board finds that the assessed valuation of all property in the 
 township is $600,000, and the total amount of revenue needed 
 is $15,000. How much tax must be levied on each dollar of 
 property valuation ? 
 
 If 115,000 must be raised on a valuation of $600,000, the 
 part taken for taxes will be sYAVtT' ^^ lo' which is 2|-%, or 
 25 mills, on the dollar. 
 
 158. Finding the Tax. Example. A man's assessment is 
 $16,000. The total rate of taxation for all purposes in the 
 city of his residence was 25 mills. What was his individual 
 tax? 
 
 If he was to pay 25 mills on each dollar, he would pay 2 J %, 
 or ^1^, of his property value as a tax. 
 ^-^ of $16,000 is $400, amount of tax. 
 
 PROBLEMS 
 
 1. If property is valued at $ 500 and the taxes are $ 5, what is the 
 rate of taxation ? 
 
TAXES AND DUTIES 187 
 
 2. If property is valued at 1 25,000 and a tax of $500 is to be paid on 
 
 it, what is the rate? 
 
 3. The property in a town is valued at $986,600 and the amount of 
 revenue to be raised |4933. What would be my tax if my property is 
 valued at $5500? 
 
 4. The assessed valuation of a man's property is $12,500, and the 
 rate of taxation is 23 mills on the dollar. What is his tax ? 
 
 5. The real estate of a city is valued at $ 9,864,000. A revenue of i\ p ^ 
 $108,640 must be raised. The tax on personal property is $12,000. ^ 
 What will be the rate on real estate ? 
 
 6. What will be a man's tax, at the above rate, who has property ; 
 valued at $6400? 
 
 7. The value of the property in a township is $ 236,600 and the amount 
 of money needed for state purposes is $709.80. What is the rate of 
 taxation for the state ? The rate fixed by the school board is 9.4 mills. 
 What amount is collected for school purposes ? 
 
 8. I have property valued at $ 9800 and the total rate of taxation is i/ 
 
 6.4 mills. How much tax do I pay ? 
 
 9. The personal property in a certain township is valued at $ 120,000 
 and the real estate at $2,400,000. The personal property is taxed at f 
 its value and the real estate at I its value. If the rate is 25.8 mills, 
 how much tax is collected ? 
 
 -' 10. The expenses of a certain town are $595,512.80. The tax rate 
 is 16 mills on the dollar. What is the assessor's valuation ? 
 
 11. The assessed valuation of a county as fixed by the board of 
 equalization is $25,395,180. The rates of taxation are as follows : State, 
 
 3.5 mills ; State University, 2 mills; State Agricultural College, 1.5 mills; 
 State Normal School, 1 mill ; County, 3 mills ; County School, 1 mill ; 
 County Bridge, 3 mills; Insane Hospital, .5 mill; County Bond, 2 mills; 
 County Poor, .2 mill; Soldier's Relief, .3 mill; County Road, 2 mills; 
 Board of Health, .1 mill; Teacher's Fund, 9.4 mills (average) ; School 
 Contingent Fund, 3.7 mills (average). What is the total tax raised in 
 the county ? What is the total rate of taxation ? What is property taxed 
 that is valued at $5400? What is the amount of tax levied for each 
 purpose? 
 
 12. A tax of $ 6649.60 is to be raised in a certain city. The valuation 
 of the real property is $898,500, which is assessed at f its valuation. 
 The personal property is assessed at $65,000. There are 1200 polls, 
 
188 ELEMENTS OF BUSINESS ARITHMETIC 
 
 assessed at 82 each. What is the rate of taxation, the rate for collection 
 being 1%! rWhat is a man's tax whose real estate is valued at $ 14,500, 
 and personal property at $ l^OO ? ^ 
 
 DUTIES 
 
 159. National Taxation. National revenue is raised largely 
 by the forms of taxation known as duties and excises. Im- 
 port duties are taxes levied upon goods imported into a 
 nation. In the United States, Congress establishes a tariff 
 or list of such duties. These duties are collected at the 
 custom houses to be found at the various ports of entry. The 
 officer in charge of the custom house is called a collector. 
 
 Import duties are of two kinds : those consisting of fixed 
 charges on certain units of measurements of commodities, as 
 the gallon, pound, or yard, called specific duties, and those 
 which form a certain per cent of the value of the goods be- 
 ing imported, called ad valorem. Specific duties only are 
 levied on certain articles, ad valorem only on others, and on 
 some, duties of both kinds are levied. A statement or in- 
 voice, showing the various articles being imported, the dis- 
 tinguishing mark, number, or quantity and value of each, is 
 termed a customs declaration or manifest. 
 
 When the goods are purchased in a foreign country for 
 import, an invoice giving descriptions and prices is filed 
 with the United States Consul in the district where the pur- 
 chase is made, and he sends a copy to the collector of the 
 port to which the goods are shipped. When the goods 
 arrive, the cases are opened and the quantity and value as 
 compared to that stated by the invoice is ascertained. For 
 too great an undervaluation a fine may be imposed or the 
 goods confiscated. Travelers' baggage is also inspected for 
 dutiable goods. 
 
 Instead of paying duty at once, goods may be stored in 
 a bonded warehouse, from which they may be withdrawn for 
 
TAXES AND DUTIES 
 
 189 
 
 export or upon payment of the duty. Such goods are said 
 to be in Bond. Importing goods without the knowledge of 
 the government officers, thus avoiding the payment of 
 duty, is called smuggling., for which heavy penalties are 
 prescribed. 
 
 Excises are taxes levied upon certain specified articles, as 
 liquors, tobacco, opium, oleomargarine, etc., which are grown 
 or manufactured in this country. For the collection of these 
 taxes, there are National Revenue districts, and collectors for 
 each district. 
 
 Table of Custom Duties on Certain Articles 
 
 Abticles 
 
 Tariff Eate 
 
 Alcohol 
 
 Brushes 
 
 Carpets, treble ingrain 
 
 Carpets, two-ply 
 
 Carpets, tapestry, Brussels 
 
 Carpets, Wilton, Axminster 
 
 Cigars and cigarettes 
 
 Cotton hosiery (made on machines and not 
 
 otherwise provided for) 
 
 Cotton plushes, unbleached 
 
 Cutlery, more than $3 per doz 
 
 Cutlery, razors, $3 or more per doz. . . 
 
 Cutlery, sword blades 
 
 Dyewoods, common extracts of .... 
 Glass, polished plate not over 384 sq. in. . 
 
 Gloves, men's kid 
 
 Glue, value not over 10 j^ per lb 
 
 Hay 
 
 Linen, wearing apparel 
 
 Silk, woven 
 
 Silk laces, wearing apparel 
 
 Skins, uncured, raw 
 
 Velvets, cotton, unbleached silk .... 
 Wines, still, in casks containing more than 
 
 14 % alcohol 
 
 Woolen or worsted clothing 
 
 ¥4^ per lb. 
 
 40% ad valorem 
 
 22 J? per sq. yd. & 40% ad valorem 
 
 18^ per sq. yd. & 40% ad valorem 
 
 28^ per sq. yd. & 40% ad valorem 
 
 60^ per sq. yd. & 40% ad valorem 
 
 $4.50 per lb. & 20% ad valorem 
 
 30% ad valorem 
 
 9^ per sq. yd. & 25% ad valorem 
 
 15^ per piece & 35% ad valorem 
 
 $1.75 per doz. & 20% ad valorem 
 
 50% ad valorem 
 
 %^ per lb. 
 
 10 j? per sq. ft. 
 
 $4 per doz. pair 
 
 2^2^ per lb. 
 
 $4 per ton 
 
 50% ad valorem 
 
 45^ per lb. and 60% ad valorem 
 
 60% ad valorem 
 
 Free 
 
 9^ per lb. & 25% ad valorem 
 
 60^ per gal. 
 
 44^ per lb. & 60% ad valorem 
 
190 
 
 ELEMENTS OF BUSINESS ARITHMETIC 
 
 PROBLEMS 
 
 1. What is the duty on 60 yd. of silk, weighing 4 oz. to the yard, 
 invoiced at $ 1.50? 
 
 2. What is the duty on a consignment of 26 doz. men's kid gloves, 
 invoiced at $22.50 per dozen ? 
 
 3. If silk laces are invoiced at 7540 francs, what will be the duty? 
 
 4. What will be the duty on an importation of |646 worth of skins? 
 
 5. What is the duty on an invoice of 642 lb. glue; 240 lb. alcohol; 
 20 doz. brushes, at 25)^ each ; and 450 lb. extract of dye wood? 
 
 6. What is the duty on 14 doz. sword blades, worth $7.40 per dozen, 
 and 16 doz. razors worth |4.50 per dozen? 
 
 7. For what must I sell hose, if they are invoiced at $2 per dozen, 
 in order to gain 20% after paying the duty? 
 
 8. A merchant imported 1420 yd. of plush (f yd. wide), invoiced 
 at 2 francs a square yard. At what price per yard must he sell it to gain 
 25%, after paying $65 freight charges and the import duty? 
 
 9. What is the duty on 650 yd. Axminster carpet, 30 in. wide, if the 
 invoice price is 40^ per yard? For what must it be sold to gain 33^ 7o? 
 
 10. 1000 boxes (100 in each) of cigars weighing 1200 lb. net cost 
 $13,675, delivered. If the freight and charges, other than the duty, 
 amount to $45, what was the invoice price of the cigars? 
 
 11. After allowing 10% for leakage, a wine merchant paid $1768 duty 
 on 24 casks of wine at $2 per gallon. How many gallons did the casks 
 originally contain? 
 
 12. What will be the duty on 250 boxes plate glass, each containing 
 25 plates, 16 x 20 in.? 
 
 13. Find the amount of the duty and the total cost of the following: 
 
 Manchester, England, Jan. 14, 19 — . 
 Messrs. A. T. BOYD & CO., 
 
 New York, N.Y. 
 
 Bought of WILLIAM & FIRTH 
 
 Terms 
 
 : 30 da. 
 
 
 
 
 
 ^ 
 
 
 565 yd. Silk Velvet 1 12s. 
 3050 yd. Linen 45. Sd. 
 1450 yd. Cotton Webbing Is. 
 4250 yd. Wilton Carpet 125. 6c?. 
 
 
 
 
 
TAXES AND DUTIES 
 
 191 
 
 14. 
 
 Manifest No. 675 Invoiced at Bradford, Eng., June 12, 19 — . 
 
 INWARD FORWARD ENTRY OF MERCHANDISE 
 
 Imported by Gates & Warman. In the Steamer Columbia. 
 
 Amos Peterman, Master. From Liverpool, Eng. 
 
 Arrived July 15, 19 — . 
 
 Marks 
 
 No. 
 
 Packages and 
 Contents 
 
 Quantity 
 
 Free 
 List 
 
 Valub 
 
 Ad 
 
 Valorem 
 
 Duty 
 
 Specific 
 Duty 
 
 TOTAI. 
 
 Duty 
 
 A.A. 
 
 1356 
 756 
 
 4 cases Woolen 
 
 Clothing 
 6 cases Cotton 
 
 Hose 
 
 2400 lb. 
 192 doz. 
 
 
 £342.12.0 
 £74.10.0 
 
 ? 
 ? 
 
 ? 
 ? 
 
 ? 
 ? 
 
 
 
 
 
 
 
 $ 
 
 •$ 
 
 $ 
 
 $ 
 
 
 
 
 
 
XVII 
 
 INTEREST 
 
 160. Loaning Money. When money is loaned, a charge 
 is made for its use. This charge or rental for the use of 
 money is called interest. Interest is always a certain per 
 cent of the amount loaned, and the rate is the rental for a 
 period of one year. 
 
 Thus, 15000 loaned at 6% would mean that 6% of the 
 $5000, or $300, must be paid for its use each year. Of 
 course, one half of that amount, or $ 150, would be the in- 
 terest for six months, and twice that amount, or $ 600, would 
 be the interest for two years, etc. 
 
 161. Notes and Mortgages. When loans are made, a 
 written instrument or contract called a promissory note is 
 signed by the persoij receiving the money. It is a written 
 promise to pay the amount named at a certain time, usually 
 with interest from date of the note, or from the date agreed 
 upon for payment, called maturity^ and contains an acknowl- 
 edgment of the receipt of value therefor. 
 
 The payment of a note is often made more secure by the 
 signature of other persons as guarantors and known as in- 
 dorsers, or by conveyance of property, real or personal, con- 
 ditioned upon the nonpayment of the note. When property 
 is thus pledged, the instrument of conveyance is called a 
 mortgage. If for personal property, it is a chattel mortgage ; 
 if for real estate, a real estate mortgage deed. In either case, 
 the title only to the property is transferred, possession to be 
 given upon nonpayment of the note, and upon the success 
 
 192 
 
INTEREST 193 
 
 of legal proceedings known as foreclosure. The one who 
 signs the note is called the maher^ and the one to whom the 
 note is to be paid is called the payee. 
 
 /i lA^-^^ C4''ri^..^.yL.d^t7-ny, '7(ayn..4'-eLJ . ///oyy f, /f 
 
 .^jz^-^ZiiCi-^ /Z'gl^k.-tr-yz^itL^ ~Aj^-tZ^^yi,^k^,^v-f' Cyt^yi.-^e6£yt<d^trny^ 
 
 /!/^a,■€U<..^^^4.C'C^^c^^ 
 
 -<^'i<2^- 
 
 
 A Promissory Note. 
 
 162. Bonds. For loans made to corporations, the note is 
 called a hond. These may be either governmental, as city 
 or county, or private, often called industrial bonds. In 
 notes or bonds extending through a period of years, the in- 
 terest is often provided for in separate notes for each interest 
 payment, which are attached to the principal note, and are 
 known as interest coupons. These bonds are usually issued 
 for the purpose of raising money for improvements, develop- 
 ment of property, etc. They pledge the property of the corpo- 
 ration as security, and are generally sold in the open market. 
 
 163. Compound Interest. In Compound Interest^ unpaid 
 interest is added to the principal at the end of each interest 
 period. The whole amount is then considered as a new prin- 
 cipal^ upon all of which interest thereafter is to be paid. 
 The interest and principal are thus compounded or united. 
 Thus, 1100 at 6^ would amount to $106 at the end of the 
 first year, to 1112.36 at the end of the second year, to $119.10 
 at the end of the third year, etc. Money loaned at compound 
 interest increases so rapidly as to more than double itself at 
 6 % in 12 years. 
 
194 ELEMENTS OF BUSINESS ARITHMETIC 
 
 164. Periodic Interest. When interest due at the close of 
 each interest period begins to draw simple interest until the 
 obligation is paid, it is said to be periodic interest. That is, 
 the interest on the principal for each time period draws in- 
 terest ; but this interest on the interest does not become due 
 until the note is paid, and therefore does not itself draw in- 
 terest ; i.e. it is not compounded. Thus, a |100 note, if paid 
 at the end of the first year, would amount to $106, at end of 
 second year to $112.36, third year to $119.08, etc. 
 
 If the interest period is one year, it is annual; if six months, 
 semiannual ; and if three months, quarterly interest. 
 
 $100 at 6% simple interest would amount in ten years to 
 $160; at annual periodic interest to $176.20; and at com- 
 pound interest to $179.08. 
 
 165. Legal Limitations to Interest. When the rate of in- 
 terest is not stipulated in the note or contract, the rate fixed 
 by law is collectable. Such rate is known as the legal rate., 
 and varies in different states and territories. 
 
 Most states establish a limit to the rate which may be 
 charged even when agreed to by both parties. This is known 
 as the contract rate. Interest charged in excess of the amount 
 allowed by law is usury. While several states have no pro- 
 visions regarding usury, most states forbid it. Various pen- 
 alties are prescribed, such as the forfeiture of principal and 
 interest, of principal, of interest, or of excess of interest over 
 the legal rate. 
 
 The collection of compound interest cannot usually be en- 
 forced by legal process, but its payment or receipt is not 
 illegal. 
 
 In some states periodic interest may be collected when in 
 the contract, while in others it cannot be enforced. When 
 each interest payment is put in the form of a note, as a coupon, 
 it may be collected in any state and is not considered usury. 
 
INTEREST 195 
 
 In order to prevent undue hardship, most states have, in 
 the past, allowed a creditor three days after the stipulated 
 date of maturity, before a note might be collected through 
 legal process. These are known as days of grace. The cus- 
 tom, however, is dying out, and most states have repealed 
 the law. Improved communication leaves less necessity for 
 days of grace than formerly. 
 
 166. Common and Exact Interest. In computing the time 
 of notes or other interest-bearing obligations, the method of 
 compound subtraction (97) is used. It is the common prac- 
 tice to count a month as uniformly 30 days and the year 
 360 days. If the note reads " days after date," the date of 
 maturity is usually computed to the exact day ; if " months 
 after date," then the same day of the month in the appropri- 
 ate month. 
 
 The method still in use by the United States government 
 and by the more conservative business men takes account of 
 the exact number of days in the intervening calendar months, 
 and the days are 365ths of a year. This is known as exact 
 interest. Exact time between given dates and exact interest 
 at any rate on various amounts for any number of dates, are 
 usually found by reference to " Exact Interest Tables," with 
 computations already made. 
 
 Interest found by the 60-day method, or by any method 
 using 30 days as a month, counts a year as 360 days. It is, 
 therefore, ^l-g-, or y^^, greater than " exact " interest for the 
 same period. Adding ^^ of itself to " exact " interest would 
 likewise equal " 30-day " interest. 
 
 167. Finding Interest. When the period of a note or other 
 obligation is an exact number of years, the problem is one of 
 simple percentage. If the interest rate is 6 % and the time 
 three years, 18 % of the amount loaned would be the inter- 
 est ; if 8 % for 4 years, 32 % of the amount loaned, etc. 
 
