Irving Stringham LIPPINCOTT'S PRACTICAL ARITHMETIC EMBRACING THE SCIENCE AND PRACTICAL APPLICATIONS OF NUMBERS BY J. MORGAN RAWLINS, A.M. // AUTHOR OF "LIPPINCOTT'S ELEMENTARY ARITHMETIC" AND " LIPPINCOTT'S MENTAL ARITHMETIC" PHILADELPHIA J. B. LIPPINCOTT COMPANY COPYRIGHT, 1899. BV T. B. LIPPINCOTT COMPANY, ELECTROTVPED AND PRINTED BY J . B. LlPPINCOTT COMPANY, PHILADELPHIA, U. S. A. PREFACE. THAT Arithmetic is both a science and an art is not only generally conceded, but emphatically affirmed. One, there- fore, investigating the methods of instruction adopted in school-rooms would logically expect to find both aspects of the subject distinctly put in evidence. What is universally acknowledged and proclaimed as essential and vital to any true system of arithmetical education the investigator, how- ever, would fail to find distinctly characterizing either the instructions given by the average teacher or the work required of the average pupil. What we mean is, that if he saw any- thing notable, he would see Art conspicuous in the foreground, while Science, if visible at all, sat mute far back. There might be seen, to be sure, remarkable skill displayed in many instances, and results brought forth with surprising facility ; but, in it all, Science, that alone imparts life to action and informs the mind, would have little or no part. The great error is that arithmetical exercises and problems are too frequently nay, almost invariably set up like so many ten-pins, to be knocked down by mechanical action, without any inquiry as to the underlying and fundamental principles upon which action is based. In a word, the schools, with little exception, are not making the best use of 800553 iv PREFACE Arithmetic as an educational force by ignoring the fact that it has principles to be explained, induction and analysis to explain them, and a philosophical reason for every step neces- sary to be taken. The text-books used may sometimes be seriously at fault, want of time may be an impediment, and other hindrances may be numerous ; but the live and intelligent teacher will find in the least scientific treatise food for quickening thought and moulding the mental state. It must be admitted, how- ever, that the teacher, in the conscientious performance of his duty, has a difficult environment, and needs all the help that text-books can furnish him. The book that we now introduce to the public we have aimed to make what it assumes to be, a Practical Arithmetic; practical, not so much by devising short processes and labor-saving schemes as by laying a scientific foundation to be studied and mastered as the essential preliminary to the intel- ligent and skilful use of any device of mere art ; practical, therefore, as a teacher's true assistant, bringing to his hand a full supply of definitions, inductive steps, illustrations, prin- ciples, analyses, syntheses, processes, rules, and suggestions, needful to him in his high vocation, a vocation that is highest when most devoted to "bright-eyed Science," and lowest when it rests content with the pretentious and empty forms of mere " mechanic art." The text-book, even in its best estate, replete with science and art-full, can have little philosophical efficiency except when intelligently used as a means to an end. In the school- PREFACE V room, where a book is expected to promote the high aims of education, the intelligent use of it must begin, if it begins at all, with the teacher ; for it is he alone whose very office it is, through voice and action, to stir into quickening force the words of the text, that otherwise may fall as good seed upon sterile ground. Every teacher ought to know what every pupil soon learns .that "to hear illustrations and explana- tions from living lips is a different thing from struggling through them on the printed page." Every page is to be learned, however, mastered, and made emphatically the pupil's own ; and, as a suggestion pertinent here, we quote the philosophic words of John Locke : " The great art to learn much is to undertake a little at a time. " The author would gladly express his thanks to all who in any way made contributions of help. To one friend, whose devotion to the work never faltered, he acknowledges lasting obligation. J. M. R. CONTENTS. PART I. Definitions 1 Notation and Numera- tion 2 Arabic Notation 3 Notation of IT. S. Money . 11 Roman Notation 13 Simple Numbers. Addition 15 Subtraction 24 Multiplication ...... 36 Division 50 Analysis 64 Indicated Solutions .... 65 General Principles of Di- vision 68 Properties of Numbers. 71 Factoring 73 Cancellation 77 Common Divisors 80 Common Dividends .... 84 Review 90 Fractions 91 Reduction 94 Addition ... 101 Subtraction 104 Multiplication 107 Division 112 Complex Fractions . . . . 115 Fractional Relations . ... 116 Review - 118 Decimal Fractions ... 125 Notation and Numeration . 126 PAGE U. S. Money 129 Reduction 130 Complex Decimals .... 132 Addition 132 Subtraction 136 Multiplication ...... 138 Division 142 Short Processes 144 Review 149 Accounts and Bills ... 152 Denominate Numbers . 156 Reduction Descending ... 157 Reduction Ascending ... 159 Measures 161 Of Length 161 Of Surface 163 Of Volume 168 Of Capacity 174 Of Weight 178 Of Time 182 Circular Measure .... 184 Reduction of Denominate Fractions (Special) . . 187 Fractional Relation .... 189 Addition 193 Subtraction 196 Multiplication 198 Division 199 Longitude and Time . . 200 Standard Time 202 Miscellaneous Problems 204 Review 206 vii CONTENTS PART II. PAGE Percentage 208 Review Exercises. . . 218 Commercial Discount . . . 221 Gain and Loss 223 Commission 227 Review 230 Stocks and Bonds 233 Insurance 240 Direct Taxes 243 Indirect Taxes 246 Interest 249 Six-per-cent. Method . . 253 Exact Interest . ... 255 Compound Interest . . . 259 Annual Interest .... 261 Promissory Notes 262 Partial Payments 266 Merchants' Kule .... 267 U. S. Kule 269 Bank Discount 273 True Discount 277 Review 279 Exchange 281 Domestic Exchange . . . 283 Ratio and Proportion . 286 Eule of Three 289 Compound Proportion . . 292 Cause and Effect 295 Proportionate Parts .... 296 Partnership 298 Averages 302 Averaging or Equating of Payments ... 303 Involution 307 Evolution 311 Square Koot 311 Cube Koot H20 Similar Figures 326 PAGE Mensuration 329 Surfaces 330 Triangles 331 Parallelograms 332 Trapezoid 333 Trapezium 334 Kegular Polygon . . . . 334 Circle 335 Miscellaneous Problems 336 Volumes 338 Prism and Cylinder . . . 339 Pyramid and Cone . . . 340 Frustums 341 Sphere 342 Circle and Largest Square 343 Sphere and Largest Cube 344 General Review 346 Appendix 399 Duodecimals 399 Metric System 402 Foreign Exchange .... 408 Arithmetical Progression . 411 Geometrical Progression . . 412 Compound Interest .... 414 Annuities 416 Circulating Decimals . . . 420 Greatest Common Divisor of Fractions -121 Least Common Dividend of Fractions 421 The Thermometer .... 422 The Clock 423 Work 424 Averaging Accounts . . . 425 Keview 427 Miscellaneous Problems . . 430 Table of Commercial Laws 434 GENERAL SUGGESTIONS. 1. There is no royal road to a knowledge of Arithmetic, and in this book no attempt has been made to preclude the necessity for laborious effort without which, it has been wisely said, life gives nothing to mortals. 2. Self-reliance is the basis of action, and " self-activity is the law of growth." To render the pupil self-reliant, self- helpful, and self-acting in the face of difficulties, is the object the teacher should keep steadily in view. 3. Two ideas are fundamental : I. Knowledge cannot be successfully built except on knowledge already acquired. The lesson to be learned to-morrow must start its growth in the lesson learned to-day. II. Lessons assigned a class must not be made too easy for some, nor too difficult for others. Care- ful judgment is, therefore, required that oral explanations and illustrations be neither too ample nor too meagre. The least active mind must be made to understand, and, at the same time, the most active brain must be required to labor. 4. In each division of the subject, as treated, will be found inductive steps, definitions, principles, processes, explanations, rules, exercises, and problems, all to be thoroughly learned and intelligently recited. Mastery of the exercises will give facility in performing the operations reojiired by the problems. x GENERAL SUGGESTIONS 5. The solution of a problem requires three distinct steps : I. The indication in arithmetical language of the opera- tions to be performed. II. The mechanical performance of the operations indi- cated. III. The statement of the reasoning by which the opera- tions as indicated were obtained, and also the elucidation of any merely mechanical step that has been taken to reach the final result. If in each subject the introduction, including principles, processes, and explanations, be systematically and thoroughly acquired, the exercises and problems that follow will seem not forbidding obstacles, but, as it were, beckoning friends. 6. Too much importance cannot be attached to the method of dealing with a problem, as pointed out above. The frequent suggestions made throughout the book attest the author's belief in the excellence of the system proposed. One advantage is that the first step the indication in arithmetical language of the work to be done, really solves the problem, and that here, in many cases, the pupil's work may be considered as satisfactorily closed. Every teacher must determine for himself, and every intelligent teacher will successfully deter- mine, how far his pupils need to work out and recite the details of a solution. He must go far enough to be con- vinced that they have got within them a conception of the truth, and are able to declare it. But how is this possible, unless he recognizes the great fact that every pupil is an indi- vidual, has a distinct individuality, and is, as far as possible, GENERAL SUGGESTIONS xi to be individually approached and trained, " not for school, but for life' 7 ? To summarize : 1. Do not go too fast; hasten slowly. 2. Assign lessons with care, keeping in mind that "too much is not good." 3. Repeat constantly ; " repetition is the mother of all learning/' 4. Require hard work ; " the harder a pupil has worked for what he knows and can do, the better for him." 5. Be methodical, enthusiastic, persistent, and patient. 6. Remember the ancient maxim, that " to the boy is due the highest reverence." PRACTICAL ARITHMETIC PART I. DEFINITIONS. 1. A Unit is a single thing or one. 2. A Number is a unit or a collection of units. 3. Arithmetic is both a Science and an Jr : as a science, it investigates the principles of numbers ; as an art, it applies those principles to practical purposes. 4. A Principle is a fundamental truth or ground of action. 5. A number is Concrete or Denominate when its unit is named, as in one man, two books, three ships. 6. A number is Abstract when its unit is not named, as one, two, three. When named, the unit of a number is one of the things expressed by the number, as one tree, one man. When not named, the unit of a number is simply one. 7. A Simple Denominate number has a single unit, as in five feet. A Compound Denominate number has two or more related units, as in three yards two feet six inches. What is the unit of the concrete number three ships ? Of the abstract number three ? l 2 PRACTICAL ARITHMETIC Tell which of the following numbers are concrete and which abstract, and what is the unit of each : 1. Ten men. 7. One apple. 13. Four horses. 2. Three. 8. Seven. 14. Six wagons. 3. Nine boys. 9. Five pounds. 15. Sixty. 4. Eleven girls. 10. Fifty-five. 16. Sixty-seven. 5. Twenty -one. 11. Twenty-nine. 17. Twenty ships. 6. Seventeen. 12. Twelve. 18. Twenty-nine. 8. Analysis (Greek, taking apart) examines the separate parts of a subject, or proposition, and their connection with each other ; it solves problems by a comparison of their ele- ments; it reasons from the given number to one, and then from one to the required number ; it reasons, also, from par- ticular instances to general principles. 9. Synthesis (Greek, putting together) unites separated parts, in accordance with their obvious relations. 10. A Rule is founded on some principle, and is a precise direction for solving a problem. 11. A Problem is a practical question requiring a solution. 12. A Solution consists of a process and an explanation made by the application of a rule or by analysis and synthesis. NOTATION AND NUMERATION. 1. Notation is the art of writing numbers. 2 . Numeration is the art of reading numbers. 3. There are three methods of notation in common use : 1. The word method. 2. The Arabic or figure method. 3. The Roman or letter method. 4. The Arabic method employs the Arabic figures : 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. NOTATION AND NUMERATION 3 5. The word method names these figures and expresses their values as follows : 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. One, two, three, four, five, six, seven, eight, nine, naught (cipher, zero). The script forms are as follows : I -234-51018^0 These figures are frequently called digits (Latin, digitus, finger) ; those preceding are called significant figures. ARABIC NOTATION. 1. Each of the first nine numbers, you perceive, is ex- pressed by a single digit ; higher numbers are expressed by combinations of the digits. One prefixed to naught (10) is ten. 2. Our system of notation is "due to the fact that we have ten fingers/ 7 and the basis of it is the first ten numbers formed into a model group. 10 is one ten, or simply ten (Latin, " decem"). 11 is eleven (Gothic, "am, one; lif, ten"), one and ten. 12 is twelve (Gothic, " tva, two; lif, ten"), two and ten. 13 is thirteen, three and ten. 14 is fourteen, four and ten. 15 is fifteen, five and ten. 16 is sixteen, six and ten. 17 is seventeen, seven and ten. 18 is eighteen, eight and ten. 19 is nineteen, nine and ten. 20 is twenty (teen becomes ty). 4 PEACTICAL ARITHMETIC 3. The numeral names that precede " teen" follow " ty" and a hyphen (-), as follows : 21 is twenty-one, that is, twice ten and one ; 30 is thirty ; 31 is thirty-one ; 40 is forty ; 42 is forty-two. 4. Any significant figure, located as four in 40, has its value increased ten-fold and denotes tens. Locate 5 thus, and name the number ; also 6, 7, 8, 9. The value of a significant figure in units 7 place is called its Simple Value. The value of a significant figure otherwise placed is called its Local Value. 5. 44 is four tens and four units, or forty- four. Which four has the increased or Local Value 1 Which has only its Simple Value ? * 99 is ninety- nine, and is the largest number that can be expressed with two figures. 6. 100 is ten tens, or one hundred. 200 is twenty tens, or two hundreds. 300 is thirty tens, or three hundreds. Any significant figure thus located expresses hundreds. 444 is four hundreds, four tens, four units, or four hundred forty-four. 404 is thus read : " four hundred four," not " four hundred and four." The naught (0) indicates the absence of tens. 7. Again, 404, or any three digits thus written together, constitute a period with units in the first place, tens in the second place, and hundreds in the third place. The left hand period may contain but one or two digits. 8. Ten units grouped make a single ten-group. Ten ten- groups make a single hundred-group. Ten one-hundred- groups make one thousand, written 1000. 9. Any single thing is a unit ; a single ten-group may, there- fore, be considered a unit ; so also, a single hundred -group. NOTATION AND NUMERATION 5 1O. On this principle a digit in the first place denotes units of the first order ; in the second place, units of the second order ; in the third place, units of the third order, etc. PRINCIPLES. 1. The first nine numbers are expressed by the nine digits (1, 2, 3, etc.), taken singly. 2. Numbers above nine are expressed by combining the digits and giving them local values. 3. Naught (O) has no value, but is used to fill a vacant place and to fix the values of the significant figures. 4. Local value increases from right to left, ten units of any order making one unit of the next higher order. EXERCISES. NOTE. Pupils should be carefully drilled in giving the digits their correct forms. Ill-formed figures often lead to erroneous results. 1. Write the Arabic numerals. 2. Write their names. 3. Write the significant figures. 4. Write figures enough to make a period. 5. Write a period and indicate the absence of tens and units. 6. Write two thousand three hundred seventy-five. 7. How many places have you written? How many orders ? How many periods ? 8. How many units of any order make one unit of the next higher order ? 9. Our system of notation puts how many units in a group ? 10. What is a unit? When may ten or a hundred be considered a unit? 11. Write units of the fourth order, and show the absence of units of the first, second, and third orders. 6 PRACTICAL ARITHMETIC 12. Express 1898 in words, remembering what was said about " and." NUMERATION TABLE. 1. Places, orders, and periods may be carried on indefinitely from right to left. 2. The whole subject may now be concisely presented in tabular form. The first six periods are as follows : NAMES OF PERIODS. 1 r^ oT c CO NAMES OP il .2 O PLACES. ' 3 2 3 rs 172 T *- C *? .0 1 S PLACES AND ] ORDERS. - 1 ( oo Hundred -j oo Ten-quac 1 oo Quadrilli 9 S C i .2 g? W H H 888, ^ a 2 S i^?. g r W H P 8 8 V Y PERIODS. GTH. 5TH. 4TH. VI c ci Q 1 oT e cc c Q C rC c oT i H3 1 c o 1 H3 | ^ 1 C W 1 1 C W 1 c H 8 v^_ 8 8, ^_-* 8 v,^ . 8 8, ^.^ 3D 2D 1ST. 3. The periods are separated from each other by commas. 4. The periods from the first to the twenty-second are named as follows : 1. Units. 2. Thousands. 3. Millions. 4. Billions. 5. Trillions. 6. Quadrillions. 7. Quintillions. 8. Sextillions. 9. Septillions. 10. Octillions. 11. Nonillions. 12. Decillions. 13. Undecillions. 14. Duodecillions. 15. Tredecillions. 16. Quatuordecillions. 17. Quindecillions. 18. Sexdecillions. 19. Septendecillions. 20. Octodecillions. 21. Novendecillions. 22. Vigintillions. NOTATION AND NUMEKATION 7 EXERCISES. 1. Write five, fifty-five, five hundred fifty-five, and state what each five expresses. 2. Write a number consisting of four digits, and point off the first period with a comma ; also, read the number. 3. Write a number consisting of two full periods ; name the periods, and read the number. 4. In 876, of what order and place is each figure? 5. In writing nine hundred seven, how will you express the tens ? Write the number. 6. Write a number consisting of three periods ; name the periods, and read the number. 7. Write numbers consisting respectively of four periods, five periods, six periods, and read each of the numbers. 8. Write a number with a significant figure located in the seventh place. What will occupy the other six places ? 9. If a number to be written omits a period, or an order, or a place, what must in all cases supply the vacancy ? 10. Write a number with 7 in the sixth place, 5 in the fourth place, and 2 in the first place. 1 1 . Write a number with five full periods and one partial period. 12. Point off into periods and read 405651320. 13. In the preceding number, how many units of the first N order? How many of the eighth order? How many ten- thousands ? How manv ten-millions ? RULE FOR NUMERATION. 1. Begin at the right and mark off the number into periods. 2. Begin at the left, read each period separately, naming each period except that of units. PRACTICAL ARITHMETIC Copy and read 89. 134. 946. 1664. 5790. 83405. 624151. 731052. 8000000. 10. 763303454. 11. 900058798. 12. 100100001. 1. 2. 3. 4. 5. 6. 7. 8. 9. EXERCISES. 13. 571320179. 14. 35627003. 15. 1000000000. 16. 4321078654. 17. 4141414441. 18. 62340007313. 19. 141662223143. 20. 700706831455. 21. 31671240630231. 22. 1987000634596521612912. 23. 1234567891011121314151. 24. 6171819202122232425262. 1 . "Write the number five hundred twenty-three. Process. Explanation. ANALYSIS. Five hundred twenty-three means five hun- 523 dreds, two tens, three units, represented by the digits 5, 2, and 3. SYNTHESIS. We, therefore, write 3 in the first place, which is the place of simple units, 2 in the second place, which is the place of tens, and 5 in the third place, which is the place of hundreds. 2. Write six hundred thirty-seven thousand one hundred six. Process. Explanation. ANALYSIS. Since the given number contains 637 thou- 637,106 sands and 106 units, there are in it two full periods. SYNTHESIS. We, therefore, write the digits of the thou- sands, 637, as the second period, and the digits of the units, 106, as the first period. NOTATION AND NUMERATION 9 3. Express in figures : 1. Five hundred sixty-four. 2. Seven hundred fifty- nine. 3. Four thousand eighty-one. 4. One thousand two hundred. 5. Twenty-five thousand seven. 6. Forty-one thousand nineteen. 7. Six thousand six hundred six. 8. One hundred thirty-one thousand. 9. Sixty-five thousand four hundred seventy -nine. 10. One million five hundred three thousand five hun- dred ninety-three. 11. Ninety-one million three hundred forty-five thousand. 12. Twelve thousand nine hundred seventy-eight. 13. Thirty- one billion three hundred thirteen million six hundred seventy-two thousand four hundred eleven. 14. One hundred sixty-four million eighteen thousand. 15. One hundred fifteen quadrillion four hundred forty- four trillion five hundred three billion four million two hundred fifty thousand one. DECIMAL PARTS OP A UNIT. 1. By placing a mark (.) called the decimal point after units of the first order, the numeration and notation table is extended to express parts of a unit, on the decimal scale : 5.5 5 5 The above number is thus read : " five and five hundred fifty-five thousandths." 10 PRACTICAL ARITHMETIC 2. The decimal point (.) is always read " and." 6.7 is read " six and seven tenths." 7.89 is read " seven and eighty-nine hundredths." .005 is read "five thousandths." The naughts are only recognized as giving local value to 5. 1.234 is read "one and two hundred thirty-four thou- sandths. 3. The decimal point never acts as a period. EXERCISES. 1. Read : .08, .75, .006, 3.079. 2. Write : Twenty-seven hundredths. Eleven thousandths. One and four hundred six thousandths. Thirteen and twenty- five thousandths. 3. Read: 17.6,1.76, .196, .144. 4. Write : Five hundred six thousandths. Five hundred and six thousandths. 5. Read : 325.72, 325, 32.5, .072. 6. Write : Five hundred four thousandths. Five hundred and four hundredths. 7. Read : 6.050, 7.200, 872.003, .409. 8. Write : Seven tenths four thousandths. Nine and seventy thousandths. 9. Express " and" by a sign. 10. What word interprets the decimal point. 11. What is the difference between a decimal point and a period ? 12. Form a number by writing the digits six, seven, eight, nine, and zero, in their natural order ; then place the decimal point in all the different positions you can ; finally read, in succession, the different numbers you have thus formed. NOTATION AND NUMERATION 11 13. State the effect of moving the point one place to the right ; one place to the left. 14. How many fold does a removal one place increase or diminish the value expressed ? UNITED STATES MONEY. 1. The currency of the United States has the decimal system. Table. 10 mills make 1 cent. 10 cents make 1 dime. 10 dimes make 1 dollar. 10 dollars make 1 eagle. 2. $ is the dollar sign, and, prefixed to an abstract number, renders it concrete : 10 becomes $10, read " ten dollars." 3. The dollar is the Unit, and the decimal point is invaria- bly placed between the dollars and dimes of any sum of money : $5.60 is read " five dollars and sixty cents," or " five dollars and six dimes." EXERCISES. 1. Express in figures nine dollars and twenty-five cents six mills. Process. Explanation. ANALYSTS. Given : nine dollars, twenty-five cents, six $9.256 mills. Twenty-five cents are two dimes and five cents. The dollar is the unit. SYNTHESIS. Write the dollar sign, 9, and a point ; and after the point 2 dimes, 5 cents, and 6 mills, in their natural order. 2. Express in figures thirty-one dollars and nine cents five mills. 12 PEACTICAL ARITHMETIC Process. Explanation. ANALYSIS. Given: thirty-one dollars, no dimes, nine $31.095 cents, five mills. PRINCIPLE. supplies a vacant place. SYNTHESIS. Write dollar sign, 31, and a point; in dimes' place; and 9 cents and 5 mills in their order. 3. Write six dollars and eighty-five cents. 4. Read $2.235, $202.025, $112.25. 5. Write five hundred dollars and nine cents five mills. 6. Write two thousand dollars. 7. Write forty dollars and four cents. 8. Read $313112.13, $20000.32. 9. Write twelve dollars and five cents six mills. 10. Write six thousand one dollars and one mill. 11. Write seven million dollars and seventy-seven cents. 12. Read $.05, $.03, $.62, $.70. 13. Copy and read the following : 1. $8.53. 5. $236.06. 9. $796.844. 2. $13.75. 6. $20000. 10. $.16. 3. $39.05. 7. $2104.083. 11. $.057. 4. $49.34. 8. $6001.102. 12. $12.500. 14. Write the following : 1. Eight dollars and fifty cents. 2. Two hundred two dollars and two cents five mills. 3. Five dollars. 4. Five hundred dollars. 5. One hundred twelve dollars and twenty-five cents. 6. Four dollars and eighty-seven cents. 7. Ninety-seven cents eight mills. 8. Six hundred twenty dollars and nine cents. 9. Twelve million seven hundred thousand dollars. NOTATION AND NUMERATION 10. Three thousand ten dollars and fifty cents. 11. Seventy dollars and ten cents. 12. Six million dollars and eighty cents. 13. Four cents. Ten cents. Nine mills. 13 ROMAN NOTATION. 1. This system of notation employs seven capital letters. I. denotes one, 1. V. denotes five, 5. X. denotes ten, 10. L. denotes fifty, 50. Table. C. denotes one hundred, 100. D. denotes five hundred, 500. M. denotes one thousand, 1,000. M. denotes one million, 1,000,000. 2. All other numbers are expressed by combining or repeat- ing these letters : I. . . 1. XIV. . .... 14. C 100. II.. . 2. XV. . . .... 15. CCCC.,orCD. . 400. III. . ... .3. XVI. . .... 16. D 500. IV. . 4. XVII. . .... 17. DCCCC.,or CM. 900. V. . . 5. XVIII 18 Ml 000 VI. . 6. XIX. . .... 19. MD 1500. VII. . 7. XX. . . .... 20. MDCLXV. . . . 1665. VIII. 8. XXI. . . . 'L . 21. MDCCXLIX. . . 1749. IX. . 9. XXX. . .... 30. MDCCCLXXIX, 1879. X. . 10. XL. . . .... 40. V. 5,000. XI. . . . 11. L. . . . .... 50. L 50,000. XII. . 12. LX. . . .... 60. C 100,000. XIII. 13. XC. . . . . . .90. M 1,000,000. From the repetitions and combinations observable above, we derive the following 14 PRACTICAL ARITHMETIC PRINCIPLES. 1. Repeating I., X., C., or M. repeats its value. XX. denotes 20; CO. denotes 200. V., L., and D. cannot be thus repeated. 2. Prefixing I., X., or O. to a letter of greater value diminishes that value by L, X., or C. 3. Afllxing L, V., X., L., C., or D. to a letter of greater value increases that value by L, V., X., L., C., or D. 4. Inserting L, X., or C. between two letters,* each of greater value, diminishes the united value of the two by I., X., or O. * The first of the two must not be of less value than the second. XIV., not VIX., denotes 14; XIX. denotes 19. 5. A bar placed over a letter, except I., increases its value a thousand-fold. C. denotes 100,000. 6. IIII. is sometimes used instead of IV., as on the dials of clocks and watches. 4OO may be expressed by COCO, or by CD. EXERCISES. 1. Read the following combinations : 1. XV. 10. XLV. 19. DCCXC. 2. IV. 11. XCIX. 20. MXXIX. 3. XIV. 12. LXV. 21. VDLV. 4. XXIV. 13. CIX. 22. DLDC. 5. XIX. 14. CXI. 23. CCXDVL 6. XXXIX. 15. XCI. 24. VIII. 7. XXXIII. 16. DCXC. 25. CCXC. 8. XXIX. 17. CCCXXXIX. 26. CXLIX. 9. XLIX. 18. DCCXXXIV. 27. HMD. 28. LXXDCCCXCIX. 29. MDXCVDCCCLXIV. NOTATION AND NUMEKATION 15 2. Write in Roman characters the following 1. 2. 3. 4. 5. 6. 15. 36. 87. 56. 49. 99. 7. 1050. 8. 5010. 9. 789. 10. 1898. 11. 18. 21. 27. 12. 42. 22. 81. 13. 66. 23. 95. 14. 86. 24. 40. 15. 63. 25. 45. 16. 100. 26. 534. 17. 3600. 27. 5000. 18. 587. 28. 436. 19. 207. 29. 999. 30. 76,959. 20. 8004. 3. Which of these are correct expressions and which in- correct ? DD. CCC. XCC. MMXL. XLIX. CIXXVII. W. LV. XIL. LXIX. XLX. XCIX. VXX. VLC. VDC. CIXYIIX. LXXXVIIL DMCC. REVIEW'. 1. Define the following terms : 1. Unit. 2. Number. 3. Arithmetic. 4. Principle. 5. Concrete number. 6. Abstract number. 7. Analysis. 8. Synthesis. 9. Eule. 10. Problem. 11. Solution. 12. Notation. 13. Numeration. 14. Word method. 15. Arabic method. 16. Roman method. 17. Simple value. 18. Local value. 19. Zero. 20. Period. 16 PRACTICAL ARITHMETIC 21. Decimal point. 25. Dime. 22. United States Money. 26. Dollar. 23. Mill. 27. Eagle. 24. Cent. 28. Significant figures. 2. Repeat the four principles of notation. 3. Name the periods from the 1st to the 22d. 4. Repeat the five principles of the Roman notation. 5. Repeat the rule for numeration. ADDITION. INDUCTIVE STEPS. 1. How many units are 5 units and 3 units? 2 tens and 7 tens ? 4 thousands and 6 thousands ? 2. A certain field has 7 acres, and an adjoining field 8 acres. How many acres in both fields ? Process. Explanation. 7 acres Since one field contains 7 acres and the other 8 acres, the 8 two fields contain 7 acres and 8 acres, which are 15 acres, acres 15 acres . If on one shelf there are 9 books and on another shelf 5 books, how many books are on both shelves ? "Write and explain the process. 4. How many pounds are 8 pounds and 6 pounds ? Write and explain. 5. The process of thus uniting quantities in a single quan- tity is called adding. 6. What is the unit of 8 pounds? Of 6 pounds? Of 14 pounds ? 7. Can quantities having like units be added ? ADDITION 17 8. Add 6 pounds and 5 dollars. Can you show a process? If you can, is your result 11 pounds or 11 dollars? 9. What, then, does addition require as to the units to be added ? 10. What does addition require as to the unit of the result or sum ? 11. Numbers having like units are called Like Numbers. DEFINITIONS. 1. Addition is the process of finding the sum of two or more like numbers. The sum is, therefore, the result of addition. 2. A Sign indicates some process or condition. The sign of addition is an upright cross, -{-. It is read " plus." 3. The Sign of Equality is two short horizontal lines, =. It is read " equals," or " is equal to." 3 -f 2 = 5, is read " 3 plus 2 equals 5." 3 + 2 = 5, being an expression of equality, is called an Equation. PRINCIPLES. 1. Only like numbers and orders can be added. 2. The numbers added and their sum are like numbers. EXERCISES. 1. Find the sum of 120, 331, and 478. Process. Explanation. 120 ANALYSIS. There are three numbers to be added, each con- oo-i taining units, tens, and hundreds. ^ PRINCIPLE. Only like orders can be added. SYNTHESIS. Hence we write the numbers with the units' 929 figures (0, 1, 8) in the first column on the right, the tens' figures (2, 3, 7) in the second column, and the hundreds' figures (1, 3, 4) in the third column. The sum of the first column is 9 units ; the sum of the second column is 12 tens = 1 hundred -[- 2 tens. We write the 2 tens, and add the 1 hundred to the hundreds' column, making 9 hundreds. Hence the sum required is 929. 2 18 PRACTICAL ARITHMETIC 2. Find the sum of 25, 206, and 9837. Process. Explanation. 25 ANALYSIS. 206 25 = 2 tens -f 5 units. 9 837 206 = 2 hundreds -f tens -f 6 units. 9,837 = 9 thousands -f 8 hundreds 4- 3 tens -4- 7 units. -i r\ AQ 9 PRINCIPLE. Only like orders can be added. SYNTHESIS. We, therefore, write the units' figures (5, 6, 7) in the first column; the tens' figures (2, 0, 3) in the second column; the hundreds' figures (2 and 8) in the third column ; and the 9 thousands alone in the fourth place. Adding the first column we have 18 units = 1 ten and 8 units. We add the 1 ten to the tens' column and have 6 tens. Adding the third column we have 10 hundreds = 1 thousand and hundreds. We say finally 1 thousand -j- 9 thousand = 10 thousand. RULE FOB ADDITION. 1. See that the numbers to be added are like numbers. 2. "Write units of the same order in the same column. 3. Begin at units' column, and find the sum of each column separately. 4. Write the units of a sum, but add the tens with the next column. 5. "Write the entire sum of the last column. EXERCISES. 1. Find the sum of: (1.) (2.) (3.) (4.) (5.) 234 134 712 473 535 365 542 314 321 213 (7.) (8.) (9.) $6.10 $31.12 $231.25 $2.11 $41.23 $542.30 $1.34 $20.44 $210.44 ADDITION 19 (10.) (11.) (12.) (13.) 4134 2460 3782 469 8104 3782 1856 7206 3910 3673 1916 39 45 418 3061 6 2. What is the sum of 2213, 1123, 3201, 2112? 3. What is the sum of: 1. 3210 -f- 2136 + 3752 + 2331 ? 2. 3561 -|- 5103 + 6385 + 5632? 3. 73,250 + 3102 + 16,287 + 1210 + 7542? 4. 50,673 + 520 + 16,302 -f 2531 + 7204? 5. 154,632 + 54,231 + 16,302* + 2120 + 8023? 4. Find the sum of: (1.) (2.) (3.) (4.) (5.) (6.) 1. 888 + 777 + 666 + 555 + 543 + 735 = ? 2. 444 + 333 + 222 + 111 + 210 + 141 =? 3. 000 + 999 + 234 + 423 + 924 + 287 = ? 4. 578 + 287 + 342 + 760 + 553 + 765 = ? 5. 504 + 167 + 359 + 578 + 751 + 432 = ? 6. 105 + 483 + 142 + 263 + 351 + 109 = ? Add the foregoing both vertically and horizontally. 5. What is the sum of 37 + 375 + 3754 + 37,546 + 64 + 645 + 4573 -f 57,373? 6. Add $317.50, $610.10, $514.085, $6.16. 7. What is the sum of four hundred sixty-two, three thousand two hundred fourteen, seventy-nine thousand six hundred fifty-nine, two hundred eighty-four? 8. What is the sum of eighteen dollars and five cents, fifty- one dollars, fifty-one cents, ten dollars and ten cents, eighteen dollars and twenty-four cents, thirty-five dollars? 20 PRACTICAL ARITHMETIC 9. Write the following numbers with Arabic numerals and find their sum: DCCCCXXXVL, MDXVL, MMMMCCIV., CLIV., XCVIL, CLXIX. 10. 4682 + 19,783 + 100 + 6402 + 178 + 19 = ? PROBLEMS. Let the pupil first indicate the solution of each problem by using the signs, -(- and =. 1. A. owns 345 sheep, B. owns 295, C. owns 436, and D. owns 524. How many sheep do all own ? Process Indicated. 345 -j- 295 -1- 486 -f 524 = number of sheep required. Process. Explanation. 345 ANALYSIS. There are four flocks of sheep : A.'s = 345. 295 B -' s = 295 ' C.'s=436. D.'s == 524. 524 In each number the unit is 1 sheep ; hence the numbers 1600 are &&e and may be added. SYNTHESIS. 345 + 295 -f 436 -f 524 = 1600. Hence all own 1600 sheep. 2. How many acres are in three fields, containing respect- ively 23 acres, 34 acres, and 38 acres ? 3. A man bought a horse for $250, a carriage for $175, a harness for $74.50, a whip for $1.25, a carriage blanket for $3.45. What did he pay for all ? 4. A. bought 7590 pounds of pea coal, 3765 pounds of nut coal, 6834 pounds of stove coal, and 2505 pounds of bituminous coal. How much coal did he purchase? 5. In a primary school there are 386 children in first grade, 258 in second grade, 237 in third grade, and 184 in fourth grade. How many pupils in the four grades ? ADDITION 21 6. Spain has an area of 195,773 square miles; France, 204,091; Switzerland, 15,922; Italy, 112,622. How great is the area of the four countries ? 7. The battle-ship "Oregon" sailed from San Francisco to Callao, 4,012 miles; from Callao to Sandy Point, 2,666 miles ; from Sandy Point to Eio, 2,228 miles ; from Rio to Bahia, 745 miles ; from Bahia to Barbadoes, 2550 miles ; from Barbadoes to St. Thomas, 346 miles ; from St. Thomas to Key West, 1040 miles. Find the total number of miles she sailed. 8. The monthly pay of a major-general in the United States army is $625 ; of a brigadier-general, $458.33 ; of a colonel, $291.67; of a lieutenant-colonel, $250; of a major, $208.33; of a captain, mounted, $166.67; of a captain, not mounted, $150; of a chaplain, $125. Find total monthly pay of the eight officers. 9. In 1897 the organized military strength of the State of New York was 13,894 men ; of Pennsylvania, 8521 ; of Illinois, 6260; of Ohio, 6004; of Massachusetts, 5154; of New Jersey, 4297 ; of California, 3909 ; of Georgia, 4450 ; of South Carolina, 3127; of Texas, 3023. What was the entire military strength of the ten States in 1897? 