196 ELEMENTS OF BUSINESS ARITHMETIC 
 
 But most interest computations in business have to do with 
 short-time loans and discounts. Hence they involve chiefly 
 months and days. On account of the frequent occurrence 
 of 6 % as a rate of interest in business, and because of the 
 simple relation 6 bears to 12 (the number of months in the 
 year) and the easy fractional parts that relation makes in 
 computing time, interest on any sum for a short period is most 
 easily and accurately calculated as though at 6 9^, and then 
 changed to the required rate. 
 
 168. Six Per Cent Interest for 60 Days. If every 12 months 
 6 % of the amount of the loan must be paid as interest, every 
 two months -f^ (e) ^^ ^^^^' or 1 %, must be paid. Then 1 % 
 of the amount loaned would be the interest for 2 months, or 
 60 days. 
 
 Thus, 1840 at 6% for 60 days is 18.40; and 11726 at 6% 
 for 60 days is f 17.26. From this it will be seen that the 
 interest on any sum at %% for 60 days is as many cents as 
 dollars loaned ; the same is true of 4 % interest for three 
 months and 3% interest for four months. These rates are 
 usable in savings banks and for interest on government 
 bonds. 
 
 169. Six Per Cent for Any Time. 
 
 Problem. — Find the interest on $840 for 1 yr. 5 mo. 6 da. 
 
 $8.40 , interest for 2 mo. 
 $67.20 ($8.40 X 8), interest for 16 mo. 
 4.20 (^ of $8.40), interest for 1 mo. 
 
 ^ (-ji^ of $8.40), interest for 6 da. 
 
 $72,24, interest for 1 yr. 5 mo. 6 da. 
 
 To find the interest for years and months, first reduce to 
 months. Then take such a simple multiple of the interest 
 for two months as will make the required time, if for even 
 months. If for odd months, add \ of the two months' interest 
 
 J?, r^ 
 
INTEREST 197 
 
 to the interest for one less than the required number of 
 months. 
 
 For days, fractional parts of the interest for 2 months 
 should be taken, using simple fractional parts wherever prac- 
 ticable and doing as much mentally as possible. Thus, the 
 interest for 15 days is ^ that for 60 days; for three days J of 
 ■f-Q of it ; for 5 days -^^ of it ; for 7 days -^q of it plus J of the 
 latter (6 days plus 1 day), etc. 
 
 170. Interest at Any Rate. If any other rate had been 
 given in the problem in the above section, the process would 
 be identical to the point of finding the interest at 6 % for 
 the given time. If the rate were 5%, the required interest 
 would be 1 less than $72.24, or f 60.20 ; if 7%, 884.28, or 1 
 more; if 4%, $48.16, or 1 less; if 8%, $96.82, or 1 more; 
 if 4J %, $54.18, or { less, etc. 
 
 Hence to find interest at any rate : Find the total interest 
 at Q% and increase or decrease it hy the fractional 'part that 
 the required rate is greater or less than 6 %. 
 
 171. To find the Face of a Note. 
 
 Problem. — The interest on a note for 8 mo. at 5 % was 
 $40. Find the face of the note. 
 
 $40, interest at 5 % for 8 mo. 
 
 8, interest at 1 % for 8 mo. (| of $40). 
 
 $48, interest at 6 % for 8 mo. 
 
 $12, interest at 6% for 2 mo. (\ of $48). 
 
 Then, $1200 ($12 x 100) is the face of the note. 
 
 When the interest is given at any rate, first change it to 
 what it would amount to at Q%. This being the interest at 
 6 % for the given time, such a fraction of it should he taken as 
 will give the interest for two months only. Since this would 
 represent as many cents as dollars loaned, 100 times that 
 amount would he the face. 
 
198 ELEMENTS OF BUSINESS ARITHMETIC 
 
 172. To find Note Period. 
 
 Pkoblem. — The interest on |680 at 4| % was 120.40. 
 How long did the note run ? 
 120.40, interest at 4J%. 
 
 6.80 , interest at li % Q of 120.40). 
 127.20, interest at 6%. 
 
 6.80, interest on S680 for 2 mo. 
 
 3.40, interest on $680 for 1 mo. 
 f 2T.20 -h 13.40 = 8, the length of note period in months. 
 
 The interest given is changed to the interest at 6fo- The 
 interest on the given principal for 1 month atQ^fo is then found, 
 and divided into the interest for the whole term at Qf/o- The 
 quotient is the number of months the note runs. 
 
 173. To find the Rate of Interest. 
 
 Problem.— If a note for 1660 draws 138.50 in 1 yr. 2 
 mo., what is the rate of interest? 
 
 1660 for 1 yr. 2 mo. would earn 146.20 at 6 %, and l of 
 that, or $7.70 at 1%. The required rate is as many per 
 cent as $7.70 is contained times in $38.50, or 5 %. 
 
 The interest at l^o for the given time, divided into the given 
 interest, will give the rate required. 
 
 174. To find what Sum will produce a Given Amount. 
 Problem. — What will produce $3955.95 at 4 % for 1 yr. 
 
 6 mo. 18 da. ? 
 
 $.01, interest $1 will produce in 2 mo. at 6%. 
 .09, interest on $1 for 1 yr. 6 mo. (18 mo.). 
 .0025, interest on $1 for 15 da. 
 .0005 , interest on $1 for 3 da. 
 .093, interest on $1 for 1 yr. 6 mo. 18 da. 
 .031, interest on $1 at 2 % (i of $.093). 
 .062, interest on $1 for 1 yr. 6 mo. 18 da. at 4 %. 
 
 $1 will amount to $1,062 in 1 yr. 6 mo. 18 da. at 4%. 
 
 J 
 
INTEREST 199 
 
 13955.95 --11.062 = 3725, or the number of dollars which 
 will amount to $3955.95 at 4% for the given time. 
 
 The amount of ^1 for the given time and rate is first found. 
 The number of times that is contained in the given amount is 
 the number of dollars required to produce that amount. 
 
 175. Other Methods. Interest tables, with the interest 
 already calculated for any number of dollars for any number 
 of days at given rates, are largely used by banks, trust com- 
 panies, insurance offices, etc. The calculated interest is 
 arranged in tabulated form for convenient reference, and 
 the work of computing interest is thereby greatly lessened. 
 Many different arrangements of such tables are published. 
 
 The formal 6 % method is often used by accountants, par- 
 ticularly when the time is more than 1 year. It is based on 
 finding the interest on $1 for the given time, and multiply- 
 ing that by the number representing the dollars loaned. 
 Thus, $1 could bear $.06 interest in one year, $.005 interest 
 in one month, 1.000 J interest for one day, *etc. 
 
 The interest at any other rate would then be found in the 
 same way as by the 60-day method. (Sec. 169.) 
 
 176. Partial Payments. Payments on long-term notes, 
 made from time to time, are known as partial payments. 
 These payments, with their dates, are usually indorsed on 
 the back of the note. 
 
 By what is known as the United States Rule^ each payment 
 is deducted from the amount of the debt at the time of the 
 payment, and the remainder or mqw principal will bear simple 
 interest until the next payment. If a payment is not as large 
 as the accrued interest, it cannot reduce the debt. It is 
 therefore disregarded, and added to the next payment. 
 
 By the Merchants' Rule^ the sum of all the partial pay- 
 ments, together with the interest on each of them from its 
 date until maturity or date of settlement of the note, is sub- 
 
200 ELEMENTS OF BUSINESS ARITHMETIC 
 
 tracted from the sum the original principal would amount 
 to, at simple interest, on the date of settlement. 
 
 Because merchants and bankers prefer short-term notes, 
 with renewal (new note) for part of debt remaining unpaid, 
 long-term notes, except when in the form of mortgage notes 
 or bonds, are disappearing from business. Partial payments 
 on mortgage notes and bonds may sometimes be made, but 
 only in stipulated amounts and at specified dates, while the 
 interest is covered by coupon notes, which are treated as 
 separate obligations. On account of its occasional use in 
 some sections of the country, the following illustration of 
 partial payments is given : 
 
 Problem. — Omaha, Neb., Jan. 1, 1920. 
 
 $500.00 
 
 On demand, I promise to pay' C. L. Webster, or order. 
 
 Five Hundred and 00/100 Dollars, value received, with 
 interest from date at 6 %. J. L. Fernby. 
 
 If endorsed March 1, 1920, $105; July 1, 1920, |108; Jan. 
 1, 1922, 120. What is due at settlement, Nov. 1, 1922? 
 1500, original debt. 
 5^ interest 2 mo. (Jan. 1 to March 1, 1920). 
 
 505, due March 1, 1920. 
 
 105, payment Jan. 1, 1920. 
 
 400, unpaid debt or new principal. 
 8, interest 4 mo. (March 1 to July 1, 1920). 
 
 408, due July 1, 1920. 
 108, payment July 1, 1920. 
 300, unpaid debt July 1, 1920. 
 
 $27, interest for 18 mo. (July 1, 1920 to Jan. 1, 1922), 
 120, payment Jan. 1, 1922, deferred. 
 42, interest for 28 mo. (July 1, 1920 to Nov. 1, 1922). 
 842, due Nov. 1, 1922. 
 20, payment Jan. 1, 1922. 
 $ 322, net amount due at settlement. 
 
INTEREST 201 
 
 PROBLEMS 
 
 1. Mr. Williams has the use of $ 240 for 1 yr. at 6 %. What amount 
 of interest does he pay ? 
 
 2. A merchant borrows $260 for 1 yr. at 5%. How much does he 
 pay for its use ? 
 
 3. At 4 % what must I pay for the use of $ 600 for 1 yr. ? 
 
 4. Mr. Constable borrowed $750 for 1 yr. at 8%. How much inter- 
 est did he pay ? 
 
 5. A note of $ 324 for 2 mo. at 6 % will draw how much interest ? 
 
 6. A banker loans $ 240 for 4 mo. at 6 %. How much interest does 
 he receive ? 
 
 7. A farmer sold a horse, taking a note of $ 175 for 6 mo. at 6 % in 
 payment. Find amount due at the end of the time. 
 
 8. A manufacturer borrows $ 1250 for 3 mo. at 6 %. How much 
 interest does he pay ? 
 
 9. B borrows $ 75 for 1 mo. at 6 %. What amount is due at the end 
 of the time ? 
 
 Find Interest: 
 
 10. On $ 150 for 1 mo. @ 6 %. 
 
 11. On 1720 for 4 mo. @ 6%. 
 
 12. On 1480 for 3 mo. @ 6%. 
 
 13. On 170 for 5 mo. @ 6 %. 
 
 14. On $65 for 10 da. @ 6 %. 
 
 15. On $120 for 45 da. @ 6 %. 
 
 16. On $160 for 6 da. @ 6%. 
 
 17. On $ 142 for 3 da. @ 6 % 
 
 18. On $85 for 6 mo. @ 8 %. 
 
 19. On $460 for 8 mo. @ 5 %. 
 
 20. On $ 540 for 10 mo. @ 7 %. 
 
 21. On $60 for 2 yr. 6 mo. @ 6%. 
 
 22. On $54 for 1 yr. 2 mo. 15 da. @ 6%. 
 
 23. On $480 for 75 da. (60 + 15) @ 6 %. 
 
 24. On $76 for 63 da. (60 + 3) @ 6 %. 
 
 25. On $960 for 90 da. (60 + 30) @ 6 %. 
 
202 ELEMENTS OF BUSINESS ARITHMETIC 
 
 26. On 124 for 85 da. (60 + 10 + 15) @ 6 %. 
 
 27. On $36 for 123 da. (2 60's + 3) @ 6%. 
 
 28. On $48 for 54 da. (60 - 6) @ 6 %. 
 
 29. On $54 for 36 da. (30 + 6) @ 6 %. 
 
 30. On $64 for 42 da. (30 + 12) @ 6%. 
 
 31. On $480 for 18 da. (3 6's) @ 6%. 
 
 32. On $730 for 72 da. (60 + 12) @ 8%. 
 
 33. On $75.60 for 87 da. @ 8%. 
 
 34. On $ 1244 for 8 mo. 20 da. @ 7 %. 
 
 35. On $450 for 1 mo. 20 da. @ 5 %; 4 %. 
 
 36. On $860 for 3 mo. 15 da. @ 4%; 8 %. 
 
 37. The interest on a note was $40, the rate 6%, and the time 8 mo. 
 Find the face. 
 
 38. What was the face of a note, if the time was 5 mo., the interest 
 $16, and the rate 6 %? 
 
 39. The time was 7 mo., the rate 6 %, and the interest $ 13.30. What 
 was the face of the note ? 
 
 40. I paid a bank $3.44 interest. I had the money 8 mo. at 6 7o' 
 How much did I have ? 
 
 41. The interest at 6% on a note for 1 yr. 3 mo. was $ 3.15. What 
 was the face ? 
 
 42. What was the face of a note given for 5 mo. at 6 %, if the interest 
 was $19.50? 
 
 43. The interest on $ 680 at 6 % was $ 27.20. How long did the note 
 run? 
 
 44. How long did a note for $ 800 at 6 % run, if the interest was $38? 
 
 45. Mr. Wood paid $20.90 interest at 6 % on a note of $380. How 
 long did he have it ? 
 
 46. If the principal was $ 120, rate 6 %, and interest $ 7.80, what was 
 the time ? 
 
 47. A note of $ 150 drew $ 8 interest at 8 %. How long had it run ? 
 
 48. The interest on a note of $ 180 at 8 % was $ 12. How long had it 
 run? 
 
 49. If a note for $650 draws an interest of $39 in 1 yr., what is 
 the rate? 
 
INTEREST 203 
 
 50. If I pay 1 60 on $ 1000 for 1 yr., what is the rate ? 
 
 51. At what rate would $ 710 produce $ 17.75 in 5 mo. ? 
 
 52. The interest on $ 640 for 4 yr. was $ 179.20. What was the rate? 
 
 53. What rate was charged if a merchant paid $ 43| for the use of 
 $ 650 for 16 mo. ? 
 
 54. A farmer deposited $ 425 in the bank and at the end of 8 mo. 
 drew out $ 437.75. What rate of interest did he receive ? 
 
 55. The amount due on a note of $ 120 at the end of 8 mo. was f 124. 
 What rate of interest was charged? 
 
 56. Principal $200, time 3 yr., interest $ 30. Rate? 
 
 57. Principal $ 180, rate 4 %, interest $ 14.40. Time ? 
 
 58. Time 8 mo., rate 6 %, interest 1 7. Principal? 
 
 59. Principal 1 76, time 3 yr., amount $ 87.40. Rate? 
 
 60. Principal $ 138.40, rate 4%, time 2 yr. 6 mo. 15 da. Interest? 
 
 61. Amount $3961.26, rate 4 %, time 1 yr. 6 mo. 18 da. Principal? 
 
 62. Amount $ 502.09, rate 6 %, principal $ 460. Time ? 
 
 63. Time 4 yr. 6 mo., rate 5 %, amount 1 7181.80. Interest? 
 
 64. Principal $7548, time 3 mo. 5 da., interest $119.51. Rate? 
 
 65. Principal $ 900, rate 5 %, interest $ 56.25. Time ? 
 
 66. Principal $ 72, rate 6 %, time 2 yr. 6 mo. Interest? 
 
 67. Principal $460, rate 4 %, time 1 yr. 5 mo. 6 da. Interest? 
 
 68. Principal $ 126, rate 7| %, time 2 yr. 3 mo. 18 da. Amount? 
 
 69. Principal $ 640, rate 6 %, interest $ 86.40. Time ? 
 
 70. Rate 6%, time 1 yr. 6 mo. 24 da., interest $ 338.40. Principal? 
 
 71. Principal $ 3200, time 2 yr. 6 mo. 6 da., amount $ 3602.67. Rate ? 
 
 72. Principal $324, rate 5 %, time 1 yr. 4 mo. 15 da. Interest? 
 
 73. Principal $ 960, rate 5%, interest $114. Time? 
 
 74. Principal $ 370, time 1 yr. 6 mo. 15 da., interest $ 34.23. Rate ? 
 
 75. Rate 8%, time 2 yr. 8 mo. 24 da., interest $125.73. Principal? 
 
 76. Principal $2125, time 11 mo. 24 da., amount $2250.38. Rate? 
 
 77. Principal $ 1440, rate 6 %, amount $ 2100. Time ? 
 
 78. A note for $450, dated April 15, 1914, payable on demand, with 
 interest at 6 %, bears the following indorsements : June 1, $ 50 ; Oct. 10, 
 $ 100 ; Nov. 15, $ 35 ; Dec. 10, $ 75. What is due Feb. 12, 1915 ? 
 
 ^Ni 
 
 
204 ELEMENTS OF BUSINESS ARITHMETIC 
 
 79. What is due May 15, 1916, on a note for $ 600, dated July 12, 
 1914, bearing 5 % interest, with the following indorsements : Sept. 21, 
 1914, $75; Feb. 18, 1915, |200; May 1, 1916, $65? 
 
 80. A note for $ 1500, dated Nov. 14, 1915, and due Nov. 14, 1917, 
 with interest, has the following indorsements: Jan. 27, 1916, $340; 
 May 13, 1916, $ 250 ; Sept. 16, 1916, $25 ; Feb. 17, 1917, $20. What is 
 due at maturity ? 
 
 Compute exact interest on the following. (See Sec. 166 and Prob. 1, 
 at close of Chapter IX.) 
 
 81. $ 720 for 73 da. at 8 %. 
 
 82. $460 for 63 da. at 5%. 
 
 83. $860 for 1 mo. 20 da. at 8 %. 
 
 84. $800 from Jan. 14, 1900 to June 18, 1900, at 8 %. 
 
 85. $ 760 from May 16 to July 27, at 7 %. 
 
 86. $ 154.80 from Jan. 15 to Feb. 16, at 5%. 
 
 87. $ 172.50 from Nov. 12, 1891 to July 6, 1892, at 4%. 
 
 88. $465.20 from Jan. 29, 1901 to Jan. 1, 1903, at 5 %. 
 
 89. $900 from June 15, 1901 to Jan. 15, 1904, at 4|%. 
 
 90. $ 140 from May 1 to Sept. 25, at 8 %. • 
 
 91. $ 560 from Jan. 24 to Dec. 16, at 5 %. 
 
 92. $37.50 from April 3 to May 1, at 8%. 
 
 93. $ 184.50 from June 4, 1901 to July 16, 1903, at 7%. 
 
XVIII 
 
 BANKING AND DISCOUNT 
 
 177. Banking. A hank is a business institution for 
 the receiving and safe-keeping of money, and the making 
 of loans. It also deals in credits, discounts negotiable 
 paper, collects accounts, and makes payments in other 
 cities. 
 
 A bank is usually a corporation, chartered by law, and is 
 subject to supervision by the federal or state authorities, as 
 a means of safeguarding the interests of depositors. 
 
 178. National Banks. Banks organized under " The Na- 
 tional Banking Act " are entitled to the use of the word 
 "National" in their name. Besides transacting a general 
 banking business, they may issue "bank notes," which are 
 secured by the deposit of United States Government bonds 
 with the United States Treasurer. The government imposes 
 certain limitations as to the modes of doing business, the 
 amount of the reserve, etc. It requires uniform reports, 
 makes regular examinations into the bank's business, and 
 prescribes penalties for violation of the requirements of the 
 bank law. 
 
 179. State and Private Banks. State banks are organized 
 under the provisions of the statutes of the various states, 
 and are subject to such restrictions as those statutes impose. 
 State banks do not issue circulating notes and are not re- 
 quired, therefore, to hold United States bonds. They are 
 not usually so carefully and frequently examined, are not 
 
 205 
 
206 ELEMENTS OF BUSINESS ARITHMETIC 
 
 required to make such rigid reports, and are not restricted 
 in so many ways in the conduct of their business, as are the 
 national banks. 
 
 Individuals or firms may operate private banks and trans- 
 act a general banking business, i.e. receive deposits, make 
 loans, deal in exchange and other commercial paper, etc. 
 They are usually subject to statutory provisions designed to 
 insure the safety of funds deposited. 
 
 180. Savings Banks. Savings banks are also organized 
 under state laws. Receiving small as well as large deposits, 
 and paying interest thereon, they are designed to promote 
 economy and encourage thrift. 
 
 The money deposited begins to draw interest on the first 
 day of the following month or of the following quarter, 
 according to the rules of the particular bank. The days 
 on which deposits begin to earn interest are known as interest 
 days. When interest days are quarterly, they are January 1, 
 April 1, July 1, and October 1. 
 
 Interest is usually allowed only on those sums which have 
 been on deposit for the full time between interest days. 
 Thus, the lowest daily balance in the month or quarter, not 
 counting fractional parts of a dollar, is the amount upon 
 which interest is computed. The interest is usually declared 
 semiannually. If not paid, it is credited to the depositor's 
 account and draws interest thereafter. 
 
 181. Savings Bank Accounts. 
 
 Problem 1. — The dividends of interest at a savings 
 bank are declared semiannually. A customer deposited 
 June 5, 1300; Aug. 10, l|150; Oct. 1, $100; Dec. 10, $20. 
 No withdrawals having been made, what was due Jan. 1, 
 following, if interest be reckoned on the deposits from the 
 first of each quarter at 4 % ? 
 
BANKING AND DISCOUNT 
 
 207 
 
 Dates 
 
 Deposits 
 
 Daily 
 
 Balances 
 
 Interest 
 Days 
 
 Smallest 
 Quarterly 
 Balances 
 
 Quarterly 
 Interest 
 
 June 5 
 Aug. 10 
 Oct. 1 
 Dec. 10 
 Jan. 1 
 
 $300 
 
 150 
 
 100 
 
 20 
 
 $300 
 450 
 550 
 570 
 570 
 
 July 1 
 Oct. 1 
 Jan. 1 
 
 $300 
 
 550 
 
 $3.00 
 
 5.50 
 
 8.50 
 
 570.00 
 
 $578.50 
 
 Solution. — Since interest begins on the first of each 
 quarter, the deposit of Aug. 10 will not begin to draw 
 interest until the beginning of the second quarter. The 
 interest on $300 for one quarter is $3. The deposits of 
 Aug. 10 and Oct. 1, together with the balance for the first 
 quarter ($550), will draw interest for the second quarter. 
 The deposit of Dec. 10 will not draw interest until the 
 beginning of the third quarter. 
 
 If interest days are monthly, the above account would 
 balance as follows : 
 
 Dates 
 
 Deposits 
 
 Interest 
 Days 
 
 Smallest Monthly 
 Balances 
 
 Monthly 
 Interest 
 
 June 5 
 Aug. 10 
 Oct. 1 
 Dec. 10 
 
 $300 
 
 150 
 
 100 
 
 20 
 
 July 1 
 Aug. 1 
 Sept. 1 
 Oct. 1 
 Nov. 1 
 Dec. 1 
 
 $300 
 300 
 
 450 
 
 550 
 550 
 
 $1.50 
 1.50 
 2.25 
 2.75 
 2.75 
 
 
 
 Jan. 1 
 
 570 
 
 2.75 
 
 13.50 
 
 4.50 
 
 9.00 
 
 570.00 
 
 $579.00 
 
208 
 
 ELEMENTS OF BUSINESS ARITHMETIC 
 
 The aggregate interest on the smallest monthly balances at 
 6 % is found to be $13.50. At 4 % it is §9. $570 + $9 = 
 $579, the balance due Jan. 1. 
 
 Problem 2. — What is the balance due July 1, on the fol- 
 lowing account ? Deposits : Nov. 20, $300 ; Jan. 14, $200 ; 
 June 10, $150. Withdrawals: March 20, $150; June 20, 
 $100. Interest is declared every 6 months, and 3% per 
 annum is allowed from the first of each quarter. 
 
 Dates 
 
 Deposits 
 
 "WiTIIDEAWALS 
 
 Daily 
 
 Balances 
 
 Interest 
 Days 
 
 Smallest 
 Quarterly 
 Balances 
 
 Qttarterly 
 Interest 
 
 Nov. 20 
 Jan. 14 
 Mar. 20 
 April 1 
 June 10 
 June 20 
 
 $300 
 200 
 
 150 
 
 $150 
 
 100 
 
 $300 
 500 
 350 
 350 
 500 
 400 
 
 Jan. 1 
 
 April 1 
 July 1 
 
 $350 
 
 400 
 
 $5.25 
 
 6.00 
 
 11.25 
 
 5.63 
 
 5.62 
 
 400.00 
 
 $405.62 
 
 The smallest balance for the first quarter is $350, and for 
 the second quarter, $400. The quarterly interest on these 
 balances aggregates $5.62. $405.62 is the balance due 
 July 1. 
 
 PROBLEMS 
 
 1. A. B. Wilson made the following deposits in a savings bank : Dec. 
 20, $60; June 30, $80; Sept. 1,$100; Oct. 15, $200; Nov. 5, $100; Dec. 
 25, $200. The interest term is 6 months, and interest is allowed on 
 balances from the first of each quarter at 4 % per annum. What is the 
 balance due Jan. 1 ? 
 
 Note. — The interest is compounded at end of each interest term. 
 
BANKING AND DISCOUNT 209 
 
 2. A. G. Thomas made the following deposits in a savings bank : 
 Dec. 15, 1^300; Jan. 14, $300; Feb. 25, $150; June 5, $100; July 1, 
 $ 120. The interest term is 3 months, and interest at the rate of 3% is 
 computed from the first day of each quarter. What amount is due 
 Julyl? 
 
 3. A. M. Peabody deposits in a savings bank as follows : Jan. 1, $400 ; 
 Feb. 20, $200; March 10, $150; April 10, $60; May 15, $5.50. He 
 withdrew during the same time, as follows : Jan. 15, $ 100 ; Feb, 5, $150 ; 
 April 20, $ 80 ; June 30, $ 120. The rate of interest is 4 % per annum, the 
 interest term 6 months, and the interest is computed from the first of 
 each quarter. Find the amount due July 1. 
 
 4. A customer makes deposits in a savings bank as follows : Nov. 25, 
 $ 600 ; Jan. 1, $ 200; May 20, $ 50 ; June 25, $ 100. If the interest term 
 is 6 months, and interest is computed from the first day of each month 
 at 4%, what is due July 1 ? 
 
 NEGOTIABLE PAPER 
 
 182. Checks. Money on deposit in a bank in an "open 
 account" is subject to check. That is, the bank will pay 
 out any part or all of it upon a written order from the de- 
 positor. This order is called a cJiech, Checks are usually 
 made payable to a certain person known as the payee^ or to 
 his order, although sometimes they are made payable to the 
 bearer. When the check is paid by the bank, it is stamped 
 " paid " and is finally returned to the depositor who signed 
 the check, known as the maker. To withdraw money from 
 deposit, the check is made payable to "self." 
 
 If there are no funds on deposit, a check is worthless. Ob- 
 taining money from a third party on such a check is against 
 the law. 
 
 For certain purposes, it is desirable to have a check show 
 on its face that it represents actual value. To do this, the 
 maker or payee takes it to the bank upon which it is drawn, 
 and the paying teller or cashier of the bank writes across its 
 face, and over his signature, that it is " good when properly 
 indorsed." This makes of it a certified cheek. Enough 
 
210 ELEMENTS OF BUSINESS ARITHMETIC 
 
 money belonging to the maker is set aside by the bank to 
 pay it when presented. This amount will be paid, or re- 
 leased for regular account, only upon presentation of the 
 check itself. 
 
 Checks signed by individual depositors of a bank are known 
 as individual eheclcs. Those signed by the cashier of a bank 
 are known as cashier's cheeks. The latter are used in paying 
 bank expenses, or to pay the proceeds of a note purchased, 
 and are sometimes issued to customers to be used instead of 
 drafts. 
 
 183. Certificates of Deposit. Money may be deposited for 
 special purposes, in a "closed account," i.e. not subject to 
 check. The bank issues a certificate of deposit, therefore, 
 certifying that a certain sum is on deposit, which will be 
 paid to the holder of the certificate when properly indorsed. 
 This may be either a demand or a time certificate. Time cer- 
 tificates usually draw interest. 
 
 184. Notes. When promissory notes (Sec. 161) are made 
 payable to a particular person, without having either of the 
 two clauses, " or order " or " or bearer," they are payable to 
 no person other than the payee, and are therefore not trans- 
 ferable. Such notes are non-negotiahle. If either of the 
 clauses are embodied, the note may be bought and sold, and 
 it is, therefore, negotiable. Checks, notes, drafts, certificates 
 of deposit, or any other papers representing value, which 
 permit of being bought and sold, are likewise termed nego- 
 tiable papers. 
 
 185. Exchange. Bank Drafts. The use of bank drafts as 
 a means of exchange has already been treated in Sec. 118. 
 As there stated, a bank usually maintains credit with an- 
 other bank, its correspondent, in some of the larger cities 
 and sells drafts on that credit for a certain fixed charge for 
 
BANKING AND DISCOUNT 211 
 
 small amounts, and at so much per hundred dollars for 
 larger amounts. 
 
 Sometimes the amount charged for exchange is expressed 
 in per cents, and when so, it is fixed variously at between 
 Jq % and ^ % of the amount sold. The price would depend 
 upon the size of the draft and sometimes upon the demand 
 for exchange on the particular city in which the paying 
 bank is located. Formerly it was the custom to quote ex- 
 change at either a premium or a discount, dependent upon 
 whether the amount of credit on a given city was greater or 
 less than the demands of business. Of late years, this finds 
 very little use, and drafts are sold at par, the only charge 
 being that for the service. 
 
 186. Collection by Draft. Individual drafts form a very 
 common means of collecting delinquent accounts. If a 
 debtor has paid no attention to repeated statements or let- 
 ters, he is notified that unless heard from by a certain date, 
 he will be drawn upon for the amount of the debt. 
 
 If no word comes, a draft is drawn on him in favor of the 
 creditor, and made payable "at sight," or some number of 
 days after sight. This is usually deposited in the bank with 
 which the creditor does business, and it is sent to a bank 
 in the city of the debtor's residence indorsed "for collec- 
 tion," and the latter bank will present it and endeavor to 
 have it paid. If successful, the bank will remit the amount, 
 less a small fee for the service. If the draft be not paid, 
 it will be returned with a memorandum giving the reason, 
 should one be given by the debtor, for its nonpayment. 
 Repeated refusal or failure to pay such drafts leaves one 
 open to suspicion of unreliability. 
 
 If the draft is for a period of time after sight, the drawee 
 who wishes to honor the draft writes the word accepted^ 
 with date and signature, across the face of the draft, which 
 
212 ELEMENTS OF BUSINESS ARITHMETIC 
 
 thereupon to all intents and purposes becomes a promissory 
 note, and .the time period is counted from the date of the 
 acceptance. If a draft reads "after date," then the time 
 period begins with the date of the draft. 
 
 The signer of a draft is the drawer^ the person or bank on 
 whom it is drawn is the drawee^ and the person to whom it is 
 to be paid is the payee. 
 
 In much the same way shippers may use individual drafts 
 to secure payment on a shipment of goods to an unknown or 
 unreliable customer. The goods are billed to the shipper at 
 the address of the customer, and the bill of lading, together 
 with the draft for the amount of the invoice, is sent to a 
 bank for collection. Upon payment of the draft, the cus- 
 tomer is given the bill of lading, which entitles him to the 
 delivery of the goods shipped. 
 
 187. Indorsement. When one wishes to cash a check, he 
 
 writes his name on the back of the check and presents it 
 
 for payment. This is known t>i i • j ^ 
 
 ^ -^ 1 . P Blank indorsement 
 
 as an indorsement^ and its ef- A M H* k' 
 
 feet is to transfer ownership ry /• i - j * 
 
 ^ Particular indorsement 
 of the check, or the money it p ^^ ^^^^^ ^^^^^^^^^ ^^_ 
 
 represents, to whoever may j^^^^ -^^^^^^ 
 
 hold the check. 
 
 When one wishes to sell a negotiable note, the payee, like- 
 wise, indorses it upon the back. This may be done "in 
 blank" by simply writing his ^^u indorsement 
 
 name, when it becomes payable p^^ ^^ ^^^ ^^^^^ ^^ 
 to the bearer of the note; or it ^^^^^^ & Sons 
 
 may be indorsed " in particular " ^ ^ Dawson 
 
 by writing "pay to " 
 
 and signing it, when it is payable only to the one whose name 
 is written ; or it may be indorsed " in full," by writing " pay 
 to the order of " and signing it. This latter in- 
 
BANKING AND DISCOUNT 
 
 217 
 
 V Find the date of maturity of the following drafts ; 
 
 Date Accepted 
 
 Time 
 
 Date Accepted 
 
 Time 
 
 5. May 9 
 
 20 da. 
 
 7. Jan. 30 
 
 30 da. 
 
 6. June 12 
 
 1 mo. 
 
 8. Sept. 6 
 
 3 mo. 
 
 Find date of maturity and term of discount: 
 
 
 Date of Note 
 
 1 Time 
 
 Date of Discount 
 
 9. May 15 
 
 3 mo. 
 
 
 June 1 
 
 10. Sept. 20 
 
 30 da. 
 
 
 Sept. 29 
 
 11. March 18 
 
 2 mo. 
 
 
 March 31 
 
 Date of Draft Time after Sight 
 
 When Accepted 
 
 "When Discounted 
 
 12. Feb. 16 
 
 10 da. 
 
 Feb. 17 
 
 Feb. 18 
 
 13. March 20 
 
 30 da. 
 
 March 22 
 
 March 25 
 
 14. Julyl 
 
 60 da. 
 
 July 5 
 
 July 20 
 
 Date of Draft Time after Date 
 
 When Accepted 
 
 When Discounted 
 
 15. July 6 
 
 3 mo. 
 
 July 10 
 
 Aug. 4 
 
 16. Aug. 8 
 
 1 mo. 
 
 Aug. 9 
 
 Aug. 9 
 
 17. Sept. 15 
 
 30 da. 
 
 Sept. 15 
 
 Sept. 18 
 
 
 PROBLEMS 
 
 
 Find bdnk discount and proceeds: 
 
 
 
 Face . Date of Note Time 
 
 Date of Discount 
 
 Rate of Discount 
 
 1. $1200 
 
 June 2 30 da. 
 
 June 6 
 
 6% 
 
 2. 12500 
 
 April 8 60 da. 
 
 April 20 
 
 6% 
 
 3. $3000 
 
 Aug. 10 2 mo. 
 
 Aug. 15 
 
 5% 
 
 4. $ 600 
 
 Sept. 16 2 mo. 
 
 Sept. 20 
 
 8% 
 
 5. 17200 
 
 Jan. 14 90 da. 
 
 Jan. 25 
 
 ^% 
 
 6. $3500 
 
 May 20 30 da. 
 
 May 21 
 
 7% 
 
 7. May 15, A. B. Kittridge & Co. borrowed of the First National 
 Bank $1600 on their note at 60 days. Find the proceeds, the rate being 
 
 7%. 
 
 8. I borrowed $450 of the Atlas Bank on my note for 70 days. 
 Write the note, and find the proceeds, the rate of discount being 6 %. 
 
218 
 
 ELEMENTS OF BUSINESS ARITHMETIC 
 
 Find the date of maturity, the term of discount, the discount, and the 
 proceeds of the following notes and drafts : 
 
 9. $1800.00 Columbus, O., April 12. 
 
 Sixty days after date I promise to pay to the order of O. E. Chase & 
 Sons, Eighteen Hundred Dollars, at the Commercial National Bank. 
 Value received. 
 
 Discounted May 8, at 5%. A. B. Grindle. 
 