10. In 1890 the population of Cincinnati was 216,239 ; of Cleveland, 92,829 ; of Toledo, 31,584 ; of Columbus, 31,274 ; of Dayton, 30,473. How many inhabitants had these cities altogether in 1890? 11. A merchant received money for goods as follows : On Monday, $357.15; on Tuesday, $463.87; on Wednesday, $279.19; on Thursday, $318.67; on Friday, $687.27; on Saturday, $348.48. Find the total receipts. 12. A builder bought a lot for $650, built upon it a house costing $5845, a barn and carriage-house costing $1075.50; he paid for fencing $215.75, for grading $87.50. For what must he sell the property to gain $640 ? 22 PRACTICAL ARITHMETIC 13. The provinces of Cuba, with the population of each, are as follows : Province. White. Colored. Havana 344,417 107,511 PinardelRio. . . . 167,160 58,731 Matanzas ..... 153,169 116,401 Santa Clara .... 249,345 109,777 Puerto Principe . . . 54,232 13,557 Santiago de Cuba . . 157,980 114,339 Find the total population of Cuba. 14. The land forces of Japan are as follows : infantry, fifty- six thousand thirty-seven ; cavalry, VDCCLX ; artillery, seven thousand 818 ; engineers and train, IVCCCXXVI. What is the total land force ? 15. January has 31 days, February 28, March 31, April 30, May 3t, June 30, July 31, August 31, September 30, October 31, November 30, December 31. How many days in a year ? 16. If a school session closes on the 29th of June and opens again on the 10th of September, how many days' vaca- tion will there be ? 17. What are the expenses of a factory for a year, if the manager receives $1850, the engineer $850, the fireman $650, the bookkeeper $800, the fuel costs $1600, the raw material $111,110, and the pay-roll of the other employees amounts to $55,000 ? 18. Add 2, 6, 8, 7, 1, 2, 8, 5, 3, 2, 8, 9. 19. Add IV., VIL, II., V., V., II, IX., VI., II., VIII., VII., V. 20. A capitalist made the following deposits in a bank : August 4, 1897, $484.50; August 7, $985.25; August 10, $436.75. In a second bank as follows : August 14, $2657.76 ; ADDITION 23 August 18, $1386.25; August 22, $2096.65. How much did he deposit in each bank ? How much in both banks ? (1.) (2.) (3.) (4-) 21. 7651 5005 10,475 $453.48 8923 4567 72,482 4,938.78 4554 2299 46,552 85,473.89 5421 9900 62,651 3,457.96 6432 8877 62,272 835.47 9888 7788 67,286 53.49 8797 6655 40,025 9.87 5032 5566 82,827 82.75 8060 4433 40,050 875.39 2134 3344 24,165 48.34 (5.) (6.) (7-) (8.) 8410 6546 4828 595 9836 3210 3424 579 984 785 293 8574 543 156 788 3250 9758 5634 2763 386 8574 7654 5612 984 451 696 942 5849 876 321 397 6546 7864 3288 5945 429 5849 2188 8020 451 762 785 694 8765 321 564 432 5634 3250 7688 6131 543 8765 5861 9876 762 688 978 750 3210 642 643 976 3288 24 PRACTICAL ARITHMETIC (9.) (10.) (11.) .85 $901.09 .3789 463.27 91.85 .7398 39.99 387.24 4.217 1.58 19,877.46 3.95 6,598.86 19.90 45.007 9,005.79 104.99 4.256 95,783.04 3,972.87 3.520 2,469.98 79,841.24 23.3 956.83 18.72 29.317 14,816.00 3,120.50 343.28 3,947.25 14.12 1899. REVIEW. 1. Define the following terms : 1. Like numbers. 5. Sign of Addition. 2. Unlike numbers. 6. Equation. 3. Addition. 7. Indicated process. 4. Sum. 8. Process. 2. Repeat the principles of Addition. 3. Repeat the rule for Addition. 4. Invent five problems in Addition and indicate their solution. SUBTRACTION. INDUCTIVE STEPS. 1. How many are 6 units less 3 units ? 7 tens less 5 tens ? 8 millions less 4 millions ? 2. If you have $9 and spend $5, how many dollars do you retain? SUBTRACTION 25 Process. Solution. $9 If I have $9 and spend $5, I retain the difference between AK $9 and $5, which is $4. $4 Is that explanation analytical or synthetical ? 3. $5 -f $4 = how many dollars? 4. Was it analysis or synthesis that gave you the $9 ? 5. Does the synthesis, then, prove the correctness of the analysis ? 6. Robert is 10 years of age and Richard is 8. What is the difference of their ages ? Write and explain the process. Prove the correctness of the result. 7. There were 7 bunches of ripe grapes on a vine; a fox took 2 bunches. How many bunches remained ? 8. Have you been finding the difference between like numbers ? 9. Finding the difference between two numbers is called Subtracting. 10. What is the difference between 6 horses and 3 sheep ? 11. Subtraction of numbers makes what requirement as to their units ? DEFINITIONS. 1. Subtraction is the process of finding the difference between two like numbers. 2. The greater number is called the Minuend; the less number is called the Subtrahend; the result is called the Difference or Remainder. 3. The Sign of Subtraction is a short horizontal line, , called minus (less), and is always placed after the minuend and before the subtrahend. 7 5 = 2 is read " 7 minus 5 equals 2." The form, 7 5 = 2, is called what? 26 PRACTICAL ARITHMETIC PRINCIPLES. 1. Only like numbers and orders can be subtracted. 2. Subtrahend -f- Remainder = Minuend. 1. From 54 subtract 33. Process. Explanation. 54 The minuend, 54 = 5 tens -|- 4 units ; 00 the subtrahend, 33 = 8 tens -f- 3 units. 2 tens -f 1 unit = 21. Z* \ PRINCIPLE. Only like orders can be subtracted. We therefore write the 3 units under the 4 units and the 3 tens under the 5 tens. We then say " 4 units 3 units = 1 unit ; 5 tens 3 tens = 2 tens. Hence the remainder is 21." Proof. PRINCIPLE. The subtrahend -f- the remainder = the minuend. 33 + 21 = 54. 2. From 469 subtract 327. Process. Explanation. 469 469 = 4 hundreds -(- 6 tens -|- 9 units. 007 327 = 3 hundreds + 2 tens + 7 units. 1 hundred -f 4 tens -f- 2 units = 142. PRINCIPLE. Only like orders can be subtracted. We therefore write the 4, 6, and 9 of the minuend, and under them the 3, 2, and 7 of the subtrahend, with units under units, tens under tens, and hundreds under hundreds. We now say " 9 units 7 units = 2 units; 6 tens 2 tens = 4 tens ; 4 hundreds 3 hundreds = 1 hundred. Hence the difference is 142." Show proof. EXERCISES. Copy, subtract, explain, prove : (1.) (2.) (3.) . (4.) (5.) (6.) 824 569 997 965 896 8953 413 245 743 752 544 3420 SUBTRACTION 27 (70 (8.) (9.) (10.) $59.86 $75.39 $56.89 $52.90 $34.24 $40.30 $45.76 $31.50 (11.) (12.) (13.) (14.) 62,979 98,316 945.791 $798.945 30,825 71,004 523.150 $653.620 PROBLEMS. NOTE. Let the pupil first indicate the solution of each problem by using the minus sign, . 1. An army went into battle with 6878 men, and came out with only 4345 men. How many men were missing? Process Indicated. 6878 men 4345 men the number missing. Process. Explanation. 6878 ! Since the army went into battle with 6878 men and 4345 came out with only 4345, the number missing was 6878 . minus 4345. 2533 2. Since the unit of both the numbers is one man, the numbers are like and can be subtracted, the less from the greater; units from units, tens from tens, etc. Therefore we say " 8 uniti 5 units = 3 units, 7 tens 4 tens = 3 tens, 8 hundreds 3 hundreds = 5 hundreds, 6 thousands 4 thousands = 2 thousands. Hence the number of men missing was 2533." Proof. PRINCIPLE. Subtrahend - -j- Remainder = Minuend. 4345 + 2533 = 6878. 2. A grain dealer, having 7890 bushels of wheat, sold 6370 bushels. How many bushels had he remaining ? 28 PRACTICAL ARITHMETIC Process Indicated. 7890 bushels 6370 bushels = the bushels remaining. 3. Watches were invented at Nuremburg in 1477. How many years ago ? 4. If I borrow $6798, and afterwards pay $3534, how much do I still owe? 5. Under a call for volunteers, California's quota was 3237 men; Arkansas's quota, 2025 men. Find the difference? 6. The population of Spain in 1820 was about 11,000,000 ; at present (1899) it is 17,550,216. Find the increase. 7. The exports of the United States from the Philippine Islands last year amounted to $4,982,857 ; their imports, to $162,446. Find the excess of the exports over the imports. 8. The population of Havana is 198,720, of Santiago, 71,300. Find the difference. 9. The telescope was invented in 1610. How many years between that date and 1899 ? 10. Harvey discovered the circulation of the blood in 1619. How many years after the invention of the telescope ? CHIEF DIFFICULTY OF SUBTRACTION. 1. From 594 take 368. Process. Explanation. 594 ANALYTIC AND SYNTHETIC. 368 594 = 5 hundreds -f 9 tens -f 4 units. 368 = 3 hundreds -f 6 tens -f- 8 units. The difficulty is that 8 units cannot he taken from 4 units. But one of the 9 tens = 10 units ; 10 units -f- 4 units = 14 units. Hence we write : {QfvS 594 = & hundreds + 8 tens + 14 units 1 Subtract . 368 = 3 hundreds -f 6 tens + 8 units / 594 ing we have 2 hundreds -f- 2 tens -f- 6 units = 226. SUBTKACTION 29 2. What is the first principle of subtraction ? 3. On what principle does the proof depend? 4. From 703 take 549. Process. Explanation. e 9 is 703 = 7 hundreds -f tens -f- 3 units 703 549 _ 5 hundreds + 4 tens -f 9 units 549 The difficulty is that we cannot take 9 units from 3 units, nor 4 tens f rom o tens. But one of the 7 hundreds = 10 tens ; one of the 10 tens = 10 units ; 10 units -j- 3 units = 13 units. Hence we may write 703 = 6 hundreds + 9 tens + 13 units j gubtracti we 549 = 5 hundreds -f 4 tens -f 9 units J have 1 hundred -f- 5 tens -[- 4 units = 154. 5. Give the principles of subtraction and prove the work. 6. From 367.280 take 298.356. Process. Explanation. 367.280 367.280 = 367 units + 280 thousandths 1 gub 298.356 298 356 = 298 units + 356 thousandths / 68.924 tracting we have 68 units -f 924 thousandths = 68.924. 7. Where must the point always be placed in the remainder ? RULE FOR SUBTRACTION. 1. See that the numbers to be subtracted are like numbers. 2. Write the subtrahend under the minuend, units under units, etc. 3. Beginning at the right, subtract each lower figure from the one above it. 4. When necessary, increase the upper figure by 1O and diminish by 1 the next upper figure on the left. 30 PKACTICAL ARITHMETIC EXERCISES. 1. Copy, subtract, explain, prove : (1.) (20 864 1095 559 867 (3.) 937 645 (4.) (5.) 865 2537 593 658 (6.) 954 893 (7.) (8.) 2957 -2794 1038 2406 (9.) 3908 2609 (10.) (11.) 4002 8923 3962 2095 (12.) 9114 6983 (13.) (14.) $35.56 $40.19 $32.49 $38.02 (15.) $58.19 (16.) (17.) $82.99 $53.44 $58.03 $19.78 (18.) $6.12 $5.125 (19.) 618.724 9 529.728 8 (20.) ,651,782 ,241,509 (21.) 69,503.48 38,298.75 (22.) 8888.88 7890.10 2. What is the value of: 1. 81,214 53,467? 2. 104,321 58,461 ? 3. 831,408 337,529? 4. 740,037 357,320? 5. 862,493 - 6. 998,765 7. 9,327,325 8. 4,986,384 729,603? 567,890? 3,586,143? 2,998,796? PROBLEMS. 1. If a man owes $97.66 and pays $70.89, how much does he then owe ? Process Indicated. $97,66 $77.89 = how much he then owes. SUBTKACTION 31 Process. Explanation. $97 66 Since he owes $97.66, and pays $70.89, lie still owes the r-n'oq difl'erence between $97.66 and $70.89. Since the numbers have the same unit, one dollar, they $26.77 are like numbers, and their difference can be found. It is $26.77. NOTE. In each problem let the process be indicated first, and then performed and explained. 2. A man bought some land for $8765, and sold it for $1 0,890. What was his gain ? 3. The first line of telegraph was established in the United States in 1844. How long ago? 4. In 1890 the population of the United States was 62,622,250, and in 1840 it was 17,063,353. How much did it increase in the 50 years ? 5. A man was born in 1785 : what was his age in 1830? 6. How old was George Washington at the time of his death? He was born in 1732, and died in 1799. 7. 29,400 feet is the greatest depth of water measured. 37,000 feet is the greatest height reached by a balloon. Find by how much the greatest height reached exceeds the greatest depth reached. 8. The displacement of the battle-ship "Alabama" is 11,525; of the cruiser " Charleston," 3730. How much does the displacement of the " Alabama" exceed that of the " Charleston" ? 9. The estimated population of the United States in 1800 was 5,308,483 ; in 1898 it was 74,500,000. Find the growth in population in the 98 years. 10. How many dollars must be added to $4872 to make $8021 ? 11. How many dollars increased by $74,015 make a million dollars? 32 PRACTICAL ARITHMETIC 12. What number must be taken from $6412 to leave $5366 ? 13. Find the value of $8.052 $3.687. 14. John has $20.19 and James has $40. How much more money has James than John ? 15. A bankrupt has $6456 assets, and owes $33,860. How much more does he owe than he can pay ? 16. Mt. Everest is 29,062 feet high; Mt. Whitney is 14,900 feet high. How much higher is the former than the latter? 17. The height of Mt. Cenis, an Alpine peak, is 11,792 feet ; the height of the pass over it is 6884 feet. How much higher is the mountain than the pass ? 18. At an election 3245 persons voted, and the candidate elected received 1808 votes. How many did the defeated candidate receive ? 19. Benjamin Franklin died in 1790, and was 84 years old at his death. When was he born ? 20. Subtract MMMIX. from LXVIIXI. ADDITION AND SUBTRACTION IN COMBINATION. EXERCISES. 1. What is the value of 20,324 -f 4756 13,186? Process. Explanation. 20 324 ANALYTIC AND SYNTHETIC. 4 755 The sign -f signifies that I must add 20,324 and 4756. Adding, the sum is 25,080. 25,080 The sign signifies that I must subtract from that sum 13,186 13,186. Subtracting, the remainder is 11,894. 11,894 2 what ig the yalue of . 1. 23,732 9478 -f 9273 ? 2. 25,657 -f 10,898 2597 ? SUBTRACTION 33 3. 20,201 9022 + 2002 ? 4. 132,571 90,798 + 78,318 ? 5. $238.70 $53.36 + $22.27 ? A Parenthesis, ( ), or Vinculum, , Indicates that all the quantities it incloses are to be considered as a single quantity ; as, (2 + 5 + 10 + 13), or 2 + 5 + 10 + 13. 2. What is an Equation ? 3. Prove the following equations to be correct : First perform the operations indicated within the parentheses. 1. 40 (2 + 5 + 10 + 13) = 10. 2. (355 + 637 + 403) 977 418. 3. 2543 504 + 600 + 725 = 714. 4. 10,000 (275 + 220 + 35 + 3675) = 5795. 5. (300 + 100 + 95 + 60 + 125) 125 + 25 + 40 490. 4. Complete the following partial equations : 1. (350,000 + 225,100 + 4000 + 96,000) 450,120 2. 23,191,876 3,204,313 + 434,495 = ? 3. (367 + 875 + 1012) 423 + 912 = ? 4. (36 + 200 + 150) 331 = ? PROBLEMS. NOTE. The indicating of a solution often facilitates the completing of it. The pupil should be faithfully drilled in the use of signs to indicate the actual solution to be made. 1. Mr. A. gave his note for $6000. He paid at one time $3586 and at another time $2000. How much remained to be paid ? Process Indicated. $6000 ($3586 -f $2000) = debt remaining. 34 PRACTICAL ARITHMETIC Process. Explanation. Paid $3586 Since he paid $3586 at one time and Paid 2000 $2000 at another time, he paid at both times $3586 + $2000, or $5586. Total paid . 5586 Since his note, or debt, was $6000, he still owes the difference between $6000 and Note $6000 15586, or $414. Paid 5586 2. A man finished a journey of Balance due, $414 972 m il es in 3 days; the first day he travelled 398 miles; the second day, 409 miles. How many miles did he travel the third day ? Process Indicated. 972 miles (398 miles -f 409 miles) = miles travelled third day. 3. A produce dealer had in bank $6032, and checked out on one day $2360, and on the next day, $2307. How much had he left in bank ? 4. A cattle dealer had 982 cattle, bought 621 more, lost by disease 32, sold 416. How many remained? 5. How much does the sum of 3694 and 5005 exceed the difference of 10,532 and 3903? 6. In 1880 there were 16,120 Indians and 75,025 Chinese in California. How many were there of both, and how many more Chinese than Indians ? 7. A. sells a house to B. for $3486 ; B. sells it to C. at a gain of $360 ; C. sells it to D. at a loss of $285. What does D. pay for the house ? 8. I have a yearly income of $10,000. I pay $450 for rent, $230 for fuel, $50 for medical attendance, and $4786 for all my other expenses. What have I saved at the end of the year ? 9. A tract of land containing 2753 acres was divided among four persons, A., B., C., D. A.'s share was 679 acres, SUBTRACTION 35 B.'s was 47 acres more, C. had 75 acres less than B., and D. had the remainder. What were the shares of B., C., and D. ? 10. A man has an income of $1845 ; he spends $645 for board, $456 for clothing, and $297 for other expenses. What has he saved at the end of the year ? 11. A man deposits in bank $2374. At one time he draws out $897, at another, $543, and at a third time, $689. How much has he remaining in bank? 12. I bought 24 shares of bank stock for $2863, and paid a broker $22 for purchasing the same; afterwards sold it for $3000. What was my profit? 13. A farmer invests $18,975 as follows : in land, $11,893 ; in horses, $1575; in mules, $4297; in stock, $937; the remainder in tools. How much did he expend for tools? 14. A man bought three houses; for the first he gave $3585 ; for the second, $5260 ; for the third, as much as for the other two. He sold them all for $15,280. Did he gain or lose, and how much ? 15. From the sum of 874 and 398 subtract their difference. 16. I have a bin that holds 936 bushels. I put into it 383 bushels, and again 457 bushels. How much more will the bin hold? 17. Some excursionists made a journey that cost them $492.97. Railroad fares cost $203.26 ; hack hire cost $48.36 ; steamboat fare cost $72.46. The remainder was expended for food. What did their food cost? 18. A merchant bought silk for $486, muslin for $286, linen for $346, and sold the whole for $1200. How much did he gain ? 19. An estate worth $23,460 was bequeathed to a wife and two children. The widow received $7820 ; the son received $3400 less ; and the daughter, the balance. Find the daughter's share. 36 PKACTICAL ARITHMETIC REVIEW. 1. Define the following terms : 1. Minuend. 6. Analysis. 2. Subtrahend. 7. Synthesis. 3. Difference 8. Solution. 4. Remainder. 9. Parenthesis. 5. Minus. 10. Vinculum. 2. Repeat the principles of Subtraction. 3. Repeat the rule for Subtraction. 4. Invent five problems involving both Addition and Sub- traction, and indicate the process of solution. MULTIPLICATION. INDUCTIVE STEPS. 1. How many are 3 + 3 ? 2. How many are 3 -f 3 + 3 ? 3. How many are two times 3 ? 4. How many are three times 3 ? 5. 3 -f 3 -f- 3 = 9, and 3 times 3 = 9. The first operation is addition ; the second is multiplication. Which is the shorter ? 6. Is Multiplication a short kind of addition ? 7. What will 4 apples cost at 2 cents apiece ? Process. Explanation. ADDITION. MULTIPLICATION. Since 1 apple costs 2 cents, 4 apples will cost 4 times 2 cents, which are 8 2 cents cents. In the preceding process, is 2cente 2 cents concrete? Is 8 cents 2 cents J^ concrete? Is the 4 concrete or 8 cents 8 cents abstract? MULTIPLICATION 37 8. What i's the cost of 5 yards of ribbon at 10 cents a yard ? Write and explain the process. In the preceding process, did you write 5 yards or simply 5 ? What kind of number is 5 ? You took 10 cents five times ; hence 5 is called the Multi- plier, and the process is called Multiplication. DEFINITIONS. 1. Multiplication is a short process of finding the sum of two or more equal numbers ; or of taking a number as many times as there are units in another number. 2. The Multiplicand is the number to be taken or repeated. 3. The Multiplier is the number which shows how many times the multiplicand is to be taken or repeated. 4. The Product is the result of the multiplication. 5. The Multiplicand and Multiplier are called Factors of the product. 6. The Sign of Multiplication is an oblique cross, X- 5 x 4 == 20, is read " 5 times 4 are 20" ; or, " 5 multiplied by 4 equals 20." 7. What are the factors of 20 ? Is 4x5 5x4a correct equation ? Proof. ***** ***** ***** ***** The 20 stars as arranged equal 5 stars in a line taken 4 times, or 4 stars in a column taken 5 times. Hence the product is the same in whatever order the factors are taken. 38 PRACTICAL ARITHMETIC 8. What is a parenthesis and what does it signify? A vinculum ? 134 - (9 + 6) X 3 signifies that you must take 9 + 6 three times and subtract the result from 134. Process. (9 + 6) X 3 = 45; 134 45 = 89. PRINCIPLES. 1. The multiplier must be considered an abstract number. 2. The product and multiplicand are like numbers. 3. Either factor may be taken as the multiplier. EXERCISES. FOR ANALYTIC AND SYNTHETIC EXPLANATION. 1. What is the product of 347 multiplied by 3? Process. Profess. Process. Explanation. 347 IST PROCESS. "We say, "Apro- o duct is the result of multiplication. Since multiplication is a short process 347 21 of adding equal numbers, we can find 347 1 2 347 ^ e P r duct by addition ; adding, we 347 9 3 have 1041 -" 2o PROCESS. We say, " Since 1041 1041 1041 34 7 must be taken 3 times, each order of units must be taken 3 times. 3 times 7 units = 21 units ; 3 times 4 tens = 12 tens ; 3 times 3 hundreds = 9 hundreds; adding, we have 1041." 3D PROCESS. The shortest process is generally the best in practice. We say, "3 times 7 = 21 ; we reserve the 2; 3 times 4 = 12, 12 and 2 reserved are 14 ; we reserve the 1 ; 3 times 3 = 9, 9 and 1 reserved 10. Hence the product is 1041." Proof. The first and second processes are proof of the accuracy of the third. MULTIPLICATION 39 2. Complete, explain, and prove the following : (1.) (2.) (3.) (4.) (5.) 365 674 756 327 408 24365 (6) (70 (8.) (9.) (10.) $5.09 $7.95 $12.75 $55.06 $43.60 5 8 7 9 12 3. How many places in the product must be pointed off for cents ? 4. Multiply : 1. 8692 by 8. 6. 13,896 by 3. 2. 5328 by 7. 7. 52,209 by 5. 3. 10,318 by 5. 8. 68,387 by 7. 4. 7289 by 9. 9. 79,588 by 4. 5. 17,345 by 4. 10. 91,983 by 9. 5. Find the value of: 1. 420 x 9. 5. 40,527 X 4. 2. 9059 x 2. 6. 305,238 X 5. 3. 78,059 X 3. 7. 40,597 X 6. 4. 1,790,478 x 7. 8. 910,362 X 8. The parts of an equation, right and left of the sign of equality, are called its members. 6. Find a second member for each of the following : 1. 127 + (2 + 8) X 9 + 85 = 2. 209 (27 + 4) X 5 = 3. 3300 -f 86 X 6 + 4 = 4. 3246 329 -f 524 x 3 = 5. 9203 6 X (350 239) = 6. (275 + 262) X 3 2 X (68 39) = 7. 1935 195 + 186 X 4 = 40 PRACTICAL ARITHMETIC PROBLEMS. Let the pupil indicate the solution of each problem by using the sign X . 1. If sound moves 1092 feet in a second, how far does it move in 5 seconds ? Process Indicated. 1092 X 5 = how far it moves. Process. Explanation. 1092 feet since sound moves 1092 feet in 1 second, in 5 seconds it ^ moves 5 times 1092 feet ; 1092 feet X 5 = 5460 feet. 5460 feet What principles are involved ? 2. When wheat is worth $1.25 per bushel, what is the value of 9 bushels ? 3. What will 7 horses cost at $175.35 each? 4. Since in 1 mile there are 1760 yards, how many yards are there in 9 miles ? 5. Find the cost of 4886 sheep at $6 a head. 6. Each workman in an iron-foundry is paid $605 a year : what do 1 1 men receive at that rate ? 7. A bushel of corn weighs 56 pounds : find the weight of 12 bushels. 8. The distance to the moon is 240,000 miles : what is 10 times that distance ? 9. If the distance of the earth from the sun is about 91,430,000 miles, how many miles is 9 times that distance? 10. If an army major receives monthly $208.33, what is the monthly pay of 1 2 majors ? 1 1 . Mr. White owns 3 houses, and the first house is worth $3872 ; the second, 3 times as much ; and the third, 7 times as much. Find the cost of the 3 houses. Process Indicated. 3872 + (3872 X 3) + (3872 X 7) = cost required. MULTIPLICATION 41 12. A farmer's wife took to a store 3 pounds of butter worth 33 cents a pound, and bought 12 yards of calico at $.08 a yard. Find the balance due her. 13. Which are worth more, 7 cows at $35 apiece or 3 horses at $75 apiece? 14. A lady bought a bicycle for $100 ; she rented it to a friend for 5 months at $3 a month, and finally sold it for $75. Did she gain or lose, and how much ? 15. A man sold three houses ; for the first he received 3575 dollars ; for the second, $950 less ; for the third, three times the difference between the price of the first and second. What did he receive for the three ? CHIEF DIFFICULTY OF MULTIPLICATION. 1. Multiply 438 by 234. 1st Process. 2d Process. 438 438 234 234 1752 1752 13140 1314 87600 876 Explanation. IST PROCESS.- f 102,492 102,492 Proof. 234 438 1872 702 936 102,492 4 units = 4 units. 234 = ! 3 tens = 30 units. 1 2 hundreds = 200 units. Therefore, we are to multiply 438 firstly by 4 units, secondly by 30 units, thirdly by 200 units, and then find the sum of the three partial products. Multiplying by 4 we have 1752 units; multiplying by 30 we have 13140 units; multiplying by 200 we have 87600 units. The sum of these products is 102,492 units. 2D PROCESS. Since 13,140 units = 1314 tens, and since 87,600 units = 876 hundreds, we omit the ciphers, and, writing 1314 as tens, and 876 as hundreds, the significant figures keep their relative positions, and the result of the addition is the same as before. PROOF. State the principle on which the proof depends. 42 PRACTICAL ARITHMETIC 2. Find the product, explain the process, and show proof of accuracy : (1-) (2.) (3-) (4.) (5.) 317 483 847 657 343 15 16 23 26 37 (6.) (7-) (8.) (9.) (10.) $13.35 $17.85 $30.23 $42.93 $55.66 33 43 72 81 29 Put a decimal point in each product. 3. Multiply: 1. 382 by 19. 11. 619 by 96. 2. 384 by 35. 12. 777 by 86. 3. 405 by 53. 13. 910 by 52. 4. 534 by 37. 14. 732 by 62. 5. 645 by 73. 15. 839 by 93. 6. 843 by 54. 16. 327 by 67. 7. 917 by 46. 17. 931 by 95. 8. 903 by 74. 18. 733 by 123. 9. 593 by 91. 19. 981 by 234. 10. 578 by 83. 20. 891 by 345. 4. How many are 837 X 38 ? 5. How many are $57.30 X 665 ? 6. 2537 X 47 = how many? 7. 5386 X 65 = how many? 8. 4860 X 9574 = ? 9. $357.63 X 47 = ? 10. Find the second member of the following unfinished expressions : 1. $9.672 X 635 - 3. 51,526 X 527 = 2. 5732 X 891 = 4. 97,601 X 987 = MULTIPLICATION 43 11. Multiply five thousand nine hundred sixty-five by six thousand nine. 12. Multiply four hundred sixty -two thousand six hundred nine by itself. 13. Multiply eight hundred forty-nine million six hundred seven thousand three hundred six by nine hundred thousand two hundred four. 14. Multiply 704 million 130 thousand 496 by three thou- sand three hundred one. 15. Multiply one hundred twenty-three and 45 hundredths by 804. 16. Multiply 415 and 5 hundredths by 367. 17. Multiply 113 dollars and 41 cents by 613. 18. Multiply XLVIII. by XIX. 19. Multiply CDLXIV. by CDIV. 20. Form an equation of 220,056,121, and 26,626,776. 21. Find the value of (3467 X 7004) (3467 X 704), and form an equation. PROBLEMS. 1. Find the cost of a farm of 202 acres at $102 per acre. Process Indicated. $102 X 202 = cost. 2. A farmer had 105 rows of trees, each row containing 105 trees. How many trees had he ? 3. If horses are worth $117 each, and oxen $85.50 a pair, what must I pay for 18 horses and 5 pairs of oxen? 4. If 786 yards of cloth can be made in one day, how many yards can be made in 1252 days? 5. A grocer's sales average $19 a day for the month of March ; leaving out 5 days for Sundays, how much money did he receive during the month ? 6. John takes 1434 steps in going to school ; if he goes and returns twice a day, how many steps will he take in 24 days ? 44 PRACTICAL ARITHMETIC 7. There are 5280 feet in a mile. How many feet are there in 18 miles? 8. If James sells 57 papers a day and Thomas 65 papers, how many more does Thomas sell than James in 54 days ? Indicate the process by using the signs, , X f =- 9. If I buy 17 tons of iron at $38.75 a ton, and 26 tons at $40.25 a ton, how much shall I gain by selling the whole at $42.50 a ton? 10. (? -[- ?) x ? (? X ? + ? X ?) = ? Substitute a number for each of the interrogation marks in the first member, solve, and state your problem. SHORT PROCESSES. "When there are ciphers on the right of multiplicand, or of multiplier, or of both. 1. Multiply 2 by 30. Process. Explanation. 2 The factors of 30 are 3 and 10. We say " 2 X 8 = 6 ; o rv and, by annexing a cipher to 6, we multiply it by 10, and have 60." 60 2. Multiply 30 by 4. 3 Q The factors of 30 are 3 and 10. We say " 3 X 4 = 12 ; annexing a cipher to 12 multiplies it by 10, and we have 120." Does the order in which factors are used in multiplying affect the result ? 40 3. Multiply 40 by 500. 5 00 40 = 4 x 10. ^7^ 500 = 5 X 100. 4 X 5 X 10 X 100 = 20,000. We say "4 X 5 = 20; and, annexing one cipher, we multiply by 10 and have 200 ; annexing two ciphers to that result, we thus multiply it by 100, and have 20,000." MULTIPLICATION 45 BULB. Out off and reserve the ciphers on the right; then mul- tiply, and to the product obtained annex the ciphers reserved. EXERCISES AND PROBLEMS. 1. Multiply 486 by 10. By 100. By 400. 2. Multiply 9560 by 40. By 80. By 1000. 3. Multiply 2870 by 600. By 800. By 900. 4. Multiply 2490 by 300. By 3000. By 4400. 5. Multiply 59,700 by 360. By 4300. By 7600. 6. Multiply 42,300 by 320. By 3700. By 57,000. 7. Multiply 4,871,000 by 270,000. By 304,000. 8. Multiply $7849.93 X 400. By 5000. 9. Multiply 600,700 X 6000. By 4,004,000. 10. Multiply CDXL. by M. By LIX. Process H* ^ tne 7 ear ty P av ^ a rear-admiral is $6000, how much will he have received in 20 y^t 12. One mile contains 5280 feet. How $120,OOC many feet in 60Q mileg ? 13. There are 350 rows of trees in an orchard, 120 trees in a row, and 3000 apples on each tree. How many apples in the orchard ? 14. One acre contains 160 square rods. How many square rods in 300 acres ? 15. One pound avoirdupois contains 7000 troy grains. How many grains in 230 pounds ? 16. A short ton == 2000 pounds. How many pounds in 570 tons? 17. The circumference of the earth = about 25,000 miles. One mile = 1760 yards. How many yards around the earth? 46 PRACTICAL ARITHMETIC 18. If one bushel of corn costs $.65, what will 1000 bushels cost? 19. At $160 an acre, what will 500 acres cost? 20. One hour = 60 minutes ; one minute = 60 seconds. How many seconds in 24 hours ? MULTIPLICATION BY FACTORS. 1. You have learned that the multiplicand and multiplier are called the factors of the product. What factors will produce 4 ? 6? 8? 10? 12? 15? 16? 18? 2. All numbers that can be thus factored are called Com- posite numbers. PRINCIPLE. Multiplication may be performed by using the factors of the multiplier. EXERCISES. 1. Multiply 5 by 6, using the factors of 6. Process. Explanation. 6=3x2. 6 = 3x2; therefore, we say " 6 times 5 = 2 times 3 times 5 ; 3 times 5 = 15, and 2 times 15 = 30." -^,30 5 Proof. 3 X- -x- X- X- -X- -X- X- -X- -X- X- -X- -X- * ) x- > 1 * ) 15 2 x- x- * * * 1 x- -x- -x- PROOF. Six rows of 5 stars each = 5X6 =30, the whole number. 3 rows of 5 stars each = 5x3 = 15. 2 groups of 15 each = 16 X 2 = 30, the whole number. MULTIPLICATION 47 2. Multiply 438 by 15. Process. Explanation. 5 __ 5 x 3. Since the factors of 15 are 5 and 3, we first, for convenience, multiply by the larger factor, 5, and 438 that result by 3, and obtain 6570. 5 Proof. 2190 438X15 = 6570. 6570 3. Multiply, using factors : 1. 6809 by 49. 5. 91,849 by 36. 2. 435,261 by 63. 6. 4953 by 81. 3. 310,204 by 48. 7. 14,953 by 144. 4. 97,387 by 45. 8. 2348 by 21. PROBLEMS COMBINING- ADDITION, SUBTRACTION, AND MULTIPLICATION. First indicate the process. 1. I have 10 bags of coffee, each containing 50 pounds. How many pounds of coffee have I ? 2. If hay is worth $14.50 per ton, and oats $.56 a bushel, what will be the cost of 27 tons of hay and 200 bushels of oats? 3. A drover bought 43 cows at $22 each, 64 sheep at $13 each, and 16 horses at $135 each, and sold them all for $4010. How much did he gain ? 4. A freight train consists of 28 cars, and each car contains 136 casks of lime, weighing 240 pounds each. How many pounds of lime in the whole cargo ? 5. Sound travels at the rate of 1092 feet in a second. If between the flash of lightning and the clap of thunder there were 9 seconds, how far distant was the cloud that produced the flash ? 48 PKACTICAL ARITHMETIC 6. If 250 pounds of charcoal are used in making a ton of gunpowder, how many pounds will be used for 1280 tons of gunpowder? 7. What sum of money will be required to pay a regi- ment of 987 men for a year's services, at $18 a month for each man ? 8. A merchant sold 324 barrels of apples at $4.75 a barrel, and gained $162 on the transaction. What did his apples cost him ? 9. A cattle train is made up of 17 cars, and each car con- tains 53 sheep. The average weight of the sheep is 115 pounds. How much do they all weigh ? 10. A farmer bought a farm containing 10 fields; 3 of the fields contained 9 acres each; 3 other of the fields, 12 acres each ; the remaining 4 fields, each 1 5 acres. How many acres in the farm ? What was the cost of the farm at $21 an acre ? 11. If it requires 1716 pickets to fence one side of a square lot, how many pickets will be required to fence 13 lots of the same size and shape ? 12. Mrs. Brown bought 12 yards of oilcloth at 65 cents a yard, and 32 yards of ingrain carpet at 75 cents a yard. What did she pay for all ? 13. One day = 86,400 seconds ; one year = 365 days. If light moves at the rate of 186,000 miles in a second, how far distant is a star whose light is one year in reaching the earth ? 14. An army lost in battle 315 killed, 417 wounded; the enemy lost in killed and wounded 17 times as many. How many soldiers were killed and wounded in this battle? 1 5. If 1327 barrels of flour will feed the inhabitants of a city for one day, how many barrels will supply them for two years ? 16. The Erie Railroad is about 425 miles long, and cost sixty-five thousand dollars a mile. When 9,645,635 dollars were paid, what was the balance due ? MULTIPLICATION 49 17. Two vessels are 4500 miles apart, and travel toward each other; one at the rate of 91 miles a day, and the other at the rate of 85 miles a day. How far apart are they at the end of 24 days? 18. Two vessels start from New York for Liverpool; one sails at the rate of 138 miles a day ; the other, at the rate of 215 miles a day. How far will they be apart at the end of nine days? 19. What is the product of three hundred eleven million two hundred twenty-one thousand multiplied by two hundred three thousand one hundred five ? MISCELLANEOUS EXERCISES. 1. Complete these equations: (16 11+2)X5 = ? (4 + 15) X (15 4) X 6 = ? 2. Use the signs, (),+,, X, =, in forming an equation of your own. 3. An unfinished equation is : 63,915 + (?) = one million. Find the required part. 4. (?) + 4872 = 8021. Complete the equation. 5. 5301 (?) = 4255. Complete the equation. 6. Multiply three million three by one hundred thousand one. 7. Write the immediately preceding numbers in Roman numerals. 8. Multiply 3008 by 132, using the two factors of 132 whose difference is 1. 9. Find the value of 6145 3408 + 1931 X 3400 (33,600 X 105). 10. A drover had 690 sheep ; he sold 340 to one man, 324 to another, and then bought enough to make his number 700. How many did he buy ? 50 PRACTICAL ARITHMETIC REVIEW. 1. Define the following terms : 1. Multiplication. 7. Parenthesis. 2. Multiplicand. 8. Yinculum. 3. Multiplier. 9. Proof. 4. Product. 10. Members. 5. Factors. 11. Composite number. 6. Sign of Multiplication. 12. Equation. 2. Repeat the principles of Multiplication. 3. Repeat the rule for Multiplication when the factors are composed of significant figures with ciphers on their right. 4. What is the principle respecting factors of the multiplier ? 5. Illustrate the principle. 6. Invent five problems that will involve Addition, Sub- traction, and Multiplication. Indicate the solution by the signs, +, , X. DIVISION. INDUCTIVE STEPS. 1. Since 2X3 = 6, the 2 and 3 are called what? 2. Then if 2 is one factor of 6, what is the other ? 3. Why must 3 be the other ? 4. Because 2 times 3 is 6, we say that 2 is contained in 6 three times. 5. How many times is 4 contained in 8 ? Process. Explanation. 4)8(2 W e say " 4 is contained in 8 two times, because 2 times s_ 4 = 8 -" 6. How many times is 5 contained in 10? Write the process and explain. 7. The process of finding how many times one number is contained in another is called Dividing. DIVISION 51 8. Divide 16 by 8, and show that division is a short method of subtraction. 1st Process. 2d Process. Explanation. 16 \ 8)16(2 IST PROCESS We say " Sub- j- Subtract. tracting 8 once, we have 8 remaining ; subtracting 8 a second time, we have Subtract remaining; therefore, 8 is contained 8 ' in 16 two times. 2o PROCESS. Since 8 times 2 = 16, 8 is contained in 16 two times. 9. One factor of 24 is 3, what is the other factor? 10. Dividing by 7 separates a number into how many equal parts ? One of seven equal parts is called one-seventh, written \. | of 28 equals what ? 11. If a farmer pays $28 for 7 sheep, how much is that apiece ? Process. Explanation. 7 ) 28 ( 4 Since he pays $28 for 7 sheep, he pays for each one- 2g seventh of $28, or $4. Did you divide by 7 sheep, or simply by 7 ? Is 7, then, an abstract or a concrete number ? Is your answer 4 or $4 ? 12. How many sheep at $4 apiece can a farmer buy for $28? if $4 is one factor of $28, is $7 the other factor ? Is 7 sheep ? Is 7 ? What, however, does the 7 indicate? DEFINITIONS. 1. Division is the process of finding how many times one number is contained in another, or of finding one of the equal parts of a number. NOTE. This latter operation is called Partition. 52 PEACTICAL AEITHMETIC 2. The Dividend is the number to be divided. 3. The Divisor is the number by which we divide. 4. The Quotient is the result obtained. 5. The number which is sometimes left after dividing is called the Remainder. When the remainder is 0, the division is said to be exact. 6. The Sign of Division is -*-, 21 H- 7 = 3, is read " 21 divided by 7 equals 3." Take notice that the dividend is written before the sign ; the divisor after the sign. In practice it is found convenient to indicate division thus : 7)21 7)21(3 21 or thus : v or thus : = 3. PRINCIPLES. 1. Dividing a number by one of its factors gives the other factor for the quotient. 2. When the divisor is an abstract number, the divi- dend and quotient are like numbers. 3. When the dividend and divisor are like numbers, the quotient is an abstract number. 4. The divisor multiplied by the quotient reproduces the dividend. EXERCISES FOR ANALYTIC AND SYNTHETIC EXPLANATION. The Divisor not exceeding 12. 1. If 6 is one factor of 24, what is the other? Process. Explanation. Divisor. Dividend. Quotient. "\y e sa y u gi nce 6 is one factor of 24, and 6 ) 24 (4 since 6 times 4 = 24, 4 is the quotient, or 24 other factor. ' ' /-v State the principle. DIVISION 53 2. 474 has a factor, 2 ; find the other factor. Process. Explanation. 2)474(237 We sa y "474 = 4 hundreds, 7 tens, 4 units. 4 4 hundreds -=-2 = 2 hundreds. Bring down 7 tens ; 7 tens -=-2 = 3 tens and 1 ten remaining. 1 ten and 4 ' units = 14 units ; 14 units -=-2 = 7 units. The quo- 6 tient is 2 hundreds, 3 tens, 7 units, or 237. 14 Proof. 14 i 237 X 2 = 474. State the principle. 3. 5 is one factor of 35 ; find the other. 4. 8 is one factor of 48 ; find the other. 5. 7 is one factor of 49 ; find the other. 6. 9 is one factor of 72 ; find the other. 7. 12 is one factor of 108 ; find the other. 8. 11 is one factor of 132 ; find the other. 9. 12 is one factor of 144 ; find the other. 10. Divide 144 by 8. We may express the division in four different ways : (1.) (2.) (3.) 8)144(18 8)144 144 8_ ~18~ ~8~ 64 M (4.) 144 -5- 8 == 18 The first is called Long Division ; the second, third, and fourth, Short Division. Explanation. 144 = 1 hundred 4 tens 4 units. 1 is not divisible by 8, hence we say " 1 hundred -(- 4 tens = 14 tens ; 14 tens -?- 8 = 1 ten, with 6 tens remain- ing ; 6 tens -f- 4 units = 64 units ; 64 units -=-8 = 8 units, with units remaining. Therefore 18 is the exact quotient." Proof. 18 X 8 = 144 54 PRACTICAL ARITHMETIC 11. Solve by Long Division, explain, and prove the fol- lowing : I 1 -) (2.) (3.) (4.) 3)849( 5)940( 7)497 8)992( Process. 5)7506(1501 5_ 25 25 06 _5^ 1 rem. (5.) (6.) (7-) 9) 7506 ( 10) 41,690 ( 11) 103,961 ( (8.) (9-) (10.) 12) 113,820 ( 11) 57,893 ( 12) 74,856 ( (11.) 8 ) 38,496 ( (12.) 9) 43,281 ( (13.) 12) 2964 ( (14.) 3) 12,414 ( (15.) 5 ) 32,795 ( (16.) 4) 374,864 ( (17.) 3)629,274 (18.) 5) 947,860 ( (19.) 6)$1589.10( (20.) 7) $6472.69 ( (21.) 8) $1025.68 ( (22.) 9) $1999.98 ( Place a decimal point in the quotient. 12. Solve by Short Division the following: (1.) (2.) (3.) (4.) 6)1698 10)1980 7)994 8)1984 (5.) (6.) (7.) (8.) 9)15,012 10)41,690 11 ) 103,961 12)113,820 DIVISION 55 (9.) (10.) (11.) (12.) 11)57,893 12)74,856 8 ) 38,496 9)43,281 (13.) (14.) (15.) (16.) 7)193,760 12)29,640 3 ) 24,828 5 ) 32,795 (17.) (18.) (19.) (20.) 4)374,864 3)629,274 5)947,860 6)158,910 (21.) (22.) (23.) $48.56 _ 9 $6.44 . $976.50 = 879 Place a decimal point in the quotient. (24.) (25.) (26.) Pounds. t- {y Q Bushels. Rods. ' - = ? 94,000 -H- 8 = ? 760,344 -=- 12 = ? In (24), (25), (26), are your quotients abstract or concrete? Give the principle. The Divisor exceeding 12, but less than 1OO. 1. Divide 34,028 by 13. Process. Explanation. th. h.t.u. th.h.t.u. 34,028 = 34 thousands hundreds 2 tens 8 13)34,028(2617 un it s . 34 thousands -=- 13 = 2 thousands, with 26 8 thousands remaining ; 8 thousands -f- hun- ~8Q~h. dreds = 80 hundreds ; 80 hundreds -=-13 = 6 7 hundreds, with 2 hundreds remaining; 2 hun- dreds -f 2 tens == 22 tens ; 22 tens -=-13 = 1 22 ' ten, with 9 tens remaining ; 9 tens -f- 8 units 1 3 98 units ; 98 units -=-13 = 7 units, with 7 units go u. remaining. 91 2. Divide 5684 by 14. 7 3. Divide 6480 by 15. 56 PRACTICAL ARITHMETIC 4. Divide 2672 by 16. 13. 1504 47 = ? 5. Divide 2928 by 17. 14. 5289 43 = ? 6. Divide 4147 by 18. 15. 9464 52 = ? 7. Divide 8797 by 19. 16. 5612 61 ? 8. Divide 6872 by 24. 17. 6336 72 = ? 9. Divide 8519 by 27. 18. 4557 21 = ? 10. Divide 9672 by 31. 19. 3264 24 = ? 11. Divide 28,490 by 15. 20. 2664 37 = ? 12. Divide 18,476 by 42. 21. 3465 99 = ? PROBLEMS. 1. 3 feet = 1 yard. How many yards in 927 feet? Process. Explanation. LONG DIVISION. Since 3 feet = 1 yard, 927 feet equal as many feet. feet. yards as 3 is contained times in 927. 927 -e- 3 = 3 ) 927 ( 309 3 09. Therefore 927 feet equal 309 yards. _ How is the in the quotient obtained ? In the process, is 309 an abstract or a ^' concrete number ? State the principle. Is the process long or short division ? Write the process in short division and explain the steps. 2. I paid 285 cents for a railroad ticket at 3 cents a mile. How many miles did I ride ? 3. If you buy 12 pounds of soap for 96 cents, how much do you pay for a pound ? 4. If the circumference of a wheel is 12 inches, how many times will it revolve in moving 1728 inches ? 5. If it takes 5 bushels of wheat to make a barrel of flour, how many barrels can be made from 65,890 bushels? 6. A merchant has 1620 yards of calico, which he wishes to cut into 15-yard patterns. How many patterns will he have? DIVISION 57 7. How many times must you take 7 dollars to make 567 dollars ? 8. A boat sails 1872 miles, going at the rate of 18 miles an hour. How many hours does it sail ? 9. How many sacks, each containing 55 pounds, can be filled from 2035 pounds of flour? 10. If I divided $570 equally among some men, gi\'ng to each man $8.00, how many men were there? 11. If 4 weeks make a month, how many months are there in 264 weeks ? 12. Into how many lots of 39 acres each can a tract of land containing 6318 acres be divided? 13. The circumference of a wheel of a bicycle is 7 feet. How many revolutions will it make in going 18,480 feet? 14. How many sheep at $9 a head can be bought for $1377? 15. A man bought land at $87 an acre, paying $31,755 for it. How many acres did he buy? 16. How many feet are there in a mile, if 42 miles contain 22 1,760 feet? 17. Divide four hundred eighteen thousand six hundred forty-eight by twenty-four. 18. If the post-office sends 13,125 pounds of mail-matter in bags, each holding 75 pounds, how many bags will it require ? 19. An estate worth 2943 dollars is to be divided equally among a father, mother, 3 daughters, and 4 sons. What will be the portion of each ? 20. Solve the following equation : 1 798 (5280 1760 -f 144) x -^ -*- 12T= ? i 21. I bought 15 horses at $75 a head; at how much per head must I sell them to gain $210? 58 PRACTICAL ARITHMETIC EXERCISES. The Divisor exceeding- 1OO. 1. Divide 145,260 by 108. Process. 108)145,260(1345 108 372 324 486 432 540 540 Explanation. 145 thousand -f- 108 == 1 thousand, with 37 thousand remaining; 37 thousand -f- 2 hundred = 372 hundred; 372 hundred -f- 108 = 3 hundred, with 48 hundred remain- ing ; 48 hundred -f 6 tens = 486 tens ; 486 tens ~ 108 4 tens, with 54 tens remain- ing ; 54 tens -j- units = 540 units; 540 units -f- 108 = 5, with remaining. 1. 1,874,774 by 162. 2. 1,206,528 by 192. 3. 815,898 by 421. 4. 199,864 by 301. 5. 315,008 by 428. 3. Find the value of: 1. 395,630 750. 2. 683,537 987. 3. 900,503 173. 4. 456,007 560. 5. 881,881 700. 6. 341,517 529. 7. 237,607 837. 2. Divide: 6. 503,652 by 564. 7. 705,776 by 728. 8. 892,696 by 839. 9. 902,260 by 916. 10. 683,537 by 987. 8. 1,056,566 -f- 326. 9. 10,365,051 -r-3021. 10. 2,159,450 -T- 2465. 11. 496,839,715 1047. 12. 9,325,814 ~ 2042. 13. 27,227,704 -f- 6472. 14. 47,254,149 -f- 4674. Find the value of: 1. 352,107,193,214-4-210,472. 2. 558,001,172,606,176,724 -f- 2,708,630,425. 3. 1 23,456,789,102,345,678 -r- 1,234,567,890. 4. 987,654,321,000,000,000 -f- 9,876,543,210. 5. 2,016,722,783,975,663,729 -4- 41,927,081. DIVISION 59 PROBLEMS. 1. A man has 12,000 dollars to invest in land. How many acres can he buy at 125 dollars an acre? 2. There are 47,520 yards in 27 miles. How many yards are there in one mile? 3. There are 640 acres in a square mile. How many square miles are there in the District of Columbia, which con- tains 38,400 acres ? 4. If the earth in its revolution round the sun moves 1,641,600 miles a day, how far does it move in one second, a day containing 86,400 seconds? 5. If 4671 building lots are worth 1,985,175 dollars, how much is one building lot worth? 6. What number of dollars must be multiplied by 124 to produce 40,796 dollars ? 7. There are 31,173 verses in the Bible. How many verses must be read each day, that it may be rend through in a common year ? 8. Pennsylvania contains 45,125 square miles, and Dela- ware contains 2050 square miles. How many states the size of Delaware could be made from Pennsylvania? 9. How long can 125 men subsist on an amount of food that will last one man 4500 days? 10. If 1988 hogsheads of molasses cost 115,304 dollars, what will one hogshead cost? 11. A balloon is said to have ascended 37,000 feet. How many miles ? (One mile = 5280 feet.) 12. If one of two factors of 4,312,695 is 1205, what is the other factor ? 13. A man has 8000 dollars ; he buys two houses for 4500 dollars, and invests the remainder in land at 140 dollars an acre. How many acres of land can he buy ? 60 PRACTICAL ARITHMETIC 14. If the distance from the earth to the sun is 91,430,000 miles, how long will it take light from the sun to reach us, if it moves 186,000 miles a second? 15. How many years will it take a man to save $5475, if his savings average one dollar per day, reckoning 365 days to the year ? 16. A railroad that cost $4,076,500 was divided into 8153 equal shares. What was the cost of each share ? 17. There are 231 cubic inches in a gallon. How many gallons in a tank that contains 139,755 cubic inches? 18. The salary of the President of the United States is $50,000 a year. How much does he receive each day ? 19. If a pound of cotton can be spun into a thread 70 miles long, how many pounds of it must be spun to reach around the world, a distance of 25,000 miles ? 20. Two trains on the same railway are 689 miles apart. If they start at the same time and run toward each other, one averaging 27 miles per hour, and the other 26 miles, in how many hours will they meet? 21. Find the value of 15 X 37,153 73,474 67,152 -s- 4 -f 40,734 X 2. Suggestion : Use X and -=-- first. 22. Find the value of (7854 4913) X 3 (20,352 5194) _|_ 53 _ 6 -f (395,456 2364) -*- 556. 23. Find the value of (12 -f 7 9) X 5 -f- 10. 24. Find the value of (5 + 7 3) X 3 + (3 -f 5 4) --4. 25. Find the value of (828 475 325) + (982 620 82). 26. Find the value of 849 X 4 ~ 3 714 X 4 -5- 3 135 X 4 -=- 3. 27. Find the value of (LI. III. + I.) -5- VII. + (III. X V. IX.) -5r III. DIVISION 61 28. Find the value of (XXVII. + XXII. XIX) x VI. 29. Find the value of (CCCLXI. CGI.) X (CCCXX. CCCXIL). 30. Find a second factor of 4807, taking 11 as the first factor. MISCELLANEOUS EXERCISES. 1. The minuend is 900,000 and the subtrahend is 323,456. What is the difference ? 2. The minuend is 300,400 and the difference 197,325. Find the subtrahend. 3. The subtrahend is 204,054 and the difference is 9735. What is the minuend? 4. The product of two numbers is 567,204, and one of the numbers is 141,801. Find the other number. 5. The multiplier is 3007 and the multiplicand is 3007. What is the product ? 6. The product is 24,483 and the multiplier is 3. What is the multiplicand? 7. The product is 24,402 and the multiplier is 21. Find the multiplicand ? 8. The product is 20,692 and the multiplicand is 739. Find the multiplier. 9. The divisor is 437, the quotient is 730, and the re- mainder is 89. What is the dividend? 10. The divisor is 954, the quotient is 840, the remainder 227. Find the dividend. 11. What number divided by 573 will give a quotient of 205 and a remainder of 89 ? 12. Of what number is 623 both the divisor and the quo- tient? 13. The sum of two numbers is 21,000,000 ; one of the numbers is 12,113,141. Find the other number. 14. Divide 18,490,700 by 73,000. 62 PRACTICAL ARITHMETIC 15. Multiply 5690 by 3008. Prove by division. 16. Show that (26 X 26 15 X 15) -5- (26 + 15) = 26 - 15. 17. How many times in succession can 3589 be subtracted from 241,462 ? What will be the remainder ? 18. A certain number is contained 41 times in 1043, with 1 8 as a remainder. What is the number ? 19. What number is that which, divided by 12, the quo- tient multiplied by 8, and 580 added to the product, equals 740? 20. Divide 9,999,999 by 33,300. MISCELLANEOUS PROBLEMS. 1. If a ship sails 10 miles an hour, in how many days will it cross the Atlantic Ocean, 2880 miles? Process Indicated. 2880 -+- 10 24 = the number of days required. Process. Explanation. 2880 _ ~ . 1. Since the ship sails 10 miles an hour, 10 it will sail 2^80 miles in 2880 -=- 10 = 288 24) 288 (12 days. 2. Since 24 hours == 1 day, 288 hours = 288 , 48 24" **** = 12 da - vs> NOTE. Carefully indicate each solution. 2. How many barrels of apples, at $2.75 a barrel, must be given for 6 barrels of cranberries, at $8.25 a barrel ? 3. How many pounds of coffee, worth $.12 a pound, must be given for 368 pounds of sugar, worth $.09 a pound ? 4. A young farmer earns $60 a month and spends $25. In what time can he save enough to pay for a farm of 50 acres, at $28 an acre? DIVISION 63 5. A grocer bought 250 pounds of coffee for $82.50, and sold it at $.37 a pound. What did he gain? 6. (309 76) + (4426 309) + (6375 4426) -f 76 = 9375 is a defective equation to what extent? 7. There were 24,012 public schools in Pennsylvania in 1893, with 994,407 pupils. How many pupils, on an average, in each school ? 8. Multiply the sum of 276 and 347 by three times their difference ? 9. A park is 48 rods long and 32 rods wide. How many times must a boy go around it on his bicycle to travel 45 miles, there being 320 rods in a mile ? How many times must he go around the park to travel one mile? 10. A man dyiug, left three tracts of land to be divided equally among his six children. The first tract contained 1118 acres; the second, three times as much lacking 193 acres; the third, twice as much as the other two lacking 105 acres. What was each one's share? 11. A/s house cost $7825, which was $4218 less than the cost of the farm. What was the cost of both ? 12. The diameter of the earth at the poles is 41,707,620 feet, and at the equator, 41,847,426 feet. How much does the equatorial diameter exceed the polar diameter? 13. What will 53,000 bricks cost at $7.25 per M.? 14. Mr. Gill, a drover, purchased 36 head of cattle, at $64 a head, and 88 sheep, at $5.00 a head. He sold the cattle for $40.00 a head, and the sheep for $4.00 apiece^ Did he lose, and how much ? 15. Of two boys, one was lazy and did not rise till nine o'clock, while the other was active and rose every morning at six. Allowing 365 days to the year, how many hours did the lazy boy lose in five years ? 16. There are two numbers, the greater of which is 25 64 PRACTICAL ARITHMETIC times 670, and their difference 55 times 81. Find the less number. 17. I bought 87 acres of land at $50 an acre, and paid $3150 in cash, and the balance in labor at $240 a year. How many years of labor did it take ? 18. A farmer has 1000 head of cattle in five fields. In the first he has 315 head; in the second, 175 head ; in the third, 300 head ; and in the fourth, the same number as in the fifth. How many has he in the fifth ? 19. If a man sells 19 bushels of potatoes at $.55 a bushel, 23 bushels of oats at $.53 a bushel, and with the proceeds buys 8 yards of broadcloth, how much does he pay a yard for the broadcloth ? 20. If a newsboy buys papers at $.08 a dozen, and sells them at $.01 apiece, how much can he clear in March, if he averages 1 20 papers a day ? Suggestion: (.01 X 12 .08) X ~j X 31 = ? ANALYSIS AND REVIEW. "Analysis reasons from the given number to one, and from one to the required number." 1. A man bought 13 horses for $2405. What would he pay for 37 horses at the same rate ? Process Indicated. * $2405 X 37 = BUmreceived . 13 Process. Explanation. 13 horses cost $2405. * since 13 horses cost $ 2405 > l horse -, i *-, uf - will cost $2405 -=- 13, or $185. 1 horse costs $185. g? horses win 37 horses cost $6845. cost $185 x 87 or $6845 . DIVISION 65 Let the pupil indicate the solution hy using the appropriate signs. 2. If 25 pounds of sugar cost $2.50, what will 36 pounds cost? 3. If I exchanged 40 barrels of flour for 61 yards of cloth at $4 a yard, how much did I get per barrel for the flour? Indicate the process and explain. 4. A carriage maker sold 15 carriages for $1875. How much would he receive for 25 carriages, selling them at the same rate? 5. 190 bushels of corn cost $100.70. At what rate must it be sold to gain 1 3 cents a bushel ? 6. If 93 oranges cost $5.58, what will 75 oranges cost? 7. If 12 yards of cloth cost $48.00, what will 7 yards cost? 8. If 16 horses cost $1952, what will 22 horses cost at' $6 less a head ? 9. A. paid $27,144 for a farm, at the rate of 15 acres for $3510. How many acres did he buy? 10. If 46 acres of land produce 2484 bushels of corn, how many bushels will 120 acres produce? INDICATED SOLUTIONS. As we have already attempted to show, the solution of any problem should first be indicated by means of signs, and afterwards carried to completion as the signs direct. In completing a solution indicated, a parenthesis or a vin- eulum must be removed first. The other signs, whether within a vinculum or not, may be safely used in the following order: X, -=-, , -f. (12 H- 3) X 2 = 4 X 2 = 8 ; but 12 -=- 3 X 2 = 12 -*- 6 = 2. 5 66 PRACTICAL ARITHMETIC 1. Perform the operations indicated in (48 X 2 84 -^ 6 X 2) + 7 - 3. 1. (48 X 2 84 -r- 6 X 2) -f- 7 3. 2. By removing sign X, (96 84 -+- 12) -f 7 3. 3. By removing sign -4-, (96 7) + 7 3. 4. By removing sign , 89 -j- 4. 5. By removing sign -f-> 93. 2. Find the value of 48 X 2 84 -*- 6 X 2 + 7 3. 3. If 6 men can do a piece of work in 10 days, how long will it take 5 men to do the work? We may let x stand for the required number of days, and write an equation thus : Process. Explanation. ANALYSIS. Since 6 men require 10 days, 1 man will require 6 X 10 days. Hence, 5 x - r men will require days. Performing 1 v fi the operations indicated, we have ^ = x = 12 days. fi0 5 J 2 = 12 days. 5 4. If 12 men can build a school-house in 25 days, how long will it take 25 men to build it? 5. How many pounds of butter, at $.23 a pound, must be given for 5 pounds of raisins at $.11 a pound, 2 pounds of tea, at $.63 a pound, and a barrel of sugar, at $9 ? 5 x $.11 4- 2 X $-63 4- 9 Suggestion: ^- $.23 (j. Find the value of: 1. 28 X 6 ~ 14 + 9 X 8 -r- 12 + 42 -f- 7 X 3. 2. 99 X (8 + 51) X 10 (7 X 104 -f 26). 3. (105 -r- 21 + 80 -+- 5) X (81 + 36 -=- 9). 16 X 3125 127 X + (380 -*- (100 4- 50) 239 (125 X 30) -4r (25 X 25) X (32 21) 55 -f- 5 n DIVISION 67 7. A lady paid a store bill of $784, giving 30 twenty- dollar bills, 4 one-dollar bills, and the remainder in five-dollar bills. How many five-dollar bills did she use ? $784 (30 X 120 + 4 X $1) $5 8. Two trains leave New York for Chicago, 900 miles, at the same hour, one averaging 30 miles an hour, the other 45 miles an hour. How long will the second train be in Chicago before the first arrives ? 9. How many men will it take to do a piece of work in 26 days that 39 men can do in 76 days ? 10. How long can 125 men subsist on an amount of food that will last 3 men 4500 days ? 11. A quantity of provisions lasts an army of 2500 men 72 days. How long would it last 18,000 men ? 12. I bought a carriage for $140, a horse for $125, and a set of harness for $18; kept them a month at an expense of $17.25, and then sold the team for $300. Did I gain or lose, and how much ? 13. A pedler sells beets, six in a bunch, at 10 cents a bunch, and gains one cent on each bunch. Find the cost per C. Suggestion : 10 ~ 1 x 100 = ? 6 14. I paid $86.40 for 1440 blocks of granite. What was the price per M ? Suggestion: ^~~ X 1000 = ? 15. If 8 acres of land cost $656, what will 35 acres cost at $4 more per acre ? 16. From 126 -f (T6~+ 4) x 2 take (48 -f- 2) + 34 X 6 - (17 - 15). 68 PRACTICAL AKITHMETIC GENERAL PRINCIPLES OF DIVISION. If 24 is the dividend (D.), 4 the divisor (d.), and 6 the quotient (.), we have 4, d. We will now notice the effect upon Q , if we multiply and divide D. and d. by 2 ? as follows : 2. 4 6. " Analysis reasons from particular instances to general principles." Reasoning from the particular instances above, we derive the following PEINCIPLES. 1. Multiplying- D. multiplies Q. 2. Multiplying d. divides Q. 3. Dividing D. divides Q. 4. Dividing d. multiplies Q. 5. Multiplying both D. and d. does not change Q. 6. Dividing both D. and d. does not change Q. Let D. = 1728 and d. = 144. Find ., and illustrate each of the above six principles. 4 24-=- 2 4 12 3 n Q is divided by 2. Q. is divided by 2. Q. is multiplied by 2. Q. is unchanged. Q. is unchanged. 4 24 24 -3 4X2 24 8 - 24 10 2^ 2 2 48 24 -*-2 12 fi 4 '>. n DIVISION 69 SHORT PROCESSES IN DIVISION. When there are ciphers at the right of the divisor, the process of division is readily simplified. The Divisor 1 with Ciphers annexed. 1. Divide 539 by 10. Process. Explanation. 1 ) 53J9 Cutting off the digit 9 from the dividend, and the coin from the divisor, we have 53 tens -j- 1 ten 53, Quo. Kern. with 9 remainin S- [Principle 6.] RULE. For each cipher in the divisor cut off a digit from the right of the dividend. 2. Divide: By 10. By 100. By 1000. 1. 6327. 6. 3267. 11. 6173. 2. 5327. 7. 5327. 12. 5432. 3. 9732. 8. 9273. 13. 8650. 4. 9267. 9. 5533. 14. 3000. 5. 2567. 10. 1234. 15. 5678. The Divisor any Significant Figure with Ciphers annexed. 1. Divide 7436 by 3000. Process. Explanation. 3|000 ) 7 [436 Cutting off 436 from D. and 000 from o?., we have 2 1 4-^P ^ t k usancls -*- 3 thousands = 2, with one thousund remaining. 1 thousand -f- 436 = 1436. Hence Q. Quo. Bern. = 2; and R = 1436 Repeat the principle involved. 2. Divide: 1. 673 by 20. 5. 1074 by 80. 2. 957 by 30. 6. 1096 by 90. 3. 686 by 40. 7. 5736 by 200. 4. 790 by 50. 8. 7300 by 300. 70 PRACTICAL ARITHMETIC 3. Complete the following : 1. 873 -*- 600 = 7. 10,432 * 4000 = 2. 1052 -4- 700 - 8. 10,037 *- 5000 = 3. 1095 -4- 800 = 9. 9396 -*- 6000 == 4. 1073 -4- 900= 10. 9116-r- 7000 = 5. 5327 -h 2000 11. 10,370 -*- 8000 = 6. 8645 -* 3000 12. 10,573 -5- 9000 - The Divisor any Number -with Ciphers annexed. 1. Divide 5658 by 3200. Process. Explanation. 32lOO^ 56lo8 ( 1 Q uo - Cutting off 58 from D. and 00 from d., we have 56 hundreds, quotient, 58 units re- maining ; 56 hundreds -=- 32 hundreds 1, 2458 Rem - quotient, with 24 hundreds remaining; 24 hundreds -j- 58 units = 2458, entire remainder. 2. Find the value of: 1. 97,658 ~ 3300 = 7. 500,896 11,000 = 2. 59,625 -&. 4600 8. 485,432 23,400 = 3. 78,695 + 5300 9. 306,959 30,500 = 4. 89,765 -4- 4400 == 10. 940,938 24,500 5. 68,543 -4- 6400 = 11. 768,448 32,300 = 6. 954,000 -4- 350 = 12. 533,337 38,000 = REVIEW. 1. Define the following terms : 1. Division. 7. Long Division. 2. Divisor. 8. Short Division. 3. Dividend. 9. Analysis. 4. Quotient. 10. Solution. 5. Remainder. 11. Principle. 6. as remainder. 1 2. Parenthesis. 2. What are the principles of division ? DIVISION 71 3. In a solution indicated by the signs you have learned to use, in what order is it always safe to use these signs ? 4. Invent five problems whose solution may be indicated by five different signs. PROPERTIES OF NUMBERS. DEFINITIONS AND INDUCTIVE STEPS. 1. A Factor (Latin, " maker") of a number is one of the numbers which, multiplied together, produce the number, as in 2 x 3 X 4 = 24. 2. Write two factors that will produce 24. Write four factors = 24. 3. Form an equation, putting five factors = 300. 4. An Exact Divisor of a number is one of its factors. What are the exact divisors of 6? 5. Since 2 X 3 X 5 = 30, are 2, 3, and 5 factors of 30, or exact divisors of 30 ? 6. If you have the equation 2 X ? = 6, how can you obtain the required factor? 7. Since 2 X 3 X ? = 30, how can you obtain the re- quired factor? 8. Then if a number and all its factors are given except one, how do we find that one? 9. Has 2. or 3, or 5, any factors except itself and 1 ? 10. A number that has no factors or exact divisors except itself and one is a Prime number, as 2, 3, 5, 7, 11, etc. 11. A number that has factors or exact divisors other than itself and one is a Composite number, as 4, 6, 8, 9, etc 12. The Prime numbers between 1 and 100 are as follows : 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. 72 PRACTICAL ARITHMETIC 13. All the other numbers between 1 and 100 are called what? [See 11.] 14. Why are they so named ? Ans. : Because they are composed of factors. 15. 12 = 3X4 is a correct equation. What kind of factor is 3? What kind is 4? What are the two equal prime factors of 4? Re- write the equation with three prime factors in the second member. 16. An Even number is exactly divisible by 2. 17. An Odd number is not exactly divisible by 2. 18. Is 3 an exact divisor of 12 ? Of twice 12 ? Of three times 12? Of any number of times 12 ? 19. 3 is an exact divisor of 12 and 21. Is it an exact divisor of their sum ? Illustrate. Is it an exact divisor of their difference? Illustrate. 20. Since numbers are either prime or composite, factors are either prime or composite. PRINCIPLES. 1. Every composite number is the product of its prime factors. 2. Every prime or composite factor of a number exactly divides that number. 3. Every exact divisor of a number is one of its prime factors or the product of two or more of its prime factors. 4. Every exact divisor of a dividend exactly divides any number of times that dividend. 5. A common divisor of two numbers or dividends ex- actly divides their sum. 6. A common divisor of two numbers or dividends ex- actly divides their difference. 7. Any factor of a number becomes a quotient when the number itself becomes a dividend, and its other factor, or the product of its other factors, becomes a divisor. SUGGESTION. Pupils should be required to illustrate each of the fore- going principles. PROPERTIES OF NUMBERS 73 EXERCISES. 1. Write: 1. Three prime numbers exceeding 100. 2. The composite numbers between 75 and 100. 3. An equation, using three composite numbers as factors. 4. An equation, using three prime numbers as factors. What kind of number is the second member 4 ? Men- tion the exact divisors of it. 2. Write: 1. Three even numbers. 2. Three odd numbers. 3. The prime numbers between 1 and 50. 4. The prime numbers between 50 and 100. 5. The single even prime number. 6. Three odd numbers that are not prime. 3. Finish the following equations : 1. 1 X 2 X 89 X 97 = ? 2. 3X5X7X31=? 3. 47 X ? X 53 = 12,455. 4. ? X 1 X 67 X 89 = 59,630. 5. 41 X 43 X 47 X = ? 4. Why are not 49, 51, and 63 prime numbers? FACTORING. Factoring is the process of obtaining the factors or exact divisors of a number. The number factored is, therefore, a dividend. The most important problem in this connection is to find the prime factors of a number, as a sure means of obtaining certain required divisors or dividends. 74 PRACTICAL ARITHMETIC EXERCISES FOR ANALYTIC AND SYNTHETIC EXPLANATION. 1. What are the prime factors of 108 ? Process. 2 2 3* 3 108 54 27 9 108 = 2 X 2 X 3 X 3X3. Explanation. Since the prime factors of a number are exact divisors of the number, we find all the prime numbers that exactly divide 108. 108 being an even number, is divisible by 2 ; 54 being even, is divisible by 2. Dividing by 3, and again by 3, the quotient is 3, a prime number. Hence the prime factors of 108 are 2, 2, 3, 3, 3. NOTE. In finding the prime factors of a number, use the prime numbers as divisors in order of their values, begin- ning with the lowest one that will divide the given number. 2. What are the prime factors : 1. Of 72? 17. Of 168? 33. Of 798? 2. Of 35 ? 18. Of 231? 34. Of 484? 3. Of 64? 19. Of 178? 35. Of 1280? 4. Of 46? 20. Of 180? 36. Of 1898? 5. Of 336 ? 21. Of 144? 37. Of 5460? 6. Of 111? 22. Of 315? 38. Of 3420? 7. Of 385? 23. Of 420? 39. Of 1470? 8. Of 429 ? 24. Of 660? 40. Of 1492? 9. Of 925? 25. Of 740 ? 41. Of 2310? 10. Of 492? 26. Of 945 ? 42. Of 2772? 11. Of 1320? 27. Of 1728? 43. Of 1600? 12. Of 8424? 28. Of 4284 ? 44. Of 8364 ? 13. Of 7698? 29. Of 1682? 45. Of 2585? 14. Of 743? 30. Of 997 ? 46. Of 1997? 15. Of 3675? 31. Of 4620? 47. Of 4851 ? 16. Of 4536? 32. Of 5250? 48. Of 7623? PROPERTIES OF NUMBERS 75 3. Find the composite factors of 40. Process. Prime factors. 40^ 20 10 Prime factors combined. 1 2X2 =4 2X2x2= 8 2X5 =10 2 X 2 X 5 = 20 Composite factors. 1. Of 102, 105, 108, 221, 715, 845. 2. Of 84, 250, 735, 9800, 11,165. 3. Of 231, 78, 415, 852, 452, 1227. REVIEW OF PRINCIPLES. [See page 72.] 1. Factor 54, and illustrate Principle 1. 2. Factor 36, and illustrate Principle 2. 3. Factor 108, and illustrate Principle 3. 4. Factor 144, and illustrate Principle 4. 5. Factor 231 and 154, and illustrate Principle 5. 6. Factor 360 and 320, and illustrate Principle 6. 7. Factor 1728, and illustrate Principle 7. MULTIPLICATION BY FACTORS. 1. What will 24 carriages cost at $257 each? Process. 24 = 4 X 6 ; $257 X 6 = 1542 ; 1542 X 4 = $6168. 2. In like manner find the cost of : 1. 35 cows at $53 each. 2. 22 violins at $10.35 each. 3. 72 cords of wood at $4.65 a cord. 76 PRACTICAL ARITHMETIC 4. 99 books at $2.18 apiece. 5. 123 hats at $5.65 apiece. 6. 51 acres of land at $125 an acre. 7. 49 barrels at $1.25 apiece. 8. 63 bags of salt at $1.875 a bag. 9. 21 shot-guns at $55.50 apiece. 10. 121 paper-weights at $.555 apiece. 11. 132 horses at $132 a head. 12. 144 rifles at $87.50 apiece. 13. 34 yards of silk at $2.56 a yard. 14. 81 bushels of wheat at $.95 per bushel. DIVISION BY FACTORS. When the divisor is a composite number, division may sometimes be readily performed by using factors of the divisor. 1. Divide 3598 by 14. Process. Explanation. 3598 Tne factors of 14 are 2 and 7. Dividing 3598 into two 1 7QQ equal parts, and each of those 2 equal parts into 7 equal parts, we thus obtain 7 times 2 or 14 equal parts, each 257 equal to 257. 2. Divide, using factors : 1. 8445 .*. 15. 7. 9345 -f- 105. 2. 7776-^24. 8. 1152-4-72. 3. 23,296 + 32. 9. 3648 -4- 96. 4. 1152-4-64. 10. 42,336-^-49. 5. 1855 -4- 35. 11. 37,464 -4- 42. 6. 16,340^-38. 12. 153,160-^56. The chief difficulty in dividing by factors is to find the true remainder. Notice the following explanation : PROPERTIES OF NUMBERS 77 3. Divide 4753 by 140, using factors. Process. Explanation. 414753 140 = 4 X 5 X 7. 5 1188 1 unit of 4753 remaining. 237 . . 8 units of 1188 remaining = 3 X 4 = 12 units of 4753. 33 . . 6 units of 237 remaining = 6x5x4 = 1 20 units of 4753 1 -j- 12 -|- 120 = 133, true remainder. The true remainder must be a part of 4753. Why must partial remainder 3 be multiplied by 4 ? Why must partial remainder 6 be multiplied by 5 X 4 ? 4. Divide, using factors : 1. 7304 by 24. 5. 2184 by 49. 2. 4104 by 45. 6. 3824 by 32. 3. 3276 by 27. 7. 3548 by 72. 4. 3275 by 56. 8. 1299 by 56. CANCELLATION. Cancellation abridges the process of division by striking out a common factor from dividend and divisor. Striking out a common factor is in effect dividing both dividend and divisor by the same number. [State the prin- ciple, page 68.] 1. Divide (96 X 9 X 8) by (12 X 16). Process. Explanation. 4 Cancelling 12 and 96, we have $ 8 as the result in the dividend ; * Hence 5 and 7 are all the factors com- Q J) == 5 y 7 == 35 mon to all the numbers, and 5 X 7 or 35 is the G. C. D. What is meant by "prime to one another" f 3. Find the G. C. D. of the following : 1. 28, 42, 70. 5. 16, 48, 80. 2. 84, 126, 210. 6. 84, 126, 210. 3. 45, 105, 135. 7. 120, 240, 600. 4. 60, 100, 200. 8. 44, 154, 110. 82 PRACTICAL ARITHMETIC 9. 51, 105, 243. 17. 180, 300, 900. 10. 36, 84, 132. 18. 360, 288, 720, 648. 11. 36, 81, 135. 19. 290, 435, 232. 12. 42, 54, 60. 20. 17, 27, 36. 13. 75, 300, 450. 21. 30, 42, 63. 14. 144, 576, 720. 22. 296, 407. 15. 13, 91, 143. 23. 2121, 1313. 16. 14, 98, 112. 24. 1326, 3044, 4520. Nos. 22, 23 and '24 may be reserved and factored by the next process. "When the numbers are not readily factored, a method founded on principle 6, page 72, is adopted. 1. What is the G. C. D. of 169 and 195? Process. Explanation. 169)195(1 By Principle 6, a common divisor of -| />q 169 and 195, divides the remainder, 26, - and consequently 156 and the remainder 13. Since 13 exact'y divides itself and 156 26, it is a common divisor of 169 and 195. G. C. D. 13) 26 ( 2 Tne G - C. D must also divide 26 and i> G ^c., are ca ll e( l fractional units. 92 PRACTICAL ARITHMETIC 5. f denotes how many fractional units ? 6. What is the value of f ? 7. Read the following fractions : }, f, $, -&, Q, fl if. Which has a decimal divisor? Which is equal to one? 8. Write two-thirds, four-ninths, seven-twelfths, ten-seven- teenths. Write a fraction whose value is one. DEFINITIONS. 1. An Integral unit is a whole or undivided unit. 2. An Integer is a whole unit or a collection of whole units. 3. A Fractional unit is one of the equal parts into which the unit is divided. 4. A Fraction is a fractional unit, or a collection of frac- tional units. As we have seen, a fraction is the expression of a division that cannot always be performed, and is written with the number to be divided (dividend) above a horizontal line, and the divisor below that line. 5. The number below the line is called Denominator, because, as we have seen, it names the fractional unit. 6. The number above the line is called Numerator, be- cause it numbers the fractional units. 7. The Numerator and Denominator are called the Terms of the Fraction. 8. A Common fraction is expressed by writing both numerator and denominator, as in -|, J-J-, ^f . 9. A Decimal fraction is usually expressed by simply writing a point before the numerator, as in .5, .37, .25. 10. A Proper fraction has the numerator less than the denominator, as J, --, T 9 ^, etc. 11. An Improper fraction has the numerator equal to or greater than the denominator, as -JJ-, ^ ^, etc. FRACTIONS 93 12. A Mixed Number consists of an integer and a frac- tion, as 2^-, 7f, etc., read, "two and one-third, seven and three-fifths." EXERCISES. Analysis examines the separate parts of a subject and their connection with one another. 1. Analyze J. 9 is the denominator, is a divisor, makes and names the fractional unit. 7 is the numerator, is a dividend, and numbers the fractional units. | is a proper fraction, its terms are 7 and 9, and its value is less than one. 2. Analyze : f , f f, y, f, , y, f, f , f, ^ Hi A, ft, 3. Name the proper and improper fractions and mixed numbers among these: f, 3|, fltf, 4 A, f, A, A. "t. 3 W> A 9 o, , H, iH, W, 2?AV, I*. > i*. 3 *- 4. Write with figures : 1. Five-ninths. 11. Sixty-five hundredths. 2. Ten-elevenths. 12. 110 ninetieths. 3. Seven-twenty-firsts. 13. 211 eighths. 4. Six and two-thirds. 14. Thirty and seven-eighths. 