 10. $660.00 Minneapolis, Minn., Sept. 1. 
 Three months after date I promise to pay to the order of Freeman & 
 
 Comstock, Six Hundred Sixty Dollars, at the Bank of Commerce. Value 
 received. 
 
 Discounted Oct. 10, at 4^%. C. M. Dunlap. 
 
 11. $850.00 Cleveland, O., May 1, 19—. 
 Six months after date I promise to pay to the order of A. Douglas & 
 
 Co., Eight Hundred Fifty Dollars, with interest at 5 %. Value received. 
 Discounted June 6, at 6 %. C. L. Trueblood. 
 
 12. $875.00 Lead, S. Dak., May 29, 19—. 
 Eight months after date I promise to pay to the order of L. M. Gittin- 
 
 ger, Eight Hundred Seventy-five Dollars, with interest at 6%. Value 
 received. 
 
 Discounted June 25, at 7%. Oliver C. Ditson. 
 
 13. $2750.00 Kansas City, Mo., March 13, 19—. 
 At sixty days' sight pay to the order of ourselves 
 
 Twenty-seven Hundred Fifty §-g- Dollars. 
 
 Value received, and charge the same to the account of 
 
 To Ensign Bros., G. W. Patchell & Co. 
 
 Sandusky, O. 
 
 Accepted March 30. Discounted April 4, at 7 %. 
 
 14. 
 
 
 ^^l'6^~^^^:^/i^^i^'y?^^i^'^ 
 
BANKING AND DISCOUNT 219 
 
 15. 
 
 16. $675.50 Cincinnati, O., Dec. 23, 19—. 
 Ninety days after date pay to the order of ourselves. Six Hundred 
 
 Seventy-five and -^^^ Dollars. Value received, and charge to the 
 account of 
 
 To A. B. HiMES, Antwerp & Bragg. 
 
 Indianapolis, Ind. 
 
 Accepted Jan. 2. Discounted Jan. 4, at 7%. Collection charges xV%« 
 
 Note. — Collection is charged on the face. 
 
 17. I wish to borrow $600 at the bank. For what sum must I issue 
 a 60-day note to obtain the amount, discount being 6%? 
 
 18. I owe $960, and have my note discounted at the bank for 75 days 
 at 6% iov such a sum that the proceeds will pay the debt. What was the 
 face of the note ? 
 
 19. A merchant purchased goods for $875 on 3 months' credit, 5% be- 
 ing offered him for cash. He accepted the cash offer and borrowed the 
 money at the bank, giving his note for 60 days at 6%. What was the 
 face of the note, and what did he gain or lose by so doing? 
 
 20. Abankdraftfor$7500 was bought for $7496.25. What was the 
 rate of exchange ? At the same rate, what would be the cost of a draft 
 for $14,500? one for $125,455.60? one for $12,367.50? 
 
 21. An agent sold a carload of 26 cattle, averaging 1125 lb., at $5.60 
 per hundred weight. He paid $ 135 freight, $ 26.75 for feed, and charged 
 2 % commission for selling. He buys a draft at ^ % preinium with the 
 proceeds. What is the face of the draft? 
 
 22. Snyder & Co. of New Orleans drew a draft on A. M. Hawkins of 
 Boston, Mass., for $7865.50, which they sold at the bank at f% discount. 
 What M'^ere the proceeds ? 
 
220 
 
 ELEMENTS OF BUSINESS ARITHMETIC 
 
 23. Complete the following letter of advice. The rate of collection 
 on the first two items is ^^%, and on the others \%. 
 
 SECOND NATIONAL BANK 
 
 Richmond, Ind., June 20, 1917. 
 Mr. WM. J. DOYLE, Cashier, 
 Atlas Bank, 
 
 Cincinnati, Ohio. 
 Dear Sir: 
 
 We credit your account this day for the proceeds of collec- 
 tions as stated below. 
 
 Respectfully yours, 
 
 JAMES W. KING, Cashier. 
 
 YOITE No. 
 
 Payer 
 
 Amount 
 
 Charges 
 
 Peoceeds 
 
 620 
 415 
 930 
 560 
 
 748 
 
 A. M. Pierson 
 F. T. Davis 
 Murphy, Grant & Co. 
 Richmond Chemical Co. 
 S. F. Carroll 
 
 600 
 
 560 
 
 3545 
 
 12345 
 
 800 
 
 00 
 75 
 10 
 80 
 00 
 
 
 
 
 
 
 
 
 
 
 
XIX 
 
 STOCKS AND BONDS 
 
 196. Organizations for Business. If an individual engages 
 in business by himself, he is entitled to all the profits and 
 assumes personal liability for all debts. Should it be de- 
 sirable to have the capital or services of more than one per- 
 son, a partnership is formed. The business is then done 
 under a firm name, e.g. Merritt & Saunders, John N. Bald- 
 win & Co., Parlin, Orendorff & Co., etc. 
 
 There is usually a written agreement for such a partner- 
 ship, specifying the services, money, or property contrib- 
 uted by each to the business, and stating what part of the 
 profits each is to receive. Each partner may bind the 
 firm by his acts, and each is personally liable for the firm's 
 indebtedness. 
 
 197. Corporations. Whenever large capital is needed for 
 a business, or investors wish to limit their financial responsi- 
 bility to the amount invested, or when the range of the 
 business is wide, or individuals wish to invest in, but not to 
 give their personal attention to a business, and for various 
 other reasons, a stock company or corporation is organized. 
 
 Such an organization is usually created under, and must 
 conform to, state laws. Its affairs are conducted by officers, 
 selected in a prescribed way. When organized it becomes 
 before the law a body corporate., vested with the same rights 
 as an individual ; to have and to hold property, to contract 
 debts (within the limits of the law), to sue and be sued, etc. 
 
 221 
 
222 ELEMENTS OF BUSINESS ARITHMETIC 
 
 198. Articles of Incorporation. In order to form a corpo- 
 ration, the investors or incorporators sign, file in some gov- 
 ernment office, and publish their Articles of Incorporation. 
 These articles set forth the purposes of the organization, the 
 amount of money subscribed by each, the capital stocky the 
 total amount that may be subscribed or authorized capital^ 
 the number of shares or parts into which the capital is to be 
 divided, the face or par value of each share, the name of the 
 company, its place of business, its officers, etc. 
 
 199. Certificates of Stock. When the legal requirements 
 are satisfied, the subscribers pay into the treasury the stipu- 
 lated price and receive a certificate of stocky stating the num- 
 ber of shares bought, the par value of each share, etc. The 
 owner of such certificate is entitled to a part in the control 
 of the business corporation, and to participate in its property 
 and its profits in proportion to the number of shares owned. 
 
 200. Dividends and Assessments. At stated periods, the 
 condition of the business is ascertained. If profits are 
 shown, a portion is distributed among the stockholders as a 
 dividend. Gains remaining are termed undivided profits. 
 
 If a loss is shown, which the necessities of the business 
 require should be made good, it is apportioned among the 
 stockholders as an assessment to be paid by them. Dividends 
 and assessments are usually expressed in per cents of the 
 face value of the stock. 
 
 202. Kinds of Stock. Corporations often issue both pre- 
 ferred and common stock. Dividends up to a certain limit, 
 usually 5 % to 7 %, are first paid on preferred stock. Profits 
 remaining may then be divided on common stock. Holders 
 of preferred stock thus have first chance for dividends, but 
 their dividends are limited. Law or charter also sometimes 
 limits dividends on common stock. 
 
STOCKS AND BONDS 223 
 
 202. Premium and Discount. When dividends declared 
 by a given concern are higher than the prevailing rates of 
 interest, the stock of such a company will naturally sell for 
 more than the face value of the shares. It is then said to 
 be above par^ or at a premium. Should the dividends be less 
 than current interest rates, the stock will not bring its face 
 value and is said to be below par^ or at a discount. 
 
 The market value of stock is the amount it will bring in 
 the market, and is usually stated as a per cent of the par 
 value. Thus, stock quoted at 26, 84, or 120 means that a 
 hundred dollars in stock of a company is worth |26, 184, or 
 1120, respectively. 
 
 203. Liability of Stockholders. In general, a holder of 
 stock in a business corporation is liable for the debts of the 
 corporation only to the extent of the par value of the stock. 
 The National Banking Law, however, makes stockholders in 
 national banks liable to the extent of the par value in addi- 
 tion to what they have paid for the stock. 
 
 Some stock is, by the terms of the charter or the by-laws, 
 non-assessable. In such companies, the entire risk assumed 
 by the holder of the stock is the amount paid for it. 
 
 204. Bonds of Corporations. Corporations may borrow 
 money, pledging their property as security, in the same way 
 as individuals. If some particular property is pledged, it is 
 upon an ordinary promissory note, with real or chattel 
 mortgage. 
 
 When the amount of money to be raised is large, it is 
 done by formally issuing bonds., and placing them upon the 
 market for sale. A bond, then, is a mortgage note, upon 
 which the corporation pays interest, and to the payment of 
 which the entire property and business of the corporation is 
 pledged. 
 
224 ELEMENTS OF BUSINESS APITHMETIC 
 
 205. Government Bonds. National and state governments, 
 counties, townships, cities, school districts, etc., are by law 
 declared to be "bodies corporate." As such they may issue 
 bonds, within certain limits, and agree to pay a limited rate 
 of interest. For the redemption of such bonds a tax is 
 levied to create a sinking fund for their payment when due. 
 In default of payment, the courts may issue judgment and 
 cause a special tax to be levied and collected for their pay- 
 ment. 
 
 206. Kinds of Bonds. When bonds are made payable to 
 the owner or his assignee, they are termed registered bonds. 
 The names of owners are registered, and the interest is sent 
 directly to them. 
 
 When the bonds are made payable to bearer, the interest 
 is provided for in attached notes, termed coupons. A 
 coupon is surrendered when each interest payment is made. 
 Bonds of this sort are termed coupon bonds. 
 
 207. Bond Values. The value of a business corporation 
 bond depends upon the amount of property or business of 
 the company. If the property is large or the business pros- 
 perous, then the bonds are reliable. Their market value 
 also depends upon the rate of interest the bonds bear. If it 
 is higher than the current interest rates, the bonds form a 
 profitable investment, and will, therefore, tend to sell at a 
 premium. If the interest rate is less than current rates, or if 
 the bonds are not absolutely secure, they will probably sell 
 at a discount. 
 
 208. Stock Quotations. The price of stocks or bonds is 
 usually quoted at a certain per cent of their par value. So 
 many elements enter into the question of the value of stocks 
 and bonds in the open market, and so easily is the confidence 
 of the investing public weakened or strengthened, that 
 
STOCKS AND BONDS 225 
 
 the market quotations of stocks or bonds often fluctuate 
 widely. 
 
 The necessity for an intimate knowledge of corporation 
 and market conditions gives rise to brokerage firms. These 
 brokers advise their clients, buy and sell stocks, bonds, etc., 
 and charge a small commission or brokerage^ usually about 
 \ of one per cent, for the service. Brokerage is always a per 
 cent of the par value, whether for buying or for selling. 
 
 209. Stock Exchanges. So important are stocks and bonds 
 that special organizations of dealers are formed, known as 
 stock exchanges, boards of trade, etc. At these exchanges, 
 brokers buy and sell for investors. If for speculation, in- 
 vestors usually buy because of an expected rise in the market 
 value, expecting to sell at an advanced price. They do not 
 always pay the full price of the stock, but a part only, leav- 
 ing the certificate of stock with the broker as security for 
 the remainder. Thus, stock may be bought upon a 20 % 
 margin, by paying 20 % of its value, depending on profits 
 to pay the remainder or using the amount paid to cover 
 losses if the stock goes down. 
 
 There is usually a group of operators^ who are interested 
 in forcing the price of certain stocks upward. These are 
 known as bulls. There are others who wish to see the price 
 lowered, and these are called bears. Bears may either wish 
 to purchase good stock at a cheap price, or in the belief that 
 the stock was certain to go down, they may have sold largely 
 of the stock without owning it, and wish, therefore, to force 
 it down so they may purchase cheaply what they have bar- 
 gained to deliver. In the latter case they are said to have 
 sold short. 
 
 210. Market Quotations. The following quotations show 
 the highest and the lowest market quotations for a given 
 year as furnished by Bradstreet's Commercial Agency : 
 
226 ELEMENTS OF BUSINESS ARITHMETIC 
 
 Stocks 
 
 High Low 
 
 Adams Express 250 236 
 
 Amalgamated Copper 91 70 
 
 American Beet Sugar 34^ 23 
 
 American Express 246 209^ 
 
 Atchison, Topeka & Santa Fe . . . 93| 71J 
 
 Brooklyn Rapid Transit 91 1 56| 
 
 Chicago, Burlington & Quincy . . . 250 201 
 
 Chicago, Milwaukee & St. Paul . . . 187i 168^ 
 
 Consolidated Coal 73 24| 
 
 Erie 52| 37| 
 
 General Chemical 72| 37i 
 
 Illinois Central 183 152| 
 
 National Biscuit 66f 52 
 
 National Biscuit Pfd 120| 110 
 
 National Lead 77^ 24| 
 
 Pittsburg, Ft. Wayne & Chicago . . 185 182^ 
 
 Pressed Steel Car 53| 34 
 
 Quicksilver 1| | 
 
 Rubber Goods Mfg. Co 39 25 
 
 Union Pacific 138f 113 
 
 United States Leather 16 10| 
 
 United States Steel 39^ 24| 
 
 Bonds 
 
 High Low 
 
 Am. Hide & Leather 6's 97^ 97| 
 
 Chesapeake & Ohio 6's, 1911 .... 110 110 
 
 Denver & Rio Grande 4's 100| 97^ 
 
 Illinois Central 4's, 1952 108 107^ 
 
 Missouri Pacific 4's 96 95 
 
 Seaboard Air Line 5's 104^ 104 
 
 United States reg. 4's, 1907 .... 104^ 104| 
 
 Pennsylvania 4|'s 109 108^ 
 
 United States coupon 4's 104^ 104 
 
 211. Illustrative Problems. In all problems in this text, 
 the par value of a share of stock will be taken as f 100, unless 
 otherwise stated. 
 
STOCKS AND BONDS v 227 
 
 1. A broker sells 145 shares of stock at 125| ; brokerage 
 ■|%. What should Im principal receive? 
 
 125| % - 1 % = 125| %, proceeds of each share. 
 
 f 100 X 145 = 114,500, par value of 145 shares. 
 
 114,500 X 1,25J = 118,161.25, proceeds. '^,; 
 
 II 
 
 2. A broker sold Union Gas stock for 126,250 at 75% 
 premium. How many shares did he sell? What was the 
 par value of the stock? 
 
 I = selling price. ^\'\*>Y 
 Jof par value = 126,250. 
 
 ^ = |of 126,250, or 13750. 
 I = $3750 X 4, or $15,000, par value of stock. 
 $15,000 -^ $100 = 150, number of shares. , v 
 
 3. What sum must be invested in 6 % bonds at 120 to 
 yield an annual income of % 2820 ? 
 
 6 % of face value of bonds bo,ught = $2820.;^ 
 100 % = -ig^ of $2820 or |47,0i)0 face value. 
 120 %=f. ' 
 
 I of $47,000 = $56,400, investment. 
 
 4. What per cent profit does an investor make on stock 
 that pays a dividend of 6 %, if he buys at 75 ? 
 
 $ 6 = income on one share. / v^ 
 
 $75 = cost of that share. 
 A = 8%, profit. N 
 
 5. A year's net profits of the Plymouth Milling Co. were 
 $15,275.50. The capitalization of the concern is $200,000, 
 divided into 2000 shares. A dividend of 6| % was declared, 
 and the remainder of the profits was carried to surplus fund. 
 Find the amount of dividend and the amount carried to sur- 
 
228 ELEMENTS OF BUSINESS ARITHMETIC 
 
 plus fund. What amount will a man receive who owns 60 
 shares of stock? 
 
 6-1- % of 1200,000 = 113,000, the dividend declared. 
 115,675.50 - f 13,000 = $2675.50, surplus fund. 
 
 6i % of $6000 = 1390, dividend on 60 shares of stock. 
 
 6. A manufacturing company is capitalized at $200,000. 
 The gross earnings for a year are $25,185, and the expenses 
 are $6785.50. After setting aside 2% for surplus fund, 
 what even per cent of dividend may be declared ? 
 
 $25,185 - $6785.50 = $18,399.50, net earnings. 
 2 ojo of $18,399.50 = $367.99, amount for surplus fund. 
 $18,399.50 - $367.50 = $18,031.50, amount to be divided. 
 1% of $200,000 = $2000, amount of 1% dividend. 
 $18,031.51 ^ $2000 = 9, the rate per cent of dividend, with 
 $31.51 additional, carried to undivided profits. 
 
 PROBLEMS 
 
 Find the market value at the highest and lowest price of the following 
 stocks and bonds, by use of the above market quotations : 
 
 1. 75 shares Adams Express. 3. 155 shares National Lead. 
 
 2. 68 shares Erie. 4. 85 shares Quicksilver. 
 
 5. 150 shares Union Pacific. 
 
 6. 7 Pennsylvania 4|'s (Denom. ^1000). 
 
 7. 9 Am. Hide and Leather 6's (Denom. $1000). 
 
 8. 245 shares National Biscuit pfd. 
 
 9. A man who holds 170 shares of stock receives a dividend of $1275. 
 What was the rate of dividend declared ? 
 
 10. An assessment of $306 is made on 72 shares of mining stock. 
 What is the rate of assessment ? 
 
 > 11. A broker sold for me 360 shares of gas stock at 145 ; brokerage \%. 
 What sum should I receive? 
 
 12. A railroad declares a dividend of 5%. How many shares does a 
 man hold who receives a dividend of $435, if the par value of stock is 
 $100? 
 
STOCKS AND BONDS 229 
 
 ^13. What sum must be sent a broker that he may buy 150 shares of 
 gas stock at 105; brokerage |%? 
 