5. Seventy-eightieths. 15. Five twenty-fifths. 6. Ninety-one ninetieths. 16. Fifteen -sixteenths. 7. 314-tenths. 17. Nine thirtieths. 8. 1898-millionths. 18. Four and two-filths. 9. Ten and five-sixths. 19. Three-thirds. 10. Ten-tenths. 20. Two and twelve- twentieths. 5. Write: 1. A common fraction. 3. A proper fraction. 2. A decimal fraction. 4. An improper fraction. 5. An improper fraction equal to one. Point out the terms of the fractions you have written. 94 PRACTICAL ARITHMETIC REDUCTION OF FRACTIONS. Reduction changes the terms of a fraction without changing its value. The change is to Higher Terms, to Lower Terms, or to Lowest Terms. Reduction of Fractions to Higher Terms. Is not $! = $? May we not multiply the terms of \ by 2 and thus obtain f ? PRINCIPLE. Multiplying both terms of a fraction by the same number does not change the value of the fraction. (See p. 68. ) NOTE. The pupil should perceive that fractions are, by their very nature, subject to the principles of division. EXERCISES. 1. Change f to twentieths. Process. Analysis. Explanation. nrj . A f ~L = 2-Q- ^ ne division shows that the terms of 1 _ _g the fraction must be multiplied by 5 to 5. x 5 = IA change fourths to twentieths. Multiply- 4" = ~ 2T ing both 3 and 4 by 5 we have J. RULE. Divide the required denominator by the given denom- inator, and multiply both terms of the fraction by the quotient. 2. Reduce: 1. 1$ to 60ths. 4. T f- - to lOOOths. 2. || to SOths. 5. fj- to 270ths. 3. J- to 40ths. 6. ff to 1 SOths. EEDUCTION OF FRACTIONS 7. ft to 90ths. 14. % to 74ths. 8. ft to HOths. 15. 1 to 42Dds. 9. f to 99ths. 16. if to 38ths. 10. f to 49ths. 17. f to 30ths. 11. J- to GOths. 18. fo to lOOths. 12. f to 24ths. 19. f to lOtbs. 13. 7 to 70ths. 20. J& to 12ths. Reduction of Fractions to Lower Terms. Is not $==$? May we not divide both terms of | by 2 and obtain % ? PRINCIPLE. Dividing both terms of a fraction by the same number does not change the value of a fraction. (See page 68.) EXERCISES. 1. Reduce f| to eighths. Process. Explanation. i p _._ o o ^e division f 1C by 8 shows that both terms of the \_ __ fraction must be divided by 2 to change sixteenths to eighths. Dividing both 12 and 16 by 2, we have f . RULE. Divide the given denominator by the required denom- inator, and divide both terms of the fraction by the quotient. 2. Reduce: 1. |f to 15ths. 6. fj to 16ths. 2. if to 9ths. 7. UJ- to lOOths. 3. |4 to lOths. 8. ^t to 12ths. 4. j|f to 12ths. 9. f|J to 9ths. 5. to 4ths. 10. i to 949ths. 96 PRACTICAL ARITHMETIC Reduction of Fractions to Lowest Terms. Eeduction to lowest terms requires the terms of the fraction to be divided by their greatest common factor (G. C. D.). EXERCISES. Process. 1. Reduce fjf$ to lowest terms. 1760)5280(3 5280 Explanation. 7T The G. C. D. of 1760 and 5280 is 1760. Di- viding both terras of the fraction by 1760 we 176 orHflF = i obtain |, the lowest terms. (See Principle, p. 95.) RULE. Divide both terms of the given fraction by their G. C. D. Continued division by a common factor will secure lower or lowest terms. 4 )JL ULQ. 4 ) 440 11)110 10)10 . J. 4)5 280 4)1320 ~~ TT)"3~3~(F ~~ 1~0")30 3' The terms are the lowest when they are prime to each other. 2. Reduce to lowest terms : 00 ( 2 -) (3-) (4.) (5.) (6.) o rr Q Q /> rj (70 M (8.) it (9.) 232 3~7T (10.) m (11.) Ht (12.) (13.) 42 (14.) If (15.) AV (16.) 150 180 (17.) 210 2T2 (18.) m (19.) Mi / O O (20.) II* (21.) Iff O 1 O (22.) m (23.) m (24.) m (25.) 650 780 (26) 1769 192Q (27.) 288 '864 (28.) 648 T2T (29.) T\ 8 2\ (30.) 86 4^ 1286 REDUCTION OF FRACTIONS 97 (31.) (32.) (33.) (34.) (35.) (36.) 726 1680 1694 TtT2 T1T12 1848 (37.) (38.) (39.) (40.) (41.). 10605 6161 1710 5040 4692 11445 7171 T4364 17160 T6 1 9 7 6 Reduction of Integers and Mixed Numbers. We have learned that -|, f , f , or f , etc., equal one. How many halves in one whole thing ? How many thirds ? Sixths ? Tenths ? How many thirds in two ? In 2f ? EXERCISES. 1. Reduce 8J to fourths. Process. Explanation. 8 = -^- Since 1 = , 8 = - 3 ? 2 - ; and 8 + f = ^ + 8J = ^t + } = ^. f = -V- BULB. Multiply the integer by the denominator, to the product add the numerator, and -write the sum over the denom- inator. 2. Reduce: 1. 61 to fourths. 11. 15 to fifths. 2. 2 to thirds. 12. 13 to sixths. 3. 121 to halves. 13. 18^- to elevenths. 4. 9f to sevenths. 14. 5^- to ninths. 5. 16J to fourths. 15. 5^- to eighteenths. 6. 13f to eighths. 16. 272^- to elevenths. 7. 31 4^- to twenty-firsts. 17. 278$ to ninths. 8. 673 T 8 2- to twelfths. 18. 946 T % to thirteenths. 9. 702|f- to elevenths. 19. 615f to fifths. 10. 122 T V to fifteenths. 20. 24 1 -fa to twenty-firsts. Have your results been proper or improper fractions ? 7 98 PRACTICAL ARITHMETIC 3. Reduce to improper fractions the following : 1. 9|. 6. 223 T V 11. 21 Of 16. 15 T y 2. 17|. 7. 13f 12. 16&. 17. 108 3. 28^. 8. 504f. 13. 62^-. 18. 51 T 3 7 -. 4. 27f. 9. 114 T V 14. 159 T V 19- 40 ff. 5. 49f 10. 312fi 15. 67 20. Reduction of Improper Fractions. How many dollars in $|-? In $-^ 8 - ? How many units in - ? In ^ ? In ^L ? J n /- ? In & ? What kind of numbers are your results? EXERCISES. 1. Reduce ^^ to an integer and ^ a mixed number. Process. Explanation. 0-4- = 72 Since -^ indicates the division of 504 by 7, we 50 5 __ YOI divide and obtain the integer 72. RULE. Perform the division indicated. 2. Reduce the following improper fractions : 1. |. 10. ff 19. 4^6, 28. 2. ff 11. |f. 20. -^ 29. 3. J$jL. 12. ^I_L. 21. *$-. 30. 3||ji 4. *& 13 - H- 22 W- - Mtt- 5. ii. 14. ||. 23. - 3 ^ 4 -. 32. 6. ff. 15. *& 24 - W- ;33 - 7. -V 5 /. 16. ^. 25. *$?-. 34. 8. ff 17. -V/. 26. ^f^. 35. 9. ||.. is. ^|8, 27. ^ffi. 36. REDUCTION OF FRACTIONS 99 REDUCTION OF UNLIKE FRACTIONS. 1. Have \ and f like or unlike fractional units ? 2. By reduction to higher terms, \ equals how many sixths ? | equals how many sixths ? 3. Are and like fractions? Why ? 4. What, then, is the difference between Like and Unlike fractions ? 5. Have and % a common denominator? 6. What is the least common dividend of the denominators 2 and 3 ? DEFINITIONS. 1. Like fractions have the same fractional unit. 2. Unlike fractions have not the same fractional unit. 3. Like fractions have a Common denominator. 4. Like fractions may have a Least common denomi- nator. (L. C. D.) PRINCIPLES. 1. A common denominator of two or more fractions is a common dividend of their denominators. 2. The least common denominator of two or more frac- tions is the least common dividend of then: denominators. EXERCISES. 1. Reduce f and -J to like fractions. Process. Explanation. 3 y g __ 24 A common dividend of 3 and 8 is 24 ; there- fore 24 is a common denominator of ^ and |. To 4 r= A * 8 = 40 reduce | to twenty-fourths we multiply hoth terms 7 7x3 = 21 by 8 ; to reduce | to twenty-fourths we multiply ^ = ~ x 3 = 2T both terms by 3. 100 PRACTICAL ARITHMETIC 2. Reduce |-, -J, and ^ to fractions having their least common denominator. Process. Explanation. L. C. Dd. of 6 9 r ^^ ie l eas t common denominator of the frac- 12 is 36 tions is tlie least common dividend of their denominators. The L. C. Dd. of 6, 9, 12 is 36. |- |. x 6 = 3JJ. -yyr e therefore multiply the terms of f by 6, the 1_ __ 7 x 4 = 2_8 terms of | by 4, the terms of {\ by 3. HI 1 x 3 = 33 The same results may be obtained by reason- ing thus: Since 1 = ff, \= &, and f == fg. Reduce the two other fractions in like manner. RULE. Find the L. C. Dd. of the denominators, divide it by each denominator, multiply both terms of each fraction by the quotient obtained by its denominator. State the principle involved. (See page 68.) Brief directions are : 1. Find the L. C. Dd. 2. Divide by the denominators. 3. Multiply the numerators by the quotients. 4. Place the products over the L. G. Dd. Before applying the rule reduce mixed numbers to improper fractions and fractions to their lowest terms. 3. Reduce -f-, -|, -^ to like fractions having their L. C. D. Process. Introductory, -f% -J-. 1. L. C. Dd. of 7, 8, 2 is 56. 2. >> = 8, 7, 28. 3. 3 X 8, 7 X 5, 1 X 28 = 24, 35, 28. 4. |f, H, H . 4. Reduce in similar manner the following : i. i i I- 3 - T> A. W- 2. i i A- 4. |, f , f . ADDITION OF FRACTIONS 1 01 5. f,f,ff. 14. 3f,i 7, 11 6. f, |, ||. 15. 91 f , A, |. 7. f, T 9 o, if- 16. 2J-, 4$, 4, f 8. }, f, A- 17. 8, ?i, f, f 9- T 8 2> H, TV 18. i, t, i, t, * 10. |, f, 2f 19. A, H, A; iWr, 11. 81 21 41 20. A, 6J, T 9 I7> 7, f, 12. f, A, H. 21. if, if, f 13. tt,A,- 22. i, A, A, ADDITION OF FRACTIONS. INDUCTIVE STEPS. 1. What is the sum of 2 books and 3 books? 2. What is the sum of f and f ? 3. Of f and |? Of -^ and ^? 4. What is the fractional unit of ^-? Of ^ T ? Of &? Are these, then, like or unlike fractions ? 5. What kind of fractions can be added ? 6._Can you directly add -f^ and T 6 ^-? 7 V If these fractions kad a like or common denominator, could you add them ? 8. How do you reduce unlike fractions to like fractions? PRINCIPLES. 1. Only like fractions can be added. 2. Unlike fractions can be reduced to like fractions and then added. EXERCISES. , 1. Find the sum of ^-, y 7 ^ and -f^. Process. State the principle involved. 102 PRACTICAL ARITHMETIC 2. Find the sum of f , f, f . Process. 1. L. C. Del. of 8, 6, 9 = 72. 2. ,, = 9 > 12 > 8 - 3. 3 X 9, 5 X 12, 4 X 8 = 27, 60, 32 4. T + " Explanation. Since the fractions are unlike, we render them like by reducing them to fractions having the L C. D. 72. t + I + t = H -f ff + ft = W = iff 3. Find the sum of 4J, 3J, 4| and Process. Explanation. Introductory. -A- = -I- " The numbers to be added are composed "? ~ : ' "2"IT of integers and fractions. We therefore 3^- = 3 -j- -^j- add the integers and fractions separately, 41 4 _j_ JJL and then unite their sums. 4 -f 3 -f 4 -f 5 = 1 6. After reduction to twentieths the sum of the fractions is |, or 16 -f 1& = 17 2 3 -, the sum total. RULE. 1. Reduce the fractions, giving them a common denomi- nator. 2. Add the integers and the fractions separately, and unite their sums. 4. Find the sum of the following : 1. *,*,A- 7. 3|, 4J, If, 2. 2. 4fc 3J, 4*, 5fL 8. H. A. 10, ff. 3. i, f , f , I- 9. A, A, f 4- I, f , i, t, H- 10 - H- H. 5- A, A, A- H- *i. 2 i. 3f, 7i, HI- 6. 8^, lOf, 14|, 12. |,|,Ai ADDITION OF FRACTIONS 103 o.A.A- 17.6i4,|,8. 14. A, fV, ft, if 18. 15fcl7f,f 15. 8 ft, 6 T 8 f , 514, H- 19- 900 r V, 450f, 75^. 16. f, H. A, A, it- 20 - f i i, t. i f i- 5. What is the value of: 1- l + t + r? 6. 5f + 18^ + 25^? 2- f + I + if? 7 - 187 i + ly7 * + 746 ? ? 3. 4i +' 3f + H? 8 - 1 7(54 I + 7867 I + /T- 4- f + f + i + fl? 9. 211 + 331 + 6-Z& + 7 5- A + ^ + l+Vr? 10. f + i 6. Answer the following inquiries ; 1- A +/f +"!+*=? 2- A-H T 7 3-HA^? .-A + A-f*=? 4. 187^ + 1976|-f 7461=? 5. 8|+9|-f 12^=? 6- 7H + 8A + 9*f =? 7. ll| + 10 r V 8. 81 + 61 + 2^. 9. 51 + 6f + 711 10. 9} + lOf + llf + 51J .+ 7 T 8 r + 18f =? PROBLEMS. 1. I bought 3 pieces of cloth containing 125^, 96f, and 48-| yards. How many yards in the three pieces ? 2. A merchant sold a customer 22^- yards silk, 3J yard, paper muslin, 11 yards silesia, 5f yards cambric, and 5^ yards ruffling. How many yards were sold ? 3. A farmer divides his farm into 5 fields. The first con- tains 26-^ acres, the second 40^-f- acres, the third 5 If acres, the fourth 59^ acres, and the fifth 62|- acres. How many acres in the farm ? 104 PRACTICAL ARITHMETIC 4. A bicycler rode 27f miles on Monday, 33^ miles on Tuesday, 37f miles on Wednesday, and 42| miles on Thurs- day. How far did he ride in the four days ? 5. A dry-goods merchant sold a lady 18 J yards of flannel, 21-J yards of silk, and as many yards of calico as of both the other goods. How many yards in all did he sell ? SUBTRACTION OP FRACTIONS. INDUCTIVE STEPS. 1. From |- subtract -J. 2 - A A = what? 3. If you have $-J (of a dollar) and spend $-|, how much have you left? 4. If you have $-J and spend $J, how do you find the remainder ? 5. What kind of fractions can be subtracted without re- duction. 6. What kind require reduction ? Reduction to what ? 7. Give four brief directions for such reduction. 8. What introductory step is sometimes necessary ? PRINCIPLES. 1. Only like fractions can be subtracted. 2. Unlike fractions can be reduced to like fractions and then subtracted. EXERCISES. 1. Find the difference between -^ and ^-. Process. Explanation. Since T 8 T and T 5 T are like fractions, having a Ijf -ff IT common denominator, 11, their difference is 8 elevenths 5 elevenths, or 3 elevenths. SUBTRACTION OF FRACTIONS 105 2. What is the difference between ^- and f ? Process. Explanation. Since T \ and f are unlike frac- 2 45 32--13 tions, we reduce them to eightieths, tV ~ ""80" "SIT making them like fractions. -^ f = If- ft = tt- 3. Subtract 7| from llf. Process. Explanation. We subtract integers and fractions sepa- ^ 8 rately. f cannot be taken from f ; but 1, 7| = _7j[_ taken from 11, equals f ; f + f == -^ 5 - I o 03 = | or |. 10 7 = 3. Uniting the two re- sults, we have 3|, the remainder. RULE. 1. Reduce unlike fractions to a common denominator. 2. "Write the difference of the numerators over the com- mon denominator. 3. Subtract integers and fractions separately, and unite the results. 4. From f take |. From f take . 5. From f take . From f take f . 6. From | take ^-. From ft take ^. 7. From |f take ^. From take ^-. 8. From f take -3^. From fj take ^. 9. From lOJf take if. From 112 take 75f 10. From 606f take 70J. From 506| take 418f 11. What is the value of: l-ff-ff- 7. 1198|-149|. 2- m - A- 8- 589| - 67|. 9. 10. 72 II. 6- ^V - TT- 12. 42 106 PRACTICAL ARITHMETIC 12. Find the value of: 1. 51 20ft. 8. 48ft 22 15.76^- 2. 66 36^. 9. 35f 29f 16. 3. 64 59 f. 10. 44| 27|. 17. 4. 38 37^. 11. 48f 9. 18. loif 5. 59 32-J-f 1 2. 73-i- 27|. 1 9. 28|f 1 6|f 6. Ill 31|f 13. 22| 7f 20. 56^ 29f 7. 36f 27f 14. 88 T 4 T 53^. 21. 65|J 30f 13. Answer the following inquiries : 3. 2i|-2^ ' 4K Q__ _ 10 5. 13^4 6. 9|- 8. -^Q- 9. 3f + 9^ + 6| = ? 10. 36 21 4| 61 ? 11. 20 84 6ft f=? 12. 200 30f 17^ 26|f == ? 14. 53-^-21-9^ = ? The teacher will suggest the shortest method of answering the above inquiries. PROBLEMS. 1 . 3 \ yards, 4f yards, and 1 2^ yards were cut off from a piece of silk containing 30 yards. How many yards re- mained ? 2. A man spent ^ of his income for rent, ^ for food, and ^ for other expenses. What part of his income remained ? MULTIPLICATION OF FRACTIONS 107 3. A farmer sold -J of his corn to one man, f to another, and had 50 bushels remaining. How much corn had he at first? 4. Show that the fraction fj- is greater than and less than f . 5. If I pay my grocer $18 j, my coal dealer $271, and my tailor $22|, how much will I have left out of four 20- dollar bills? 6. ^ of a pole is in the mud, ^- of it is in the water, and the rest of it is in the air. What part of it is in the air ? 7. Show that 13f 2^ 6-ft + 3 1 T % + 8| f f ] o^| = 4f-| is a correct equation. 8. Find the second members of these ; 1. 450 + (12 X 5) 86fo 2. gi 3. 59 2 4. 52 -f (87 5. 231 62| + 101^- | = ? 6. 453 (32-^ + f 10) = ? 7. LXXVII. iV + CLXIX. 11^. =? MULTIPLICATION OF FRACTIONS. INDUCTIVE STEPS. 1. How much is 2 times 3 dollars? 2. How much is 2 times 3 sevenths? 3. How much is 2 times ^-? 4. How much is 5 times |-? 5. Multiply ^ by 10. & by 3. 6. Have you been multiplying numerators or denomi- nators ? 7. Then what effect has multiplying the numerator? 8. What is 3 times f ? What are the lowest terms of f ? 108 PRACTICAL ARITHMETIC 9. Then 3 times -| = |. How could you have obtained | more directly than by multiplying the numerator ? 10. What effect, then, has dividing the denominator? 11. Is that effect in agreement with principle 4, page 68 ? Why? PRINCIPLE. Multiplying the numerator or dividing the denominator multiplies the fraction. EXERCISES. 1. Multiply Jg- by 4. Process. Explanation. 7 vx 4 __ _T_ According to the principle we may multiply the numerator or divide the denominator. Since the denominator, 16, is divisible by 4, we divide and obtain the result, |. 2. Multiply 4 by -fr. Process. Explanation. 4 X 7 7 = 111 Since the denominator, 17, is not exactly divisible by 4, we multiply the numerator by 4 and obtain the result, \ f 1||. RULE. To find the product of an integer and a fraction divide the denominator or multiply the numerator by the integer. 3. Multiply: 1. A by 7. 8. A by 4. 15. if by 18. 2. A by 5. 9. # by 11. 16. # by 6. 3. if by 7. 10. fj by 14. 17. j|| by 18. 4. JL. by 6. 11. T 3 T by 5. 18. ^ by 10. 5. ^ by 8. 12. -fa by 3. 19. -fa by 28. 6. U by 3. 13. if by 9. 20. -^ by 19. 7. l| by 13. 14. # by 14. 21. -^ b y 12. MULTIPLICATION OF FRACTIONS 109 4. Multiply 8f by 4. Process. Explanation. 4 times f = f = 2. 4 times 8 = 32. 2-J- 32 + 2 = 34. 34^ 5. Find the value of: 1. 9| X 6. 8. 28|f X 10. 15. 45ff X 60. 2. 7f X 9. 9. |fj X 48. 16. 19f X 14. 3. 8^ X 5. 10. -fffs X 144. 17. 25^| X 15. 4. 18$ X 8. 11. 18| X 10. 18. 46|f X 13. 5. 21f X 4. 12. 9f X 21. 19. 54f| X 35. 6. 6| X 13. 13. 8f X 24. 20. 65|f X 68. 7. 171 X 9. 14. 63f X 56. 21. 77 T Vr X 77. 6. Multiply: 1. 9 by ^ 8. 100 by ^. 15. $406 by T 3 T . 2. 57 by |f. 9. 144 by |f 16 - * 718 b 7 if 3. 88 by |. 10. 51 by -^. 17. $825 by f|. 4. 17 by 5 8 T . 11. 75 by $f. 18. $fff by 49. 5. 12 by if. 12. 90 by ^. 19. $^| by 26. 6. 124 by -jV 13. 91 by ^J-. 20. $400 by f. 7. 153 by ^. 14. $318 by ^-. 21. $|-fi by f STEPS TO GENERAL RULE. 1. Both expressions, 3 X 4 and 4 X 3, = ? 2. What principle do you find established on page 37 ? 3. How much is -J- X 6 ? ^XlS? 4. How much is 6 X |? 18 X i? 5. How much is ^ of 6 ? \ of 18 ? 6. "Of" between a fraction and a following number is equivalent to what sign? 7. Express 27 X | by using "of." 110 PRACTICAL ARITHMETIC 8. In how many ways can you indicate the product of 16 and |? PRINCIPLES. 1. Fractions, as factors, may be used in any convenient order. 2. A fractional multiplier may be used as expressing the part of the multiplicand to be taken. EXERCISES. 1. Multiply f by f. Process. Explanation. l of | = ^ |X | = | of |, Principle 2. ioff = F 7 f of f 4 1! = ^. See Principle, page 68. 4 GENERAL RULE. "Write the integers and mixed numbers in fractional form; cancel common factors, and find the product of the remaining factors of the numerators for a new numerator, and of the denominators for a new denominator. 2. Find the value of: 1. fofiJ. 6.* of AX A of ft- 2. of . 7. |f of |f X U of - 3- tfoftf 8.#of W of&of#. 4. of|f 9. foffoff Xfoff. 5. ttofffr. 10. fXAofHoff. 3. Reduce: 1. f X 31 5. Jfr X f 9. X 3|. 2. AX 12f 6. X A- 10. lAXltt- 3. 91 X f 7. A X f. 11. X A. 4. 2^ X If 8. % X 2f. 12. f x # X 6f MULTIPLICATION OF FRACTIONS 11] 4. Reduce : 1. f of If X of ^ of 4. 1 &X 4r X &.x f x tf 3- ttx^xAx^xtt- 4. Hx^xttxiixf & A x H x tf x f x &. x & x tf x A x If ' 1T6" X T5" ^ T2TF X 9-3 X TTT* 8. f x | x> x $ x f x f 9-AxAx-fxlxfxM- 10. * x x If x # x & x f 5. What is the value of : 1. f of f of 5 X fVof | of 3? 2. | of ^ of 8 X f of ^ of 15? 3. 3^ X f X 4 X f of 7 ? 4. 5^ times ^ X 18 X f of 3 times | of 4? 5. & of 15 X | of -^ of ^ of 6? 6. ^of 3f of if X |of 49? 7. if of if XH of 8- o PBOBLEMS. Required the cost of: 1. 45 pairs of shoes at $1 J- per pair. 2. -^j- of a yard of cloth at $|- a yard. 3. 1 20 yards of ribbon at 1 6f cents a yard. 4. 4f tons of hay at $16| per ton. 5. 465 Rochester lamps at $7-|- apiece. 6. 250 tons of coal at $6| a ton. 7. 12^ cords of wood at $5f a cord. 8. If 16^ feet make a rod ? how many feet are there in rods? 112 PRACTICAL ARITHMETIC 9. There are 24f cubic feet in a perch of stone. How many cubic feet in 5^- perches ? 10. Mr. Lipmann bought a lot of crockery, of which the retail price was $576-|, but he got a reduction of ^ for whole- sale and ^ for cash. What amount did he pay ? DIVISION OP FRACTIONS. INDUCTIVE STEPS. 1. 1 divided by 1 equals what? 4 divided by 1 equals what ? - divided by 1 equals what ? -1- divided by 1 equals what ? 2. If -| ~ I = -J, |- -f- ^ equals how many times -J- ? 3. If -J- -7- -J- = 3 times -J-, -r- -- = f times -J-. Hence, -l==iX|or|of 1 4. Divide in like manner -J by |^ and -|- by ^., What principles on page 68 cfo'c/ yo?/, apply f What change in the form of the divisor do you observe? EXERCISES. 1. Divide f by f. Process. Explanation. become X, and | has become |, i.e., has become inverted. 2. Divide 4 by . Process. Explanation. 4_i_7. 4.><8=J[2 44 4 = f- Inverting | and writing sign X, we have f X f = - 3 / = 4f DIVISION OF FRACTIONS 113 3. Divide T 9 7 by 3. Process. Explanation. 3 3 _^-_?-l_A - + 3 = lof 9 - = x which by ~ 10 ' 10 3 10 cancellation gives T 3 ^. RULE. 1. Give integers and mixed numbers fractional form. 2. Invert all divisors. 3. Cancel factors common to numerators and denomi- nators. 4. Find the product of the remaining factors. 4. Divide : 1. 15 by f 2. 18 by f 6. 75 by |. Process. 3. 63 by f 7. 32 by f . ' 4. 25 by f 8. 45by. f X |= 21 5. 49 by 9< 64 bv i^ 5. Divide : 1. if by 6. 2. ftf by 5. 6. ^ by 12. Process. 3. ft by 8. 7. & by 15. /, , 2 4. W-by6. 8. fby5. is X ? = Is 5. af. by 60. 9. M* by 9. 6. What is the value of: i- H- 8 -*? 4 - *V-? 7. ft-*? 2- If -*- 1 ? 5. if -.- 1 ? 8. ff - il ? _3- H-i-}? 6. If-l? 9. M-H? 7. Find the value of: 1. f of | of 16 -s- f of f of 5J, Process. 3__25__-,9 " 16 ~ 16* 114 PRACTICAL ARITHMETIC 2. of f of 51 -h 41 times l of 17, 3. I of 21-51 4. l of | of ^ by 7 times | of f 5- (^ + T 3 o) X T V 6. I of 41 - | of 3f . 7. f of f of 15 -h | of | of 6. 8. 21 of 21 - ^ of 3f . 9. f of -fr ot 22 -f- T % of f O f i 6 . 10. | of -f- of if - 6. 11. ^ of 3| of 6 -f- -J of 6 times If 12. fof 3iof^-5i 13. 81 times of 7 -=- f of f of 5. 14. ^ - I of 21 of If 15. I of 351 -4- f of 8|. 16. A oflf-fof|. 17 - A of ir -*- A of II- 18. W ofi|-2Vof3^. 19. |offof|-|ofi|of ff. 20. Joffof|-|of T Vofiloflf. PROBLEMS. 1. If 7f yards of cloth cost $47^, what is the price per yard ? 2. If a man spends $f per day for cigars, in how many days will he spend $17-^? 3. If -- of a ton of hay costs $15, what is the cost of one ton? 4. A man has 229^- pounds of honey, which he wishes to pack in boxes containing 8^- pounds each. How many boxes will he require ? 5. A man owning ^|- of a ship, sold -| of his share, and divided the remainder equally among his three sons. What part of the ship did each son own ? COMPLEX FRACTIONS 115 6. The product of two numbers is ^f, and one of the numbers is 1^. What is the other number? 7. What number multiplied by 1-| will produce 14^? 8. How many yards of cloth at $3f per yard can be bought for $317f? 9. When wheat is selling at $1-J per bushel, how many bushels can be bought for $3168? 10. For $8^ how many thousand feet of gas at $1J per thousand can be bought ? COMPLEX FRACTIONS. A Complex Fraction has a fraction in one or both of its 1. Simplify S. Process. Explanation. ^ - 6 4 = : " ( A ^ H Since || signifies that 5^ is to be divided by 6, we proceed according to the rule for division, and obtain ft. 2. Simplify : 116 PRACTICAL ARITHMETIC 16. ' n /\ o 4.' A 3 i T ~~~ T!" ^" 9 Q FRACTIONAL RELATIONS. 1. In the equation, -J- of 4 = 2, the ^ expresses the relation of 2 to 4. If the question is asked, " What is the fractional relation of 2 to 4 ?" the answer simply reverses the equation, " 2 = 1 of 4." This equation may be derived analytically, thus : Since 1 = J- of 4, 2, being twice 1, = f or J of 4. 2. In like manner show the fractional relation of 3 to 9. Of 5 to 8. What part of 8 is 5 ? Does the answer show the relation of 5 to 8? 3. What part of $5 is $1? Since $1 == i- of $5, $1, being 1 of $1, = of of five dollars, or ^ of five dollars. 4. In like manner find the fractional relation of $f to $6. 5. What part of 7 acres is |- of an acre? 6. Is $5 any part of 10 acres? 7. What is the fractional relation of 7 men to 9 trees ? PRINCIPLE. Only like numbers can have fractional relation to each other. FRACTIONAL RELATIONS 117 EXERCISES. To find the Fractional Relation between Two Numbers. 1. Form an equation to show the fractional relation of: 1. 8 to 24. 11. | to 5. 21. 6f to 40. 2. 13 to 26. 12. f to 10. 22. 6J to 425. 3. 12 to 18. 13. f to 7. 23. 2 to 42. 4. 10 to 15. 14. -f to 9. 24. 6 to 128. 5. 9 to 27. 15. f to 16. 25. 6f to 75. 6. 35 to 40. 16. f to 26. 26. 12^- to 180. 7. 16 to 24. 17. | to 7. 27. 1 of 3| to 84. 8. 15 to 35. 18. f to 15. 28. f of f to 75. 9. 19 to 95. 19. | to 16. 29. 8^ to ^ of 90. 10. 20 to 110. 20. -& to 3. 30. J of 2| to of 18. 2. Find the fractional relation of: 1. ftof. Suggestion : f == j and f = & ; 10 = ty of 9. 2. $i to $. 12. 2J- to f . 3. $f to $f . 13. 1 to 2|. 4. 1^ to $. 14. 2| to 7|. 5. $| to $1. 15. 74- to 2|. 6. ^tofrfr. 16. 3Jto8f 7. $f to $|. 17. $6 to $100. 8. $f to $ 18. $8f to $100f . 9. $f to $f 19. f of If to 3. 10. $f to $f 20. 9^P- to 12^-. 11. $ T 3 g- to $f. 21. If to 31 X f of f To find a Number from its Fractional Relation to Another Number. 1. Tn the equation, 3 = -^ of 9, the ^ expresses the relation of 3 to 9. 118 PRACTICAL ARITHMETIC 2. If the question arise, 3 is ^ of what number ? what is the answer? 3. The analytical answer is what? Suggestion : % of the number 3 ; f, the whole of the number, = what ? 4. Answer the following inquiries : 1. 12 is ^ of what number? 2. 24 is of what number ? Suggestion : of the number = ^ of 24. 3. 24 is -| of what number ? 4. 28 is f- of what number? 5. 48 is % of what number ? 6. TT- is f of what number ? Suggestion : ^ of the number = ^ of ffi. 7. -Jf is -^ of what number? 8. !~| is if- of what number? 9. ^2. i s HI of what number? 10. l of 3 is -f- of what number? REVIEW. 1. The sum of two fractions is f , and their difference is \. Required the fractions. Suggestion : Were the difference 0, the fractions would be | and ^. Hence the greater fraction = f -f i of \ ; the less = f | of | . 2. If a man can cut in one day ^ of a field containing 7 acres of wheat, how many acres can he cut in ^ of a day ? 3. Reduce -J-, -| , f and ^ to equivalent fractions whose denominators shall be 24. 4. Add ^, ff, 4f, 15|, and explain fully. , 4 of 4 of 7f 5. Find the value of a - & . FRACTIONAL RELATIONS H9 6. The product of three numbers is % ; two of the num- bers are 2-J- and % what is the third ? 7. A housekeeper bought 6 mahogany chairs at 3|- dollars each, and gave for them 2 ten-dollar bills and one five-dollar bill. What change ought she to receive? 8. Find the sum of 1^- + 3J4- -f 4lf. 9. A box contains 345 eggs. What is their value at $.16f a dozen? 10. If |- of an acre of land cost 101 dollars, what will f of an acre cost? 11. Reduce 816 T 5 T to an improper fraction. 12. Subtract l of T 9 from 8 ^ "^ 2 4\ 16 1 21 13. Simplify ^ ; also, p; y 14. If T 3 jj- of an acre of land is worth $79^, what is 1 acre worth ? 15. Reduce ^ 1 to a simple fraction. i 16. From what must 6f be subtracted to leave 1 of 3|? 17. At -2- of a dollar per bushel, what will be the cost of of a bushel of potatoes? 18. | of 27 is f of what number? 19. From f of f take of |. 20. Simplify ^_X 18 X 75 X 6^ 7 25 X 17 X 14 X 9 21. A merchant paid 85^ dollars for 15^- tons of coal. How much did the coal cost him per ton ? 22. Simplify 8^ - 2 - 3J + 6^ - 5|. 23. 150 is ^ of what number? 24. $150 was paid for a horse and saddle. If the cost of the saddle was ^ of the cost of the horse, what was the cost of each ? 120 PRACTICAL ARITHMETIC 25. 26. Find the quotient of 20f -t- -ft of 27. How many lemons, at -f^ of a dollar a dozen, will pay for 80 oranges at 2-J- cents each ? 28. Four loads of hay weigh respectively 1723f, 231 7f, 1547f, and 1357^ pounds. What is the total weight of the hay? 29. Reduce ifff to its lowest terms. 30. A farmer had ^ of his sheep in one pasture, \ in another, and the remainder, which were 77, in a third pasture. How many sheep had he? 31. If I give A. \ of my money, B. ^ of it, and C. \ of it, what part of my money have I left ? 32. The divisor is -ffg, and quotient -f-|-|. What is the dividend ? 33. A man bought land for $5130, and sold it so as to gain y^ of the cost, the gain being $3 per acre. How many acres did he buy ? 34. After buying a suit of clothes for $60 I found I had ^ of my money left. How much had I at first ? 35. What number, diminished by the difference between J and of itself, leaves a remainder of 34 ? 36. Divide f of 3 by f of {%. 37. If 3J bushels of oats will sow an acre, how many bushels will it take to sow 7-^ acres? 38. Reduce *** X * 2* i of i 39. Find the value of 5 + 6f 7 T 7 F + ||. 40. The circumference of a bicycle wheel is 7^ feet ; the circumference of another bicycle wheel is 7^ feet. How many more times will the smaller wheel turn than the larger ingoing 5280 feet? 41. Divide (| - |) by (f- |). FRACTIONAL RELATIONS 121 42. Reduce ^ to 786ths. 43. If -f- of a cord of wood cost $6|-, what will 10 cords cost? 44. Find the sum and product of |-, -J and -f . 45. What is the value of (f of f of 3f + 8|) -^ (10 - 7 A)? 46. A shepherd, being asked how many sheep he had, answered that -f- of f of the whole number was 45. How many had he? 47. Find the G. C. D. and the L. C. Dd. of 833, 1127, 1421, 343. 48. What part of f is ? 49. If 3|- yards of cloth cost 84 cents, how much is that per yard ? 50. The reciprocal of -| is 1 -f- -J. What are the sum, the difference, and the product of -J and its reciprocal ? 51. Reduce to a common denominator and add -| X X |-j A, I. and -ft- 52. A lot which cost $400 was sold for $480. What part of the cost was gained ? 53. How much is the sum of ^, |-, ^ greater or less than | of the sum of 11 1? 54. If bricks cost $8.50 a thousand, what is the cost of one brick? 55. Simplify -jf- Give the principles involved. 12 56. A regiment lost in battle 250 men, which was f of the regiment. What was the number of men before the battle ? 57. Divide 98 by 11-^, and multiply the quotient by f of 8f. 58. Reduce T 7 F , ii, fj to their L. C. D. 122 PRACTICAL ARITHMETIC 59. A barrel of beef, which holds 200 pounds, was |- full. How many pounds would there be left in it after 53f pounds were taken out ? 60. A man having $5^ bought a knife, and then had left $4 T 9 g-. How much did the knife cost? 61. Mr. Gould sold a cow for $30, which was f of what she cost him. How much did he lose? 62. At 9^ dollars a barrel, how many pounds of flour can be bought for $3? [One barrel =196 pounds.] 63. When hay is worth $9J a ton, what will f of 3|- tons cost? 64. A. and B. kill an ox. A. takes f and B. the re- mainder. If B.'s share weighs 361^ pounds, what is the weight of the ox ? 65. What fraction of 18f is 6f ? 66. If 2f acres of land cost $220, what will be the cost of 17-J acres? Indicate the work and cancel. 67. If 15 men do a piece of work in 10-| days, how long would it take 5 men to do the same work ? 68. If 5 be added to both terms of the fraction |-, will its value be increased or diminished ? 69. If A. can do a piece of work in 5 days and B. in 8 days, how long will it take both to do it ? 70. There are two numbers whose sum is 140, one of which is f the other. What are the numbers ? 71. The product of two numbers is 6, and one of them is 1846. What is the other? 72. If 3 dozen lemons cost $1|, what will be the cost of 56 lemons? 73. If 7^- pounds of rice cost $.90, how many pounds can be bought for $1.10? 74. A clerk earns $lf a day, and spends $-- a day. How much does he save in a year? FRACTIONAL RELATIONS 123 75. Multiply by TO. (f| x ff) -5- ( x f|)- 77. Margaret, in attempting to divide a fraction by |~|, in- verted the dividend instead of the divisor, and obtained a quotient of ^J-. What was the given fraction ? 78. If a man's brain is ^ of his weight, and weighs 3-^ pounds, what is his weight? 79. Which is the greater, -fifa or -^ ? 80. When land is worth 1 00 dollars per acre, what part of an acre will be worth 26f dollars ? 81. A cistern can be filled by one pipe in 15 hours, and by another in 20 hours. In what time can the two pipes fill it flowiug together ? 82. What part of " is ? t i 83. What is the quotient of 1^ divided by its recip- rocal? NOTE. The reciprocal of a quantity is 1 divided by that quantity. 84. Change f to a fraction whose denominator shall be 35. 85. Find the least number of apples that, arranged in groups of 8, 9, 10, or 12, will have just 6 over in each case. 86. Three times a number plus -| of it, plus 4f times the number plus ^ of it, are how many times the number ? 87. If f of a steeple casts a shadow 83-| feet long, how long is the shadow cast by f of it? 88. A man has 4^ bushels of potatoes, which is -| of the quantity that he planted. How many did he plant ? 89. A man who received -J- of his father's property gives to his own son J of what he received. Who then has -fa of the whole ? 124 PRACTICAL ARITHMETIC 90. Two men require 8^ days to take account of a stock of goods. Six men would need what time ? 91. What fraction is the quotient of -fffc -=- ^? 92. In sowing a field, one kind of seed is used at the rate of 12^ bushels to 5 acres. What will be required to sow 22f acres, using -f- as much to the acre as before ? 93. When oysters yield 1J gallons to the bushel, a 25- gallon barrel can be filled from how many bushels in the shell? 94. l of a bushel of berries is picked ; ^ of them are sold to one man, 1 of the remainder to another. What fractional part remains unsold ? 95. Oranges are bought at 3 for $.05 and sold at 4 for $.09. What is gained on a box of 9 dozen, 1 in 12 of which are worthless. 96. 21 f | | | = ? 97. Find the G. C. D. and the L. C. Dd. of 45, 90, 100, and 200. 98. A., B., and C. can do a piece of work in 10 days. A. can do it in 25 days, and B. in 30 days. In what time can C. doit? 99. A. and B. together had $5700. f of A.'s money was equal to -J of B.'s. How much had each ? 100. Define: 1. Fraction. 10. Reduction. 2. Decimal fraction. 11. Higher terms. 3. Common fraction. 12. Lower terms. 4. Fractional unit. 13. Lowest terms. 5. Denominator. 14. Like fractions. 6. Numerator. 15. Unlike fractions. 7. Proper fractions. 16. Common denominator. 8. Improper fractions. 17. Least common denominator. 9. Mixed number. 18. Fractional relation. DECIMAL FRACTIONS 125 101. Kepeat: 1. The principles of Addition of Fractions. 2. The rule for Addition of Fractions. 3. The brief directions for finding L. C. D. of Fractions. 4. The principles of Subtraction of Fractions. 5. The rule for Subtraction of Fractions. 6. The principles of Multiplication of Fractions. 7. The rules for Multiplication of Fractions. 8. The principles of Division of Fractions. 9. The rules for Division of Fractions. 10. The principle of Fractional Eelation. 102. Invent and solve : 1. Five problems in Reduction of Fractions. 2. Five problems in Addition of Fractions. 3. Five problems in Subtraction of Fractions. 4. Five problems in Multiplication of Fractions. 5. Five problems in Division of Fractions. 6. Five problems in Relation of Fractions. 7. Five miscellaneous problems in Fractions. DECIMAL FRACTIONS. DEFINITIONS. 1. A Decimal Fraction denotes one or more of the decimal divisions of a unit. 2. Decimal Fractions are usually called decimals (Latin, decem, "ten"). 3. A Pure Decimal consists of decimal figures only, as .234. 126 PRACTICAL ARITHMETIC 4. A Mixed Decimal consists of an integer and a decimal, as 23.005. 5. A Complex Decimal has a common fraction on the right of the decimal, as .06f . NOTATION AND NUMERATION. 1. By placing a mark (.), called the decimal point, after units of the first order, the numeration and notation table is extended to express parts of a unit on the decimal scale. 2. The relation of decimals and integers to each other is clearly shown by the following Numeration Table. . H 3 I I , * | ,; s | s ^H 1 ^ N 6 CO .2 g 1 1 1 1 1 1 1 s CO o> 1 P O 1 Is 1 G w G o> H 1 G w G -1 H ^ c C/3 c H s P 1 c w 1 G o> H c w s 1 C W 9 8 7 6 5 4 3 2 i . 2 3 4 5 6 7 8 9 INTEGERS. DECIMALS. By examining this table we see that : Tenths are expressed by one figure. Hundredths are expressed by two figures. Thousandths are expressed by three figures. Ten thousandths are expressed by four figures. Hundred thousandths are expressed by five figures. Millionths are expressed by six figures. 