 14. A stockholder meets an assessment of $167.50, which is levied at 
 2^ % on his stock. How many shares has he ? 
 
 15. How many shares of Atchison, Topeka, and Santa Fe stock can 
 be bought for |8100 at 89|; brokerage ^%? What will the dividend on 
 this stock amount to at 5%? What rate of interest would this be on the 
 investment ? 
 
 16. What amount of stock must be sold at 41^ to yield $8275, if 
 brokerage is i%? 
 
 17. How many shares of American Sugar at 31 1 can be bought for 
 $202,400; brokerage i%? If this stock pays a dividend of 2|%, what 
 rate of interest will a man receive on his investment? 
 
 18. What sum must be invested at 93 to bring an income of $4800, if j 
 the rate of dividend is 4%'^ 
 
 19. What is the quotation of 7% stock that brings an income of 10%? 
 
 20. Wliat rate per cent is realized on an investment by investing in 
 5% stock at 80? 
 
 21. A banker sold through a broker 150 shares of stock at 124|, pay- 
 ijQg i% brokerage. What amount did each receive? 
 
 22. A speculator bought 2500 shares of the United States Steel stock 
 at 33| and sold it at 39f. What was his net profit after allowing \% 
 brokerage each for buying and selling ? 
 
 23. A man buys 120 shares of stock at 76^ and six months later sells 
 it for 85. In that time he received a dividend of lf%. If money is 
 worth 6% interest and he paid \% brokerage for buying and for selling, 
 did he gain or lose, and how much? 
 
 24. The capital stock of a company is $200,000. \ of this is preferred 
 stock, entitled to 6% dividend. What rate of dividend is paid on com- - 
 mon stock, if $9000 is distributed in dividends? 
 
 25. An investor buys 604 shares of stock, par value $ 50, for $ 35 a 
 share; brokerage \%. Six months later he sells for $58 a share. In the * 
 meantime he had received a dividend of 5%. Money being worth 6%, 
 what did he gain or lose? 
 
 26. A bank with capital stock of $150,000, declares a semi-annual j 
 dividend of 4^%. What is the amount of the dividend, and how muchy 
 will a man receive annually who owns 275 shares? ^ / 
 
230 ELEMENTS OF BUSINESS ARITHMETIC 
 
 27. A corporation has a capital stock of $100,000. Its net earnings 
 for the year are 114,256.32. 4 % of the net earnings is set aside as a sur- 
 plus fund to cover losses, 7 % dividend is declared, and the remainder is 
 carried to undivided profits. What are the amounts carried to surplus 
 fund, undivided profits, and to dividend accounts? 
 
 28. A merchant sold his business for $245,000. He invested $196,000 
 in Pullman stock at 195|, and the remainder in Erie preferred at 69 J. 
 Pullman stock pays 12% and Erie 6% dividend; brokerage ^% in each 
 case for buying. What was his annual income ? 
 
 29. A broker purchased 500 shares Amalgamated Copper at 70 ; 250 
 shares C. B. & Q. at 201; 400 shares National Biscuit Co. at 66| ; 475 
 shares General Chemical at 72|, and 150 shares Western Union at 113. 
 What is the total cost to his principal, if brokerage is ^%? 
 
 30. A speculator purchased 200 shares United States Steel at 24 J; 
 600 shares Pressed Steel Car at 34 ; 700 shares Western Union Telegraph 
 at 95^. He sold the Steel stock at 39^, the Pressed Steel Car at 53|, and 
 the Western Union at 92, brokerage being \% for buying and for selling. 
 What was the net gain or loss ? 
 
 31. A certain county, on Jan. 1, 1907, issued $250,000 worth of 5% 
 10-year coupon bonds. If these bonds were sold through a broker at 
 102|, how much was received by the county? Brokerage |%. If the 
 interest is payable semi-annually, what is the amount of each interest 
 coupon? How much must be levied in taxes each year to pay the 
 interest and provide a sinking fund sufficient to pay the bonds in full at 
 maturity? What would be the annual rate to be levied, if the assessed 
 valuation of the county averages $46,875,000? 
 
 32. A broker bought for a customer 800 shares of United States Steel 
 common, at a total cost of $30,100; brokerage \%. Find market quota- 
 tion of stock. 
 
 33. How much must be paid, including brokerage at ^%, for a suffi- 
 cient number of United States 4*s at 123| to obtain an annual income of 
 $1200? 
 
 34. My broker, after selling 500 shares of Philadelphia Gas stock and 
 deducting the usual commission, remitted $ 534,312.50. What was the 
 market quotation ? 
 
 35. What income will a man receive from an investment of $6448 in 
 United States coupon 4's at the lowest market price as per above table ? 
 
XX 
 
 INSURANCE 
 
 212. Nature of Insurance. Some losses or damages to 
 property, such as by fire or tornado, are unavoidable. Such 
 losses are serious, and they are often likely to bring finan- 
 cial ruin to owners. To avoid such calamities, owners of 
 property subject to losses of some particular kind have fre- 
 quently banded themselves together to share losses among 
 themselves. By agreeing to assist in making good the loss 
 to whomsoever it might fall, they secured themselves against 
 disaster. From such beginnings has grown the institution of 
 modern insurance. Insurance^ then, consists of a contract 
 guaranteeing to make good a loss from a certain cause. 
 The agreement is known as an insurance policy. 
 
 213. Kinds of Companies. When an agreement for insur- 
 ance is made between those mutually interested, to mutually 
 share losses from a particular cause, it is known as mutual 
 insurance. 
 
 When, as a business investment, a company is organized 
 which undertakes, for a given fee, to make good a loss during 
 a specified time, it is a stock insurance company. Sometimes 
 a stock company agrees to divide all earnings of the company, 
 above a specified dividend on their stock, among the insured. 
 It thus partakes of the nature of both stock and mutual 
 companies. 
 
 214. Property and Personal Insurance. Whenever it is a 
 loss in property which is insured against, it is property insur- 
 ance. Whenever the insurance is against a loss due to sick- 
 ness, accident to, or death of a person, it is personal insurance y, 
 
 231 
 
232 ELEMENTS OF BUSINESS ARITHMETIC 
 
 PROPERTY INSURANCE 
 
 215. Forms of Property Insurance. If property is insured 
 against loss from fire, it is fire insurance; if against loss from 
 wind or storm, it is tornado insurance; if against loss or 
 damage while being transported by land or by sea, it is 
 transit insurance^ that for ship or cargo lost at sea being 
 marine insurance ; if against loss or damage to live stock by 
 death, disease, lightning, or other casualty, it is live-stock 
 insurance. These are among the principal forms of property 
 insurance. 
 
 216. Kinds of Policies. If the value of the property 
 insured or the amount of the indemnity in case of loss is 
 determined and agreed upon, at the time the policy is issued, 
 it is known as a closed or valued policy. If the real value of 
 the property loss is open for determination after the loss 
 occurs, regardless of the face of the policy upon which the 
 insurance premium has been paid, it is an open policy. 
 Many states, by law, have declared that all policies must be 
 valued policies. This requires the insurance company to 
 pay the full face of the policy in case of a total loss, regard- 
 less of the actual value of the property. 
 
 217. Cost of Insurance. In mutual companies, the cost of 
 insurance depends upon the amount of losses suffered by the 
 different members of the company. This, together with the 
 actual cost of carrying on the business of the company, is 
 apportioned among the members, in the form of assessments. 
 In stock companies, a definite fee is charged for insurance 
 during a given pieriod of time, and this fee is called a premium. 
 The amount of the premium is usually a per cent of the face 
 of the policy, or the amount of loss which it is agreed to 
 make good. This is known as the rate of insurance. It 
 varies with the kind of buildings, their location with refer- 
 ence to other buildings, efficiency of fire protection, etc. 
 
INSURANCE 
 
 233 
 
 Usually, a given district is plotted, and the rate for each 
 building is fixed. 
 
 Below is a schedule of rates for such a plotted district, 
 expressed as a certain sum for each $100 of insurance. 
 The diagram on which this is based is on page 234. 
 
 Table of Eates 
 
 District No. 
 
 City of 
 
 KiSK 
 
 Plot No. 
 
 Annual Ratb 
 PER $ 100 
 
 Frame Dwelling and Contents 
 
 Brick Store and Contents 
 
 Brick Church and Contents 
 
 Brick Business Block and Contents . . . 
 Frame Handle Factory and Contents . . . 
 Brick Livery Barn 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 • 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 $.35 
 
 .25 
 
 .50 
 
 .50 
 
 1.75 
 
 1.2j5 
 
 Frame Store and Dwelling, and Contents . 
 
 Frame Barn and Contents 
 
 Brick Dwelling and Contents . . . . . . 
 
 Brick Schoolhouse and Contents .... 
 
 Brick Store and Dwelling, and Contents . . 
 
 .40 
 1.00 
 .18 
 .50 
 .30 
 
 PROBLEMS 
 
 1. Find the cost of insuring property valued at $5000, at 1^%. 
 
 2. At 2%> how much insurance can I procure for $148? 
 
 3. I paid $30 for insuring a house worth $3200 at | its valuation. 
 What was the rate ? 
 
 4. Find the cost of insuring each of the buildings in the table above 
 at I the following valuation : 
 
 Frame Store & Dwelling . $2,450 
 
 Frame Barn on Above Lot . 500 
 
 Brick Dwelling 6,700 
 
 Frame Barn on Above Lot . 800 
 
 Brick Schoolhouse .... 75,000 
 
 Brick Store & Dwelling . . 7,500 
 
 Frame Dwelling . . . 
 
 $2,400 
 
 Brick Store 
 
 10,500 
 
 Brick Church .... 
 
 12,000 
 
 Brick Business Block 
 
 32,000 
 
 Frame Handle Factory . 
 
 4,500 
 
 Brick Livery Barn . . 
 
 3,600 
 
234 
 
 ELEMENTS OF BUSINESS ARITHMETIC 
 
 _J 
 
 l_ 
 
 133aiS 
 
 133yiS 
 
 ~i 
 
 r 
 
INSURANCE 235 
 
 5. The contents of the frame dwelling of the plot are valued at 
 $1100; the contents of the brick store, at $12,450; the contents of the 
 brick church, at $5500; the contents of the brick business block, at 
 $25,600. If these goods are insured for | of their valuation, what would 
 be the premium on each lot ? 
 
 6. An agent's premium is $25 for insuring a house at |%. What is 
 the face of the policy? 
 
 7. At \%, what is the annual premium, at | valuation, on a house 
 worth $16,000? 
 
 8. A merchant pays $ 150 for insurance on his stock of goods at f %. 
 What is the amount of the policy ? 
 
 9. An agent receives $112.50 for insuring a house for 80% of its 
 valuation at |%. What is the value of the house? 
 
 10. A man has a house valued at $ 24,000, and furniture valued at 
 $6000. He insures the house at | its valuation, and the furniture at | 
 its valuation. What is the annual premium, |% for the house and f% 
 for the furniture ? 
 
 11. If it cost $663 to insure a building valued at $132,600, what will 
 it cost at the same rate to insure a building valued at $ 105,000 ? 
 
 12. The premium on a house valued at $10,500, insured at f its 
 valuation, is $ 47.25. Find the rate of insurance. 
 
 13. If a house and its contents are valued at $6500, for how much 
 must it be insured, at 1^%, to cover loss and premium in case of total 
 destruction ? 
 
 14. A cargo of coffee valued at $35,000 was insured for $20,000, in a 
 policy containing an average clause. It was damaged to the amount of 
 $15,000. How much should the company pay? 
 
 Note. — Under an " average clause " such a part of the loss is paid, as 
 the policy is of the real value of the property insured. 
 
 15. A steamer is insured for $75,000. Its value is $100,000. If it is 
 insured at 2^%, what will be the loss to the company in case of total 
 destruction ? 
 
 16. An agent insures a cargo of cotton costing $ 9775, at the rate of 
 2\%, for an amount that will cover the cost of cotton and premium. 
 What is the face of the policy ? 
 
 17. A speculator bought 4000 bbl. of flour, and had it insured for 
 80% of its cost, at 3|%,- He paid a premium of $980. At what price 
 per barrel must he sell it in order to gain 10 % on the total cost ? 
 
236 ELEMENTS OF BUSINESS ARITHMETIC 
 
 18. A manufacturer insured his factory for 1 27,000, and its contents 
 for $66,000. He paid $700 for premium and policy. If the policy cost 
 $2.50, what was the rate per cent premium? 
 
 19. A dealer in New York ordered his Chicago agent to buy 4000 bu. 
 of wheat at 70^; 3000 bu. at 30^; 7500 bu. corn at 37^ ^ ; paying 2^% 
 commission for buying. The grain was shipped by boat, and a policy at 
 1-J% was taken out to cover the cost of grain and commission. What 
 was the amount of the policy, and what the amount of premium? 
 
 20. The Atlas Insurance Company insured a block of buildings for 
 $150,000, at 75 }2^ per $100. Thinking the risk too great, it reinsured 
 $ 50,000 in the ^tna, at | % and $ 65,000 in the Manhattan, at | %. How 
 much premium did each company receive? What was the gain or loss 
 to the Atlas? What per cent premium did it receive for the part of the 
 risk not reinsured ? 
 
 21. A house cost $ 8000 ; it was insured for | its valuation at 1^ % for 
 3 years. What would be my loss and that of the company, if the house, 
 were totally destroyed by fire ? 
 
 22. A residence valued at $4-500 is insured for | its value at | % per 
 annum. The company will insure the house for 3 years on a payment 
 of 2| times the annual premium in advance. What will it cost to insure 
 the house for 3 years ? What will it cost to insure for 5 years, if the 
 company will accept 4 annual premiums in advance as payment for 5 
 years ? 
 
 23. How much will it cost to insure a factory valued at $75,000 at 
 1%, and the machinery valued at $25,000 at |%? 
 
 PERSONAL INSURANCE 
 
 218. Life Insurance. When insurance is upon the hazard 
 of life, it is life insurance. Upon the death of the insured it 
 is paid to an heir or other person named in the policy as the 
 beneficiary. 
 
 Life insurance may be paid for either by assessments when 
 deaths occur, by a stated number of assessments in a year, or 
 by a fixed premium^ payable monthly, quarterly, semi-annu- 
 ally, or annually. When a fixed premium is paid, the policy 
 may be either participating or nonparticipating in the profits 
 of the association. The premiums may be payable annually 
 
INSURANCE 237 
 
 until death of the insured, a life payment policy ; or may be 
 payable annually only for a period of 10, 15, or 20 years, a 
 limited payment policy^ when the policy is said to be " fully 
 paid up." The latter are termed 10 payment life, 20 pay- 
 ment life, etc. 
 
 What is known as a term policy may also be purchased for 
 a period of 1, 5, 10, 15 or 20 years. The holder is insured 
 for the term only, and, if desirous of continuing the insur- 
 ance, must purchase another policy at increased rates, because 
 of increased age. 
 
 When a fixed premium is charged, the insurance is said to 
 be " old line." If the policy is a participating one, the profits 
 of the company may operate to lessen the annual premium, 
 or may be deferred until the policy matures and added to 
 the face value. 
 
 Fraternal organizations, with a side feature of insurance 
 or with insurance as their chief purpose, offer one method 
 of life insurance, often with additional features of health or 
 accident benefits. They usually employ the assessment plan, 
 and often assess for more than enough to pay the death 
 losses, in order to use the surplus in establishing a reserve 
 fund to assist in paying death losses when the membership 
 becomes older, and the percentage of deaths increase. 
 
 219. Endowment Insurance. Endowment insurance is a 
 modern outgrowth of the life insurance idea. At the end of 
 the stated endowment term, e.g. 5, 10, 15, or 20 years, the in- 
 surance is to be paid to the insured himself, should he be 
 alive at that time. Should he die before that time, the full 
 face of the policy is to be paid to the beneficiary named in 
 the policy. While retaining the life insurance idea, it thus 
 adds to it a savings or investment feature. 
 
 220. Accident and Health Insurance. Accident or casualty 
 insurance companies pay an indemnity when one is injured 
 
238 
 
 ELEMENTS OF BUSINESS ARITHMETIC 
 
 by an accident, in travel or otherwise. The amount paid is 
 usually graduated to the extent of the injury. 
 
 Insurance against ill health may also be had, the insured 
 receiving a weekly or monthly payment while sick. 
 
 221. Cost of Life Insurance. The price of a life insurance 
 policy is a stated annual amount or premium per thousand 
 
 Annual Premium Rates for $1000 Insurance 
 
 WHOLE LIFE 
 
 ENDOWMENT PREMIUM 
 
 Aqb 
 
 Life 
 
 20 Yeaes 
 
 15 Yeabs 
 
 10 Years 
 
 Age 
 
 In 15 Yeabs 
 
 In 
 20 Yeaes 
 
 21 
 
 $19.47 
 
 129.59 
 
 §35.65 
 
 $48.11 
 
 21 
 
 $67.03 
 
 $49.07 
 
 22 
 
 19.91 
 
 30.06 
 
 36.20 
 
 48.85 
 
 22 
 
 67.13 
 
 49.17 
 
 23 
 
 20.36 
 
 30.55 
 
 36.78 
 
 49.61 
 
 23 
 
 67.23 
 
 49.27 
 
 24 
 
 20.84 
 
 31.06 
 
 37.38 
 
 50.40 
 
 24 
 
 67.33 
 
 49.39 
 
 25 
 
 21.34 
 
 31.58 
 
 38.00 
 
 51.22 
 
 25 
 
 67.44 
 
 49.52 
 
 26 
 
 21.86 
 
 32.12 
 
 38.63 
 
 52.06 
 
 26 
 
 67.56 
 
 49.65 
 
 27 
 
 22.41 
 
 32.69 
 
 39.63 
 
 62.93 
 
 27 
 
 67.69 
 
 49.79 
 
 28 
 
 22.99 
 
 33.27 
 
 39.98 
 
 53.83 
 
 28 
 
 67 83 
 
 49.95 
 
 29 
 
 23.59 
 
 33.88 
 
 40.70 
 
 54.76 
 
 29 
 
 67.97 
 
 60.11 
 
 30 
 
 24.23 
 
 34.51 
 
 41.43. 
 
 65.73 
 
 30 
 
 68.12 
 
 50.28 
 
 31 
 
 24.87 
 
 35.14 
 
 42.17 
 
 56.70 
 
 31 
 
 68.29 
 
 50.48 
 
 32 
 
 25.65 
 
 35.81 
 
 42.94 
 
 57.71 
 
 32 
 
 68.46 
 
 50.69 
 
 33 
 
 26.30 
 
 36.52 
 
 43.76 
 
 68.78 
 
 33 
 
 68.66 
 
 50.91 
 
 34 
 
 27.08 
 
 37.26 
 
 44.62 
 
 59.88 
 
 34 
 
 68.87 
 
 51.16 
 
 35 
 
 27.91 
 
 38.04 
 
 45.51 
 
 61.03 
 
 35 
 
 69.09 
 
 51.42 
 
 36 
 
 28.78 
 
 38.85 
 
 46.43 
 
 62.21 
 
 36 
 
 69.34 
 
 51.72 
 
 37 
 
 29.70 
 
 39.70 
 
 47.39 
 
 63.44 
 
 37 
 
 69.60 
 
 52.04 
 
 88 
 
 30.68 
 
 40.59 
 
 48.39 
 
 64.71 
 
 38 
 
 69.90 
 
 52.40 
 
 39 
 
 3L71 
 
 41.51 
 
 49.43 
 
 66.03 
 
 39 
 
 70.21 
 
 52.79 
 
 40 
 
 32.81 
 
 42.49 
 
 50.52 
 
 67.40 
 
 40 
 
 70.56 
 
 53.22 
 
 41 
 
 33.94 
 
 43.50 
 
 61.63 
 
 68.80 
 
 41 
 
 70.95 
 
 63.70 
 
 42 
 
 35.14 
 
 44.55 
 
 52.79 
 
 70.24 
 
 42 
 
 71.37 
 
 54.22 
 
 43 
 
 36.45 
 
 45.69 
 
 64.04 
 
 71.77 
 
 43 
 
 71.85 
 
 54.81 
 
 44 
 
 37.83 
 
 46.90 
 
 55.34 
 
 73.36 
 
 44 
 
 72.37 
 
 65.46 
 
 45 
 
 39.30 
 
 48.17 
 
 56.71 
 
 75.02 
 
 45 
 
 72.96 
 
 56.17 
 
INSURANCE 239 
 
 dollars of insurance. The lowest in cost is in purely mutual 
 companies, like fraternal organizations. 
 
 The price depends, too, upon the age of the applicant and 
 upon the conditions of the contract. The term policy, or the 
 straight life-payment nonparticipating policy may be pur- 
 chased at the least cost, while a limited payment or endow- 
 ment policy will cost proportionately more. Rates vary, but 
 the table on the opposite page gives a fair approximation of 
 the cost of the common kinds of policies in standard com- 
 panies at different ages. 
 
 PROBLEMS 
 From the table find the annual premium required for : 
 
 1. A life policy of $3000, age 30. 
 
 2. A twenty-payment life policy of $ 5000, age 27. 
 
 3. A twenty-year endowment policy for $4000, age 32. 
 
 4. A ten-payment life policy for $ 3500, age 35. 
 
 5. A man takes out a twenty-payment life policy for $ 3000 at the 
 age of 25. If he dies at the age of 40, how much does the face of the 
 policy exceed the premiums paid ? 
 
 6. If money is worth 6%, what do the premiums in problem 5 
 amount to? How much does the face exceed that amount? (Annual 
 interest.) 
 
 7. A man at the age of 28 takes out a straight life policy and a 
 twenty-year endowment policy, each for $ 2000. If he dies at 40, which 
 gives the greater returns ? 
 
 8. A man 30 years of age took out an endowment policy for $ 3000, 
 payable in 15 years, and died after making six payments. How much 
 less would a life policy have cost ? 
 
 9. A man aged 35 years takes out an endowment policy for $ 15,000, 
 payable to himself in 20 years, or to his heirs if he dies before that time. 
 What annual premium will he have to pay ? If death occurs at the end 
 of the ninth year, how much would he have paid out in premiums? 
 How much less would a twenty-payment life policy have cost ? 
 
 10. At the age of 32 a man takes out a .|3500 life policy, and at 
 the age of 35 a $ 1000 twenty-year endowment. How much does the 
 insurance exceed the premiums paid, if he dies at the age of 45? 
 
XXI 
 
 PROPORTION 
 
 222. Ratio. The relative size of two numbers, as shown 
 by division and expressed by their quotient, is called their 
 ratio. Thus the ratio of 8 to 4 is 2 ; 96 to 60 is If ; and of 
 25 to 50 is J. Instead of using the sign of division (-?-) to 
 express a ratio, the horizontal line is left out, making the 
 sign simply a colon (:), e.g. 8 : 2 or 96 : 60. The meaning, 
 however, remains the same, and the ratio may always be 
 found by dividing the first number or term by the second. 
 
 223. Comparison of Like Quantities Only. Since ratio 
 shows comparative size, it may only exist between quantities 
 like in kind. Thus, 8 ft. and 4 ft., or 8 houses and 4 houses, 
 may be compared, while 8 boxes and 4 horses may not. 8 ft. 
 and 4 yd. may be compared, but, to do so, they must first be 
 reduced either to feet or yards, e.g. the ratio of 8 ft. to 12 ft., 
 orf 
 
 When the numbers refer to units of measurement, the 
 quantities being measured must also be like in kind, if the 
 ratio of the quantities being measured is desired. Thus, 
 8 doz. and 2 doz. may be compared, if it is 8 doz. chairs and 
 2 doz. chairs, but not if 8 doz. chairs and 2 doz. horses ; or 
 8 ft. and 2 ft. may have ratio if it is 8 ft. high and 2 ft. 
 high, but not if 8 ft. high and 2 ft. wide, etc. 
 
 224. Order of Terms. The number to be treated as a 
 dividend is always written first. The ratio of 6:3, then, 
 
 .would be 2 and not |. Since the number of which it is 
 
 240 
 
PROPORTION 241 
 
 desired to know the comparative size is always written first, 
 it is called the antecedent^ i.e. the one which goes before. 
 The number with which it is desired to compare the first is 
 always written second, and because it follows the other, it is 
 called the consequent. The antecedent and its consequent 
 together form a couplet. 
 
 225. Proportion. The ratio of two different couplets may 
 be the same. Thus, the ratio of 8 horses to 2 horses is 4, 
 and the ratio of 1 400 to f 100 is 4. Likewise, the ratio of 
 3 ft. in height to 6 ft. in height is J, and the ratio of 3 days 
 to 6 days is ^. 
 
 Whenever two ratios are equal, they are in proportion. 
 Thus, as the ratio of 8 horses to 2 horses is 4, the ratio of 
 their cost ($400 to 8100) would also be 4. We would, 
 therefore, say that the ratio of 8 horses to 2 horses equals 
 the ratio of 1400 to $100. 
 
 The equality of related ratios is usually expressed by the 
 double colon, e.g. 8 horses: 2 horses :: $400 : $100. This 
 proportion would be read as follows : 8 horses is to 2 horses 
 as $400 is to $100. The equality sign may also be used, 
 giving the above proportion the form of 8 horses : 2 horses 
 = $400: $100. 
 
 226. Means and Extremes. As there must be two ratios 
 whose equality forms a proportion, every proportion must 
 have four terms. When written formally as a proportion, 
 the first and fourth terms are called the extremes^ meaning 
 the outside numbers. The second and third terms of a pro- 
 portion are called the means, meaning the middle numbers. 
 
 Since the antecedents of each ratio are the dividends, and 
 the consequents are divisors, the extremes and means each 
 consist of one dividend and one divisor. The quotient of 
 each couplet being the same, the product of the dividend of 
 one couplet and the divisor of the other is equal to the prod- 
 
242 ELEMENTS OF BUSINESS ARITHMETIC 
 
 uct of the other dividend and divisor. In other words, the 
 product of the means is equal to the product of the extremes. 
 
 The above being true, it is only necessary to know any 
 three of the terms of a proportion in order to find the fourth. 
 Thus, in the proportion, 2 hats : 5 hats : : 13 : fa;, the product 
 of the means (treating the terms of the first ratio as multi- 
 pliers and, therefore, abstract numbers) is f 15. Since this 
 $15 is the product also of the extremes, and one of them is 
 2, the other must be ^ of $15, or $7.50. The completed pro- 
 portion, then, would be 2 hats : 5 hats : : $3 : $7.50. 
 
 The use of three terms to find the fourth, has given rise 
 to the phrase " the rule of three," which is a name formerly 
 applied to solutions by proportion. 
 
 227. Directly and Inversely Proportional. Assuming that 
 each man does an average day's work, the larger the number 
 of men employed on a given piece of work, the more work 
 is done. Thus, if 2 men can wrap and pack 150 boxes of 
 oranges in a day, 4 men can wrap and pack 300 boxes. In 
 other words, if the number of men is doubled, twice the work is 
 accomplished. Whenever two quantities increase or decrease 
 together in this way, and with a constant ratio, they are said 
 to vary directly or to be directly proportional. We may also 
 say that the amount of work done would bear a direct ratio 
 to the number of men employed. 
 
 On the other hand, if the number of men is increased, 
 the same amount of work would require less time for its 
 completion. Thus, if 2 men can wrap and pack 600 boxes 
 in 4 days, 4 men could do it in half the time. In other words, 
 if the number of men is doubled, the time required is but one 
 half as much. Whenever one quantity increases as another 
 decreases, or decreases as another increases, keeping the ratio 
 constant, they are said to vary inversely., or to be inversely 
 'proportional. Thus, the time required to do a given piece 
 
PROPORTION 243 
 
 of work bears an inverse ratio to the number of men 
 employed. 
 
 Query. — Which of the following are directly and which 
 inversely proportional : 
 
 1. The weight of coal and its cost? 
 
 2. The height of buildings and their shadows? 
 
 3. The number of workmen and the amount of work 
 done in a given time? 
 
 4. The number of workmen and the time required to do 
 a given amount of work? 
 
 5. The speed of an automobile and the time required to 
 travel a certain distance? 
 
 6. The weight of freight and the freight charges? 
 
 7. The area of squares and the length of their sides? 
 
 8. The amounts loaned and interest earned? 
 
 9. The attraction one body has for another and their dis- 
 tance apart? 
 
 10. The greater the capital stock in a company and the 
 size of the dividend which can be declared with given profits? 
 
 11. The amount of assessable property and the tax levies 
 to raise a given amount? 
 
 12. The amounts loaned and the rates to earn the same 
 amount of interest? 
 
 228. Statement and Solution. The direct proportion given 
 in Section 227 would be stated thus : 
 
 4 men : 2 men : : 300 boxes : 150 boxes. 
 
 In the second couplet, 300 boxes, or what is done by 4 
 men, is the first term, just as 4 men is the first term of the 
 first couplet ; and 150 boxes is, likewise, the second term 
 of the second couplet, as 2 men is the second term in its 
 couplet. 
 
244 ELEMENTS OF BUSINESS ARITHMETIC 
 
 The inverse proportion would be stated thus : 
 4 men : 2 men : : 4 days : 2 days. 
 
 This order is necessary that the ratio of each couplet be 
 the same (in this case, 2). It will be noticed, however, in the 
 second couplet, that 2 days, the time required by 4 men, is 
 the second term in its couplet, although the 4 men is the first 
 term in its couplet. Likewise, the 4 days is the first term 
 of its couplet, although the two men working is the second 
 term. In other words, in direct proportion the order of the 
 terms in the second couplet is the same as in the first. 
 
 In inverse proportion the order of the terms in the second 
 couplet is the inverse of the order in the first. 
 
 Note. — As a matter of convenience and simplicity in solution, it will 
 be found best to use the couplet which contains the unknown or required 
 term as the first couplet and the unknown term as its second term. 
 
 1. 2 men : ? men : : 150 boxes : 300 boxes. 
 300 boxes x 2 = 600 boxes. 
 
 600 boxes -s- 150 boxes = 4, or the number of men required. 
 
 2. 2 men : ? men : : 2 days : 4 days. 
 
 4 days x 2 = 8 days. 
 
 8 days -^ 2 days = 4, the number of men required. 
 
 In general, to solve a proportion when it is correctly 
 stated : 
 
 1. Divide the product of the two given means hy the one given 
 extreme^ or 
 
 2. Divide the product of the two given extremes hy the one 
 given mean. 
 
 229. Compound Proportion. Whenever two or more ratios 
 are equal to another ratio, the proportion is said to be com- 
 pound. Compound proportion, therefore, involves three or 
 more couplets, all having the same ratio. 
 
PROPORTION 245 
 
 The statement of problems containing a compound pro- 
 portion requires the comparing of every couplet to one basal 
 couplet (usually taken as the first couplet), ascertaining 
 whether the proportion be direct or inverse, and arranging 
 the terms of the second couplets accordingly. 
 
 230. Arranging in Couplets. As an aid to comparing the 
 couplets and to stating the proportions involved, it will be 
 found helpful to arrange first all terms given in the problem, 
 in their proper couplets. 
 
 Problem. — If 90 men have completed the construction 
 of 3 miles of railroad in 80 days, how many men should be 
 engaged to fulfill a contract to build 10 miles of road in 
 150 days ? 
 
 1. Write all the terms belonging together in a column, 
 
 thus, 
 
 90 men 
 
 3 miles 
 80 days 
 
 2. On the other side of a vertical line drawn to the right 
 of the column, write the remaining terms, arranging like 
 opposite like, forming couplets, thus, 
 
 90 men 
 
 ? 
 
 3 miles 
 
 10 
 
 80 days 
 
 150 
 
 231. Statement of a Compound Proportion. Using the 
 couplet containing the unknown term as the first couplet, 
 the following couplets should be compared to it and the 
 terms of each written as a part of the second or compound 
 couplet, their order depending on whether the statement is 
 of direct or inverse proportion. Thus, in the example given, 
 if the number of men were increased, the number of miles 
 that could be built would also increase. It is, therefore, a 
 
246 ELEMENTS OF BUSINESS ARITHMETIC 
 
 direct proportion, and the order in both couplets would be 
 the same, e.g. 
 
 90 men : ? men : : 3 miles : 10 miles. 
 
 But if the number of men were increased, it would take a 
 le%8 number of days to do the work. This proportion would, 
 therefore, be inverse, and the second couplet would be 
 written in inverse order, thus, 
 
 90 men ; ? men : : 150 days : 80 days. 
 
 Combining these two proportions into a compound propor- 
 tion, we have : 
 
 Statement 
 
 ^^ o 3 miles : 10 miles 
 
 90 men : ? men : : ., r^ j qa j 
 
 150 days : 80 days. 
 
 232. Solving Compound Proportion. 
 
 10 miles X 80 X 90 = 72,000 miles. 
 3 miles x 150 = 450 miles. 
 72,000 miles -f- 450 miles = 160, or the number of men required. 
 
 Solution by Cancellation 
 
 2 
 
 AA o ? miles : 10 miles 
 
 ^ men : ? men : : -•L -, qV, , 
 
 ^^ }.p^ days : 80 days. 
 
 2 men x 80 = 160 men. An%. 
 
 Note. — All factors removed by cancellation must be taken out of 
 hoik the means and the extremes. 
 
 PROBLEMS 
 
 1. What effect does the multiplying or dividing of both terms of a 
 ratio have on its value ? 
 
 2. What effect does multiplying or dividing the antecedent have on 
 the ratio ? Multiplying or dividing the consequent? 
 
PROPORTION 247 
 
 What is the ratio of : 
 
 3. $8 to $2? $|to|i? $.50 to $.12^? $26to|5.20? 
 
 4. 100 to 25? 100tol4f? 33itol00? 100 to 6f ? 
 
 5. $ .50 to $ .15 ? 2 m. to 40 cm. ? 15 hr. to a day? 
 
 6. What number has to 40 the ratio of 2 ? Of ^? To 5 the ratio of 
 5? To 15 the ratio of I ? Of 3 ? To 84 of 7? Of }'} 
 
 7. 28 has the ratio of 2 to what number ? 12^ has the ratio of | to 
 what number? | has the ratio of 3 to what number? 
 
 8. A pound of coffee costs 40^, and of butter 25 j^. What was the 
 ratio of their costs ? 
 
 9. The diameter of a circle is 7 ft. and the circumference is 22 ft. 
 What is the ratio of the circumference to the diameter? 
 
 10. If a map is drawn on a scale of 1 in. to 1 mi., in what ratio are 
 the dimensions diminished? 
 
 11. One door is 6 ft. 6 in. by 3 ft. 8 in. ; another is 7 ft. 6 in. by 4 ft. 
 What is the ratio of the first to the second? Of the second to the first? 
 
 Examine each of the problems from 12 to 30, and tell whether the 
 ratio is direct or inverse. Solve. 
 
 12. If 6 horses cost $ 1200, what will 10 horses cost at the same rate ? 
 
 13. If 12 yd. of cloth cost $ 20, what will 35 yd. cost? 
 
 14. If 6 horses cost $ 300, how many can be bought for $900? 
 
 15. If 15 yd. of silk cost 1 22.50, how many yards can be bought 
 for $36? 
 
 16. If it takes 48 yd. of carpet 1 yd. wide to cover a floor, how 
 many yards will it take of carpet | yd. wide ? 
 
 17. $200 earns $ 12 interest. How much interest will $350 earn? 
 