3. The decimal point is a separatrix, not a period ; it is read "and." DECIMAL FRACTIONS 127 Remember that the name of the 6th decimal order is Millionths, and give orally the names of the following orders : 6th order, 5th order, 4th order, 3d order, 2d order, 1st order, 3d order, 5th order, 4th order, 6th order, 1st order, 5th order, 2d order, 4th order, 3d order, 6th order, 5th order, 4th order, 3d order, 2d order, 1st order, 6th order. In what decimal place do you find : Millionths ? Thou- sandths? Tenths? Hundredths? Ten-thousandths ? Hun- dred-thousandths? Ten-millionths ? Hundredths? Mil- lionths ? Thousandths ? 4. Eead the following: 1.2, 1.03, 1.004, 1.0005, 1.00006, 1.000007, 2.008, 3.09, 4.0001, 5.000002, 6.00003, 7.0004, 8.9, 9.10. PRINCIPLES. 1. Decimals and integers are subject to the same law of local value. 2. Each cipher inserted between the decimal point and the first figure of a decimal diminishes the value of the decimal ten-fold. 3. Annexing ciphers to a decimal does not alter its value. .05 = .050, for thousandths add nothing to 5 hundredths. 4. The denominator of a decimal, when expressed, is 1 with as many ciphers annexed as there are orders, or places, in the decimal. Read 7.039. ANALYSIS. 7 is an integer representing 7 units, and is read " seven." The decimal point is read "and " denotes the absence of tenths, and is not read. 3 hundredths + 9 thousandths is read " 39 thousandths." Hence 7.039 is read " 7 and 39 thousandths " RULE. Bead the decimal as an integral number, and add the decimal name of the right-hand figure. 128 PRACTICAL ARITHMETIC EXERCISES. 1. Read the following : 1. .7. 2. .36. 3. .625. 4. .025. 5. .0005. 6. .12345. 7. .789123. 8. .405607. 9. .890123. 10. .456789. 11. 8.54. 12. 85.4. 13. 9.213. 14. 7.389. 15. 12.3601. 16. 19.0032. 17. 25.00081. 2. Write decimally 13 thousandths. 18. 29.15625. 35. 6.839. 19. 341.63456. 20. 1001.000089. 21. .6305. 36. .24|. 37. 3.70& 38. 7.039. 22. .446|. 23. .00371. 39. 8.1367. 40. 7.0308f. 24. .0506. 25. .087345. 41. 9.1007& 42. 146.0302056. 26. 6.00056. 43. 376.932474. 27. 11.04735. 44. 2.234006. 28. 63.04048. 45. 487.000081035. 29. 100.000001. 46. 586.0004003256, 30. 734.819181. 47, .5. 31. 341.63456. 48. 5.078. 32. .684. 49. 8.008. 33. .084. 50. 6.2040. 34. .004. 51. 37.40253. ANALYSIS. 13 thousandths = one hundredth -f- 3 thousandths. tenths are given. As the number is a pure decimal, the expression of it must begin with the decimal point. Hence 13 thousandths expressed decimally is .013. RULE. "Write the number as an integer, and give the right-hand figure the place indicated by the decimal name of the number. 3. Express decimally : 1. Seven tenths. 2. Nine tenths. Twelve hundredths. Seventeen hundredths. DECIMAL FRACTIONS 129 3. Four hundredths. 42 hundredths. 4. 125 thousandths. 22 thousandths. 5. 20 hundredths. Eight thousandths. 6. 30 thousandths. 206 thousandths. 7. 3027 ten-thousandths. 8. Three hundred ten-thousandths. 9. Forty-two ten-thousandths. 10. 145 hundred-thousandths. 11. Fifty-one hundred-thousandths. 12. One hundred seven million ths. 13. 306 ten-millionths. 14. 3259 hundred-thousandths. 15. 429 ten-millionths. 16. 4268 hundred millionths.. 17. 13,760 millionths. 18. Three hundred forty-two millionths. 19. One hundred forty-five hundred thousandths. 20. 703,205 millionths. 4. Express as mixed decimals the following : 1. 5&. 8- 24^V 15. 2- 7JV. 9. 27^. 16. 3. 8 T ^. 10. 54^^. 17. 4. ftrfo. 11. 74 T | W . 18. 5. 12^. 12. M&M,. 19. 6. 16^Hhr. 13. 48. 20. t. I^^^' ^* UNITED STATES MONEY. 1. Read $12.925 as a mixed decimal, and as dollars, cents, and mills. It is read u 12 and ^-thousandths dollars," or "12 dollars, ninety-two cents, five mills." 130 PRACTICAL ARITHMETIC 2. Read in like manner the following : 1. $89.06. 5. $59.375. 9. $1.375. 2. $94.254. 6. $86.047. 10. $0.876. 3. $69.045. 7. $344.002. 11. $0.093. 4. $195,005. 8. $20.25. 12. $0.001. 3. Express decimally $ T 4 8 o, $20J, $35^ $ r jfo, $4^, five cents, five dimes, five mills, five dollars five cents five mills. REDUCTION. To Like Decimals. $ T T = $^0- Therefore $.06 = $.060. PRINCIPLE. Annexing ciphers to a decimal does not alter its value. EXERCISES. 1. Reduce .7, .05, and .304 to like fractions. Process. Explanation. ^ : .700 Thousandths is the lowest order given, hence all QK _ Q-Q the fractions must be reduced to thousandths. Since annexing ciphers to a decimal does not alter its value, - .olH we annex two ciphers to .7, thus rendering it 700 thousandths ; one cipher to .05, thus rendering it 50 thousandths. RULE. By annexing ciphers give each decimal the same number of decimal places. 2. Reduce to like decimals the following : 1. .25, .025, .37. 2. .523, 4.36, 5.0315. 3. .4036, .5, .375. DECIMAL FRACTIONS 131 4. .06, .008, .4267, .026. 5. .409, 3.61, .75, .10055, 19.6. 6. 7.07, 5.0909, 1.9090, 19.099. 7. .12, .8, 306.973, .004, 48.56. 8. .0436, .04506, .82. 9. .8104, .0008, 8000.4. 10. 8.1, .43, .68, 3.96. To a Common Fraction. 1. What is the denominator of .125 ? 2. What is its numerator ? 3. Write .125 as a common fraction. 4. What part of the expression .125 did you omit? EXERCISES. 1. Reduce .375 to a common fraction. Process. RULE. "Write the decimal, omitting the decimal point; supply the decimal denominator, and reduce the fraction to its lowest terms. 2. Reduce the following decimals according to the rule : 1. .45. 8. 4.0125. 15. 23.075. 2. .027. 9. .4355. 16. .354. 3. .72. 10. 10.25. 17. .00625. 4. 1.39. 11. .0005. 18. .05375. 5. .375. 12. .5000. 19. 15.064. 6. .625. 13. 10.25. 20. .005396. 7. 4.75. 14. 15.725. 21. .0007890. 132 PKACTICAL ARITHMETIC COMPLEX DECIMALS. EXERCISES. 1. Reduce .9^ to a common fraction. Process. Explanation. ,9 = |i- = ff = If Multiplying both terms of *$ by 3, we obtain f $. 2. Reduce in like manner : 1. .16f. 8. .04|. 15. $66.66f. 2. .3}. 9. .0371 16. $25.14f 3. .561 10. .5621 17. $50.061 4. .33t. 11. $5.9f 18. $100.871 5. .121 12. $12.18. 19. $700.371 6. .16f. 13. $33.031 20. $1000.111 7. .871 14. J55.83J. 21. $33.621. COMMON FRACTIONS. 1. What is the denominator of a common fraction that may be directly expressed as a decimal? 2. If ^ be reduced to a decimal, what is the smallest denominator it can have? 3. | = how many lOths? 4. How is ^5- written decimally ? How -f ? 2 units _ 20 tenths _ 4 5 5 5 units 5000 thousandths ^5 8 8 5. How does the number of places in the quotients agree with the number of ciphers annexed ? DECIMAL FRACTIONS 133 EXERCISES. 1. Reduce f to a decimal. Process. Explanation. JLJLQJL .375 We find by trial that three ciphers must be an- nexed to 3 to secure a complete quotient. The three ciphers annexed require the pointing off of three decimal places in the quotient. RULE. Annex ciphers to the numerator and divide by the denominator. Point off in the quotient as many decimal places as there are ciphers annexed. 2. Reduce the following to decimals : 1. i. 9. |if. 17. Jf 25. 2. f, 10. f. 18. |f. 26. 3. f. 11. f. 19. |f. 27. 4. f 12. f. 20. ff 28. ^5 1Q 1 91 51 9Q o. yj. M. YJ-. zi. f T . z. -nnnnr- 6. A- 14. A- 22. fft. 30. 7. if. 15. if 23. tfff. 31. 8 1 7 . 16 * 3 24 7 32 NOTE. It is not possible in every case to render the division exact by annexing ciphers. Frequently a remainder occurs, which may be used as the numerator of a fraction ; or it may be disregarded, and the sign -(- employed to denote the incompleteness. 3. Reduce f to a decimal. Process. JLJLO_Q_10 + ^75^ 3. (155.006 .32) -f (80.0032 + 55.1). 4. 12.07 11.432765. $-97 - ($ + $.621 + $|). MULTIPLICATION. ! A X rk = what? - 3 X .07 == what? 2. How many decimal places in both factors ? 3. How many decimal places in their product ? PRINCIPLE. The product of decimal factors has as many decimal places as the factors. EXERCISES. 1. Multiply 9.06 by .045. Explanation. 9.06 = fft ; .045 = T fe ; foe x ^ = _y^ O r, since the factors have 2 -f- 3 or five decimal places, the product must have five decimal places (Principle). .40770 RULE. Multiply -without regarding the decimal point, but in the product point off from the right as many places for deci- mals as there are decimal places in the factors. NOTE. Should the result of a multiplication not contain as many figures as the factors contain decimal places, we must supply the deficiency by prefixing ciphers, as in .02 X -003 = .00006. MULTIPLICATION OF DECIMAL FRACTIONS 139 2. What is the value of : 1. 13.2 X 2.475. 2. .132 X 2.475. 3. .236 X 12.13. 4. 9.06 X .045.. 5. .008 X 751.1. 6. 70 X 387.45. 7. 70.07 X 387.45. 8. 4.2 X .065. 9. 2000 X .075. 10. .436 X .46. 11. .579 X .035. 12. 3.94 X 3.84. 13. 5384 X .0064. 14. .014 X 6.2 X .007. 15. 200 X 3| X .006. 16. 947.36 X .00423. 17. 6| X 7f X .81 18. .305 X .00046. 19. 10000 X 8.6213. 20. 8.47 X 9.432. 21. .84 X 9.60. 22. 3.468 X 2.008. 23. 81 x 5.076. 24. 28.8 X 41 25. 8.375 X 61 26. | X 2.5. 27. 1561 X .625. 28. 1.776 X .24. 29. 1.603 X 2.564. 30. .0069 X 95.6. 31. 2000 X .075. 32. 8000 X .0755. 33. .785 X .0191. 34. .00432 x .00037. 35. 81 X .071 x 10. 36. .37 X 10000. 37. 161 X 14.55. 38. 277| X 12.004. 39. 3 hundredths X 3 thousandths. 40. Four hundred thousand two hundred sixty-eight ten-millionths by two hundred sixty and two hundred seventy-five thousandths. THE DECIMAL POINT AS A MULTIPLIER. 1. .0004 X 10 = what? 0.004 X 10 = what? 00.04 X 10 == what? 000.4 X 10 = what? 2. Since .0004 X 10 = 0.004 and 0.004 X 10 = 00.04, how does the decimal point become a multiplier? 3. To become a multiplier, does it move toward the right or the left? 140 PRACTICAL ARITHMETIC 4. Its removal one place to the right multiplies the number by what ? Its removal two places multiplies by what ? Three places ? PRINCIPLE. Every removal of the point one place toward the right multiplies the number by ten. RULE. To multiply by a number consisting of 1 with ciphers an- nexed, remove the decimal point as many places towards the right as there are ciphers in the multiplier. EXERCISES. 1. Multiply .394 by 100. Process. Explanation. ~ . Since the multiplier is one with two ciphers annexed, we remove the decimal point two places towards the right, and have 39 and 4 tenths as product. 2. Multiply: 1. 8.7 by 10. 7. 9.2 by 10. 2. .0069 by 10. 8. 7.49 by 100. 3. 95.6 by 100. 9. .036 by 100. 4. .0453 by 100. 10. 854.3 by 1000. 5. 4.069 by 1000. 11. 1.00182 by 10,000. 6. .000094 by 10,000. 12. 76.541 by 1,000,000. PROBLEMS. 1. Find the value of: 1. 57 horses, at $86.375 each. 2. 200 barrels of flour, at $8.53^- each. 3. 251 yards of cloth, at $5-1- a yard. 4. 236 bushels of oats, at $.515 a bushel. 5. 36f bushels of clover seed, at $4.52 a bushel. 6. 1000 pounds of wool, at $.375 per pound. 7. 280 barrels of apples, at $3f a barrel. DIVISION OF DECIMAL FRACTIONS 141 8. 100 cords of wood, at $5.47 a cord. 9. 305^ acres of land, at $82f an acre. 2. A lady made the following purchases : 47 yards of sheeting, at $.14^ per yard ; 9 yards of ribbon, at $.45-|- per yard ; 38 yards of silk, at $3.46 per yard. What did her purchases cost her ? 3. Multiply six hundred twenty-five ten-millionths by three and eight thousandths. DIVISION. 1. .6 X -9 = what? Since one factor of .54 is .6, what is the other factor ? Since .54 -r- .6 = .9, how does the number of decimal places in the dividend compare with the number in the divisor and quotient ? 2. .054 = .09 X .6. Assuming .054 to be a dividend and .09 to be a divisor, what is the quotient? Since the dividend has 3 decimal places and the divisor 2, how can you operate with 3 and 2 to find the number of places in the quotient? PRINCIPLE. The number of the decimal places in the quotient equals the number of places in the dividend minus the number in the divisor. EXERCISES. 1. Divide 82.32 by 2.1. Process. Explanation. 2.1 ) 82.32 ( 39.2 2.1 = ft, divisor ; 82.32 = -^3_2 . _ 8 _?jL2 x |o (J3 = - 3 T 9 o 2 - = 39.2. Or, dividing without regard to -j QO the decimal point, we have 392 as quotient. -. Q Since the dividend has two decimal places and the divisor one, the quotient has one; hence the 42 quotient sought is 39.2. 42 142 PRACTICAL ARITHMETIC BULB. Divide without regard to the decimal point, but finally point off from the right of the quotient as many figures for decimals as the number of decimal places in the dividend exceeds the number of those in the divisor. NOTES. 1. "When the 'quotient does not contain as many figures for decimals as the rule requires, supply the deficiency by prefixing ciphers. 2. Before beginning to divide, it is best to make the number of decimal places in the dividend at least equal to the number of decimal places in the divisor. 3. When the process of division has used only as many decimal places of the dividend as equal the number of decimal places of the divisor, the quotient will be an integer. 2. Divide: 1. 21.6 by .006 (Apply Notes 2 and 3). 2. .4913 by 1.7. 14. 3.2572 by 3.4. 3. 2.1952 by .028. 15. 467.37 by 100. 4. .5964 by 35 (Note 1). 16. .003125 by .125. 5. 26.01 by 51. 17. .03759 by .01253. 6. .456 by .06. 18. .13 by .026 (2 and 3). 7. 4375 by .25 (Note 2). 19. .75 by .025. 8. 89.756 by 8. 20. 7 by .007. 9. 36.792 by 4.2. 21. .4 by .008. 10. 44.98 by 1.3. 22. .005 by .0015. 11. .0002 by .02. 23. .0003 by 3.75. 12. 325.72 by 34. 24. .018 by 3600. 13. 10,864.2 by 5432.1. 25. 2.0064 by 2.09. 3. Divide : 1. 1235.434256 by 20.074. 2. 195.388698 by 6.0708. 3. 273.2879688 by 6.0708. 4. 3.859243392 by 3.5702. 5. .00020596611 by .03507. 6. 625 ten-thousandths by 25 millionths. DIVISION OF DECIMAL FRACTIONS 143 THE DECIMAL POINT AS A DIVISOR. 1. 4000 -s- 10 = what? 400.0 -7- 10 = what? 40.00 ~- 10 = what? 4.000 -5- 10 = what? 2. Since 4000 -r- 10 = 400.0 and 400.0 -s- 10 = 40.00, how does the decimal point become a divisor ? 3. To become a divisor, does the decimal point move towards the right or the left ? 4. Its removal one place to the left divides the number by what? Its removal two places divides the number by what? Three places? PRINCIPLE. Every removal of the point one place toward the left divides the number by ten. RULE. To divide by a number consisting of 1 with ciphers an- nexed, remove the decimal point as many places toward the left as there are ciphers in the divisor. EXERCISES. 1. Divide 48.26 by 100. Process. Explanation. 4826 Since the divisor is 1 with two ciphers annexed, we re- move the decimal point two places toward the left, and have .4826 as quotient. 2. Divide: 1. 534.79 by 100. 6. 4956.74 by 10,000. 2. 492.568 by 1000. 7. .038649 by 100,000. 3. 24.9653 by 1000. 8. 82.253 by 1,000,000. 4. 5.908 by 100. 9. $9.391 by 10. 5. ,07156 by 1000. 10. 785.437 by 10,000. 144 PRACTICAL ARITHMETIC PROBLEMS. 1. Find the value of a single one if: 1. 144 eggs cost $2.88. 2. 20 francs = $3.86. 3. 20 shillings = $4.8665. 4. 25 dress patterns 102.50 yards. 5. 125 bushels of oats cost $36.50. 6. TOO acres of land cost $3156J. 7. 72.50 C. cigars cost $84.10. 8. 1.440 M. bricks cost $10.44. 9. 22 days' work = $29.70. 10. .62 of a ton of hay cost $11.47. 11. 7| acres of land cost $70.125. 12. .7} yards of cloth cost $.73625. 13. 5 weeks' provisions cost $47.31 J. 2. Find the cost of 8.25 tons of hay when 2.2 tons cost $311 3. Find the value of (6.25 -f- 3J-) -r- (3J- .275). SHORT PROCESSES. When the Multiplier approximates 1OO, 1OOO, etc. 1. How much less than 10 times a number is 9 times that number ? 2. How much less than 100 times a number is 98 times that number ? 3. Multiply 4965 by 99. Process. Explanation. 496,500 496,500 = 100 times 4965. 4965 4965 1 time 4965. 491,535 491,535 = 99 times 4965. SHORT PROCESSES 145 4. Multiply : 1. 4993 by 99. 6. 597,076 by 995. 2. 4967 by 98. 7. 575,854 by 98. 3. 59,678 by 999. 8. 954,367 by 96. 4. 98,849 by 98. 9. 697,547 by 996. 5. 457,836 by 997. 10. 5,064,367 by 97. 5. What cost 496 bushels of wheat at $.98 per bushel? 6. Find the cost of 240 acres of land at $96 per acre? 7. What is the value of .0755 X 997? 8. 2484 pounds of tea at $.96 = what? 9. Find the value of When One Part of the Multiplier is an Exact Divisor of Another Part. 1 . 6 is how many times 3 ? 2. When you have taken a number 3 times, how many times that product will 6 times the number be? 3. Multiply 1728 by 63. Process. Explanation. 1728 6 tens = 2 tens X 3. 1728 X 3 = 5184 units. 6184 X g3 2 tens = 10,368 tens. Adding the two products we have "5184 108,864. 10368 NOTE. Be careful in placing the first figure of each 108,864 partial product. 4. Multiply : 1. 4795 by 124 [12 = 4 X 3]. 9. 7495 by 735. 2. 4936 by 93 [9 = 3 X 3]. 10. 5349 by 927. 3. 7935 by 123. 11. 23,894 by 756. 4. 25,384 by 142. 12. 47,523 by 918. 5. 39,764 by 246. 13. 47,596 by 2505. 6. 79,546 by 328. 14. 39,864 by 8024. 7. 57,324 by 217. 15. 49,975 by 9045. 8. 4793 by 945 [45 = 5 X 9]. 16. 35,656 by 642. 10 146 PRACTICAL ARITHMETIC To Multiply Process. 4)189800 47450 Process. 1900 4 76.00 Process. 3)189900 6330Q Process. 1728 3 51.84 Process. 8 ) 569600 71200 Process. 3.14156 8_ .2513248 Process. 6)78.54 J3.09 or Divide when the Multiplier or Divisor is an Aliquot Part of 1O, 1OO, or 1OOO. 1. Multiply 1898 by 25. Explanation. Since 25 = i, we multiply by 100 and divide the product by 4. 2. Divide 1900 by 25. Explanation. if inverted =r T * T ; hence we multiply by 4 and point off' two places to the right 3. Multiply 1899 by 331 Explanation. Since 33| = -MJ-&, we multiply by 100 and divide the product by 3. 4. Divide 1728 by 331 Explanation. J-f^ inverted = T ^ ; hence we multiply by 3 and point off" two places to the right. 5. Multiply 5696 by 121. Explanation. Since 12J i-jp, we annex two ciphers and divide the product by 8. 6. Divide 3.14156 by 121 Explanation. if inverted = T ^ ; hence we multiply by 8 and point off five plus two places to the right. 7. Multiply .7854 by 16|. Explanation. Since 16| = ^{p, we multiply by 100 and divide the product by Q, SHORT PROCESSES 147 Process. 8. Divide 1492 by 16f 1 499 Explanation. f i$ inverted T 7 ; hence we multiply by 6 and 89.52 point oft' two places to the right. 25, 331, 12|, an d 16f are aliquot parts of 100, i.e., they are such parts as exactly divide 100. Other aliquot parts of 100 may be dealt with similarly ; also aliquot parts of 10 and 1000, etc. EXERCISES. 1. Multiply : 1. 2556 by 25. 6. 40002 by 16f. 2. 7.36 by 50. 7. 205.59 by 166f. 3. 72.06 by 33J. 8. 380.087 by 12J. 4. 207.27 by 3. 9. 5908 by 14f 5. 9.087 by 333^. 10. 390.8 by 2f 2. Divide: 1. 404 by 25. 6. 399099 by 333f 2. 5005.09 by 2f 7. 7906.73 by 16|. 3. 407.709 by 50. 8. 970008 by 166f. 4. 33659 by 33f 9. 5227.38 by 12|. 5. 9304.75 by 3. 10. 470058 by 14f PROBLEMS. Finding the Cost of Articles sold by the 1OO, 1OOO, and Ton. 1. If 100 articles cost a certain price, how many times that price will 500 articles cost? 2. If 1000 articles cost a certain price, how many times that price will 6000 articles cost? 3. How many articles does C, represent ? M. 7 how many ? 148 PEACTICAL ARITHMETIC Process. 4. What will 2673 feet of timber cost at 26.73 $2.25 per C. ? 2.25 Explanation. 13365 2673 feet == 26.73 C. feet. Since C. feet cost $2.25, 5346 26.73 C. feet cost $2.25 X 26.73 = $60.1425. 5346 $60.1425 5. What is the value of 1262 fence pickets at $12^ per M. ? Suggestion : 1262 pickets = 1.262 M. pickets. 6. I sold 6000 cigars at $4.20 per C. Find the amount received therefor. 7. I paid $10.44 for 1440. What was the price per M. ? (1440 = 1.440 M.) 8. If the price of gas be $1.75 per M., find the amount of a man's bill when 12,240 cubic feet have been consumed. 9. What cost 23| M. feet of pine at $55? 10. What cost 4| M. brick at $8 ? 11. A contractor furnished the following materials for a house : 34,600 bricks at $9.60 per M., 7960 feet of lumber at $16 per M., 9050 feet of flooring at $22.50 per M., 7600 shingles at 40 cents per C. Find the total cost of the materials. 12. If 2000 pounds cost $10.00, what will 1000 pounds cost? 13. At $5.00 per M., what will 3200 pounds cost? 14. 2000 pounds = one ton. If one ton of iron costs $30, what will 1000 pounds cost? What will 4000 pounds cost? 15. What must I pay for 7850 pounds of stone at $2.20 a ton? Process. Explanation. 2 )$2.20 200 lbs - cost $ 2 - 20 - -, -JQ 1000 lbs. cost fl.10. " 7850 lbs. = 7.850 M. 7 - 85Q Since M. cost $1.10, 7.850 M. cost $1.10 X 7.850 = $8.63500 REVIEW OF DECIMAL FRACTIONS 149 1 6. What will 3426 pounds of plaster cost at $3.48 per ton ? 17. How much must be paid for 6745 pounds of clay at $15.25 a ton? 18. For 7890 pounds of hay at $16.60 per ton? 19. For 27,936 pounds of fertilizer at $18.50 per ton? 20. For 7330 pounds of wool at $5.50 per ton ? 21. For 9041 pounds of iron at $125 a ton? REVIEW. Indicate the processes. 1. What is the quotient when 3 is divided by 3 thou- sandths ? 2. If the divisor is 207, dividend 4776, quotient 23, find the remainder. 3. The product of two numbers is |-, and one of them is ^ of 2. What is the other ? 4. Reduce .094t to a common fraction. o 5. Reduce |~f-| to a decimal. 6. Add 3.5 tons, 2.25 tons, 5 tons, 5.486 tons, 2.986 tons, 3.6 tons, 2.336 tons, and 2.376 tons. 7. Add .0273 and -fff^. 8. Subtract .00976 from 2.03. 9. Find the value of .21 of f X 50 X .Ollf 10. When pork is selling at $6.25 per hundred- weight, how much can be bought for $725? 11. Divide the sum of six thousandths and six millionths by their difference. Find six decimal places. 12. Multiply 732.89 by 33J. [Use short process.] 13. Multiply 92.5674 by 333. 14. Divide 96.325 by 121 15. How many pounds of coffee can be bought for $16.25 if 5 pounds can be bought for $1.78J? 150 PRACTICAL ARITHMETIC 16. (2.04 -r- 17 + .235 X 5000) = ? 17. Divide 250,000 by .00005. 18. I paid .33 of a sum of money for a slate, .17 for a book, and .375 for a pair of skates. What fractional part of the sum was lelt? 19. Simplify ' 3 ? j, expressing the result as a decimal. 20. What is the value of 95,150 bricks at $7.25 per M.? 21. Subtract .0507009 from .08. 22. Reduce .15f to a common fraction. 23. Find the cost of 560 pineapples at $13.35 per C. 24. Multiply 39,864 by 3609. [36 = 4 X 9.] 25. Add -fa, -|, and ^ ; subtract -fa from ^ of |-| ; subtiv.ct the second result from the first, and take -J- of the difference. 26. If 15 tons of hay cost $125.25, what will 35 tons cost? 27. A long ton = -?4M of a short ton. Reduce to a mixed O _- U U U decimal. 28. Write as decimals and as simple common fractions : 29. Find the result of 1.76 X 49.647 -=- .0088. SO. Add -|f, -J|-, -|f . Express the result as a decimal. 31. What cost 164,960 pounds of coal at $6.00 per short ton? 32. Bought 100 sheep at $3.375 a head, and sold them at $3.875. What did I gain on each, and on the whole number ? 33. What would 7f bales of cotton cost, each bale weighing 537.5 pounds, at $.1 If a pound? 34. A horse and bridle are worth $178.50; but the horse is worth 20 times the bridle. Find the value of each. 35. Reduce .7708^ to a common fraction. 36. Multiply 793.295 by .0001. REVIEW OF DECIMAL FRACTIONS 151 37. How much iron in 89,276 pounds of ore if .72 of it is pure iron ? 38. An agent charged $5.85 for collecting a bill of $260. What was his charge per dollar ? 39. If .35 of a share in a mining company is worth $31.15, what is the value of 15 shares? 40. A grocer sold 8970 pounds of sugar at $4.75 a hun- dred pounds. How much did he receive ? 41. A farmer exchanged 9 tons of hay worth $16.87^ a ton for oats at 31^ cents a bushel. How many bushels did he receive ? 42. If 49 yards of broadcloth cost $251.1 2 J, what would be the price per yard ? 43. If one acre of land costs $38.75, how much can be bought for $3560? 44. How many days' work at $1.25 a day must be given for 6 cords of wood worth $4.12^- a cord? 45. Bought a roll of carpet containing 82 yards for $45, and sold it for 75 cents a yard. Find the amount of profit? 46. Find the value of 60& + 49^ +!&& + 6} + 90|. 47. Find the cost of 3700 cedar rails at $5.75 per C. 48. If a man earns $12^- a week and spends $7-f per week, in how many weeks can he save $500 ? 49. What is the value of 86,260 bricks at $7.50 per M. 50. Express as a decimal ~ *) C* + ^ 51. What cost 49.76 pounds of raisins at 12J cents a pound ? 52. What cost 65 yards of muslin at 16| cents a yard? 53. Find, the cost of 85 bushels of apples at 33^ cents a bushel. 54. What must you pay for 50 pairs of gloves at 125 cents a pair? 152 PRACTICAL ARITHMETIC ACCOUNTS AND BILLS. DEFINITIONS. 1. A Debt is that which one person owes to another, whether money, goods, or services. 2. A Credit is that which is due from one person to another ; or, that which is paid towards cancelling a debt. 3. A Debtor is the person who owes. 4. A Creditor is the person to whom a debt is due. 5. An Account is a record of debts and credits. 6. The Balance of an account is the difference between the sums of the debts and credits. 7. A Bill describes the goods sold by giving quantity and price. 8. The Footing of a bill is the total cost. 9. A Receipt acknowledges the payment of a bill at its foot, thus : " Received payment, " JAMES JOHNSON." Common Abbreviations. @, at. Cwt., hundred weight. Mdse., merchandise. a / c , account. Do., the same. No., number. Acc't, account. Doz., dozen. Pay't, payment. Bal., balance. Dr., debtor. Pd. , paid. Bbl., barrel. Fr't, freight. Per, by Bo't, bought. Hhd., hogshead. Rec'd, received. Bu., bushel. Inst., this month. Ult., last month. Co. , company. Int., interest. Yd., yard. Cr., creditor. Lb., pind. Yr., year. ACCOUNTS AND BILLS 153 Bills are usually written in the following form : LANCASTER, PA., July 31, 1898. Bought of BAIR & STEINM AN. MR. R. B. RISK, 2500 ft. Boards, @ $27.50 per M. $68 75 1875 ft. do. " 25.00 u " 46 88 1650 Laths, " .32 " C. 5 28 1520 Pickets, " 15.00 " M. 22 80 7500 Shingles, " 6.50 " " 48 75 $192 46 Find the footings of the following bills : (i.) MR. JOHN TODD, CHICAGO, ILL., Aug. 3, 1898. Bought of RIGGS & CARTER. 25,000 ft. Pine Boards, @ $15.00 perM. 8,500 Plank, 11,850 " Scantling, 4,970 " Timber, 6,398 " do. " 9.50 " " " 7.00 " " " 3.25 " " " 4.00 " * Received payment, RIGGS & CARTER. NOTE. Finding; the value of the different items of a hill is called making the extensions." 154 PRACTICAL ARITHMETIC (2.) DR. J. C. GOOD, TRENTON, 1ST. J., Aug. 4, 1898. Bought of STEPHEN SMITH. 35 Ibs. Coffee @$.30 $ 5 Ibs Tea " .50 30 Ibs. Mackerel " .15 5 gals. Molasses " .60 20 Ibs. Sugar " .05J 3 doz. Eggs " .18 2 Ibs. Cheese " .10 3 Ibs. Butter .20 $ (3.) MR. SHERMAN ROGERS, ATLANTA, GA., Aug. 5, 1898. To PAUL R. JONES, Dr. To 48 bbl. Pork @ $12.50 " 138 bbl. Flour " 7.15 " 4 bbl. Molasses, 169 gal. " .40 " 30 firkins Butter, 2200 Ib. " .17 " 4 boxes Raisins " 4.60 " 4 bbl. Kerosene, 164 gal. " .19 " 30 doz. cans Fruit " 2.50 " 2 bundles Tobacco " .40 " 12 doz. Spices " 1.12J i $ i DENOMINATE NUMBERS 155 Set in bill form the following purchases, find the footings, and assume that the bills were paid : 4. Mrs. T. N. Butcher bought of Hervey Martin, 15 yd. of carpet @ $1.00; 50 yd. of muslin @ 12J cts. ; 18 yd. of calico @ 9J cts. ; 5 pairs of hose @ 75 cts. ; 15 yd. of gingham @ 11 J cts. ; and 25 yd. of Canton flannel @ 10 J cts. 5. Mr. D. F. Lovett bought of S. Q. Lowrey, 8679 ft. of hemlock @ $13.85 per M. ; 9640 ft. of flooring @ $24.75 per M. ; 6709 ft. of pine @ $50.00 per M. ; 4926 ft. of oak @ $35.00 per M. ; 8457 ft. of ash flooring @ $40.00 per M. 6. Mr. H. K. Landman bought of B. A. Gross, 47 bu. of wheat @ $.87 ; 60 bu. of corn @ $.60; 50 bu. of oats @ $.33; 30 cwt. of flour @ $3.50; 160 bu. of bran @ $.18; 83 Ib. of corn meal @ $.05. 7. Mr. John Rodgers bought of William H. Cartwright, 100 Ib. of breakfast bacon @ $.10 ; 55 Ib. of lard @ $.08 ; 37J Ib. of picnic hams @ $.06 ; 45 Ib. of tallow @ $.05^ ; 25 Ib. of creamery butter @ $.17; 10 doz. Western eggs @ $.12J; 16 Ib. of fowls (hens) @ $.15; 5 Ib. of cheese @ $.10J; 12J ib. of Rio coffee @ $.171. DENOMINATE NUMBERS. Denominate Numbers are Simple or Compound. A Compound Denominate Number is composed of units of two or more denominations that have among them a certain natural relation ; as, 4 feet 6 inches, or 3 bushels 2 pecks 1 quart. Compound Denominate Numbers have their origin in the existence of the various Measures in common use. 156 PRACTICAL ARITHMETIC The Measure of Value is Money, which is also called Currency. 1: United States Money consists of Coin and Paper Money. Coin is called Specie. The Coins are : GOLD. SILVER. The Double Eagle == $20.00. The Dollar . = $1.00. Eagle = 10.00. Half-Dollar = .50. Half-eagle = 5.00. Quarter-Dollar = .25. Quarter eagle = 2.50. Dime = .10. The Nickel Coin = $.05. The Bronze Coin = .01. Other United States coins found in circulation are not now coined. Paper money is issued in the form of bills whose face value is one dollar and upward. The Unit of United States Money is the Dollar. Table. 10 Mills (m.) = 1 Cent (ct.). 10 Cents = 1 Dime (d.). 10 Dimes = 1 Dollar ($). 10 Dollars = 1 Eagle (E.). $ d. cts. m. 1 = 10 = 100 = 1000. Scale: 10, 10, 10 (Decimal). 2. Canadian Money has the denominations of the United States money, except the gold coins, which are the Sovereign and Half-Sovereign. 3. French Money has the following denominations : Cen- time, Decime, and Franc. The Unit is the Franc. Table. 10 Centimes (ct.) (son-teems) = 1 Decime (dc.). 10 Decimes (des-seems) = 1 Franc (fr.). Fr. dc. ct. 1 = 10 = 100. = $0.193. Scale: 10, 10 (Decimal). REDUCTION DESCENDING 157 4. English or Sterling- Money is the currency of Great Britain. The coins are : GOLD. SILVER. The Sovereign = 20 shillings. Crown = 5 shillings. Half-Sovereign = 10 u Florin = 2 " Guinea = 21 " Shilling. Six-penny piece. Three-penny piece. COPPER : Penny, Half-penny, Farthing (four things). The Unit is the Pound or Sovereign. Table. 4 Farthings (far.) = 1 Penny (d.). 12 Pence = 1 Shilling (s.). 20 Shillings = 1 Pound (). . s. d. far. 1 = 20 = 240 = 960 = $4.8665. Ascending Scale : 4, 12, 20. REDUCTION DESCENDING. INDUCTIVE STEPS. 1. Since 4 farthings = 1 penny, how many farthings = 2 pence ? 3 pence ? 4 pence ? 5 pence ? 2. Since 12 pence = 1 shilling, how many pence = 6 shil- lings? 10 shillings? 12 shillings? 3. How many shillings in 2? In 2 6 shillings? 4. How many pence in 2 6s. 5d. ? 5. l = how many shillings? 6. Js. = how many pence? 7. \ = how many shillings ? Reduction descending changes a denominate number from a higher to a lower denomination without changing its value. 158 PEACTICAL ARITHMETIC EXERCISES. 1. Reduce 4 12s. 8d. to farthings. Process. Explanation. 4 12s. 8d. Since 1 = 20s., 4 = 80s. 80s. + 12s. = 92s. 20 since ls - = 12d., 92s. = 92 x 12d. = 1104d. 1104d. + 8d. = -i 9 ' Since Id. = 4 far., 1112d. = 1112 X 4 far. = 4448 -I -Jo* far. Hence 4 12s. 8d. = 4448 farthings. 2. Reduce f to pence. Process. l2d. _ 4 = J^d. = 133-|d. Or, f X -^ X -^ = 4448 far. 3. Reduce ^ to integers of the lower denominations. Process. f = f of 20s. = ^s. = llfs. f s. = I- of 12d. = - 3 /d. 5id. 4d. = -fof4far. = 4- far. Hence, f = 11s. 5d. Of far. RULE. Multiply by the numbers of the scale in reverse order, beginning with that number that makes one of the highest given denomination; and, as you proceed, add to the pro- ducts the given numbers of lower denominations. 4. Reduce : 1. 24 6s. to shillings. 2. 40 9s. 6d. to farthings. 3. 35 6s. 8d. to pence. 4. 7s. 6d. 2 far. to farthings. 5. 14 18s. lid. to pence. 6. 92 15s, 8d. 2 far. to farthings,. REDUCTION ASCENDING 159 7. 56 4s. lOfd. to farthings. 8. f + fs. -f }d. to pence. 9. 3. 5s. 7|d. to farthings. 10. fs. to farthing. 11. f to pence. 12. f- to farthings. 13. f- to integers of lower denominations. 14. ^ to integers of lower denominations. 15. -f- to integers of lower denominations. 16. $150 to mills. 17. $17.28 to cents. 18. 10 eagles to mills. 19. 19 francs to decimes. 20. 19 francs 8 decimes to centimes. REDUCTION ASCENDING. INDUCTIVE STEPS. 1. How many pence in 8 farthings? In 12 farthings? In 24 far. ? 2. How many shillings in 12d. ? In 36d. ? In 108d. ? 3. 40 shillings equal how many pounds ? 60 shillings ? 4. If you take 2 out of 45s., how many shillings remain ? 5. If you reduce 45s. to pounds, what is your result? Reduction ascending changes a denominate number from a lower to a higher denomination without changing its value. EXERCISES. 1. How many pounds are there in 8365 pence? Process. Explanation. 12)8365d. Since 12d. = Is., T ^ of the number 9Q \ 07,3 _[_ 1(1 pence = the number of shillings. Since ^ ' 20s. = 1, -fa of the number of shillings = the number of pounds. Hence, a365d. 8365d, ^ 34 17s. Id, = 34 17s. Id, 160 PRACTICAL ARITHMETIC 2. Reduce | ft. J = 55- *$* I \ of 12 in. = V- m. = 1| in. 3 f t ? f yd = f of 3 ft . = y. -f- rd. = 4 yd. 2 ft. If in. ft. = 2f ft. Since 1 ft. = 12 in., f of aft. =} of 12 in. = if in. = 1^ in. Hence f rd. = 4 yd. 2 ft. If in. 2. Reduce .795 Ib. Troy to units of lower denominations. Process. Explanation. .795 Ib. Since 12 oz. =1 Ib., 12 times the number of pounds 12 = tbe number of ounces; 12 times .795 = 9.540 oz. 9 540 Since 20 pwt. = 1 oz., 20 times the number of ounces = 20 the number of pwt. .540 X 12 = 10.8 pwt. Hence 190.800 .795 lb.= 9 OZ . 10.8 pwt. 3. Reduce -g-f-g- g a l- to gills. Process. Explanation. V4v2v4 = -*n Since l gal ' = 4 qt- J qt- = 2 pt., ?$$ ' 9 8 and 1 pt. = 4 gi., we multiply by the numbers of the scale, 4, 2, 4, and ob- tain by cancellation f gills. 188 PRACTICAL ARITHMETIC EXERCISES. 1. Reduce : 1. -f$ of. a bushel to the fraction of a pint. 2. -gJ-g- da. to minutes. 3. f rd. to yards, feet, and inches. 4. .065 of a gallon to integers of lower denominations. 5. ^ of a ton to lower denominations. 6. y^j- of an acre to lower denominations. 7. .007 of a bushel as a decimal of a pint. 8. .796 of a Ib. troy to lower denominations. 9. .686 to lower denominations. 10. .436 of a ream to integers. 11. .875 of a leap year to integers. 12. .795 of a league to integers. 13. -2 9 tf cu - yd. to lower denominations. 14. .115625 Ib. troy to integers. 2. Reduce: 1. yf^ bu. to pints. 9. 12 ^ 6o T. to ounces. 2. l7 1 aa to pence. 10. l8 j[ 2Tr mi. to inches. 3. -| Ib. troy to integers. 11. -J da. to integers. 4. -J mi. to integers. 12. ^ of a rod to integers. 5. ^ bu. to integers. 13. .03125 T. to integers. 6. -f A. to integers. 14. -J yd. to integers. 7. .1845 gal. to integers. 15. -/% mi. to yards. 8. .15625 bu. to integers. 16. -^ mi. to rod. Reduction Ascending-. 1. Reduce -f- gi. to the fraction of a gal. Process. Explanation. % 4 gi. = 1 pt., therefore ^ the number of y 1 \^ 1 x 1 __ A gi. = the number of pt. ; 2 pt. = 1 qt., 9 2 4 36 therefore | the number of pt. = the number of qt. ; 4 qt. = I gal., therefore i the number of qt. = the number of gal. ; hence f gi. = f X I X \ X i = aV S al - RELATION OF ONE DENOMINATE TO ANOTHER 189 2. Reduce .375 wk. to the fraction of a year. Process. .375 wk. = | wk. I x 7 = L da. X Explanation. 375 = ^VV = f I wk. = I- of 7 da. = - 2 J- da. 365 days = 1 yr., therefore - 2 ^ da. = -V- X -sh = My yr. EXERCISES. Reduce : 1. -J gi. to gal. 2. ^| min. to da. 3. ff ft. to mi. 4. 5f oz. to T. 5. f in. to rd. 6. 2fpt. to-bu. 7. ^ sec. to deg. 8. 3^ min. to da. 9. .45 Ib. toT. 10. 4 cu. in. to cu. ft. 11. sq. yd. to A. 12. $ pt. to bbl. 13. f gi. to gal. 14. 4^ sq. in. to sq. rd. 15. 9^ in. to mi. 16. | of | "l to cong. 17. 5-| X 7^ cu. in. to cd. 18. 1 li. to mi. 19. |f Ib. to T. 20. .89725 oz. to cwt. THE FRACTIONAL RELATION OF ONE DE- NOMINATE NUMBER TO ANOTHER. EXERCISES. 1. What part of 4 ft. 7 in. is 3 ft. 4 in. ? Process. Explanation. 4 ft. 7 in. = 55 in. 1 in. = ^ of 55 in. ; therefore, 40 in. = 3 ft. 4 in. = 40 in. It of .40. = _8_ 2. What decimal part of 1 is 16s. 8d. 1st Process. Explanation. Both quantities must be reduced to the same denomination. l = 240d. 16s. 8d. = 200d. 200d. = m or I of 240d - n. reduced = T 8 T . 16s. 8d. = 200d. = f = .8333 +. . reduced to a decimal = .8333 -f-- 190 PRACTICAL ARITHMETIC 3. What fraction of: 1. 1 yd. is 2 ft. 9 in.? 2. 1 mi. is4rd. 21 yd.? 3. 1 A. is 24 sq. rd. 33 sq. yd. ? 4. 4| Ib. is 51 oz. ? 5. 3 mi. is 5 rd. 2 yd. 2 ft. 2 in.? 6. 3 bbl. are 13 gal. 3 qt. 3 pt. 3 gi.? 7. 3 bu. is 1 bu. 3 pk. 4 qt. 8. 5 Ib. troy is 6 oz. 6 pwt. 6 gr. ? 9. 4 Ib. avoirdupois is 4 Ib. troy? 10. A 6-in. cube is 6 cu. in. ? 11. 65 ch. is 1430 ft.? 12. 365 da. is 4 wk. 4 da. 4 hr.? 13. 360 is 40 40' ? 14. 1 is 18s. 5Jd.? 15. 1 cwt. is 16 Ib. 11 oz.? 4. What decimal fraction of 1 bu. is 3 pk. 6 qt. 1 pt.? 2d Process. Explanation. 1 . 