 18. A merchant pays $ 6 freight on 1000 lb. of merchandise. What 
 rate is that per 100 lb. ? 
 
 19. When coal is worth $9 a ton, what will 1200 lb. cost? 
 
 20. A man with an income of $ 1000 saved $ 300. The next year his 
 income was $ 1200 and he saved a proportional amount. How much did 
 he save ? 
 
 21. At a certain time of day, a post 4 ft. high casts a shadow 3 ft. 
 long. What is the height of a tree that casts a shadow of 15 ft. ? 
 
248 ELEMENTS OF BUSINESS ARITHMETIC 
 
 22. A pipe discharging 6 gal. a minute can fill a cistern in 4 hr. 
 How long will it take a pipe discharging 8 gal. a minute to empty it? 
 
 23. If a man sells f of his farm for $4200, what would | of it be 
 worth at the same rate ? 
 
 24. A hall was paved with tiles 9 in. square, and 640 were used. 
 How many tiles 6 in. square would it take ? 
 
 25. If a tower 40 ft. high casts a shadow 70 ft. long, how long a 
 shadow will a tower 110 ft. high cast ? 
 
 26. If it cost 1 60 to make a walk 10 ft. wide and 180 ft. long, how 
 much will it cost to make a walk 8 ft. wide and 450 ft. long ? 
 
 27. If 3000 bricks, each 8 in. long and 4 in. wide, will lay a walk, how 
 many bricks 6 in. square would it take ? 
 
 28. If it cost $ 168 to roof a space 72 ft. long and 21 ft. wide, how 
 much will it cost to roof a space 66 ft. long and 27 ft. wide ? 
 
 29. If 170 bu. of oats feed 120 horses 34 days, how long would 150 bu. 
 feed 90 horses ? 
 
 30. If 12 men, in 4 da. of 8 hr. each, earn $ 152.60, at the same rate, 
 how much will 22 men earn in 5 da. of 9 hr. each ? 
 
 31. 10 men can pave a street 30 ft. long and 48 ft. wide in 2 da. 
 How many men will it take to pave a street 400 ft. long and 36 ft. wide 
 inl2|da.? 
 
 32. If $290.70 interest accrues on $1020 at 6% for 4 yr. 9 mo., how 
 much interest must be paid on $2700 at 7^% for 3 yr. 4 mo. ? 
 
 33. If $ 675, put at interest at 8 %, earns $ 9 interest in 60 da., in how 
 many days will $1240 earn $28.80 interest at 6% ? 
 
 34. If 20 men working 12 da. of 8 hr. each can cut 400 cd. of wood, 
 how many cords should 12 men cut in 15 da. of 10 hr. each? 
 
 35. If a piece of timber 11 ft. long, 10 in. wide, and 8 in. thick weighs 
 1848 lb., find the length of another piece of timber which weighs 
 6048 lb., and which is 6 by 24 in.? 
 
 36. If 10 horses eat 16 bu. 16 qt. oats in 9 da., how many days, at the 
 same rate, will 123 bu. 28 qt. feed 34 horses? 
 
 37. If 21 men can build a wall 28 rd. long in 96 da., how many men 
 will be required to build 31^ rd. in 84 da. ? 
 
/ 
 
 PROPORTION 249 
 
 38. If 12 compositors in 60 da. of 10 hr. each set up 50 sheets of 16 
 pages each, 32 lines on a page, in how many days of 8 hr. can 18 com- 
 positors set up, in the same type, 128 sheets of 12 pages each, 40 lines to 
 the page ? 
 
 39. A contractor engaged to lay 20 mi. of road in 300 da. At the end 
 of 80 da. he finds that 90 men have laid 3 mi. How many more men 
 must he engage to finish the work in the required time ? 
 
 40. If 54 T. of anthracite coal can be stored in a bin 28 x 20 x 4 ft., 
 how many tons can be stored in a bin 45 x 18 x 8 ft. ? 
 
 41. If a mow 10 x 6 x 8 yd. holds 32 T. of hay, how deep must a 
 mow be that is 24 ft. long and 15 ft. wide, in order to hold 86 T. ? 
 
 42. If 60 men make an embankment f of a mile long, 30 yd. wide, 
 and 7 yd. high in 42 da., how many men will it take to make an em- 
 bankment 1000 by 36 yd. and 22 ft. high in 30 da. ? 
 
 43. If 50 men can do a piece of work in 48 da. working 8 hr. a day, 
 how many hours a day would 50 men have to work in order to do the 
 same work in 32 da. ? 
 
 44. If the interest on $84 at 6% for 3 yr. is |15.12, what sum must 
 be loaned at 8% for 1 yr. 6 mo., to earn the same amount? 
 
 45. If I loan $600 for 8 mo. and get $20 interest, for what time must 
 I loan $1200 at the same rate to get $ 90 interest? 
 
XXII 
 
 PROPORTIONAL PARTS AND PARTNERSHIP 
 
 233. Partnership. The association of two or more persons 
 in a business firm, or partnership^ has already been outlined 
 under Stocks and Bonds (Sec. 196). During the progress of 
 a business, it often becomes necessary to take an inventory of 
 the financial condition of the firm. For this purpose, a state- 
 ment of resources and liabilities is made. Under resources 
 are listed all property on hand and all accounts owed to the 
 business, and under liabilities all debts of the firm. This 
 casting up of accounts will show whether the profits of the 
 business exceed the expenses (net gain^ or whether the ex- 
 penses have been greater than the profits (net loss}. It will 
 also show whether the firm has sufficient resources to meet 
 all liabilities, in which case it is solvent, or if its liabilities 
 are greater than its resources, when it is insolvent. From 
 these statements, too, may be computed the present financial 
 condition of the firm (the net resources after allowing for 
 all liabilities), which is termed the present worth. 
 
 234. An Application of Proportion. The distribution of 
 the profits or apportionment of the losses of a business part- 
 nership is often so simple as to be easily resolved into frac- 
 tional parts. The conditions of partnership may, however, 
 become quite complicated, involving different forms and 
 amounts of investments, withdrawals, and increases, different 
 lengths of time, etc. In such cases the apportionment of 
 profits or losses are often easily and quickly accomplished 
 by proportion. 
 
 250 
 
PROPORTIONAL PARTS AND PARTNERSHIP 251 
 
 235. Partitive Proportion. The form of proportion used 
 for that purpose is known as partitive proportion or propor- 
 tional pojrts. The term " partitive proportion " means parti- 
 tioning a whole into parts, proportionally. The parts of the 
 profits belonging to each partner would bear the same ratio, 
 i.e, be proportional to the investments made, the time the 
 capital was used, or some other definite ratio that may be 
 ascertained. 
 
 Thus, if A invested 1500, and B invested flOOO, a profit 
 of 1750 would be divided into 8 250 and |500, respectively. 
 The sum of all the parts invested would be $1500, and A 
 would be entitled to ^^% (or J) of all the profits, which 
 would be 8250. 
 
 Stated proportionally, A's profit would be found by solv- 
 ing the proportion 
 
 1500: 11500::?: $750 
 1375,000 H- 11500 = 250, or 1 250 profit. 
 B's profit would be : 
 
 11000:11500::?: $750 
 
 ig^ = 500, or $ 500 profit. 
 
 236. Equivalent Investments. When investments in the 
 business are made for different lengths of time, the profits 
 or losses are often distributed in proportion to the equiva- 
 lent investments. By equivalent investment is meant the sum 
 which, invested for a unit of time, is equivalent to various 
 sums invested for different periods of time. 
 
 Thus, if A's $500 were invested for 6 mo. and B's $1000 
 for 4 mo., the profit of $750 would not be distributed as \ 
 and |. A's $500, invested for 6 mo., would be equivalent 
 to $3000, invested for 1 mo. ; and B's to $4000, invested 
 for 1 mo. Their total equivalent investments, then, would 
 be $7000, and A's profits would be f of $750, or $321.43. 
 
252 ELEMENTS OF BUSINESS ARITHMETIC 
 
 Proportionally solved, A's profits would be 
 
 13000 :|T000::?: 1750 
 
 12,250,000 ^ $7000 = 321.43. or 1321.43 profit. 
 
 When a partner's capital is increased or decreased during 
 the term he remains a partner, his equivalent investment is 
 found by finding equivalent investments for a unit of time 
 for each different capital, and adding such equivalent" 
 investments. 
 
 Thus, if $1000 was invested, and increased 1500 after 
 4 mo., and again increased $500 after another 2 mo., and 
 decreased $800, 6 mo. later, where it remained for 10 mo. 
 longer before there was a distribution of profits, equivalent 
 investments would be found in the following way : 
 
 $1000 X 4 = $4,000, equivalent investment for 1 mo. 
 
 1500 X 2 = 3,000, equivalent investment for 1 mo. 
 
 2000 X 6 = 12,000, equivalent investment for 1 mo. 
 
 1200 X 10 = 12,000, equivalent investment for 1 mo. 
 
 $31,000 equivalent investment for 1 mo. 
 
 237. Adjustments by Interest. Inequalities in amounts 
 and time of investments, especially when there are increases 
 and withdrawals of investments at irregular periods, are 
 often adjusted b}^ allowing interest on all investments and 
 charging interest on all sums withdrawn. The profits or 
 losses remaining after interest has been allowed or charged, 
 may then be divided equally or according to any fixed ratio. 
 
 Thus at 6 % A's $500 would bear $15 interest in 6 mo. If 
 $200 were withdrawn at that time for the remaining 6 mo., 
 his $300 would earn $9 in the remaining 6 mo., and he would 
 be charged $6 interest on such withdrawal. His net in- 
 terest earned would, therefore, be $15 + $9 - $6 = $18. 
 B's $1000 for 4 mo. would be allowed $20 interest, and 
 $400 for 8 mo., $16, and if $600 were withdrawn, he would 
 
PROPORTIONAL PARTS AND PARTNERSHIP 253 
 
 be charged $24 for the remaining 8 mo., leaving a net in- 
 terest earning of $12. A's earned interest of $18 and B's 
 of $12 must first be paid out of the $750 profit, leaving 
 $720 to be divided in proportion to the original investment. 
 Of this, A would get J, or $240, plus his interest of $18, or 
 $258, and B would get |, or $480, plus his interest of $12, 
 or $492. 
 
 Note. — When interest is allowed and charged on capital increases 
 and withdrawals, net profits ate often shared equally, after interest has 
 been paid. 
 
 PROBLEMS 
 
 1. Divide 360 into parts proportional to 3 and 6. 
 
 2. Divide $800 into parts proportional to 1, 3, and 6. 
 
 3. Divide $240 into parts proportional to ^ and ^. 
 
 4. Divide $780 among three persons, whose shares will be in pro- 
 portion to ^, ^, and ^. 
 
 5. A, B, and C engage in business for 1 yr. A puts in $5000, B 
 $3000, and C $2000. If they gain $3600, what is each man's share? 
 
 6. Divide $2400 among A, B, and C, so that A's part will be twice 
 C's and ^ B's. 
 
 7. The total receipts of a gold mining company for 1 yr. were 
 $15,750,000. The expenses were to the net earnings as 12 to 3. What 
 were the expenses ? The net earnings ? 
 
 8. Divide the simple interest on $65,000 for 1 yr. 8 mo., at 5|%, 
 among A, B, and C, so that A's part is 8 times C's and | B's. 
 
 9. A, B, and C pay $75.60 for a pasture. A puts in 10 horses, B 
 24 cows, and C 120 sheep. If 3 sheep eat as much as 1 cow and 2 cows 
 as much as 3 horses, what rent must each pay? 
 
 10. The annual earnings of a steamship company were $39,000,000. 
 Find the amounts received from freight charges, from passenger service, 
 and from other sources, if they were in the proportion of 7 : 4 : 2. 
 
 11. The holdings of the shareholders of a corporation are 32, 13, 22, 
 20, 50, 34, 42, and 72 shares, respectively. If a dividend of $12,825 is 
 divided among them, what does each shareholder receive ? 
 
254 ELEMENTS OF BUSINESS ARITHMETIC 
 
 12. C and D engaged in business and gained $3500. C's capital 
 was $8000, and D's was $6000. Find each partner's share, if the profits 
 were divided according to investment. 
 
 13. Two men owned a carriage factory. One had invested $75,000, 
 the other $45,000. The net earnings for 1 yr. were $12,200. What 
 was each partner's share ? 
 
 14. A, B, and C enter into partnership with a joint capital of 
 $130,000 ; A furnishes i, B I, and C the remainder. Their net gain is 
 30% of the amount invested. Find each man's share of the gain. 
 
 15. Dunn and McDonald formed a partnership in which Dunn in- 
 vested $5000 and McDonald invested $2500. The gains were: mer- 
 chandise, $940.25; real estate, $356.50; losses, expense, $420. What 
 was the net gain ? What was the gain of each partner ? What was each 
 partner's present worth at the close of the business ? 
 
 16. A and B formed a partnership with a capital of $10,000. A 
 furnishes $4000 and B $6000. After 16 mo. A withdrew $500, and at 
 the end of 18 mo. B withdrew $1000. At the end of 2 yr. the partner- 
 ship was dissolved and a profit of $8750 was divided. How much did 
 each partner receive ? 
 
 17. A and B engaged in a dry goods business for 3 yr. from April 1, 
 1907. Each invested $1800. July 1, 1907, A increased his investment 
 $350, and B withdrew $300; Feb. 1, 1908, each withdrew $800; Feb. 1, 
 1909, each invested $1200. There was a gain of $1800 on March 1. 
 How should it be divided ? 
 
 18. C and D formed a partnership, C investing $9000, and B $12,000. 
 It was agreed that B should take $3000 from the gains before a division 
 was made, and that the net gain or loss should then be shared equally. 
 The gains were $7580 and the losses $1275. What was the net gain of 
 each partner? The present worth at dissolution? 
 
INDEX 
 
 Reference to Pages 
 
 Acceptance, 212. 
 
 Accident insurance, 327. 
 
 Accounts, 46 ; savings bank, 206. 
 
 Account sales, 180. 
 
 Addition, 1 ; by groups, 2 ; two-col- 
 umn, 2; of decimals; 25; of frac- 
 tions, 56. 
 
 Ad valorem duties, 188. 
 
 Angles, 98. 
 
 Annual interest, 194. 
 
 Antecedent in ratio, 241. 
 
 Apothecaries', liquid measure, 121 ; 
 weight, 137. 
 
 Arabic notation, 23. 
 
 Area, 72 ; of rectangles, 73 ; non- 
 rectangles, 101-104 ; quadrilaterals, 
 101 ; trapezoids, 102 ; triangles, 102- 
 104 ; circles, 105 ; other surfaces, 
 107; metric, 151. 
 
 Articles of incorporation, 222. 
 
 Assessed valuation, 185. 
 
 Assessments, on stocks, 222; insur- 
 ance, 232. 
 
 Authorized capital, 222. 
 
 Averaging, 34. 
 
 Avoirdupois weight, 138. 
 
 Bank discount, 213. 
 
 Banks, 205 ; national, 205 ; state and 
 
 private, 205 ; savings, 206. 
 Base lines, 94. 
 
 Bears, in stocks and bonds, 225. 
 Beneficiary of insurance, 236. 
 Bills, 46 ; of foreign exchange, 145. 
 Board measure, 115. 
 Bonds, 193, 223. 
 Bricklaying, 114. 
 Brokerage and brokers, 225. 
 Bulk, measures of, 119. 
 
 Bulls, in stocks and bonds, 225. 
 Bushel, unit, 119; weights, 138. 
 
 Calendar months, 126. 
 
 Capacity, in dry units, 120 ; liquids, 
 121. 
 
 Capital stock, 222. 
 
 Carpeting, 86. 
 
 Cashier's checks, 210. 
 
 Casting out nines, 4. 
 
 Certificates of deposit, 210 ; of stock, 
 222. 
 
 Certified checks, 209. 
 
 Change, making, 10 ; memorandum, 50. 
 
 Chattel mortgage, 192. 
 
 Check on addition, 4. 
 
 Checks, 209. 
 
 Cipher, for marking goods, 177. 
 
 Circle, 101 ; area of, 105 ; measure- 
 ment of, 128. 
 
 Circumference, 101 ; ratio to diam- 
 eter, 105. 
 
 Civil service, addition method, 4. 
 
 Closed insurance policy, 232. 
 
 Coins, authorized, 141. 
 
 Collection by draft, 211. 
 
 Combinations, addition, 1. 
 
 Commercial month, 127. 
 
 Commission, 180, 225 ; on purchases, 
 181. 
 
 Common denominator, 57 ; divisor, 34 ; 
 stock, 222. 
 
 Compound interest, 193 ; proportion, 
 244. 
 
 Cone, 123 ; surface of, 107 ; volume, 
 124. 
 
 Consignment, 46, 180. 
 
 Cord, of wood, 113 ; of stone, 114. 
 
 Corporations, 221. 
 
 265 
 
256 
 
 INDEX 
 
 Correction lines, 95. 
 
 Cost of articles, 41 ; per hundred, 
 43 ; per thousand, 43 ; per ton, 45. 
 
 Couplet, in proportion, 241, 245. 
 
 Coupons, 193, 224. 
 
 Cube root, 112. 
 
 Cubes, table of, 113. 
 
 Cubic units, 109. 
 
 Cylinder, 122 ; surface of, 107 ; vol- 
 ume, 124. 
 
 Date line, international, 132. 
 
 Days of grace, 195! 
 
 Decimal equivalents ot common frac- 
 tions, 40. 
 
 Decimal fractions, 62. 
 
 Decimal notation, 20 ; digit value in, 5. 
 
 Decimal point, use of, 23. 
 
 Decimals, 23 ; square root of, 80. 
 
 Degree, 128. 
 
 Demand certificates, 210. 
 
 Denominator, 53. 
 
 Direct ratio and proportion, 242. 
 
 Discount, trade, 172 ; series in, 173 ; 
 bank, 213 ; true, 215 ; fractional, 
 173 ; is interest, 215 ; selling at, 
 223, 224. 
 