2 pt. = 1 qt. ; therefore the number of pt. = the n K number of qt. ^ of 1 .5 qt., which added to 6 qt. = 6.5 qt. 8 qt. = 1 pk. ; therefore 1 of the number of qt. = the number of pk. \ of 6.5 = .8125 .953125 pk., which added to 3 pk. == 3.8125 pk. 4 pk, = 1 bu. ; therefore i the number of pk. = the number of bu. I of 3.8125 = .953125 bu. 5. What decimal fraction of: 1. 1 S. is 6 25 / 36 // ? 2. 1 mi. is 5rd. 3ft. 10 in.? 3. 1 yd. is 31 in. ? 4. 1 T. is 3 cwt. 48 Ib. 9 oz. ? 5. 1 A. isl R. 39 P.? 6. 1 T. is 6 cwt. 75 Ib. ? 7. 1 da. is 11 hr. 55 min. 41.7 sec.? RELATION OF ONE DENOMINATE TO ANOTHER 191 8. 1 A. is 4276 sq. ft. ? 9. 1 Ib. is 14 oz.? 10. 82 mi. 70 rd. is 10 mi. 10 rd.? 11. 228 bu. 3 pk. is 8 bu. 2 pk. 6 qt? 12. 1 Ib. is 4 oz. 8 pvvt. 12 gr.? 13. 1 mi. is 765yd. 9 in.? 14. 1 cd. is 4 cd. ft. 8 cu. ft. ? REVIEW. 1. What will be the cost of : 1. 1 T. 15 cwt. 36 Ib. of sugar @ 3 cts. a pound? 2. 3 Ib. 9 oz. 13 pwt. of gold dust @ $.75 a pwt.? 3. 8 tons of coal @ $.26 a cwt. ? 4. 9 barrels of flour at $.03 a pound? 5. 16 Ib. 9 oz. butter at $.30 a pound? 6. 4 pk. 5 qt. cherries at 10 cts. a quart? 7. 40 rd. 8 ft. 9 in. fence at $.80 per ft. ? 8. 25 bu. of seed at 8 cts. a pint ? 9. 7 bu. 3 pk. 2 qt. blackberries at 7 cts. a qt. ? 10. 14 hhd. of molasses a{ 12 cts. a qt.? 2. How many bu. of wheat in 1260 Ib. ? 3. How many min. in the yr. 1898? 4. How many cords of wood in a pile 4 ft. wide, 7 ft. high, 70ft. long? 5. How many days of 12 hrs. each will it require to make a million figures if one figure is made each second ? 6. How many bu. of carrots will a 10-acre field produce if each sq. rd. produces 5 bu. ? 7. How many sec. are there in 365 da. 5 hr. 48 min. 49 sec.? 8. How many bu. of oats in 2000 Ib. ? 9. How many sec. from 7 A.M., Aug. 15th, to Dec. 7th, 7 P.M. ? 192 PRACTICAL ARITHMETIC 10. How many kegs, each holding 7 gal. 3 qt. 1 pt., can be filled from 11 hhd. of wine? 11. How many degrees in a quadrant measured on a meridian of the earth's surface? How many miles? 12. If 1 ton of phosphorus is used in making 10,000,000 matches, how many gr. of phosphorus on each match ? 13. If a cistern holds 4890 gal. of water, how many bbl. 'does it hold ? 14. If hay at $15 per T. is exchanged for flour at $5.85 per bbl., how many bbl. will a ton of hay buy ? 15. If a druggist put 83 43 59 of a medicinal substance in 2-gr. pills, how many pills did he make ? 16. If a man constructed a cistern 12 ft. long and 8 ft. wide to hold 150 bbl., how high did he make it? 17. If 10 bales of goods weigh 22 cwt. 86 lb., what will 155 bales of like size weigh? 18. If a silver dollar weighs 412J gr., what will 1,000,000 dollars weigh? 19. If a bbl. of flour costs 1 4s. 9d., how many bbl. can be bought for 275 10s. gd. ? (Reduce before dividing.) 20. If a man travels 24 mi. 7 fur. 30 rd. in a day, how long will it take him to travel 300 mi. 6 fur. 20 rd. ? tion : 40 rd = 1 fur. 21. If a cu. ft. of ice weighs 58.1 lb., how many tons will an ice-house hold that is 45 ft. long, 32 ft. wide, and 20 ft. high ? 22. Find the cost of 1 qt. of olive oil when 1 doz. pt. cost $3.50. 23. Find the number of gal. in a cistern 5J ft. square and 7 ft. deep. 24. Find the cost of covering the floor of a hall 46J ft. long and 14 ft. 9 in. wide with matting 1 J yd. wide at $.25 a yard. ADDITION OF DENOMINATE NUMBERS 193 25. If a glacier moves uniformly 100 ft. a year, how far will it go in 181 days? 26. If a man earns $3 per day and pays $6 a week for board, etc., how much can he save in 7 mo. ? 27. A square lot, having 32 chains on a side, contains how many acres? 28. How many times will the wheel of a carriage 17.5 ft. in circumference revolve in going 1 mi. 5 ft. ? 29. How many board ft. in 3 planks 12 ft. long, 9 in. wide, and 3J in. thick ? 30. What will it cost to carpet a room 18 ft. by 24 ft. with carpet f yd. wide at $1.25 per yd., the breadths to run length- wise ? 31. What decimal part of a yr. has passed with August 15th? ADDITION OF DENOMINATE NUMBERS. In the addition of simple numbers we have a uniform decimal scale ; in the addition of compound numbers we have a varying scale ; apart from this there is no difference in the process of adding. EXERCISES. 1. What is the sum of 12 Ib. 5 oz. 13 pwt, 21 Ib. 8 oz. 15 pwt., 13 Ib. 7 oz. 10 pwt., 51 Ib. 3 oz. 17 pwt. ? Process. Explanation. 12 20 Units of the same denomination must stand in Ib. oz. pwt. the same column. 12 5 13 The scale is 24, 20, 12. We use 20 and 12. )1 1 p. The sum of the pwt. is 55. 55 pwt. = 2 oz 15 pwt. We write the 15 under the column of pwt. I '^ and add the 2 oz. with the column of ounces. The 51 3 17 sum of the oz. is 25, which equals 2 Ib. and 1 oz. QQ i 15 We write the 1 oz. under the column of oz. and add the 2 Ib. with the column of Ib., making 99 Ib. 13 194 PRACTICAL ARITHMETIC 2. What is the sum of 37 A. 159 P. 25 sq. rd. 8 sq. ft. 126 sq. in., 20 A. 110 P. 30 sq. rd. 8 sq. ft. 131 sq. in., 345 Process. A. Ill P. 16 sq. rd. 7 sq. ft. 99 sq. in? Explanation. \ sq. yd. 4| sq. ft. \ sq. ft. = 72 sq. in. Adding 4 sq. ft. and 72 sq. in., we have a result free from fractions. 3. Find the sum of the following : PO rd. yd. ft. in. 140 5 2 7 225 3 9 402 4 10 160 30 9 144 A. P. sq. rd. sq. ft. sq. in. 37 159 25 8 126 20 110 30 8 131 345 111 16 7 99 404 62 12(1) = 7 68 i == ^(V 1 = 72 404 62 13 2 140 Suggestion : Reduce the i yd . occurring in the result to feet and inches. (2.) (3.) mi. rd. yd. ft. in. A. P. sq. yd. sq .ft. sq. in. 5 251 4 2 9 112 80 21 5 5 184 4 6 108 75 16 4 8 256 5 1 7 93 57 12 7 159 4 8 115 18 28 (4.) (5.) Ib. oz. pwt. gr. T. cwt. Ib. oz. 15 9 17 11 4 6 38 9 14 8 16 23 9 12 49 12 15 6 3 18 14 4 44 11 12 10 19 9 20 10 24 3 16 5 21 5 12 8 13 14 7 9 65 6 ADDITION OF DENOMINATE NUMBERS 195 6. 6 mi. 80 rd. 3 yd. 2 ft. 1 in., 4 mi. 75 rd. 1 yd. 2 ft. 7 in., 5 mi. 170 rd. 2 yd. 1 ft. 8 in. 4. Find the value of mi. + 13-J- rd. Process. f mi. = 266 rd. 3 yd. 2 ft. 13J- rd. = 13 rd. 1 yd. 2 ft. 6 in. Sum == 279 rd. 5 yd. 1 ft. 6 in. Explanation. 5 yd. 1 ft. 6 in. = 16J ft. = 1 rd. 279 rd. + 1 rd. = 280 rd. 5. Find the value of: 1. | mi. + .46 rd. -f 3| rd. 2. .oo| sq- y d - + - 4 f s q- ft - + - 0008 s q- in - 3. f -f 3.75s. + .975d. 4. .2965 T. + .8725 cwt. + .3725 cwt. + .1625 Ib. 5. | Ib. + 3f oz. + 5| pwt. 6- -h 7 r - + A wk. + -ft- hr. 7. A mi. + | rd. + | yd. 8. W + ' + ". 9. | + | of 5f s. 10. f wk. + f hr. + ft- min. 11. | A. + $ sq. rd. -f f sq. yd. 12. 274- cwt. + 26 Ib. -f 14 oz. [112 Ib. = 1 cwt.] 13. 1| hhd. -f 36 gal. 3 qt. 1 J pt. + | gal. + 2 qt. f pt. + 1.75 pt. 14. f of 13 + i of -/T of f of 2 12s. + f of 9d. 6. Add | of | of a guinea to .4 of .375 of 1, and express the sum as the decimal of a crown (5s.). 7. Express .05735 mi. -f- 46.25 yd. as the decimal of 7 fur. 8. What is the value of 1.1375 fathoms + .875 yd. -f 2.965 ft. + 9.75 in. in feet. 196 PRACTICAL ARITHMETIC SUBTRACTION OF DENOMINATE NUMBERS. EXERCISES. 1. From 2 mi. 116 rd. 4 yd. ft. 4 in. take 1 mi. 120 rd. 2 yd. 1 ft. 8 in. Process. Explanation. mi. rd. yd. ft. in. Units of the same denomination must stand 2 116 404 * n ^ e same column. Since we cannot sub- tract 8 in ' from 4 in ' we ac ^ to the ft ' ne 1 20 2 1 8 of the 4 yds. ; 1 yd. =3 ft. ; now having 3 316 118 ft ^ i ns tead of ft., we add to the 4 in. one of the 3 ft. ; 1 ft.= 12 in. ; 12 in. -f 4 in. = 16 in. ; 16 in. 8 in. = 8 in. Proceeding to the feet, we say, "1 ft. from 2 ft. leaves 1 ft." Proceeding to the yards, we say, "2 yd. from 3 yd. leaves 1 yd." One of the 2 mi. added to 116 rd. gives us 320 + 116 = 436 ; 436 rd. 120 rd. = 316 rd. (2- ) (3.) s. d. A. sq. rd. sq. yd. sq. ft. sq. in. From 37 17 9 From 18 40 25 6 100 take 29 18 10 take 9 50 13 7 140 (4.) (5.) T. cwt. Ib. oz. Ib. oz. pwt. gr. From 5 13 21 13 From 284 take 3 19 2 14 take 100 9 17 21 (6-) (7.) yr. wk. da. hr. min. sec. S. / // From 99 36 5 31 46 49 From 12 25 20 43| take 81 46 6 32 47 50 take 10 28 49 57f 8. From 1 hhd. 38 gal. 3 qt. 2 pt. take 60 gal. 2 qt. 1 gi. 9. From 8 Ib. take 1 Ib. \l 23 29. 10. From 5 T. take 10 Ib. 8 oz. 11. From -f oz. take -J pwt. SUBTRACTION OF DENOMINATE NUMBERS 197 Process. Explanation.. 3. oz> 7 pwt. 12 ST. Reduce the fractions to lower denom- Y , _ 01 inations and then subtract. 6 pwt. 15 gr. 12. From 2 oz. take | pwt. 13. From ^- da. take -| min. 14. From -^ hhd. take f qt. 15. From \ wk. take .9 da. 16. From f pk. take .0625 bu. 17. From .625 Troy Ib. take 4.25 Troy oz. 18. From ^ sq. rd. take f sq. yd. 19. From 45 sq. yd. take 45 sq. in. 20. From 360 take T 4 r of a quadrant. 21. Find the lapse of time between July 4, 1890, and August 15, 1898. Process. Explanation. 1898 8 15 J ul J J s ^e 7th month and Aug. the 8th month 1890 7 4 of the calen ^ ar - 8 1 11 22. Between Jan. 9, 1842, and Mar. 4, 1898. Process. Explanation. 1898 3 4 January is the 1st month and March the 3d 1842 1 9 month of the calendar. 1 month = 30 days in most 56 1 25 computations. 23. Between Mar. 2, 1857, and July 4, 1866. 24. Between Jan. 5, 1844, and Mar. 16, 1862. 25. Between May 3, 1804, and Dec. 16, 1871. 26. The Spanish- American war began April 21, 1898, and ended Aug. 12, 1898. Find the difference of the dates. 27. The American civil war began April 11, 1861, and ended April 9, 1865. How long did it continue? 198 PRACTICAL ARITHMETIC 28. The Revolution commenced April 19, 1775, and closed Jan. 20, 1783. How long did the war last? 29. Columbus discovered America Oct. 11, 1492. How long ago did that event occur? 30. A note dated Aug. 10., 1882, was paid Nov. 11, 1887. How long did it run unpaid ? MULTIPLICATION OF DENOMINATE NUMBERS. EXERCISES. 1. Multiply 5 gal. 3 qt. 1 pt. 3 gi. by 9. Process. Explanation. gal. qt. pt. gi. 9 times 3 gi. = 27 gi. = 6 pt. 3 gi. We reserve 5313 the 6 pt. to add to the next product. 9 times 1 pt. = 9 pt. ; 9 pt. -f 6 pt. reserved = 15 pt. = 7 qt. and 1 pt. We reserve the 7 qt. 9 times 3 qt. = 53 2 1 3 27 qt 27 ^ ^_ 7 qt reserved _ 34 qt _ 8 ga j and 2 qt. We reserve the 8 gal. 9 times 5 gal. = 45 gal. ; 45 gal. + 8 gal. reserved = 53 gal. 2. Multiply 18 Ib. 9 oz. 4 pwt. 16 gr. by 11. 3. Multiply 2 T. 2 cwt. 46 Ib. 7 oz. by 8. 4. Multiply 9 mi. 3 fur. 20 rd. 3 yd. 2 ft. by 6. 5. Multiply 6 yr. 5 mo. 15 da. 18 hr. by 12. 6. Multiply 26 cd. 3 cd. ft. 12 cu. in. by 18. 7. Multiply 9 17s. 6d. 1 far. by 28. 8. Multiply 5 T. 8 cwt. 64 Ib. 8 oz. by 37. 9. Multiply 7 mi. 4 fur. 15 rd. 3 yd. 2 ft. 8 in. by 48. 10. Multiply 5 Ib. 7 oz. 15 pwt. 19 gr. by 75. 11. Multiply 21 Ib. 93 23 19 16 gr. by 25. 12. Multiply 9 A. 3 R. 22 P. 6 sq. yd. 5 sq. ft. by 10. 13. Multiply 25 37' 51" by 16. 14. Multiply 10 18s. 7d. 2 far. by 29. 15. Multiply 8 mi. 120 rd. 4 yd. by 26. DIVISION OF DENOMINATE NUMBER 199 DIVISION OF DENOMINATE NUMBERS. EXERCISES. i. Divide 76 Ib. 10 oz. 14 pwt. 12 gr. by 6. Process. Explanation. Ib. oz. pwt. gr. To divide a quantity by 6 is to take ^ 6) 76 10 14 12 of it; - i of 7( > lb - = 12 lb - and 4 lb - re ' ~Vo q TK -To~ maining ; 4 lb. = 48 oz. ; 48 oz. -f- 10 oz. = 58 oz. i of 58 oz. = 9 oz. and 4 oz. remaining ; 4 oz. 80 pwt. ; 80 pwt. -f 14 pwt. = 94 pwt. ; $ of 94 pwt. = 15 pwt. and 4 pwt. remaining ; 4 pwt. = 9G gr. ; 96 gr. -{- 12 gr. = 108 gr. ; i of 108 gr. = 18. Hence the quotient is 12 lb. 9 oz. 15 pwt 18 gr. 2r Divide: 1. 112T. 16 cwt. 66 lb. by 7. 2. 17 bu. 3 pk. 4 qt. by 8. 3. 29 lb. 53 33 19 by 9. 4. 125S. 24 12' by 10. 5. 427 A. 131 sq. rd. by 11. 6. 342 gal. 2 qt, 1 pt. 2 gi. by 5. 7. 16 T. 1300 lb. by 12. 8. 120 mi. 313 rd. 3 yd. 2 ft. by 12. 9. 31 5s. 8d. by 4. 10. 196 cd. 4 cd. ft. 12 cu. ft. by 36. 11. 275 10s. 6d. by 1 4s. 9d. Process. 275 10s. 6d. = 66,1 26d. 1 4s. 9d. = 297d. 66,1 26d. -~ 297d. = 222ff. What rule is derivable from the process? 12. 48 T. 9 cwt. 23 lb. 8 oz. by 6 T. 1 cwt. 1 5 lb. 7 oz. 13. 200 mi. 6 fur. 18 rd. by 24 mi. 7 fur. 22 rd. 14. 31 cwt. 18 lb. by 3 lb. 8 oz. 200 PRACTICAL ARITHMETIC 15. 13 Ib. 7 oz. 15 pwt, by 2 oz. 10 pwt. 16. 5f mi. by 7 ft. 4 in. 17. 118 bu. 2 pk. by 7 bu. 1 pk. 5 qt. 18. 35 wk. 3 da. 15 hr. 25 min. by 17 wk. 6 da. 22 hr. 39 min. 19. 61 ft. 3 in. by 8 ft. 7 in. 20. A quadrant by 27 14' 45". LONGITUDE AND TIME. INDUCTIVE STEPS. 1. Does the earth revolve on its axis from west to east or from east to west ? 2 . It revolves once in how many hours ? 3. Does the sun actually revolve, or only appear to revolve around the earth ? 4. If the earth revolves from west to east, do Eastern or Western people behold the sun first ? 5. Has a place 30 east of Philadelphia later or earlier time? 6. When it is noon in New York, is it afternoon or fore- noon in Chicago? 7. Through how many degrees does the sun appear to travel in 24 hrs. ? 8. Then how many degrees of longitude and how many hours are compassed in a day ? Time. Longitude. 24 hrs. = 360. Dividing the equation by 24, we have : 1 hr. = 15. Dividing by 60, we have : 1 min. = 15'. Dividing again by 60, we have : 1 sec. = 15". LONGITUDE AND TIME 201 RULE. To reduce time to longitude, multiply by 15 ; to reduce longitude to time, divide by 15. PROBLEMS. 1. The difference of time between two places is 1 hr. 15 min. 30 sec. Find the difference of longitude. Process. Explanation. hr. min. sec. Since we are required to reduce time to longi- 1 15 30 tude, we multiply the given hours, minutes, and seconds by 15, and obtain 18 52' 20". 18 S2 30" 2. The difference of longitude between New York and Baltimore is 2 36'. Find the difference of time. Process. Explanation. 15)2 36' Since we are required to reduce lon- 10 m [ u 24 S ec gitude to time, we divide the given number of degrees and minutes by 15, and obtain 10 min. 24 sec. as the difference of time. The Meridian of a place is an imaginary line running from North Pole to South Pole through that place. A meridian divides longitude into east longitude and west longitude, making 180 of each. The meridian of Greenwich, near London, or of Washing- ton, D. C., is commonly reckoned from. 3. The longitude of Washington (from Greenwich) is 77 2' 48" W., and of San Francisco 122 24' 15" W. Find the difference of longitude and the difference of time. 4. If the difference of time between San Francisco and Philadelphia is 3 hr. 9 min. 7 sec., what is the longitude of Philadelphia? 202 PEACTICAL ARITHMETIC 5. The difference in time between Berlin and New York is 5 hr. 49 min. 35 sec. Find the difference in longi- tude. 6. If Berlin is 13 23' 43" E., find how much of the preceding difference is west longitude. 7. In travelling west my watch seemed to gain 20 min. How many degrees did I travel? 8. Constantinople is 28 59' E. When it is noon in Greenwich, what is the time in Constantinople? 9. When one place is in west longitude and the other in east longitude, do you add or subtract to find the difference of longitude? 10. New York is 74 3' west; Paris, France, is 2 20' east. Find the difference of time. 11. When it is noon at Boston (71 3' 30" west), what is the time at Paris (2 20' 22" east)? 12. Canton in China is 113 14' 30" east longitude and Washington is 77 west longitude. When it is midnight on July 4th, at Washington, what time will it be at Canton? Standard Time. Nov. 18, 1883, the United States was divided into four time-belts, each 15 wide, named respectively, Eastern, Central, Mountain, and Pacific. The local time of the middle meridian of each time-belt was adopted as the standard time of the whole belt. 1. Eastern time is that of the 75th meridian. 2. Central time is that of the 90th meridian. 3. Mountain time is that of the 105th meridian. 4. Pacific time is that of the 120th meridian. All places lying within 7 30' of the middle meridian have the time of that meridian. 204 PRACTICAL ARITHMETIC PROBLEMS. 1. St. Paul is in longitude 93 5' W. Find the difference between the local and the standard time of St. Paul. Explanation. St. Paul is within 7 30' of 90, and is within the central time-belt. Its dis- tance from the 90th meridian is 3 5', oo c/ which, divided by 15, gives 12 min. 20 12 mm. 20 Sec. sec., the difference of time required. 2. Boston is in longitude 71 3' 30". Find the difference between the local and the standard time of Boston. 3. Pittsburg is within 7 30' from Philadelphia, which lies close to the nliddle meridian of the eastern belt. When it is noon at Philadelphia, what is the standard time at Pittsburg ? 4. Galveston is in longitude 94 50' W. When it is noon there by local time, what hour is it by standard time ? 5. St. Louis is in longitude 90 15' 15" W. When it is noon there by standard time, what is the local time? MISCELLANEOUS PROBLEMS. 1. If one doz. pints of oil cost $4.00, what is the cost of one qt. ? 2. A gentleman in travelling found at a certain railroad station that his watch was 1 hr. and 25 min. slow. What direction was he travelling? How far had he travelled? 3. A note dated June 12, 1896, was paid Jan. 5, 1897. How long did the note run ? 4. How many steps |- yd. long will a man take in walk- ing 1 mi. and 580 yd. ? 5. If 10 grain bins contain 254 bu. 3 pk. 7 qt. 1 pt., what does 1 bin contain? 6. Since noon the sun has seemed to pass through 10 43' 35". What is the time of day ? MISCELLANEOUS PROBLEMS 205 7. If a cu. ft. of water weighs 1000 oz., how many Ib. avoirdupois does a cu. yd. of water weigh ? 8. When it is 1 hr. 37 min. 12 sec. P.M. at Bangor (68 47' W.), what is the time at St. Paul (93 5' W.) ? 9. A crib measuring 16 ft. X 6 ft. 9 in. X 7 ft. is full of corn in the ear. How many bu. of shelled corn will there be ? 10. In 556,688 ft. how many miles? 11. How many gal. of air in a room 16 ft. long, 11 ft. wide, and 10ft. high? 12. How many bu. of shelled corn will fill a vat that holds 6000 gal. of water? 13. A block of marble 4 ft. long and 2J ft. wide contains 12J cu. ft. How thick is the block ? 14. How many bu. in 6 tons of oats ? 15. How much is gained on 65 doz. eggs bought at $.15 a doz. and sold at the rate of 1^ doz. for $.25 ? 16. What is the cost of 4 tons and 468 pounds of hay at $12 a ton? 17. A firkin of butter weighed 61 Ib. 12 oz. How much did the vessel itself weigh ? 18. If a man can do a piece of work in 22 hr. 30 min. 25 sec., what part of it can he do in 13 hr. 11 min. 15 sec.? 19. Divide 3 gal. 2 qt. 2.03 pt, by 18, and reduce the result to the decimal of a barrel. 20. What decimal of a Ib. avoirdupois is a Ib. troy ? 21. How many bu. of potatoes in 2240 Ib? 22. The longitude of New York is 74 0' 3" W. ; of London, 5' 48" W. Find the difference of time between the two cities. Which has the earlier time ? 23. If a bicycle wheel 7 ft. 4 in. in circumference makes 3 revolutions in a second, at what rate per hour is the rider going ? 24. How many francs equal $1.00? 25. Reduce 3 8s. 4d. to dollars, U. S. currency. 206 PRACTICAL ARITHMETIC 26. The annual cost of Spanish royalty is 9,500,000 pesetas. Reduce to U. S. money. (Peseta = $.193.) 27. Latitude is distance north or south from the Equator. If a man travels due north from the Equator 2500 mi., what latitude does he reach ? (1 = 69 mile.) 28. 3780 gal. of water will fill how many barrels ? 29. If hyoscine hydrobromate is worth $12.50 a grain, what will be the cost of 12 tablets of the drug, each con- taining .01 of a grain ? 30. A owns -fj- of a farm, and B owns the remainder. | of the difference between their shares is 16 A. 80 sq. rd. Find the share of each in acres. REVIEW. Define : 1. Denominate Number. 19. 2. Compound Denomi- 20. nate Number. 21. 3. Money. 22. 4. U. S. Money. 23. 5. Sterling Money. 24. 6. Reduction. 25. 7. Reduction Descending. 26. 8. Reduction Ascending. 27. 9. Extension. 28. 10. Linear Measures. 29. 11. Surface Measures. 30. 12. Measures of Volume. 31. 13. Measures of Capacity. 32. 14. Angle. 33. 15. Rectangle. 34. 16. Square. 35. 17. Area. 36. 18. Solid, 37. Rectangular Solid. Cube. Volume. Solid Contents. Board Measure. Weight. Troy Weight. Apothecaries' Weight. Circular Measure. Circle. Circumference. Arc. Quadrant. Degree. Fractional Relation. Uniform Scale. Varying Scale. Longitude. Standard Time. REVIEW 207 2. Repeat the table of : 1. U. S. Money. 2. English Money. 3. French Money. 4. Linear Measure. 5. Surveyors' Linear Measure. 6. Surveyors' Square Measure. 7. Liquid Measure. 8. Apothecaries' Liquid Measure. 3. Name the : 1. U. S. Coins. 4. Repeat the rule for : 1. Reduction Descending. 2. Reduction Ascending. 3. Area of Rectangle. 4. Volume. 5. What is the unit of: 1. U.S. Money? 2. Canadian Money? 6. What is the unit for : 1. Land? 2. Plastering? 3. Paving? 4. Roofing, etc. ? 9. Dry Measure. 10. Avoirdupois Weight. 11. Troy Weight. 12. Apothecaries' Weight. 13. Time. 14. Months (Stanza). 15. Circular Measure. 16. Counting. 17. Paper. 18. Books. 2. English Coins. 5. Time to Longitude. 6. Longitude to Time. 7. Board Measure. 8. Fractional Relation. 3. French Money ? 4. English Money ? 5. Bricklaying, etc. ? 6. Excavations, etc.? 7. Brickwork? 8. Grain? PART II. PERCENTAGE. INDUCTIVE STEPS. 1. A man earned $5 and spent $1.00. What fractional part of the $5 did he spend? What part of $10 would he have spent ? What part of $50 ? What part of 100 ? 2. $20 out of $100 means 20 per hundred, or 20 per cent. 3. What is the meaning of 10 per cent? Of 25 per cent.? Of 50 per cent. ? Of 75 per cent. ? Of 100 per cent. ? 4. Having taken 100 per cent, of a sum of money, how much is left? 5. How much is 1 per cent, of $100? Of $200? Of $1000? 6. What is 5 per cent, of $200? Of 100 acres? Of 500 men? 7. What is 6 per cent, of $600? Of 900 sheep? Of 1200 yards? DEFINITIONS. 1. Percentage means computation by the hundred, and has 100 for its unit. One per cent, of any number is y^ of it; 5 per cent, is y^ of it. Per cent, is a contraction of the Latin per centum, by the hundred. 2. The result of computation is also called Percentage. T s^ O f $1000 = $50, the percentage. 3. The Symbol for per cent, is %. Per cent., however, may be expressed in five different ways : 6 per cent. = 6 % = .06 = yf^ = -^j-. The best way in any given case is the one that affords the shortest solution. 208 PERCENTAGE 209 4. The Rate per cent, is the number of hundredths ; T |~g- indicates that the rate is 5 per cent. 5. The number on which the percentage is computed is- the Base. Attention, therefore, must be given to Base, Rate, and Percentage. 6. Amount is the Base plus the Percentage. 7. Difference is the Base minus the Percentage. EXERCISES. 1. Use 10, 100, 16f, 125, J, .00625, in five different ways to express rate per cent. Process. 10 per cent. = 10% = .10 = ^ a = T V 100 per cent, = 100% = 1.00 = {%$ = 1. 16| per cent. = 16f % = .16f = ffl = Gfifr) = f 125 per cent. = 125% = 1.25 = ifj>- = f. | per cent. = 1% = .005 = ^ = 2W- .00625 per cent. = .00625% - yiftftWinF = TGTO^ 2. In like manner express as rate per cent, the following numbers : 15, 20, 25, 40, 50, 55, 65, 75, 96, 45, 85, 121 8|, 61 66|, 871 371., Hi 18|, 621 250, 375, fc 1 ^, ft, 1 .3, .06, .0121 .001. 3. Change ^, 1,1 J, ^ ^^ into the symbol form. Process. = W = 100 %- = m = 10 = = ^W*- = -00625%. 4. Change the following fractions into th symbol form IT> rV i i i> i ro> i> | ft i 1. 1, ft xV f ft ft A- 14 210 PRACTICAL ARITHMETIC 5. Give the following symbol forms their simplest frac- tional form: 90%, 871%, 80%, 75%, 70%, 66f%, 621%, 60%, 50%, 40%, 371%, 33^%, 30%, 25%, 20%, 16f%, To Find the Percentage. EXERCISES. 1. What is 25% of $24.00? Process. Explanation. 25% = . 1 of $24.00 = $6.00. The base is $24.00; the Or 24 00 X 25 Hlfi 00 te is 26 & ; th P ercenta S e Jr, ^4.1K. ^b.UU. is required Since 25^ ^ |, 25^ of $24.00 = of 124.00 = $6.00, the percentage. Hence the formula : Percentage = Base X Bate. NOTE. As an important preparation, let the pupil write in a table the following rates, both as common and decimal fractions, and make solution with both forms. 2. What is: 1. 70% of 30 sheep? 2. 331% of 9 books? 3. 28|% of 35ft? 4. 121% of 48 A.? 5. 50% of 600yd.? 6. 621% of 64 da.? 7. 4% of 200 gal. ? 8. 80% of 60 horses? 9. 25% of 124yd.? 10. 5% of 700 men? 11. 4% of 1000 horses? 12. 40% of 800 lb.? 13. 371% of 160 oxen? 14. 331% of $900? 15. 15% of $500? 16. 20% of 400 bu.? 17. 70% of 500 yr.? 18. 90% of 9000 sec.? 19. 871% of 1600 books? 20. 80% of 1000 horses? 21. 75% of 1000 oz.? 22. 66f% of 1500 gal.? 23. 621% of $2400? 24. 30% of 10,000 fr.? 25. 16f% of g^ of abbl.? 26. 121% O f 4 of a yr. ? 27. 10% of a million? 28. 81% of 1728 cu. in.? 29. 61% of 144sq. in.? 30. |% of $2856.00? PERCENTAGE 211 PROBLEMS. 1. Of 600 trees, 33-J% are peach trees. Find the number of peach trees? 2. A bicycle marked $90 was sold at a reduction of 12-|-%. Find the reduction and the selling price. 3. If a man owes $3564 and pays 30% of it, how many dollars does he pay? 4. Mr. A. deposited in bank $963, and afterwards drew out 5% of it. How many dollars did he draw out? 5. A farmer who had a flock of 540 sheep, sold 33-^% of them. How many did he sell, and how many had he left? 6. A merchant bought goods for $630, and sold them at a gain of 23%. How much did he gain? 7. How much is made by selling at 20% profit a house which cost $10,500 ? 8. How much is lost by selling at 8% below cost 163 tons of coal which cost $6.00 per ton. 9. I sold a horse which cost me $250 at a loss of 35%. What did I get for him? Is the difference asked for? 10. A merchant paid $.80 a yard for silk. For how much must he sell it to gain Is the amount to be sought? 11. I bought a bill of goods amounting to $986.60, from which was deducted 5%. Find the percentage allowed and the amount paid. 12. A certain mine yields 60% of metal, and of the metal |% is silver. Find how much silver and how much other metal are obtained from 1300 tons of ore. 13. In a school of 80 children 17J% are girls. Find the number of boys. 212 PRACTICAL ARITHMETIC 14. Assuming that gunpowder contains 75% of saltpetre, 10% of sulphur, 15% of charcoal, find how many pounds of each there are in a ton of powder. 15. Express as a rate per cent. .33333 J and apply it to 99,999 as a base. Change .33 J% to a common fraction in its lowest terms, and apply it to 99,900 as a base. To Find the Rate. Since Percentage is the product of Base and Kate, obviously Rate = Percentage Base EXERCISES. 1. What rate per cent, of $276 is $82.80? Process. By Analysis. 276 = 100% of B. 82.80 ^M = 30% of B. Explanation. The base is $276 ; the percentage is $82.80 ; the rate is required. Since p the rate is required, we use the formula R. = --, and obtain R. = 30$. B. 2. What per cent, of: 1. $450 is $90? 11. 812 T. is 203 T.? 2. $12 is 15 cents? 12. $5600 is $1600? 3. 15 Ib. is 5 Ib. 10 oz.? 13. 64% is 5J% ? 4. 250 head of cattle is 4 head ? 14. 4.5 % is 3f % ? 5. f of 80 is \ of 120? 15. f is ? 6. 380 pages is 120 pages? 16. 1 is .35? 7. $465 is $130.20? 17. 8 is .375? 8. $832 is $807.04? 18. 1 T. is 75 Ib. ? 9. $1041.66f is $62.50? 19. 6 A. is 5 sq. rd. ? 10, 93 yd. is 6.51 yd.? 20. 11 is 1 2s.? PERCENTAGE 213 PROBLEMS. 1. A farmer raised 5390 bu. of grain and sold 1078 bu. What per cent, of it did he sell ? 2. A merchant having 375 yd. of cloth, sold 150 yd. What per cent, did he sell ? 3. As agent, I sold a house for $5000, and received as remuneration $50. What rate per cent, did I receive ? 4. A lady having invested funds to the amount of $4750, on withdrawing the money received $4987.50. What per cent, did she gain ? 5. If 8 Ib. of an article loses 4 oz. in weight by drying, find what per cent, of water escaped. 6. A baseball team won 15 games and lost 9 games. What per cent, of its games did it win ? 7. 1-|- times a number is what per cent of it ? 8. If $56.70 is paid for the use of $1260, what is the rate per cent. ? 9. A boy misspells 55 words out of 660. What per cent, does he misspell ? 10. .875 is what per cent, of .125? 11. In one month clover seed advances from $6.50 to $7.00 per 100 Ib. What was the rate per cent, of increase? 12. In one year mixed hay advanced from $9 to $11.75 per ton. Find the rate per cent, of increase. 13. When in one year the production of wool in the United States increased from 298 million Ib. to 309 million Ib., what was the rate per cent, of increase ? 14. When tallow fell from 5 cents to 4-|- cents per Ib., what was the rate per cent, of the fall ? 15. If carpet which should be 1 yd. wide is only 34|- in. wide., what per cent, should be deducted from the price ? 16. What per cent, of 1 rd. 3 yd. 2 ft. 5 in. is 7 ft. ? 214 PRACTICAL ARITHMETIC 17. What per cent, of MMMDLXX. is 25 per cent, of MMDCCCLVI? 18 Gas is reduced from $1.50 to $1.00 per M. What per cent, of the original cost is saved ? 19. If ^ of a ton of coal is sold for what 1000 Ib. cost, what is the gain per cent. ? 20. The cost was $3486, the selling price was $4161. Find the gain and the gain per cent. Also point out the base, the amount, the percentage, and the difference. To Find the Base. Since Percentage is the product of Base and Rate, obviously _ Percentage Rate EXERCISES. 1. $82 is 121 per cent, of what base? Process. By Analysis. 100% of B. = MM = $656. Explanation. The percentage is $82.00 ; the rate is \1\% ; the base is required. -p Since the base is required, we use the formula, B. = -, and obtain B. = ---- 2. Find of what number : 1. 385 is 12^%. 7. 168 men is 8%. 2. 396 is 11 %. 8. 462 oxen is 7%. 3. 250 is 15%. 9. 12 is 16 J %. 4. 8.25 is 33^%. 10. 70 is 66f %. 5. $64.36 is 10%. 11. 300 is 33^%. 6. 38.6 bu. is 13%. 12. 100 is 62|%. PERCENTAGE 215 13. 72 is 44f%. 20. 5 cwt. is 40%. 14. 48 is 371%. 21. 75 yd. is 18}%. 15. 84 is 871%. 22. $.50 is 31J%. 16. 126 is 90%. 23. 160 is 106f %. 17. 24 is 81%. 24. 75 Ib. is 61%. 18. 10 is 61%. 25. 837 gal. is 6%. . 19. If is 14%. 26. 9006 is .06%. 3. $281.25 is 371% of what number? 4. If 28% of a number = $71.68, what is the number? 5. If 25% of a number = $324, what is 40% of that number ? PROBLEMS. 1. A man sold a horse at a gain of $15, which was 15% of the cost. Find the cost and selling price. 2. A farmer sold 384 barrels of apples, which was 96% of all he had. How many had he ? 3. A farm was sold for $536 less than cost, which was at a loss of 20%. What was the cost of the farm? 4. The immigrants of a population number 143,000 per- sons, or 1 1 % of the whole. Find the total population. 5. A farmer sold 110A. 43 sq. rd. of land, which was 20% of his land. How much land had he at first? 6. A man pays $500 rent a year ; 85% of this sum is 33 J per cent, of f of his income. Find his income. 7. On the sale of a patent $1600 was lost. What was the value, if the rate of loss was 16% ? 8. A sale resulted in a loss of $38.46, which was J% of the cost. Find the cost. 9. By selling an article for 66f % of its cost, $23.25 was realized. What was the cost? 10. At a gain of -^%, a profit of $36 was realized. What was the cost? 216 PRACTICAL ARITHMETIC 11. 30% of B.'s money is in a bank, and 50% in a farm; the remainder, $2000, is in P. R. R. stock. How much money does B. own ? To Find the Base when the Rate and the Amount or Difference are Given. EXERCISES. 1. What number increased by 6J% of itself equals 510? Process. Explanation. + = if , the number + V of the _ c-ir\ i _ on i 6 _ 480 number equals ^| of the number = 510; and since j| of the num- Or, 1 -f .06-1- -- L Q 6 1 = ber = 510, T V of the number = *in ^10-1 OR9^ - - 4SO TT of 51 = 30 5 and since T6 = 30, H = 16 times 30 or 480. Or, since once the number and .06^ times the number = 510, that number must equal 510 ^- 1.06J, or 480. Hence we have the formula : Amount A. Base = l or, B. = --. 2. What number diminished by f % of itself equals 794 ? Process. Explanation. |% = T 1F . &$ the Dumber, ^ of the 40.0 _ _3 _ .39.1 7Q4 num ' = H of the num - = 794 > and since f^ of the num. = 794, T0"0 -~2. ^ __ ^ O f 794 2; and since 4J * = 3. I sold a horse and lost $50, which was 20% of the cost. What was the cost? By Formula. By Analysis. Loss, 20% - $50. 1 % = n = i , Cost, 100% = f 4 = $250. 4. I sold wheat at $1.00 per bu., and gained 12|% of the cost. What was the cost per bu. ? By Formula. By Analysis. Gain, 121% =i. f,cost,+ n _ S- P- _ $1.00 __ i ? gain, = |^, selling price. 1 4- R. 1-1- .12j i oo 2.00 _ |- $1.00; i = ^J 2.25 8.00 . QQ8 f = -^- *= $.88|, cost. 5. I sold hay at $10 per ton and lost 10% of the cost. What was the cost of the hay ? By Formula. By Analysis. s. p. 10 Loss, 10% = T V Cost, : 1 R. " " 1 .10 " -==$11.11+. -144 = $11.11 +. 6. Cost, $10,500; rate, 20%. Gain? Selling price ? 7. Cost, $6 ; rate, 8%. Loss? Selling price? 8. Cost, $8560; selling price, $10,700. Gain? Rate? 9. Gain, $6.30; rate, 14%. Cost? Selling price? 10. Cost, $700; rate, 15%. Gain? Selling price? 11. Selling price, $19; rate, 5%. Loss? Cost? GAIN AND LOSS 225 12. Selling price, $175; rate, 30%. Loss? Cost? 13. Cost 12 cts.; selling price, 10 cts. Loss? Rate? 14. Cost, 9J cts. ; rate, 12}%. Gain? Selling price? 15. Cost, $.75; selling price, $1.00. Gain? Rate? 16. Cost, $1.00; selling price, $1.25. Gain? Rate? 17. Cost, $250; rate, 35%. Loss? Selling price? "18. Cost, $1.75; selling price, $1.25. Loss? Rate? 19. Cost, $.06 ; selling price, $.05. Loss? Rate? 20. Gain, $.12; rate, 8%. Cost? Selling price? 21. Gain, 10 cts. ; rate, 10%. Cost? Selling price? 22. Selling price, $180; rate, 20%. Cost? Gain? 23. Selling price, $230; rate, 8%. Gain? Cost? 24. Selling price, $4.56} ; rate, 17%. Loss? Cost? 25. Selling price, $49.95; loss, $4.05. Loss, %? Cost? 26. Gain, $47.25; rate, 7|%. Selling price ? Cost? 27. Loss, $38.46 ; rate, J%. Cost? Selling price? 28. Cost, $75.52 ; rate, 3J%. Gain? Selling price? 29. Selling price, $24.975 ; loss, $2.025. Rate ? Cost ? 30. Cost, $1939.50; rate, |%. Loss? Selling price? PROBLEMS. 1. A man sold his house at a profit of 15%. If he paid $3000 for it, how much did he get for it? 2. What per cent, is lost by selling tea at $.75 that cost $1.00? 3. A man sold a horse at an advance of $75, which was a gain of 25%. What was the cost of the horse? 4. A boot and shoe dealer lost 9% by selling boots at $3.75 a pair. What was the cost of the boots? 5. What per cent, is gained on goods sold at double their cost? 6. 1 bought a horse for $500 and sold it for $300. What per cent, did I lose ? 226 PRACTICAL ARITHMETIC 7. The selling price was $30, the gain 25%. What was the cost? 8. A dry-goods merchant sells goods at 12 J cts. above their cost and makes a gain of 8%. Find the cost. 9. By selling a house for $3500 I lose $500. What is my loss per cent. ? 10. How shall I mark goods that cost me $1.00 a yd., then deduct 1 5 % from that mark, and still realize 2 % ? 11. I bought 480 barrels of flour at $4.50 a barrel and sold it for $2880. Find the gain per cent. 12. A cargo of flour was bought for $690. For what must it be sold to gain 66f% ? 13. I sold tea for 114% of its cost and made a profit of $,07 a Ib. What was the selling price? 14. I paid $30 for a vase. I desire to gain 30% on it, after dropping 40 % from the asking price. What price shall I ask? 15. When 4% is lost on cheese sold at 12 cts. a Ib., what was the cost ? 16. I sold a lot of goods for $200 and thereby gained 15%. Had I sold them for $220, what per cent, would I have gained ? 17. If a wagon was purchased at 20% less than $50, and afterwards sold at 25% more than cost, at what price was it sold? 18. A merchant selling goods at a certain price loses 5% ; but if he had sold them for $20 more he would have gained 3%. What did the goods cost him ? 19. If a merchant sells goods for f of their cost, what per cent, does he lose ? 20. I sold a quantity of potatoes for $850 which cost me $970. What per cent, did I lose? 21. An agent gets a discount of 40% from the retail pnr?p COMMISSION 227 of articles and sells them at the retail price. What is his gain per cent. ? 22. When coal was sold at $4.56J per ton there was a loss of 17%. What was the cost? 23. A druggist gained 300% by retailing quinine at $3.00 per ounce. How much did it cost him per ounce ? 24. A grain dealer sold 1380 bu. of wheat at $1.00 per bu. and lost 8%. What per cent, would he have gained had he sold at $1.20 per bu. ? 25. A drover bought 100 cows at $20 a head. If 20 were killed by accident, for how much must he sell the remainder per head to gain 25% on the cost of the whole number bought? COMMISSION. 1. Commission is the percentage allowed an agent for buying or selling goods or transacting other business. 2. Commission is computed on money collected by him or on money paid out by him. 3. The Collection or the Payment is the Base. 4. The Rate per cent, is the Rate. 5. The Commission is the Percentage. 6. The Payment -\- the Commission is the Amount. 7. The Collection the Commission is the Difference. Hence the formulae on page 215 become : 1. Com. = Coll. or Payt. X R. ^ ~ Coll. or Payt.' 3. Payt. = Amount 1 + R. 4. OOII == 1 R. 228 PRACTICAL ARITHMETIC PROBLEMS. 1. An attorney collects a debt of $500 on a commission of 3%. What is his commission? By Formula. By Analysis. Com. = Coll. X R. 10 ?5 == 500 x .03 = $15.00. oyo = 2. A tax collector receives $180 for collecting taxes on a 3% commission. What is the amount collected? By Formula. By Analysis. Com. = Coll. X R. /. Coll. = SS5t. 3% = $180. Coll. = : = = $6000. 100% =$6000. 3. Find the value of the goods that can be purchased for $420, if the agent's commission is 5%. By Formula. By Analysis. $420 = the Payment + the Commis- 100% = Pay t. sion = the Amount. 5 % = Com. Amt. 420 , 105% = $420. Payt. = = ppj = w = = $400, value = R of goods. 100% = $400. 4. An agent sells goods at 21% commission. After de- ducting his commission, he remits his employer $3763.50. How much money did he collect for the goods sold ? By Formula. By Analysis. $3763.50 = the Collection 100% = Coll. the Commission the 2j-% = Com. Difference. 1\% = $3763.50. Difference 3763.50 - 3763.50 - $3860. 100% =. - $3860. COMMISSION 229 5. An agent sold $2275 worth of goods at 2% commis- sion. What was his commission ? 6. A commission merchant received $318.25 for selling $12,730 worth of bankrupt goods. What was his rate of commission ? 7. A merchant sent his agent in Cincinnati $7000 to invest in pork, after deducting his commission at 2J%. What was his commission, and how much did he invest? 8. An agent received a certain sum of money to invest in goods after deducting his commission of 3%. He invested $6250. What sum did he receive? 9. If I send my agent $4050 to invest in goods after deducting 3% commission, what sum will he invest? 10. What is the commission at 3% for selling 125 bbl. of potatoes at $2.37J per bbl. ? 11. A commission merchant receives 2J commission for buying grain for a customer. The cost of the grain and his commission = $4223. How much does the grain cost? 12. Find the amount of an agent's sales when his commis- sion at 5% = $37.65. 13. A real estate agent collects the annual rent of a house and retains $13.25 as his commission at 2J%. What is the rental of the house ? 14. My attorney collected a bill for me at a commission of 12J% and paid me a net sum of $56. How much money did he collect? 15. An agent collected 20% of an account of $860, charg- ing 4% commission. Find commission and sum paid over. 16. My agent collected 90% of a debt of $5600 and charged 7J% commission. How much should I receive from him? 17. A sale of real estate returned, as net proceeds, $2396.49, after paying $324.18 charges and a commission of 2%. For how much did it sell ? 230 PRACTICAL ARITHMETIC 18. Had sold for me 500 bbl. of apples at $4.50 per bbl., paying 2J% commission ; had bought for me with the pro- ceeds wheat at 70 cts. a bu., paying 3% commission. How many bu. of wheat did I obtain ? 19. How much commission must be paid to a collector for collecting an account of $928.75 at 3f % ? 20. An agent's commission for selling grain was $76.80, at 4%. How much did he get for the grain? 21 . A real estate agent sold a house for $7500 and charged }% commission. Find the net proceeds of the sale. 22. An agent sold goods to the amount of $8725. What was his commission at 2J% ? 23. A consignee sells $6742 worth of woollen goods, charg- ing 2J% commission and \\% for insuring payment. What sum will he pay over to the consignor ? 24. I send $10,000 to my correspondent in New Orleans to invest in cotton. His commission is \% for buying. What sum does he invest and what is his commission? 25. A man receives $1500 commission on his yearly sales. What is the amount of his sales if he is allowed J% commis- sion? 26. To be invested in cotton at 15 cents a lb., $21,630.00; commission allowed, 2J% ; marine insurance paid, 1J% ; cartage and freight paid, 1J%. Find the sum invested in cotton and the number of lb. of cotton bought. REVIEW. Formulae to be used. 1. I sold a horse, which cost me $250, at a loss of 35%. What did I get for him? 2. What is an agent's commission on the purchase of an estate for $30,000, at \\% ? COMMISSION 231 3. By selling a watch for $19, the seller loses 5% on his outlay. What would have been his loss or gain per cent, if he had sold the watch for $23.75? 4. A merchant's prices are 25% above cost price; if he allows a customer 12% on his bill, what profit does he make ? 5. If my broker buys for me goods worth $13,000, and his commission is 1J%, how much must I pay him? 6. A speculator sells at a profit of 75%, but his purchasers fail, and only pay 25 cents on a dollar. How much does the speculator gain or Jose by this venture ? 7. A man gains 15% in buying an article, and again 15% in selling it. Find the whole of his gain per cent. 8. If goods marked at 45% above cost are sold at 40% off, what is the gain or loss per cent. ? 9. If 8% be lost by selling an article for $25.50, what per cent, is gained or lost if it be sold at $38.00 ? 10. A carriage is sold for $175, which is 30% less than cost. What was the cost? 11. An army lost 18% of its men by disease and desertion, and then lost 14% of the remainder in battle; the number then remaining was 84,624. Of how many men did the army consist at first ? 12. If I sell a piano, which cost $275, for $315, what is the rate per cent, of gain ? 13. If I buy coffee at 16 cents, and sell it at 20 cents a pound, what per cent, do I make? 14. A cargo of wheat was sold for $12,500, by which a gain of 25% was made. What was the amount of net gain, after paying $150 freight and $75 for other charges? 15. If the commission is 1 J per cent., what bill will $3950 buy? 16. An auctioneer sells for me a carriage for $140, a table 232 PRACTICAL ARITHMETIC for $15, 50 yd. of carpet at 60 cts. a yd. His commission is 2J%. What will be due me for the goods? 17. A commission merchant, receiving 2J% commission, had 410 bu. of potatoes sent him, with orders to sell at 96 cents per bu. He held them until he received $492 above his commission. What per cent, was made by holding them ? 18. At what price must I sell goods that cost $f to gain 20%? 19. What per cent, of $90 is 33J% of $67.50? 20. Having purchased a farm for $9000, and spent $2500 in improvements, I sold it for $13,800. What per cent, did I make on my investment? 21. A man sold a set of harness for $15, and lost 16f %. If he had sold it at a profit of 20%, what would he have received ? 22. W^hat per cent, is gained by selling 15 ounces of tea for what a pound costs ? 23. A speculator sold 2760 bu. of wheat at $1.00 per bu., and lost 8 % . How much per cent, would he have gained had he sold at $1.20 abu.? 24. Which is the better, a discount of 25% and 10% off the remainder, or a discount of 33 J% off? 25. A broker sells 4000 bu. of wheat, and, after deducting his commission of 2%, remits by check $4900. At what price per bu. did he sell the wheat? 26. I sent a commission agent 500 bbl. of potatoes, which he sold at $2.50 per bbl. His charges were : commission, 2J% ; storage, 1J% ; cartage, $9.00. How much was due me? 27. A drummer earns $3000 annually. $1500 is guaran- teed ; the remainder is his commission, at 5 % . What are his annual sales? 28. I bought 1000 gross of screws at 27 cents, at a discount STOCKS AND BONDS 233 of 15, 10, and 5. I sold the lot at cost plus 30%. What was my gain ? 29. Offered cattle for sale at 25% above cost, but was obliged to sell them for 14% less than that mark, and gained thereby $170. What did the cattle cost? What did I ask for them? How much did I sell them for? SO. Received $2020 to buy with ; commission, 1 % Find the cost. 31. Collection, $14,000 ; commission, $420. Find the rate. STOCKS AND BONDS. DEFINITIONS. 1. Stock is invested capital, and is represented by certifi- cates which attest the ownership of a certain number of shares. 2. Bonds are written obligations, in which an agreement is made to pay a specified amount on or before a specified date, with interest. 3. The Pace-Value is the sum mentioned in certificates and bonds. When stocks and bonds sell for their face-value, they are said to be at par. When they sell for more than their face- value, they are said to be at a premium. When they sell below their face-value, they are said to be at a discount. 4. Coupons certify to interest due, and are cut off and sur- rendered when the interest is paid. 5. Bonds are issued by corporations organized under law, and take their name from the name of the corporation that has issued them. " Buffalo Railway 4 Q 84," means stock issued by the Buffalo Railway Company, rate 4%, in- terest payable quarterly, now selling at $84 per share, i.e., at a discount. 6. Stock Brokers buy and sell stocks and bonds ; their 234 PRACTICAL ARITHMETIC commission is called brokerage. Brokerage is reckoned at \% or \/o on the par value. 7. A Quotation is a published statement of the current sell- ing price of a stock. Quotations. GOVERNMENT BONDS. STOCKS. 1. IT. S. 4s, registered, 1906, 110|. 6. Adams Express 4s, 105. 2. U. S. 5s, coupon, 1904, U2|. 7. Penna. 4 p. c., 110. 3. Currency 6s, 1899, 102]. 8. Schuylkili E. R. 5s, 105. 4. Cherokee 4s, 1899, 101. 9. N. C. Railway 4^s, 104i. 5. U. S. small bonds, 105f . 10. L. V. R. R. Coal 5s, 94. 8. The Par Value is the Base. 9. The Rate of Premium, or Discount, is the Rate. 10. The Premium, or the Discount, is the Percentage. 11. The Quotation Value (ab. par) is the Amount. 12. The Quotation Value (bel. par) is the Difference. Hence the formulae on page 218 become : 1. Prem. or Disc. = P. V. X R. Prem. or Disc. 2. R. = 3. P. V. = P. V. Prem. or Disc. R. Q. V. (ab. par) Q. V. (bel. par) 1 + R. 1 R. Income is computed on the face of a bond, at the face-rate. 5 shares ($500) U. S. 4s will yield $4 per share, or $20.00 income. MODEL SOLUTIONS. 1. Find the premium on 25 shares U. S. 4s quoted at 1 10 f ; also the annual income derivable therefrom. U Of 100 = 10f. .'. R. = 10f^. Par value of 25 shares = $2500. Formula: Prem. = P. V. X R- $2500 X -1075 == $268.75. Income = $2500 X .04 = $100. STOCKS AND BONDS 235 2. The par val. of U. S. bonds = $25,000, and the premium = 500. Find the rate and the quotation. Formula: E. = ^-. rf$fa = & = *%. 100# + 2# = 102 %, the quotation. If these bonds were U. S. 5s, what income would they yield ? 3. I bought railroad stock quoted at 96, and the discount I obtained was in all 24. How many shares of stock did I buy? 100 96 = 4. .-. K. = .04 = 4%. Formula : P. V. = 5g. -^ = $600 = 6 shares. If this is 6 % stock, what is my yearly income therefrom ? 4. I bought railway stock quoted at 102^, investing $10,250. Find the number of shares I bought, and my income there- from at 5%. Formula : P. V. = p. ^ = $10,000 = 100 shares. 5% of $10,000 = $500, income. 5. If you buy railway stock quoted at 84, and invest $3360, how many shares will you buy, and what will be your income therefrom if the stock pays 4% ? Formula : P. Y. = j%^-. ^ = $4000 = 40 shares. \Vhat will be your income at 4% ? 6. How much must I invest in 6% stock at 102^- to secure me an annual income of $300, brokerage J ? Suggestions : "What will be the income from 1 share ? How many shares will yield $300 ? What will the shares cost you at 1021? What will the brokerage be at $ per share ? What will the total cost of the investment be? 7. What per cent, does an investment in Coney Island Brooklyn 6s offer me if the stock is quoted at 140? Suggestions : $140 invested yields what income? That income is what per cent, of $140 ? 236 PRACTICAL ARITHMETIC 8. If I wish to obtain 7% on my investment, wnat must I pay for a 5 % stock ? Suggestion : If T ^ 7 of a sum = $5.00, what must that sum be? Or, if 5 is the percentage and 7 is the rate, what is the base ? PROBLEMS. NOTE. The numbers in the following problems refer to the Bond and Stock Quotations found on a preceding page. Brokerage is not considered unless mentioned. 1. Find the market value of 150 shares of No. 1 ; also the income and the rate the investment pays. 2. Find the cost of investment in No. 10 to secure an income of $ 250. 3. Mr. A. owns 90 shares of No. 2. What is his income ? 4. What rate do Nos. 6, 7, 8, 9, 10 severally pay on the investments ? 5. Mr. B. has $41,309 to invest. Which will secure him the larger income, No. 3 or No. 4 ? 6. Which will give him the higher rate on his investment? 7. How many shares of No. 3 can he buy, paying ^-% brokerage ? Suggestion : Cost of a share = 102^ -(- -J-. 8. If I wish to obtain 7% on my investment, what must I pay for a 6% stock? 9. How much will 55 shares C. C. C. and I. R. R. stock cost at 28}, brokerage \% ? 10. How many shares of railroad stock at 3% discount can be bought for $2139.50, brokerage \% ? 11. Which is more profitable, and how much, to invest $6000 in 6% stock purchased at 75%, or 5% stock purchased at 60% ? 12. What sum must I invest in Louisiana 7s at 107^ to secure an annual income of $1750? STOCKS AND BONDS 237 13. Which affords the greater per cent, of income, bonds bought at 125, which pay 8%, or bonds which pay 6%, bought at a discount of 10% ? 14. At what price must I purchase 15% stock that it may yield the same rate of interest as 6 % stock purchased at 90 ? 15. What is the cost of 125 U. S. 6s at 104, brokerage i%? 16. How many shares must a broker sell to realize $10.50, commission at \% ? 17. B paid $10,989 for U. S. 6s at 110J, brokerage \%. What was his income? 18. What sum must be invested in U. S. 5s at 116J, brokerage J%, to secure an annual income of $160? 19. What is the annual income from investing $4446 in 5J% stock at 92J, brokerage \% ? 20. What will be the cost of 17 shares of canal stock at 93J and 143 shares gas stock, par value $10, at 102f ? 21. A man invested $9562.50 in the stock of a city bank at 127J. If a dividend of 3J% is declared, what amount of dividend would he get? 22. 6% bonds were sold at 118; the proceeds were in- vested in 4J% bonds. If the former and the latter incomes were the same, at what quotation were the 4J% bonds bought? 23. 200 shares of stock, par value $25, and sold at 102J, J% being retained for brokerage, how much is paid over? 24. Which is the better investment, U. S. 5s at 98|%, or U. S. 6s at 108f %, brokerage \% in each case? 25. Which is the more costly, and how much more, 15 shares of N. Y., N. H. & H., at 85, or 13 shares N. Y. & N. E., at 102, if the brokerage in each case is \% ? 26. How much money must be invested in U. S. 4Js to yield a quarterly income of $225, bonds selling at 105 \ ? 238 PRACTICAL ARITHMETIC 27. A man invested some money in bonds, at par, bearing 6% interest, and received $300 semi-annually. What was the sum invested? 28. A 5% stock is quoted at 85 J. A purchaser pays brokerage at \%. What rate per cent, does he receive on his investment ? 29. A lady would secure by investment an annual income of $650. How much 5% stock must she buy at par for the purpose ? 30. How much stock can be bought for $14,178 when the quoted price is 208 \ ? 31. Find the quoted price of railroad stock when the cost of 250 shares, including brokerage at J%, is $30,312.50. 32. What income wall $10,120 yield if invested in 4% stock bought at 115? 33. If a 6% stock is quoted at 120, what rate per cent, will an investor receive on his money ? 34. If I invest $1500 in 3% stock at 75, what is my in- come and what rate per cent, do I get on my investment ? 35. If I exchange 48 shares of a 9% stock at 176 for U. S. 4s at 116J, how much must I add to my investment to secure the same income ? 36. What sum of money must be invested in Louisville & Nashville Railroad certified gold 4% bonds at 84J to produce an annual income of $320, brokerage J ? 37. If $8000 5% stocks are sold at 90 and the proceeds invested in 3J% stocks at 60, find the increase or decrease in income ? 38. Find the price of a 3J% bond that will be as profitable an investment as a 6 % bond at par. 39. Mr. A. bought U. S. 6s for 108, kept them a year, and then sold them at 118-|. What rate of interest did the in- vestment pay him for that year ? STOCKS AND BONDS 239 40. What sum must be invested in 6% stocks, worth 95, to yield an income of $4500 ? 41. Which would yield the larger income, 11,400 invested in 7% stock, at 95, or the same amount invested in 5% stock, quoted at 57 ? 42. At \vhat rate must a 5% stock be sold to produce 8% on the investment? 43. If I buy 6% stock at 15% discount, what is the rate of interest on the investment ? 44. If I give a house and lot worth $2000 for 175 shares ($10) N. Y. Gas Co.'s stock, what is the rate of premium ? 45. How many shares of bank stock, selling at 5% dis- count, can be bought for 250 shares of insurance stock, selling at 14% premium? 46. How many shares of stock, par 25, can be bought for $2730, when quoted at 105 ? 47. A capitalist bought stock at 65, and after receiving a dividend of 5J%, sold it at 82, and made $1125. How much stock had he, and what per cent, did he realize ? 48. If stock is bought at 3J% discount, and sold at a premium of 2J%, and the gain is $258.75, what is the par value of the stock ? 49. I bought bank stock at 96J, and sold it at 112J, thereby gaining $3556. How many shares were there? 50. What must I pay for 6% bonds to realize 5J% on my investment, brokerage \/o ? 51. In order to realize 6% annually on an investment, what must I give for bonds that pay a semi-annual interest of 3%, if I immediately reinvest the semi-annual interest at 6% ? 52. If 5% bonds are bought at 90, what is the rate of income on the investment? 53. A lady desiring to invest money, considered 5s at 108, 6s at 124, and 7s at 129, Which was preferable? 240 PRACTICAL ARITHMETIC 54. A broker charges $25 at \% for buying Pennsylvania R. R. ($50). How many shares did he buy ? INSURANCE. 1. Insurance is security guaranteed for loss by fire or other specified causes. 2. Property Insurance includes : 1. Fire Insurance. ^ ^. n , , . T Jrremmm computed as Z. Marine insurance. Y , T . . T percentage. 3. Live Stock Insurance. J 3. Personal Insurance includes : 1. Life Insurance. ^ ^ , , . , T Jrremium computed at a 2. Accident Insurance. V . AI certain sum per $1000. 3. Health Insurance. J 4. The written agreement is called the Policy ; the sum named in the policy is called the Face ; the sum paid annually, semi-annually, or quarterly is called the Premium. 1. The Face (the amount insured) is the Base. 2. The Rate of Premium is the Rate. 3. The Premium is the Percentage. Hence we have the following formulae : 1. Pace X Bate = Premium. 2. Premium -*- Face = Bate. 3. Premium -r- Bate Face. MODEL SOLUTIONS. 1, How much will it cost to insure a house worth $3000 at Formula: Premium = Face X Rate = $3000 X -fa 137.50. 2. A merchant insures his store, valued at $4850, for |- of its value at %%. What is the premium? | of 4850 = $3880. |# = T fo = .00875. Formula: Face X Rate = $3880 x ,00875 = $33.95. INSURANCE 241 3. The insurance on a barn at -f-% costs $18. What is the face of the policy ? Formula: Face = Premium -=- Kate = 18 -=- .0075 = $2400. Or, \% = $18. \% = $6. \% = $24. 100^ = $2400. If I pay $30 insurance on a $3000 house, what is the rate ? Formula : Rate = ^ Or, $3000 = 100#. $1 = -tffoft = &%. $30 = f$# =1%. PROBLEMS. 1. If a man pays $30 insurance at 1J%, what amount of insurance does he get ? 2. A vessel and cargo valued at $2840 are insured at 3J%. What is the premium? 3. A man has a house worth $5600. He insures it at \\% on fy of its value. Find the cost of insurance. 4. What is the total premium on a house worth $4500 insured for 5 years at 1 J % ? 5. How much is the premium for insuring a stock of goods for $15,000 at 1J%? 6. Mr. Jacobs paid $652.50 for insuring property valued at $43,500. What was the rate ? 7. A vessel and cargo were insured for f of their value at 1J%. The premium was $2475. At what price were the vessel and cargo valued? 8. $3.75 was the premium on f the value of some furni- ture at 1 % a year. What was its insurance valuation ? 9. One company offers to take a $12,000 risk at \\% for five years, and another at J% a year. Which is the cheaper? 10. An insurance company loses $3528 by the wreck of a carload of flour which it had insured for $3600. What was the rate of insurance ? 11. A merchant imports a cargo from Liverpool, England, 16 242 PRACTICAL ARITHMETIC worth 1500 and insures it at -%. Find the premium in U. S. money. 12. For what sum must a policy be made out to cover the insurance on a property of $2100 at -f-% ? 13. If it cost $93.50 to insure a store for oue-half of its value, at lf%, what is the store worth? 14. A person insured his house for j of its value at 40 cents per $100, paying a premium of $73.50. What was the value of the house? 15. At -%, how much insurance can be effected upon a store for $108? 16. For what sum should a cargo worth $74,496 be insured at 3% so that, in case of loss, the owner may recover both the value of the cargo and the premium paid ? 17. A man has a house worth $5600. He insures it at 1^% on %- of its value. Find the cost of insurance. 18. If a tax of $12 is paid on a house and lot valued at $1200, what is the rate per cent, of tax? 19. A vessel worth $28,000 was insured at If %, and the cargo, worth $15,000, at 2^%. Both were totally lost. What was the loss to the insurer? 20. A man 25 years of age has his life insured for $6000 at $19.85 on $1000 annually. What annual premium does he pay ? 21. If a man 35 years of age takes out a life policy for $8500 at $22.70 on $1000 annually, and dies at the age of 60, how much does the amount insured exceed the sum of the premiums ? 22. If Mr. B. takes out a life policy for $8000, what is his yearly premium at the rate of $26.50 on $1000? 23. At the age of 28 years I took out an endowment policy for $10,000. What is my yearly premium at the rate of $45.15 on $1000? DIRECT TAXES 243 24. I insure my life for $8000, paying $19.80 per $1000 per year. What do I pay the company if I live 20 years after insurance? 25. If a person who is insured for $5000, at an annual premium of $28.90 per $1000, dies after 9 payments, how much more will his heirs get than has been paid in premiums? 26. A lady insures her life for $8000, at an annual pay- ment of $29.30 per $1000. If she lives 15 years, what amount will she have paid in premiums? DIRECT TAXES. 1. A Tax is a sum of money levied on persons in behalf of the public welfare. 2. A Poll Tax is levied on the person. A Property Tax is levied on property. 3. Assessors determine the value of property. 4. A Tax-Collector collects the taxes ; his salary is com- monly a percentage of the sum collected. 5. Property Tax is reckoned at some rate per cent, on the value of the property assessed. MODEL SOLUTION. A tax of $15,600 is to be raised in a town in which the taxable property is $3,200,000 ; there are 1000 persons who pay a poll-tax of $2.00 each. What is the rate of taxation ? What is A.'s tax, whose property is valued at $6000, and who pays a single poll-tax? 1. The poll-tax = $2.00 X 1000 = $2000. 2. Total tax, $15,600 $2000 = $13,600, tax to be raised on property. 3. $13,600 ~- 3,200,000 == .004. Kate = 4 mills on a dollar. 4. A.'s tax = $6000 X -004^ = $25.50, on property. 5. $25.50 + $2.00 = $27.50, A.'s entire tax. 244 PKACTICAL ARITHMETIC Hence the formulae : 1. Bate of Taxation = [Total Tax Poll Tax] -=- Total Valuation. 2. Each Citizen's Tax = His Valuation x Rate + His Poll Tax. PROBLEMS. 1. A certain town wishes to raise $1644 by taxation. The property of the town is assessed at $224,000. There are 400 polls, assessed at $0.75 each. What is the tax on $1 ? 2. At the above rate, what would be A.'s tax if he pays for real estate valued at $3655, for personal property valued at $980, and for 2 polls? 3. If a tax-collector receives $54 for collecting $1800, what is his rate of commission? 4. A tax-collector receives $180 for collecting taxes on a 3% commission. What is the amount collected? 5. How many dollars on $1000 must be levied on $597,600 to raise $5976 tax? 6. How much is a man taxed who was assessed for one poll $0.75, and on property valued at $5390, the rate being \% ? 7. In a town whose taxable property is valued at $5,463,000 a tax of $9560.25 is raised. What is the rate of taxation? 8. My property, which cost me $7800, is taxed at f of its value. If my tax is $15.60, what is the rate of taxation? 9. What sum must be assessed to raise $3750, besides paying 2% for collection ? What would be the taxable valua- tion of property to raise that sum if the rate were .003275 ? 10. A tax of $14,250 is to be assessed on a town ; the real estate is valued at $1,200,000 and the personal property at $750,000 ; there are 400 polls, each of which is taxed $1.50. What is the rate of taxation ? DIRECT TAXES 245 11. What is the assessed value of property taxed $87.50 at the rate of 5 mills on a dollar ? 12. A tax of $28.50 is to be raised on a town, and suffi- cient besides to pay for collecting at 5%. If the rate is ^ cent on a dollar, what is the property worth ? 13. In a certain district a school-house is to be built at a cost of $18,527. What amount must be assessed to cover this and the collector's fees at 3% ? 14. Find the entire tax that must be assessed in order that a town may receive $12,134 after the collector deducts his commission of 2^%. After the tax rate has been determined, the computation of a tax list is facilitated by the use of a table. Table. Rate, $0.015. PROP. TAX. PROP. TAX. PROP. TAX. $1 ... .015 $4 ... .06 &7 f I . . . .105 2 ... .03 5 ... .075 8 ... .12 3 ... .045 6 ... .09 9 ... .135 15. By using the table, find the tax on $8450. Process. Tax on $8000 = $120.00, 1000 times .12. Tax on 400 = 6.00, 100 times .06. Tax on 50 = .75, 10 times .075. $126.75. 16. In like manner find the tax of: 1. C. H. Anheier, on $910. 2. R. B. Bates, on $2356. 3. G. B. Caldwell, on $3600. 246 PRACTICAL ARITHMETIC 4. M. F. Dooley, on $9855. 5. Z. S. Eldridge, on $10,864, paying 2 polls, at $1.50. 6. J. S. Escott, on $20,200, paying 1 poll, at $1.50. 7. S. R. Flynn, on $31,750, paying 3 polls, at $1.50. 8. E. J. Graham, on $111,368, paying 2 polls, at $1.50. 9. C. P. Hatch, on $200,500, paying 5 polls, at $1.50. 10. E. J. Johnson, on $567,005, paying 2 polls, at $1.50. INDIRECT TAXES. 1. Indirect Taxes are levied upon merchandise. They consist of Duties, levied on imported goods, and of Internal Revenue, levied on domestic goods. 2. Duties are of two classes, Specific and Ad Valorem. 3. Specific Duties are levied on each yard, pound, etc., of the article. Ad Valorem Duties are levied at a rate per cent, of the cost of the article in the country in which it was bought. 4. Specific Duties are computed on the net measure or weight, Tare being allowed for the weight of box or wrappings, and for Breakage, Leakage, etc. PROBLEMS. NOTE In the examples that follow, the present tariff rates (1899) are used. 1. What is the duty on 1250 Ibs. of desiccated apples imported at the rate of 2 cents per pound ? Process. 1250 X -02 == $25.00, duty. INDIKECT TAXES 247 2. A merchant imported 1000 yd. of Brussels carpet costing in Europe 3 shillings per yd. What was the duty at 40% ? Process. 1 db $4.8665; Is. = $.243325; 3s. = .729975. .-. The cost of 1000 yd. = 1000 X .729975 = $729.975. 40% of $729.975 = $291.99, duty. 3. A merchant imported $1250 worth of silk beaded goods. What was the duty at 60% ? 4. What is the duty, at 3 cts. per lb., on 175 bags of coffee, each containing 115 lb., valued at 20 cts. per Ib. 5. Find the duty on 100 boxes of Castile soap, containing each 110 lb., costing 20 liras per cwt., at 1^ cts. per lb., tare allowed, 5%. 6. What is the duty on 400 boxes of cigars, each box containing 500 cigars, gross weight 400 lb., costing 80 cts. per lb. in Havana, at the rate of $4.50 per lb. and 25% ad valorem, together with the internal revenue tax of $3.00 per 1000 cigars? 7. If an imported piano cost in Europe $200 and was subject to a duty in New York of 45%, at what price must it be sold to gain 25%? 8. What is the duty, at 21 cts. a pound, on 3750 lb. of coffee, allowing 5% for tare? 9. What is the duty on 500 lb. of raisins, in boxes, valued at 10 cts. a pound, allowing 15% for tare, when the duty is 2^- cts. a pound ? 10. What is the duty, at If cts. per pound, on 7 T. of steel anvils, of 2240 lb. each, invoiced at 20 cts. a pound ? 11. An importer paid duties amounting to $386.75. If the duty was 25% of the value of the goods, what was their value ? 248 PRACTICAL ARITHMETIC 12. What will be the duty, at 55 cents per sq. yd., on 6 pieces of cloth, each containing 54 yd., 32 in. wide? 13. A merchant imported from Havana 25 hhd. of W. I. molasses, which was invoiced at 40 cents per gal. Allowing J % for leakage, what was the duty at 6 cents a gallon ? 14. What is the duty, at 44 cents per lb., and 55% ad valorem, on 700 yd. of cloth, invoiced at $1.60 per yd., one yd. weighing 1^ lb. ? 15. The duty on certain cotton goods is 5J cents per sq. yd., and 20% ad valorem. Find the duty on 267 pieces, 30 in. wide, each piece containing 37 yd., and costing 7 cents per yd. 16. I imported 100 tons of iron, costing IJd. per lb., on which I paid a duty of $4.00 per ton. The freight was 6s. per ton. What was the entire cost in U. S. currency ? 17. An importer bought 1000 pieces of certain goods at $40 per piece ; the duty thereon was 50% ; the freight, etc., was $1200. How must the goods be sold to gain 25% ?" 18. A quantity of bookbinders' calf-skins cost $630, in- cluding $15 for freight and $102.50 for duty. What was the rate ad valorem ? 19. If the importation of 83 J doz. of gloves doubled their cost, which was 50 fr. per dozen, what was gained on each pair, and on the entire lot by selling them at $2.00 per pair ? 20. Find the total cost of glassware on which $311.85 for duty at 45% ad valorem was paid, and 16% for breakage was allowed. 21. If the gross cost is $2630, the freight $100, the duty $330, what is the rate? 22. 100 pieces of French goods were invoiced at $40 per piece ; the duty paid was 50% ; the freight, etc., amounted to $1500. How must they be sold to gain 20% ? 23. The invoice price of goods is $1.00 per yard; the ad INTEREST 249 valorem duty is 20% ; the specific duty is $0.20. Find the gross value of a single yard. 24. The specific duty is $0.44 per Ib. ; the ad valorem duty is 60% ; the gross cost is $244. Find the invoice price of 100 Ib. 25. The ad valorem duty is 50% ; the invoice price is $500; the selling price at a profit of 25% is $1000 on 100 Ib. Find the gross value and the specific duty. INTEREST. 1. Interest is money paid for the use of money, and de- pends both upon a certain rate per cent, and the length of time the money is in use. 2. The money used is the Principal (Base). 3. The interest for one year is the Percentage. 4. The interest for a longer or a shorter time than one year is the product of the percentage and the time expressed in years or in the fraction of a year. 5. The time, when expressed in months, must be divided by 12 ; when expressed in days, by 360. 6. Since percentage equals the product of principal (base) and rate, we have the following formulae : 1. Interest = Principal X Bate X Years ; or, (Int. = Pr. X B. X Y.). 2 Interest Principal X Rate X Months . 12 or, (int. = Pr. X R. X mo/ 12 3 intercot = Principal x Rate X 36O 250 PRACTICAL ARITHMETIC MODEL SOLUTIONS. 1. What is the interest of $550 at 5% for 4 yr.? Process. Int. = Pr. X K. X Y. = $550 X .05 X 4 = $110. Explanation. Since the int. is required for exactly 4 yr., we use the formula, Int. = Pr. X B. X Y. = $550 X .05 X 4 = $110. Or, we may explain thus : Since the rate per cent, is 5, the int. for 1 yr. = .05 of $550 = $27.50. Since the int. for 1 yr. = $27.50, the int. for 4 yr. = 4 times $27.50 = $110. 2. What is the interest of $500 for 5 yr. and 2 mo. at 6% ? Process. 5 yr. 2 mo. = 62 mo. 31 Int. = : ^ = ^ = 5 X 31 = $155. ** W m Explanation. Since the int. is required for 62 mo., we use the formula, Int. = Pr - x ^ 2 xmo - = 5QOX 1 f x62 . By cancellation we have 5 X 81 = $155. Or, The int. of $500 for 1 yr. at \% = $5.00; at 6#, therefore, it = $5 X 31 6, for 1 yr. ; for 1 mo. it = ^- 5 -*- 6 ; for 62 mo. it = * 5 X f^ X ^ = $155. 1 2 3. What is the interest of $222.50 for 10 yr. 8 mo. 21 da, at 3% ? INTEREST 251 Process. 10 yr. = 120 mo. ^ 8 mo.^ 8 mo. U 128.7 mo. Int = Fr " X ^ X m '. 21 da. = 0.7 mo. j .55625 = mm*J x 128 - 7 = .55625 X 128.7 = $71.59. m Or, 10 yr. =3600 da. -) 8 mo. = 240 da. V = 3861 da. Int. = - X 360 X 21 da. = 21 da. j 22250 X__XJ81 .89 NOTE. The work may be simplified by removing decimals before can celling. EXERCISES. Int = Pr. X B. X Y. 1 . Find the interest of : 1. $100 for 1 yr. at 8%. 6. $600 for 6 yr. at 10%. 2. $200 for 2 yr. at 6%. 7. $700 for 7 yr. at 7%. 3. $300 for 3 yr. at 10%. 8. $800 for 8 yr. at 6%. 4. $400 for 4 yr. at 7%. 9. $900 for 9 yr. at 6%. 5. $500 for 5 yr. at 6%. 10. $1000 for 10 yr. at 8%. TV* - Pr - X R. X mo. ^2~ 2. Find the interest of : 1. $590 for 3 yr. 7 mo. at 7%. 2. $600 for 4 yr. 11 mo. at 10%. 3. $830 for 5 yr. 10 mo. at 6%. 4. $950 for 5 yr. 9 mo. at 6%. 5. $1070 for 6 yr. 11 mo. at 6%. 6. $470 for 3 yr. 8 mo. at 10%. 7. $2359 for 4 yr. 7 mo. at 12%. 252 PRACTICAL ARITHMETIC 8. $3597 for 6 yr. 9 mo. at 8%. 9. $2300 for 4 yr. 6 mo. at 6%. 10. $7000 for 1 yr. 7 mo. at 7%. 3. Find the interest of : 1. $10 for 1 yr. 1 mo. 3 da at 5%. 2. $121 for 2 yr. 2 mo. 6 da. at 6%. 3. $25.16 for 3 yr. 3 mo. 9 da. at 6%. 4. $36.24 for 4 yr. 4 mo. 12 da. at 7%. 5. $48.20 for 5 yr. 5 mo. 15 da. at 7%. 6. $2000 for 6 yr. 6 mo. 18 da. at 6%. 7. $590.50 for 7 yr. 7 mo. 21 da. at 6%. 8. $640.82 for 8 yr. 8 mo. 24 da. at .10%. 9. $725.83 for 9 yr. 9 mo. 27 da. at 6%. 10. $618.24 for 10 yr. 10 mo. 3 da. at 6%. T , ._ Pr. X R. X da. 360 4. Find the interest of: 1. $7000 for 1 yr. 6 mo. 7 da. at 3%. 2. $8300 for 2 yr. 5 mo. 5 da. at 4%. 3. $670 for 4 yr. 8 mo. 8 da. at 4%%. 4. $950 for 6 yr. 6 mo. 10 da. at 5%. 5. $500 for 8 yr. 7 mo. 11 da. at fy%. 6. $700 for 8 yr. 6 mo. 13 da. at 6%. 7. $3000 for 5 yr. 8 mo. 14 da. at 7%. 8. $600 for 4 yr. 9 mo. 16 da. at 8%. 9. $300 for 4 yr. 8 mo. 17 da. at 9%. 10. $536 for 3 yr. 10 mo. 19 da. at 10%. The Amount equals the Principal plus the Interest. 1. Find the amount of: 1. $1 for 3 yr. 3 mo. 3 da. at 6%. 2. $125 for 4 yr. 4 mo. 4 da. at 6%. INTEEEST 253 3. $24.50 for 5 yr. 5 mo. 5 da. at 7%. 4. $1000 for 6 yr. 6 mo. 6 da. at 10%. 5. $280.75 for 7 yr. 7 mo. 7 da. at 6%. 2. Find the interest of: 1. $2000 for 5 moat 2. $6030 for 15 da. at 4 3. $700 for 6 mo. 20 da. at 4. $60.70 for 11 mo. 27 da. at 5. $400 for 30 da. at 6%. 6. $1670 from April 1 to Dec. 25 at 7%. 7. $4440 from Feb. 4 to Jime 8 at 5%. 8. $1060 from April 13, 1897, to Dec. 21, 1898, at Six Per Cent. Method. MODEL SOLUTION. At 6% the interest of $1.00 for one year = $.06 ; for one month = -JV of $.06 == $.OOJ ; for one day = -fa of $.OOJ = $.0001 ' Hence, writing 6 cents for every year, J a cent for every month, and J of a mill for every day, we have the formula : /' $.O6 x yr. } Int. = Pr. X \ -^ X mo. L add . [ .POOj X da. J What is the interest of $236 for 3 yr. 4 rno. 18 da., at 6% ? Process. r.18 } Int. = $236 X 1 .02 [ = 236 X .203 = $47.91. 1 .003 J Explanation. For 3 yr. we write $.18; for 4 mo., $.02; for 18 da., $.003 ; the sum of these three is $.203. Since the int. of $1 is $.203, the int. of $236 is 236 times $.203, or $47.91. 254 PRACTICAL ARITHMETIC EXERCISES. 