 Dividends, 222. 
 
 Divisibility, tests of, 33. 
 
 Division, two kinds of, 18 ; long divi- 
 sion, 20 ; shortening of, 20 ; of deci- 
 mals, 26 ; by 1 with ciphers, 26 ; 
 of fractions, 56, 61. 
 
 Divisors, common, 34. 
 
 Drafts, 143, 210. 
 
 Drill cards, 1, 17. 
 
 Drill tables, addition, 1, 2 ; multi- 
 plication, 16. 
 
 Dry measure, 119. 
 
 Duties, 188. 
 
 Endowment insurance, 237. 
 
 Equation in explanations, 13, 17, 18. 
 
 Equivalent investments, 251. 
 
 Even numbers, 33. 
 
 Exact interest, 195. 
 
 Excises, 189. 
 
 Exchange, 142; bank, 143, 210; 
 
 by wire, 144 ; foreign, 144 ; by cable, 
 
 147. 
 Explanations, suggestions for, 13, 17. 
 Extremes in proportion, 241. 
 
 Factors, 33. 
 
 Fire insurance, 232. 
 
 Firms, 221. 
 
 Flooring, 85. 
 
 Foreign exchange, 144. 
 
 Fractional parts, 32 ; in division, 18 ; 
 
 finding, 32, 36; of a dollar, 41. 
 Fractions, 53 ; and decimals, 62, 
 
 square root of, 80. 
 
 Gain, 157, 250. 
 Geographical mile, 129. 
 Government bonds, 224. 
 Gram, 152. 
 
 Greatest common divisor, 34. 
 Groups, adding by, 2. 
 
 Indorsement, 212. 
 
 Indorsers, 192. 
 
 Insolvency, 250. 
 
 Inspection, reduction by, 55. 
 
 Insurance, 231. 
 
 Integers, 23. 
 
 Interest, 192; sixty-day method, 196; 
 6%, 199; at any rate, 197; tables 
 of, 199 ; days, in savings banks, 
 206 ; in partnership, 252. 
 
 International date line, 132. 
 
 Inverse ratio and proportion, 242. 
 
 Investments, 251. 
 
 Invoicing, 46. 
 
 Isosceles triangle, 99 ; area of, 103. 
 
 Kilogram, 152. 
 
 Land measure and survey, 94. 
 
 Lathing, 88. 
 
 Latitude, 128. 
 
 Least common multiple, 34, 57. 
 
 Legal, limitations to interest, 194 ; 
 
 rate of interest, 194 ; tender, 142. 
 Length, measures of, 67 ; metric, 150. 
 Letters of credit, 145. 
 Liabilities in partnership, 250. 
 Liability of stockholders, 223. 
 Life insurance, 236 ; policy, 237 ; cost 
 
 of, 238. 
 Limited payment policy, 237. 
 Linear measure, 67. 
 Liquid measure, 120; apothecaries', 
 
 121. 
 Listing goods, 177. 
 
INDEX 
 
 257 
 
 List price, 172. 
 
 Liter, 152. 
 
 Live-stock insurance, 232. 
 
 Log measure, 116. 
 
 Longitude and time, 128; diEference 
 in, 129 ; reduction in, 131 ; of lead- 
 ing cities, 134. 
 
 Long ton, 138. 
 
 Loss, 157, 250. 
 
 Lumberman's reference table, 118. 
 
 Lumber measure, 115. 
 
 Maker, of note, 193; draft, 209. 
 
 Manifest, or customs declaration, 188. 
 
 Marine insurance, 232. 
 
 Marked price, 172. 
 
 Market quotations, 225. 
 
 Marking goods, 177. 
 
 Maturity, 192 ; value, 213. 
 
 Means in proportion, 241. 
 
 Merchant's rule, 199. 
 
 Meter, 149. 
 
 Metric system, 149. 
 
 Mixed number, 53. 
 
 Money order, P.O., 143 ; express, 143 ; 
 foreign, 146. 
 
 Mortgage, 192. 
 
 Multiples, 33 ; common, 34. 
 
 Multiplication, 17; drill table, 17; 
 long, 19 ; of decimals, 26 ; by 1 with 
 ciphers, 26 ; of fractions, 58. 
 
 Mutual insurance, 231. 
 
 National banks, 205. 
 
 Nautical units, 68. 
 
 Negotiable paper, 209. 
 
 Non-assessable stock, 223. 
 
 Notes, 192; negotiable and non-, 210. 
 
 Numerator, 53. 
 
 Odd numbers, 33. 
 
 Open insurance policy, 232. 
 
 Papering, 92. 
 
 Paper money, authorized, 141. 
 
 Parallel, 99. 
 
 Parallelograms, 100 ; area of oblique, 
 
 101. 
 Parallels of latitude, 128. 
 Partial payments, 199. 
 Partitive proportion, 251. 
 
 Partnership, 221 ; by proportion, 250. 
 
 Par value, 222. 
 
 Payee, of notes, 193 ; of drafts, 209. 
 
 Pay rolls, 49. 
 
 Per capita tax, 186. 
 
 Percentage, 157; finding 50%, 25%, 
 20%). 158; 33i%, 163%, 12J%, 
 14?%,, 160; 10%, 1%, 5%, i%, 
 162 ; finding other per cents, 164. 
 
 Periodic interest, 194. 
 
 Personal property, 185 ; tax, 185. 
 
 Pi (TT), 105. 
 
 Plastering, 88. 
 
 Pointing off, in decimals, 24 ; in mul- 
 tiplication, 27 ; in division, 28. 
 
 Policy, of insurance, 231 ; partici- 
 pating, 236. 
 
 Poll tax, 186. 
 
 Polygons, 99 ; area of, 101. 
 
 Ports of entry, 188. 
 
 Preferred stock, 222. 
 
 Premimn, selling at, 223 ; on insurance, 
 232. 
 
 Present worth, 215, 250. 
 
 Prime numbers, 33. 
 
 Principal, 199 ; meridians, 94. 
 
 Prisms, 122; volume of , 110, 123; sur- 
 face, 107. 
 
 Proceeds, 215. 
 
 Profit and loss, 157. 
 
 Promissory notes, 192. 
 
 Proof, in addition, 4. 
 
 Proper fractions, 53. 
 
 Proportion, 240; in partnership, 250. 
 
 Proportional parts, 251. 
 
 Pyramids, surface of, 107; volume, 
 123. 
 
 Qtcadrilaterals, 100; area of, 101. 
 Quotient, denomination of partial, 20. 
 
 Rate, in interest, 192, 194 ; insurance, 
 232. 
 
 Ratio, 240. 
 
 Reading problems, 19 ; of decimals, 23. 
 
 Real estate, 185. 
 
 Reduction, of fractions, 54 ; by in- 
 spection, 55. 
 
 Registered bonds, 224. 
 
 Remainder, exact in division, 30. 
 
 Reserve fund, insurance, 237. 
 
 Resources, 250. 
 
258 
 
 INDEX 
 
 Right angle, 99. 
 
 Roofs and roofing, 82. 
 
 Roots, square, 76; application to 
 
 right triangle, 80 ; of decimals and 
 
 fractions, 80 ; cube, 112. 
 Rule of three, 242. 
 
 Scalene triangles, 99 ; area of, 104. 
 
 Sections of land, 96. 
 
 Short, selling, 225. 
 
 Simplified processes, in division and 
 multiplication, 37. 
 
 Sinking fund, 224. 
 
 Six per cent method, 199. 
 
 Sixty-day method, 196. 
 
 Smuggling, 189. 
 
 Solar day and year, 126. 
 
 Solvency, 250. 
 
 Specific duties, 188. 
 
 Sphere, 123 ; surface of, 107 ; volume, 
 124. 
 
 Square, 75, 100 ; root, 76. 
 
 Standard parallels, 94; time, 132. 
 
 Stationer's table, 147. 
 
 Statement, of account, 46 ; in propor- 
 tion, 243. 
 
 Statutemile,67, 129. 
 
 Stock company, 221. 
 
 Stocks and bonds, 221 ; exchanges, 225. 
 
 Study, suggestions for, 16. 
 
 Subtraction, 10; horizontal, 11; of 
 decimals, 25 ; of fractions, 56, 57. 
 
 Surface, 98; forms, 98; of cylinders, 
 pyramids, cones, and spheres, 107. 
 
 Survey of land, 94. 
 
 Surveyor's measure, 68. 
 
 Tariffs, 188. 
 
 Taxes, 185. 
 
 Term discount, 173. 
 
 Term of discount, 213; table, 214. 
 
 Term policy, 237. 
 
 Terms of ratio, 240. 
 
 Time certificates of deposit, 210. 
 
 Time, measures of, 126 ; difference in, 
 
 127 ; compound subtraction, 127 ; 
 
 relative, 130 ; and longitude, 137. 
 Tornado insurance, 232. 
 Township, 94. 
 Trade discount, 172. 
 Tradesman's table, 147. 
 Trapezium, 100, 104. 
 Trapezoid, 100, 102. 
 Traveler's checks, 146. 
 Triangles, 99 ; area of, 102 ; applicar 
 
 tion of square root to, 80. 
 Troy weight, 137. 
 True discount, 215. 
 
 Undivided profits, 222. 
 Unitate addition proof, 4. 
 Usury, 194. 
 
 Valued insurance policy, 232. 
 
 Value, unit of, 141. 
 
 Volume, 109 ; of rectangular prisms, 
 110; non-rectangular, 123 ; of pyra- 
 mids, 123 ; cylinders, cones, and 
 spheres, 124; metric, 151. 
 
 Weight, measures of, 137 ; metric, 152. 
 Wood measure, 113; metric, 152. 
 Writing decimals, 24. 
 
'HE following pages contain advertisements of a 
 few of the Macmillan books on kindred subjects. 
 
MACMILLAN'S COMMERCIAL SERIES 
 
 Edited by CHEESMAN A. HERRICK 
 
 President of Girard College, formerly Director of School of Commerce, Philadelphia 
 Central High School 
 
 Each volume i2mo, cloth 
 
 The Meaning and Practice of Commercial Education. By the Editor, xv + 
 378 pages, i^i.25 net. 
 
 The Geography of Commerce. By Spencer Trotter, M.D., Professor of 
 Biology and Geology in Swarthmore College, Pa. xxiv + 410 pages, 
 ^i.io net. 
 
 Commercial Correspondence and Postal Information. By Carl Lewis Alt- 
 MAIER, Drexel Institute, Philadelphia, xiv + 204 pages. 60 cents net. 
 
 Comprehensive Bookkeeping : A First Book. By Artemas M. Bogle, Head 
 of Department of Mathematics, High School, Kansas City, Kansas, xi + 
 142 pages. 90 cents net. 
 
 Bookkeeping Blanks. By Artemas M. Bogle. Four numbers. 75 cents a 
 set net. 
 
 Teacher's Manual to Accompany Comprehensive Bookkeeping. By Artemas 
 M. Bogle, vi + 75 pages, ^i.oo net. 
 
 Elements of Business Arithmetic. By Anson H. Bigelow, Superintendent of 
 Schools, Lead, South Dakota, and William A. Arnold, Director of Busi- 
 ness Training, Woodbine (Iowa) Normal School, xv + 254 pages. 
 
 The volumes of Macmillan's Commercial Series named 
 below are in preparation and other volumes will follow: 
 
 The History of Commerce. By the Editor of the Series. 
 
 Applied Arithmetic for Secondary Schools. By Ernest L. Thurston, District 
 Superintendent of Schools, Washington, D.C. 
 
 PUBLISHED BY 
 
 THE MACMILLAN COMPANY 
 
 64-66 Fifth Avenue, New York 
 BOSTON CHICAGO DALLAS SAN FRANCISCO 
 
MACMILLAN'S COMMERCIAL SERIES 
 
 The Idea of the Series 
 
 This series is prepared in the belief that disciplinary education can be secured 
 through the use of subject-matter of practical worth. Much that is fixed in our 
 system of education is retained and given new application; new elements are in- 
 troduced and are properly related to the old. In brief, the plan is to modernize 
 the instruments of instruction and make schools more effective as a preparation 
 for present economic life. The best from foreign books has been utilized for 
 suggestion; the best in our educational development is preserved. The plan 
 and its execution are the work of experienced teachers. The books are products 
 of specialists, working under the general supervision of the editor. Each volume 
 is adequate to its subject, authoritative, and supplied with a working equipment 
 such as illustrations, maps, and diagrams. 
 
 Elements of Business Arithmetic 
 
 By Anson H. Bigelow, Superintendent of Schools, Lead, S.D., and 
 William A. Arnold, Director of Business Training, Woodbine (Iowa) 
 Normal School. Cloth. i2mo. xv + 254 pages. 
 
 The preparation of this text was undertaken in the belief that the arithmetic 
 of the grammar school and of the commercial course of the high school should 
 teach the methods most in vogue in the business world, and that those methods 
 should be so taught as to form correct habits in those who are to attack the prob- 
 lems of real life. It is distinctly a business arithmetic, presenting the minimum 
 of theory and the maximum of practice in business methods. Various methods 
 are presented, but only those used in practical business computations. The topics 
 treated, by chapters, are : addition and subtraction, multiplication and division, 
 decimals, fractional parts (short methods), fractions, measures (length, area, vol- 
 ume, time, weight, and value), French metrical system, percentage, trade dis- 
 count, commission, taxes and duties, interest, banking and discount, stocks and 
 bonds, insurance, proportion, proportional parts, and partnership. These subjects 
 are chosen with reference to business needs and they are treated in such a man- 
 ner as to give the pupil the largest possible amount of drill in practical business 
 methods. The book purposely brings the work of the school and the needs of 
 common life into vital connection. It is suitable for use in the grammar school 
 and in the commercial courses of the high school. 
 
 PUBLISHED BY 
 
 THE MACMILLAN COMPANY 
 
 64-66 Fifth Avenue, New York 
 BOSTON CHICAGO DALLAS SAN FRANCISCO 
 
The Meaning and Practice 
 of Commercial Information 
 
 By Cheesman A. Herrick 
 
 The book above mentioned explains the idea and describes the actual work- 
 ings of commercial schools. It treats commercial education from various points 
 of view, and shows that this form of instruction is a result of present economic 
 conditions and a natural step in our educational development. The author shows 
 also that special education for the present commercial age is both possible and 
 desirable, and that such education will gradually bring about a higher form of 
 commercialism. 
 
 The work reviews the movements to furnish commercial education in leading 
 countries. For the United States a series of chapters are devoted to the Private 
 Commercial School, the High School of Commerce, the Curriculum of the Sec- 
 ondary Commercial School, and the Higher School of Commerce. Numerous 
 illustrations of men and institutions are furnished. 
 
 An appendix supplies a good number of curricula for schools of various grades. 
 The value of the work is further increased by a select bibliography of the subject. 
 
 The Geography of Commerce 
 
 By Spencer Trotter, M.D. 
 
 This book is exceptionally fortunate as well as unique in its authorship. Dr. 
 Trotter is a scientist and geographer of high standing, while the editor, Dr. Her- 
 rick, is a trained economist. Both are experienced and successful teachers. The 
 text has stood the test of work with high school students. 
 
 The Geography of Commerce gives a clear presentation of existing conditions 
 of trade. Throughout the book emphasis is laid on the relation between physi- 
 ography, climate, etc., and the activities and the organizations of men. As a re- 
 sult, the book is on the " practical side " of geography. Trade relations between 
 the United States and other countries are given special prominence. The causal 
 relations of physical environment to men, of men and environments to products, 
 and of products to trade, are treated with a unity that makes the book admirably 
 suited to class use. 
 
 A complete working equipment and a list of books for further consultation 
 are furnished. Supplementary questions and topics are also supplied. 
 
 PUBLISHED BY 
 
 THE MACMILLAN COMPANY 
 
 64-66 Fifth Avenue, New York 
 BOSTON CHICAGO DALLAS SAN FRANCISCO 
 
Commercial Correspondence and 
 Postal Information 
 
 By Carl Lewis Altmaier 
 
 Mr. Altmaier's work supplies two present needs, a text-book for school use 
 and a handbook for office use. In the first place, his book is a working manual 
 for instruction and practice in letter writing, and thus it furnishes material for 
 practical English composition. Correct forms of letters are furnished, after which 
 the learner is asked to deal with situations of the kind actually met with in busi- 
 ness correspondence. The treatment of correspondence is supplemented by a 
 somewhat detailed account of postal arrangements, both domestic and interna- 
 tional. The book is illustrated with photographs of documents, reproductions of 
 actual letters, and a postal map of the world. 
 
 Comprehensive Bookkeeping 
 
 By Artemas M. Bogle 
 
 A few of the points that commend this volume are : 
 
 I. The gradual and systematic development of the subject. 2. Preliminary 
 sets for drill followed immediately by more concrete sets for the more advanced 
 work of the student. 3. Material so arranged that the teacher may use it largely 
 in his own way. 4. The sets so arranged that short exercises or longer ones may 
 be given as may be most advantageous. 5. Provision for drill on important 
 points and at the place where needed, thus insuring the mastery of each point. 
 6. Arrangement such that at almost any stage previous points may be reviewed 
 without going back and working over the old material. 7. Clear, concise expla- 
 nations. 8. A large number of cross references, showing the connection of one 
 portion of the subject with another. 
 
 Teacher's Manual to Accompany Comprehensive 
 Bookkeeping 
 
 By Artemas M. Bogle 
 
 This book contains the results of computations required by the regular series 
 of exercises given in Bogle's " Comprehensive Bookkeeping." These tables, giv- 
 ing the " answers " which should be right, save the teacher labor in checking up 
 pupils' results. The forms are not intended for models but only as results to save 
 labor by the teacher. 
 
 PUBLISHED BY 
 
 THE MACMILLAN COMPANY 
 
 64-66 Fifth Avenue, New York 
 BOSTON CHICAGO DALLAS SAN FRANCISCO 
 
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