1. Find the interest of: 1. $560 for 3 yr. 2. $636 for 5 mo. 3. $700 for 60 da. 4. $236 for 1 yr. 10 mo. 18 da. 5. $35.60 for 4 yr. 9 mo. 24 da. 6. $2000 for 4 mo. 15 da. 7. $390.86 for 6 yr. 24 da. 8. $3000 for 5 yr. 11 mo. 25 da. 9. $6030 for 6 yr. 6 mo. 7 da. 10. $700 for 4 yr. 8 mo. 9 da. 2. Find the amount of : 1. $45.70 for 5 yr. 10 mo. 2. $443.76 for 9 mo. 12 da. 3. $1085.93 for 3 yr. 6 mo. 18 da. 4. $627.92 for 5 yr. 8 mo. 19 da. 5. $113.96 for 9 yr. 9 mo. 9 da. 6. $5090 for 10 yr. 10 mo. 10 da. 7. $3500 for 11 yr. 11 mo. 11 da. 3. Find the interest of $700 for 3 yr. 10 mo. 13 da., at Process. 3yr. =.18 10 mo. = .05 13 da. = .002| .232 X 700 = 162.517, int. at 6jg. 6)162.517 27.086, int. at 1%. $135.430, int. at 6%. NOTE. Observe how the six per cent, method is applicable when other rates are given. INTEREST 255 4. Find the interest of: 1. $760 for 3 yr. 11 mo. 12 da., at 5%. 2. $4030 for 5 yr. 3 mo. 7 da., at 7%. 3. $26.74 for 4 yr. 2 mo. 6 da., at 5|%. 4. $3000 for 6 yr. 6 mo. 6 da., at 4J%. 5. $2736 from July 12, 1897, to Sept. 15, 1898, at 5%. 6. $526 from Nov. 10, 1898, to June 16, 1900, at 7%. 7. $600 from May 15, 1890, to July 11, 1898, at EXACT INTEREST. Exactness requires that in reckoning interest for less than one year 365 days should be considered one year, and not 360 days. Hence Pr. X R- X exact No. of days Exact Int. = 365 NOTE 1. To find the exact number of days between two dates reckon each year as 365 days, and give to each month the number of days assigned it in the calendar. To be more exact, 366 days should be reckoned for each leap year and 29 days to February of every leap year, but in the fol- lowing problems leap years are not considered. NOTE 2. Since 5 days -fa of 365 days, common interest diminished by fa of itself will give exact interest for any number of days less than 365. CAUTION. This does not apply to interest reckoned for one or more entire years. Find the exact interest of $840, at 6 % y from Mar. 3, 1894, to Aug. 24, 1897. MODEL SOLUTION. From Mar. 3, 1894, to Mar. 3, 1897 = 3 yrs. = 365 da. X 3 = 1095 days Mar. Apr. May June July Aug. From Mar. 3, 1897, to Aug. 24, 1897=28+30+31+30+31+24= 174 days Exact No. of days = 1269 days 168 Exact Int. = Pr ' X K> X Exact Na of da y s = $& X .06 X 1269 = $ 175 23 365 30$ 73 256 PEACTICAL ARITHMETIC EXERCISES. 1. Find the exact interest of: 1. $960 from Feb. 5, 1898, to Dec. 26, 1898, a 2. $2370 from Apr. 10, 1887, to Aug. 15, 1890, at 5%. 3. $3500 from Jan. 1, 1891, to Nov. 20, 1895, at 1%. 4. $2670 from May 29, 1890, to Mar. 4, 1891, at 6%. 5. $4440 from Feb. 5, 1898, to Dec. 25, 1898, at 5 % . 2. Find the exact amount of : 1. $747.37 from March 22, 1896, to Aug. 5, 1896, at 6%. 2. $837.46 from April 3, 1896, to Dec. 21, 1897, at 9%. 3. $1094.94 from Sept. 2, 1896, to Sept. 2, 1898, at 10%. 4. $231.03 from Sept. 1, 1897, to Feb. 28, 1899, at 6%. 5. $556.44 from Jan. 1, 1893, to April 21, 1900, at To Find the Principal, the Rate, and the Time. FORMULAE. Since Interest = Pr. X R X yr., obviously Int " 1 Pr . = -- 2. R. = 5-. 3. Yr. = p R. X yr. Pr. X yr. Pr. X R. Again, since Pr. X R. X yr. + Pr. = Amount, and since Pr. X R. X yr. -|- Pr. = Pr. X (R. X yr. -f 1), we have, Pr. X (R. X yr. -f 1) = Amount, and 4 Pr - Amt. 4 ' Pr> ~ R. X yr. + 1' INTEREST 257 MODEL SOLUTIONS. 1. What principal will in 3 yr. 8 mo. 12 da. yield $1117.48 interest at 5% ? Process. 3 yr. = 36 mo. ^ 8 mo. = 8 mo. V= 44.4 mo. =3.7 yr. Pr. = " xv - A j x 12 da. = 0.4 mo.) 2. What principal will in 4 yr. 7 mo. 6 da. amount to $859.52 at 4% ? Process. 4 yr. = 48 mo. ^ 7 mo. = 7 mo. > 55.2 mo. = 4.6 yr. Pr. = R x ' , l 6 da. = 0.2 mo. J .04 X 4.6 -f 1 ~ Or, $1.00 in the given time will amount to .01 X 4 X 4.6 -f 1.00 $1.184. Since $1.00 amounts to $1.184, and since some number of dollars multiplied by 1.184 yields $859.52, that number must be $859.52 -=- 1.184, or $725.946. 3. At what rate per cent, will $4220 produce $503.235 interest in 2 yr. 7 mo. 24 da. ? Process. 2 yr. = 24 mo. ^ 7 mo. = 7 mo. >= 31.8 mo. 2.65 yr. R. = pr ^' r 24 da. = 0.8 mo. J 503.235 _ 503.235 _ 4220 X 2.65 ~ 11183 " ' 45 == 4 i%' 4. In what time will the interest on $4220 at 4|% amount to $503.235? 17 258 PKACTICAL ARITHMETIC Process. ^ T Int. 503.235 Yr - = P^TR: = 42.20 X 4.5 "* 2 - 65 = 2 y r - 7 mo - 24 da - EXERCISES. 1 . Find the principal that will : 1. Produce $180 int. in 6 yr. at 4%. 2. Produce $126 int. in 6 yr. at 6J%. 3. Produce $200 int. in 16 yr. 6 mo. at 5%. 4. Produce $823.30 int. in 1 yr. 11 mo. at 6%. 5; Produce $6 int. in 14 mo. at 5%. 6. Produce $25 int. in 144 da. at 4J%. 7. Produce $669.64 int. in 1 yr. 7 mo. 12 da. at 6%. 8. Produce $2624.65 int. in 2 yr. 6 mo. at 5%. 9. Produce $1680 in 6 yr. at 4%. 10. Produce $840 in 3 yr. at 4J%. 11. Amount to $45,056.92 in 2 yr. 6 mo. at 5%. 12. Amount to $3000 in 42 da. at 5J%. 13. Amount to $595.20 in 16 mo. at 6%. 14. Amount to $3189.375 in 2 yr. 2 mo. at 5%. 15. Amount to $10,523.475 in 1 yr. 11 mo. 21 da. at 4J%. 16. Amount to $360.18 in 4 yr. 6 mo. 18 da. at 5%. 17. Amount to $770.50 in 2 yr. 7 mo. 15 da. at 6%. 18. Amount to $47,187.58 in 3 yr. 8 mo. 25 da. at 19. Amount to $5133.30 in 4 yr. 6 mo. 27 da. at 6%. 20. Amount to $950 in 3 yr. 3 mo. 3 da. at 7%. 2. At what rate will : 1. $1800 gain $396 in 3 yr. 8 mo.? 2. $852 gain $106.50 in 2 yr. 6 mo.? 3. $660 gain $192.50 in 5 yr. 10 mo.? 4. $840 gain $107,80 in 2 yr. 4 mo.? INTEREST 259 5. $144 gain $128.52 in 12 yr. 9 mo.? 6. $220 gain $82.36 in 3 yr. 8 mo. ? 7. $420 gain $42.30 in 2 yr. 9 mo. 24 da. ? 8. $9.10 gain $5.115 in 9 yr. 9 mo. 9 da.? 9. $100 double itself in 3 yr. ? 5 yr. ? 6 yr. ? 10. Any principal treble itself in 7 yr. ? 8 yr. ? 20 yr.? 3. Find the time in which : 1. $500 will produce $60 interest at 6%. 2. $1200 will produce $48 interest at 8%. 3. $230 will produce $27.60 interest at 6%. 4. $25.20 will produce $8.30 interest at 7%. 5. $70.50 will produce $26.50 interest at 7%. 6. $50 will produce $50 interest at 6%. 7. $300 will double itself at 8%. 8. $200 will double itself at 5%. 6%. 7%. 9. Any principal will double itself at 4J%. 1 0. Any principal will treble itself at 6 % . 7%. 8%. PROBLEMS. 1. Find the exact interest of $680.20, at 7|%, for 73 days. 2. What sum, bearing interest at 4^-%, will yield an annual income of $1500? 3. Find the amount of $1040 for 2 mo. 3 da., at 6%. 4. How long must $1952.46 be on interest, at 6%, to amount to $2284.38 ? 5. At what rate per cent, will $6000 produce $500 interest in 1 yr. 10 mo. 7 da.? COMPOUND INTEREST. Compound Interest is interest computed, at certain in- tervals, on both the principal and unpaid interest. Such intervals are commonly 1 yr., 6 mo., or 3 mo. 260 PKACTICAL ARITHMETIC MODEL SOLUTIONS. 1. Find the amount of $70, at compound interest for 3 yr., at 6% ; also the compound interest. Process. Int. for 1st yr. = Pr. X R. X yr. = 70 X .06 = $4.20. Amt. = $74.20. Int. for 2d yr. = 74.20 X .06 = $4.45. Amt. = $78.65. Int. for 3d yr. = 78.65 X .06 = $4.72. Amt. = $83.37. (Amt.) $83.37 (Pr.) $70.00 = $13.27, compound interest. Hence the formula : Compound Int. = Final Amount Principal. 2. Find the compound interest of $630 for 2 yr. 6 mo., at 5%. Process. Explanation. Amt. for 1st yr. = $661.50. 2 yr = two fu]] intervalg . , Amt, for 2d yr. == 694.58. mo . = J an interval. We there- Ami for 6 mo. = 729.31. fore find the amount of $694.58 $729.31 $630 == $99.31. for tbe half interva1 ' 6 m ' PROBLEMS. 1. Find the compound interest of $200, at 7%, for 3 yr. 6 mo. 2. What is the amount of $458.50 for 2 yr., interest com- pounded semi-annually, at 6% ? Suggestion : Compute for four intervals at 3%. 3. Compute the compound interest of $580 for 1 yr. 3mo., interest compounded quarterly, at 8%. Five intervals, 1%. 4. Find the compound interest, at 6%, on $2000 for 1 yr. 10 mo., interest payable semi-annually. INTEREST 261 6. What is the compound interest of $525.75 for 3 yr. 4 mo., at6%? 6. Find the compound interest on $1050 for 1 yr. 6 mo., at 5%, interest being compounded quarterly. 7. Compute the compound interest of $600 for 2 yr. 3 mo., at 4%, interest being compounded semi-annually. 8. Find the compound interest of $20,000 for 6 mo., at 6%, interest being compounded monthly. ANNUAL INTEREST. Annual Interest is interest on the principal and each year's interest from the time each interest is due until settle- ment. Annual interest is computed when the words " with interest payable annually " are in the contract. MODEL SOLUTION. Find the interest of $300 for 3 yr. 6 mo. 20 days at 4%, payable annually. Process. 3 yr. 6 mo. 20 da. = 1280 da. 10 .01 Int. = m *j^- = $42.67, for the whole time. W 3 Int. for each of the 3 yr. = $12.00. The $12 -will be on interest : Firstly, for 2 yr. 6 mo. 20 da. Secondly, for 1 yr. 6 mo. 20 da. Thirdly, for 6 mo. 20 da. 4 yr. 8 mo. = total time = 56 mo. Int. == & x -M x 56 = 56 X .04 == $2.24. Total Int. == $42.67 + $2.24 = $44.91. 262 PRACTICAL ARITHMETIC Hence the following brief directions : 1. Find int. of Pr. for whole time. 2. Find int. of Pr. for one yr. 3. Find the sum of the time intervals. 4. Find int. on the one year's int., for the sum of the time intervals. 5. Find the sum of int. first found and int. last found. EXERCISES. 1. Find the annual interest of: 1. $360 for 4 yr. 5 mo. 16 da. at 6%. 2. $250 for 3 yr. 9 mo. 12 da. at 1%. 3. $3500 for 4 yr. 6 mo. at 6%. 4. $1247.75 for 3 yr. 5 mo. 10 da. at 6%. 5. $987.25 for 4 yr. 9 mo. 6 da. at 4%. 6. $1098.36 for 5 yr. 10 mo. 7 da., at 5%. 2. Find the amount, at annual interest, of: 1. $360 for 4 yr. 5 mo. 16 da. at 5%. 2. $250 for 3 yr. 9 mo. 12 da. at 7%. 3. $600 for 3 yr. 4 mo. 12 da. at 6%. 4. $840 for 4 yr. 8 mo. 18 da. at 5J%. 5. $2180 for 6 yr. 11 mo. 27 da., at 4J#. 6. $1070 for 5 yr. 10 mo. 24 da. at 4%, the interest of the first two years having been paid. PROMISSORY NOTES. 1. A Promissory Note is a promise, made in writing, to ^ay a sum of money 091 demand or at a specified time. 2. The Face of a note is the sum of money named in it. 3. The Maker of a note signs it. The Payee receives payment for it. The Holder has rightful possession of it. 4. The Endorser of a note writes his name on the back of it, and thus becomes responsible for payment of it. PROMISSORY NOTES 263 5. A Negotiable Note is one that is transferable. 6. Notes are said to be negotiable or transferable when they contain the words " or bearer/' or " or order/ 7 but no transfer of the latter can be made without the endorsement of the payee. To insure the negotiability of a note, in Pennsylvania the words " without defalcation" should be added. In New Jersey the words " without defalcation or discount' 7 should be added ; in Missouri, "negotiable and payable without defalcation or discount." 7. The words "with interest" render the note interest- bearing from its date. 8. A note not containing the words " with interest" begins to bear interest at maturity if not paid. 9. The words " value received" are proof that the note represents actual value. 10. The day of maturity of a note is the day when it be- comes due. 11. In any case, when the rate per cent, is not specified the legal rate is always understood. 12. Interest computed at a higher rate than the law allows is called usury. 13. In many States the time of payment is postponed three days, called " Days of Grace." 14. A Protest is a notice sent to the endorsers that the maker of the note has failed to pay it. The protest, to be valid, must not be sent later than the last day of grace. 15. A note signed by two or more persons, who thus become jointly and severally responsible for its payment, is called a Joint or Several Note. 264 PRACTICAL ARITHMETIC v Forms of Notes. PO CHICAGO, ILL., Sept. 1, 1898. Three months after date, I promise to pay Edward L. Baker- Four hundred eighty-six- - - - ---------- jSUL. Dollars, with interest at 5%, for value received. EGBERT H. KING. WASHINGTON, D. C., Sept. 1, 1898. Four months after date, I promise to pay Edward, L. Baker ----- or bearer, Four hundred eighty-six - ~ ----------------- ^-f^ Dollars, with interest at 7%, for value received. ROBERT H. KING. (3.) $486^ PHILADELPHIA, PA., Sept. 1, 1898. Six months after date, / promise to pay Edward L. Baker - ^^~^ __________ ^ r order, Four hundred eighty-six^- - ------ ~-f-f^ Dollars, without defalcation, for value received. ROBERT H. KING. (*) $486^ TRENTON, N. J., Sept. 1, 1898. On demand, I promise to pay Edward L. Baker Four hundred eighty-six - -------------------- ffa Dollars, with interest at 6%, without defalcation or discount. ROBERT H. KING. PROMISSORY NOTES 265 (5.) $486^ ST. Louis, Mo., Sept. 1, 1898. Four months after date, we jointly and severally promise to pay Edward L. Baker- or order, Four hundred eighty-six ff$ Dollars, with interest at 3%, for value received, negotiable and payable without defalcation or discount. ROBERT H. KING. JOHN C. TAYLOR. (6.) ATLANTA, GA., Sept. 1, 1898. Sixty days after date, I promise to pay Edward L. Baker- or order, at the Atlanta National Bank Four hundred eighty-six-^ ^"ffo Dollars, with interest, value received. ROBERT H. KING. QUESTIONS AND EXERCISES. 1. Find when the above notes will severally mature. 2. Compute the amount due on each at maturity. 3. Point out which are negotiable and which non- negotiable. 4. Point out which are interest-bearing from date and which from maturity. 5. When may a " demand" note be collected ? 6. When is a time note collectible ? 7. When no rate per cent, is specified in a note, what rate is understood ? 8. What is the legal rate in your State ? 9. Can a note be protested after its maturity ? 266 PRACTICAL ARITHMETIC 10. If the maker of a note fails to pay it, who is held responsible for payment of it? 11. Write a negotiable note, in favor of George Hudson, for $500.50, using your own name as that of maker. 12. Write a note from the following data: Face, $347.56 ; negotiable ; time, 60 days ; payee, George Jones ; maker, Hiram Smith; rate, 6% ; place, Reading, Pa. 13. Write a non-negotiable note. 14. Write a note that will bear interest from date. 15. Write a note payable at a bank. 16. Write a note with an endorsement. NOTE. Latin, dorsum, the back. 17. Write a note payable with annual interest. 18. Assume a date for settlement, and compute the amt. due on said note. 19. Find the day of maturity and amount due, having given : 1. $631.36, Feb. 13, 1898, 63 da., 6%. 2. $796.56, Apr. 23, 1898, 90 da., 5%. 3. $397.86, Sept. 6, 1898, 5 mo., 4%. 4. $1055.51, Nov. 21, 1898, 4 mo., 6%. 5. $631.36, Nov. 6, 1898, 33 da., 7%. 6. $937.72, Jan. 17, 1898, 6 mo., 8%. 7. $2632.98, Apr. 30, 1898, 1 mo., 10%. 8. $2849.65, June 23, 1898, 15 da., 6%. 9. $984.05, Aug. 11, 1898, 3 yr., 10. $1968.10, Sept. 3, 1898, 3 mo., PARTIAL PAYMENTS. 1. The payment of part of a note or other obligation is called a Partial Payment. 2. Notes on which partial payments have been endorsed are PARTIAL PAYMENTS 267 computed chiefly by two rules : The Merchants' Rule and The United States Rule. The Merchants' Rule. The Merchants' Rule applies to notes settled within a year. The method is as follows : MODEL SOLUTION. PITTSBURO, PA., Feb. 25, 1898 For value received, I promise to pay John Wayland, or order, Six Hundred Dollars, on demand, with interest from date. JAMES BROWN. On this note were made the following payments : May 25, 1898, $156.00; Aug. 25, 1898, $200.00; Nov. 25, 1898, $185.00. What was due on Feb. 20, 1899 ? Process. Date of settlement, 1899, 2, 20. Date of note, 1898, 2, 25. Interval, 11 mo. 25 da. Principal (note) $600.00. Interest for 11 mo. 25 da. 35.50. Amount $635.50. 1899, 2, 20. 1899, 2, 20. 1899, 2, 20. 1898. 5, 25. 1898, 8. 25. 1898, 11, 25. 8 mo. 25 da. 5 mo. 25 da. 2 mo. 25 da. 1st payment and int. for 8 mo. 25 da $162.89. 2d payment and int. for 5 mo. 25 da 205.84. 3d payment and int. for 2 mo. 25 da 187.62. 556.35. Balance due at settlement $79.15. Hence the formula : Balance = Amount of Pace due at time of settlement the sum of the Payment- Amounts due at time of settle- ment. 268 PRACTICAL ARITHMETIC PROBLEMS. 1. A note for $960, on demand, with interest at 7%, dated Feb. 1, 1897, was endorsed as follows : May 11, 1897, $300 ; Oct. 16, 1897, $366. How much was due Dec. 16, 1897? 2. What amount is due Nov. 27, 1898, on a note for $800, dated Jan. 15, 1898, with interest at 6%, on which are the following endorsements: May 3, 1898, $300; July 9, 1898, $400? 3. A man holds a note of $460, dated Jan. 20, 1898, on which the following payments are endorsed : $140, Mar. 26, 1898; $100, June 16, 1898; $160, Oct. 14, 1898. Settle- ment is made Dec. 22, 1898. Find the balance due, interest at 5%. 4. What is due Dec. 20, 1898, on a note for $1300, dated Feb. 10, 1898, with interest at fy%, on which is the following endorsement: June 7, 1898, $900? 5. A note of $1100, dated April 1, 1898, payable on demand, with interest at 6%, bears the following endorse- ments : June 6, $300; Aug. 5, 236.48; Nov. 19, $333. What is due Jan. 1, 1899? (6.) $696^ BIRMINGHAM, ALA., April 4, 1898. Nine months after date, for value received, I promise to pay to the order of Paul Stakeman, Six Hundred Ninety-six and^^- Dollars, with interest at 8%. ROBERT S. CAMPBELL. This note was endorsed as follows: July 9, 1898, $436; Sept. 4, 1898, $95.40; Oct. 3, 1898, $100. What was due on the note at maturity ? 7. A note of $946.36, dated Aug. 1, 1898, payable on demand, with interest at 5^%, bears the following endorse- PARTIAL PAYMENTS 269 ments : Sept, 21, $268.60 ; Oct. 22, $280.36 ; Nov. 6, $300 ; Dec. 2, $90. What remained due after Dec. 2 ? 8. A note for $4300, dated Feb. 8, 1898, has the fol- lowing endorsements on it : Mar. 20, 1898, $900 ; April 20, 1898, $700; July 25, 1898, $600; Aug. 17, 1898, $500; Nov. 25, 1898, $400. What is due Jan. 1, 1899, at 6% ? (9.) $9600-,%- HARRISBTJRG, PA., July 20, 1898. Thirty * days after date, for value received, I promise to pay Charles Davenport, or order, Nine Thousand Six Hundred Dollars, without defalcation. JAMES H. BOYD. Endorsements: Aug. 30, $200; Oct. 12, $400; Nov. 10, $600 ; Dec. 20, $800. What is due July 15, 1899 ? 10. A note for $7000, at 90 days, dated Sept. 25, 1898, has the following endorsements : Dec. 31, $300 ; Jan. 18, 1899, $500; May 20, 1899, $600; Aug. 10, $100. What was due Sept. 1, 1899? 11. On a note for $2898, dated Jan. 4, 1897, and bearing interest at 5^%, the following payments were made : Jan. 20, 1897, $600; Feb. 25, 1897, $700; June 10, 1897, $400; Sept. 30, 1897, $360. How much is due Jan. 1, 1898. The United States Rule. This rule applies to notes settled beyond the limit of a year, and forbids the deducting of a payment unless the payment equals or exceeds the interest due. The compounding of in- terest is thus prevented. * Interest must not be computed for these 30 days. 270 PRACTICAL ARITHMETIC MODEL SOLUTION. NEW YORK, May 1, 1895. I promise to pay George Jenkins, or order, Four Hundred Seventy-five^- Dollars, on demand, with interest at 7%, for value received. THOMAS WRIGHT. Endorsements: Dec. 25, 1895, $50.00; July 10, 1896, $15.75; Sept. 1, 1897, $25.50; June 14, 1898, $104.00. How much is due April 15, 1899? Process. Face of note ....................... $475.50. Date of 1st pay't, 1895, 12, 25. Date of note, 1895, 5, 1. Interval 7 mo. 24 da. Int. of face for interval .................. 21.64. Amount ............. $497.14. 1st payment (exceeding interest) .............. 5000. 1st balance ............ $447.14. As the next two payments will not equal or exceed the interest due, we compute the interval to June 14, 1898. Date of 4th pay't, 1898, 6, 14. Date of 1st pay't, 1895, 12, 25. Interval, 2 yr. 5 mo. 19 da. Int. of 1st balance for interval ............... 77.29. Amount ............. $524.43. 2d payment ................ $15.75 3d payment ................ 25.50 Sum less than int. due .......... $41.25 4th payment ............... 104.00 14525. 2d balance ...... ...... $379.18. Date of settlement, 1899, 4, 15. Date of 4th pay't, 1898, 6, 14. Interval, 10 mo. 1 da. Int, of 2d bal. for interval ................. 22.19. Balance due April 15, 1899 ..... $401.37. PARTIAL PAYMENTS 271 BULB. 1. Find the amount of the principal to the given date at which the payment, or sum of the payments, is equal to or greater than the interest. 2. From this amount deduct the payment, or the sum of the payments. 3. Consider the remainder as a new principal and pro- ceed as before. PROBLEMS. (!) $3000^ PHILADELPHIA, PA., Feb. 26, 1895. On demand, I promise to pay George Palmer Three Thousand Dollars, with 6% interest. JOHN JAY. Payments were made on this note as follows: Sept. 10, 1895, $25.00; Jan. 1, 1896, $500.00; Oct. 25, 1896, $75.00; April 4, 1897, $1500.00. How much was due Feb. 20, 1898 ? (20 $750 T % LYNN, MASS., July 15, 1893. Three months after date, I promise to pay George Mason, or order, Seven Hundred and Fifty Dollars, for value received. GEORGE PALMER. Payments were made as follows: Aug. 3, 1896, $75.00; May 1, 1897, $560.00. What was due Feb. 20, 1898 ? 3. A note was given Jan. 1, 1890, for $700. The follow- ing payments were endorsed upon it: May 6, 1890, $85; July 1, 1891, $40; Aug. 20, 1891, $100; Jan. 10, 1893, $350. How much was due Sept. 30, 1894, interest at 6% ? 4. What is due Aug. 18, 1894, on a note for $400, dated April 1, 1892, with interest at 6%, on which are the following 272 PRACTICAL ARITHMETIC endorsements : Jan. 13, 1893, $50 ; Sept. 22, 1893, $10 ; April 25, 1894, $125. (5.) $750^0- PHILADELPHIA, PA., May 10, 1895. Three years after date, for value received, I promise to pay Thomas Newbury, or order, Seven Hundred and Fifty Dollars, with interest, without defalcation. SAMUEL TOWNSEND. Endorsements: Jan. 15, 1896, $124.75; Sept. 12, 1896, $20; Dec. 16, 1897, $216.80. How much remained due May 10, 1898? 6. A note for $5600 was given May 1, 1895, and was endorsed as follows: Oct. 17, 1895, $350; Feb. 18, 1896, $455 ; July 10, 1896, $318.50. What was due May 1, 1897, interest at 1% ? 7. What is due Dec. 31, 1898, on a note for $2800, dated Oct. 10, 1894, with interest at 7%, on which are the follow- ing endorsements: April 1, 1895, $66.60; July 21, 1896, $300 ; June 15, 1898, $300 ? (8.) $4600-^ CHICAGO, ILL, April 6, 1896. On demand, I promise to pay George K. Brown, or order, Four Thousand Six Hundred Dollars, for value re- ceived, with interest at 5 % . CHARLES MOREHEAD. Endorsements : July 10, 1897, $1360 ; Oct. 4, 1897, $500 ; Jan. 16, 1898, $660 ; June 21, 1898, $700. How much was due Jan. 21, 1899? 9. A note for $8580 was given July 12, 1894. Endorsed : Jan. 8, 1895, $300; April 26, 1896, $500; July 16, 1897, BANK DISCOUNT 273 $335; Oct. 8, 1897, $250. What was due at settlement, Jan. 1, 1898? 10. A note was drawn in Michigan for $2774.65, payable in 2 years, with interest, and dated March 15, 1896. Pay- ments were made as follows: July 30, 1896, $100; Dec. 8, 1896, $200; Jan. 5, 1897, $250; May 17, 1897, $600; Jan. 1, 1898, $600. How much remained unpaid, April 1, 1898? BANK DISCOUNT. 1. A Bank is an institution established for the purpose of receiving, loaning, and issuing money. NOTE. All banks do not issue money. 2. For cashing notes in advance of their maturity, banks make a deduction from their face value. This deduction is called Bank Discount. 3. Bank discount depends upon Face, Rate, and Time, and is computed precisely like simple interest. 4. The Time of Discount of a note is the interval between the day of its presentation and the day of its maturity. This interval is commonly called time to run. NOTE. In some States the time to run is increased by 3 days, called "Days of Grace." 5. The Proceeds of a note equal its Face less the Discount. MODEL SOLUTION. $2360^ PHILADELPHIA, PA., Feb. 26, 1897. Three months from date, I promise to pay to the order of George Gross, at the West Philadelphia Bank, Twenty-three Hundred and Sixty Dollars, for value received. JAMES JENKINS. 274 PRACTICAL ARITHMETIC This note was presented at bank for discount April 1, 1897. Find : 1. The day of maturity. 2. The time to run. 3. The discount. 4. The proceeds. Process. Feb. 26, 1897 + 3 mo. = May 26, 1897, the day of maturity. Day of maturity, 1897 5 26 Day of presentation, 1897 4 1 1 mo. 25 da., time to run. rv x r cr j 236 X -06 X 55 A 01 nA Discount for 55 da. = - 36Q - = $21.64. Face of note = $2360.00 Discount = 21.64 Proceeds = $2338.36 That is, the bank took the note and paid in cash for it $2338.36. Hence the brief directions are : 1. Find the day of maturity. 2. Find the time to run. 3. Find the discount (simple interest). 4. Find the proceeds (subtract discount from face). EXERCISES. Find the discount and proceeds of: 1. $350 for 30 da. at 5%. 2. $400 for 90 da. at 6%. 3. $540 for 60 da. at 7%. 4. $600 for 60 da. at 8%. 5. $2000 for 3 mo. at 10%. 6. $80.60 for 90 da. at 5J%. 7. $5000 for 18 da. at 6J%. 8. $780 for 40 da. at 7J%, with grace. 9. $600 for 2 mo. 12 da. at 8J%, with grace. 10. $1000 for 90 da. at 10%, with grace. BANK DISCOUNT 275 PROBLEMS. Apply the brief directions to the following notes : 00 $5003% SAN FRANCISCO, CAL., Feb. 20, 1898. Sixty days after date, I promise to pay James Warner, or order, Five Hundred Dollars, value received. JOHN GORDON. Discounted Mar. 15, 1898, at 7%. (2.) $800^ BALTIMORE, MD., Feb. 1, 1898. Ninety days after date, I promise to pay to the order of Peter Welsh Eight Hundred Dollars, for value received. HENRY BRYCE. Discounted April 1, 1898, at 6%. (3.) PHILADELPHIA, PA., Jan. 5, 1898. Ninety days after date, I promise to pay Charles Garrett, or order, Four Hundred Dollars at the Girard Bank, for value received, without defalcation. JOHN WATERMAN. Discounted Jan. 10 at 6%. (4-) WASHINGTON, D. C., April 20, 1898. Six months after date, for value received, I promise to pay Alfred Rickert, or order, Four Hundred Sixty-five 1 2 ^j- Dollars, at the First National Bank. WESLEY EVANS. Discounted June 23 at 7%. 276 PRACTICAL ARITHMETIC 5. Find the bank discount of a note for $3600, dated March 6, 1898, and payable 3 mo. after date, with interest at 5%, if discounted May 13, 1898, at 6%. 6. Find the proceeds of a note for $2400, dated Aug. 26, 1898, payable 90 days after date, with interest, at 5J%, and discounted Oct. 1, 1898, at 6%. To find the Face of a Note. It is sometimes necessary to determine what face to give a note in order to secure a certain sum as proceeds. Find the face of a note that, discounted for 60 days at 6%, will yield $500 as proceeds. Process. Explanation. Discount of $1 .00 = .01 . Since S 1 - 00 ' as face > Proceeds of $1.00 - 1.00 - .01 = .99. Counted for 60 da. $500,-. 99 = $505.05. many dollars as face will yield $500 as proceeds ? Obviously, $500 -=- .99 = $505.05. Hence the formula : Face = Given Proceeds .-=- Proceeds of $1.OO. Face = Given Discount * Discount of $1.OO. EXERCISES. Find the face in each of the following instances : 1. Proceeds, $800; time 60 da. ; rate, 6%. 2. Proceeds, $989.50; time, 2 mo.; rate, 6%. 3. Proceeds, $3000 ; time, 90 da. ; rate, 6%. 4. Proceeds, $15,000; time, 2 mo.; rate, 7%. Grace. 5. Proceeds, $240; time, 3 mo.; rate, 5%. Grace. 6. Proceeds, $975; time, 2 mo.; rate, 1%. Grace. 7. Discount, $40; time, 90 da. ; rate, 6%. 8. Discount, $4.18 ; time, 60 da. ; rate, 6%. TRUE DISCOUNT 277 9. Discount, $8.48; time, 60 da. ; rate, 5%. 10. Discount $17.50; time, 2 mo. 12 da.; rate, 7%. PROBLEMS. 1. I wish to borrow $400 at a bank. For what sum must I draw my note, payable in 60 da., so that when discounted at 6% I shall receive the desired sum ? 2. What is the face of a note at 60 days which yields $780 when discounted at a bank? Rate, 5%. 3. Suppose you buy goods in Philadelphia to the amount of $1248.50, and give your note in payment drawn at 6 mo. What must be the face of the note ? 4. For how large a sum must a note be drawn, payable in 3 mo., that the net proceeds may be $7500 after deducting the bank discount at 8% ? 5. A Chicago merchant sold goods, and received in payment for them a 6-mo. note, which he had immediately discounted at 7%. If he received $1898 in cash for the note, for what sum had he sold the goods ? 6. For what amount must a note be made payable in 3 mo., so that when discounted in Baltimore at the legal rate (6%), the proceeds may be $1420. 7. For what sum must a note be drawn, payable in 3 mo., so that when discounted in Montana at the legal rate (10%), the proceeds may be $1000? 8. In Oregon I suffered a discount of $6.18 on a 6-mo. note at the legal rate (8%). Find the face of my note? TRUE DISCOUNT. 1. The Present "Worth of a debt is a sum which, put at interest, amounts to the debt when due. 2. True Discount is the difference between the present worth and the debt. Finding the present worth is the same as finding 278 PRACTICAL ARITHMETIC what principal will in a given time, and at a given rate, amount to a given sum. Hence we have, from page 256, Pr. R x tj , 1? which becomes : T-> TTT Amt. or Debt RW '=R. xyr. + 17 MODEL SOLUTION. What present worth, or principal, will amount to $1000 in 8 mo. at 6 % ? Also, find the true discount. Process. p W Amt. HOOP = $1000 = ftofi-i 54 ~ K. X yr. + 1 .06 X f + 1 ' 1-04 " $1000 $961.54 = $38.46, true discount. Explanation. Since the P. W. stands to the Amt. in the relation of principal, we use the formula, P. W. = R x ^ + 1? and obtain $961.54. $1000 $961.54 = $38.46, true discount. Or, we may say : $1.00 amounts to $1.04 ; therefore, it will require the quotient of $1000 -=- 1.04 to amount to $1000. Hence the P. W. = $961.54. EXERCISES. Find the present worth and true discount of: 1. $400, due 1 yr. hence, at 6%. 2. $200, due 1J yr. hence, at 6%. 3. $180, due 1 yr. 5 mo. hence, at 5%. 4. $600, due 2 yr. 3 mo. hence, at 8%. 5. $350, due 2 yr. 6 mo. 9 da. hence, at 7%. 6. $1500, due 2 mo. 21 da. hence, at 6%. 7. $2000, due 2 yr. 3 mo. 6 da. hence, at 6%. 8. $487.75, due 3 yr. hence, at 7%. 9. $422.00, due 2J yr. hence, at 6%. 10. $479.37 J, due 3 yr. hence, at 5%. TUE DISCOUNT 279 PROBLEMS. 1. Find the present worth and true discount of $200, due in 3 yr. 8 mo. 16 da., at 5J%. 2. Find the bank discount on $1000, due in 9 mo., without grace, money being worth 6%. 3. Find the difference between the bank discount and the true discount of $1000, due in 9 mo., rate 5%. 4. Find the true discount on $980, due in 6 mo., money being worth 4J%. 5. Money being worth 6%, find the difference between the true discount and the bank discount of a note for $525, due in 10 mo., without interest. 6. If I buy goods for $3000 on 3 mo. credit, what dis- count should I receive if I pay cash, money being worth 5 J % ? 7. If I pay a debt of $9450 2 yr. 6 mo. 15 da. before it is due, what discount should I receive, money being worth 8%? 8. What is the aggregate present worth of two notes, each for $800, due at the end of one and three years respectively, the rate of bank discount being 7% ? 9. I wish to place at 6% interest a sum that will amount to $832.50, from Jan. 9, 1897, to Nov. 9, 1898. What is the sum? 10. If you owe $500, to be paid in 1 yr., without interest, what ought you in equity to pay now in order to cancel the debt, if money is worth 7 % ? REVIEW. 1. On property worth $15,000, fire caused a loss of $3840. Find the rate per cent, of loss. 2. An agent makes 20% by selling a book for $2.88. Had he sold it for $4.00, what per cent, would he have made? 280 PRACTICAL ARITHMETIC 3. Find the interest on $960 for 7 yr. 6 mo. 27 da., at 4J%. ^Also, find the interest at 9%. 4. Find the rate per cent, when $1758 amounts to $1869.34 in 8 mo. 5. Find the time when the principal, at 7%, is doubled. 6. What principal will amount to $2222.22 in 2 yr. 2 mo. 2 da., at 5%? 7. The face of a note is $1975 ; its date, Sept. 12, 1898 ; its time, 3 mo. ; its day of discount, Sept, 26, 1 898 ; its rate of discount, 5J%. Find its day of maturity, etc. 8. Find the compound interest of $4000 for 2 yr. 6 mo., at 5% per annum. 9. Find the annual interest of $1600 for 4 yr. 8 mo., at 6%. Also, find the annual interest at 4%. 10. Find the present worth of $6450, due in 6 mo., with- out grace, money being worth 6%. 11. Find the proceeds of a note for $2500, payable in 90 da., without grace, discount, 5J%. 12. How much greater is the interest on $25,000 for 3 yr. 6 mo., at 6%, at compound interest, than at annual interest? 13. What was due Jan. 1, 1898, on a note for $1150, dated Sept. 1,1894, at 7% ? 14. The interest on a note from Aug. 3 to Dec. 27, at 10% per annum, was $33.00. What was the face of the note ? (15.) $850 T 4j&^&ez-s <^sa-. , May 10, -/*" 1 Ib. = 5 cts. ; x = 100, the cost of 20. 20 Ib. = 100 cts. NOTE. The pupil will observe that because x is to be greater than its antecedent, the second term must be greater than its antecedent. It is only in this way that the equality of ratios can be preserved. RATIO AND PROPORTION 289 2. If 6 men can do a piece of work in 5 days, in how many days can 10 men do the work ? Process. We find that the answer, or x, will be less than the third quantity, for 10 men will not require so long a time as 6 men. We have, therefore : 10:6-6:*. Or, 6 : 10 = * : 5. Analy8iS ' If 6 men = 5 days, ** 1 man = 30 days ; X = 3 da. 10 men = 3 days. THE RULE OP THREE. 1. Let x represent the required term or answer. 2. With x and the quantity of like denomination form a ratio. 3. Compare the two terms of the ratio, and determine from the conditions of the problem whether x is greater or less than the other term. 4. "With the two given like quantities form a ratio equal to the first. 5. Express the equality of the ratios, and apply formula A or B, as the case may require. PROBLEMS. NOTE. Solve the following problems both by analysis and proportion. Suggestion : Let x represent the fourth term. 1. If 31 yd. of cloth cost $62, what will 21 yd. cost? 2. How long will it take 24 men to do a piece of work that 8 men can do in 12 days ? 3. How far can a certain load be carried for $34, if $64 will carry it 100 miles? 4. If 231 men have provisions for 8 mo., how long will the same provisions last 308 men ? 5. If 95 cents will buy one bushel of wheat, how many bushels will $11.75 buy? 19 290 PRACTICAL ARITHMETIC 6. A man owes $2500, and can pay only $1000. How much does he pay on a dollar ? 7. If 28 yd. of oil-cloth, .875 yd. wide, cover a certain floor, how many yards 1.25 yd. wide will cover the same floor ? 8. If 22 bu. 3 pk. of corn be produced on one acre, how many acres will produce 546 bu. ? 9. If two men earn $72 in 6 da., how much will 30 men earn in the same time ? 10. If 12J tons of hay cost $180.25, what will 81J tons cost? Suggestion : Let x represent the third term in the following problems. 11. A regiment of 960 men has provisions for 40 days. How long will it last if the regiment is reinforced by 240 men? 12. A field can be mowed in 4 days of 11 hours each ; how many days of 9 hours each will it take ? 13. At the time when a man 5 ft. 9 in. in height casts a shadow 4 ft. 6 in. long, what is the height of a tree that casts a shadow 52 ft. 6 in. long ? 14. If a locomotive runs 96f miles in 3J hours, how many miles will it run in 5J hours? 15. A wheel makes 75 revolutions in 5 min. How many does it make in an hour ? 16. A. can do a piece of work in 6 days, B. can do it in 7 days. If B.'s wages are $2.10 per day, how much should A. receive per day ? 17. If a 5-cent loaf of bread weighs 8 ounces when flour is worth $5, what should such a loaf weigh when flour is at $6 ? 18. $ yd. cost $. Find the cost of f yd. 19. If a cistern containing 3000 gal. leak 1 gal. 2 qt. a min., how long will it take to empty it? 20. If 42 yd. of carpet 2 ft. 3 in. wide are required for a room, how many yd. of carpet 2 ft, 4 in, wide will be required ? RATIO AND PROPORTION 291 Suggestion : Let x represent the second term in the following problems. 21. If a train, at the rate of ^ of a mile per min., requires 3J hr. to make a certain distance, how long will it require at the rate of -fa of a mile a min. ? 22. If a train travels of a mile in 18 sec., how many miles an hour does it travel? 23. A. gains 4 yd. on B. in running 30 yd. How many yd. will he gain while B. is running 97J yd. ? 24. If a man spends $276 in the three summer months, how much will he spend in a year at the same rate per day ? 25. If 28 men mow a field of grass in 12 days, how many men will be required to mow it in 8 days ? 26. If 17 men can mow a field in 9 days, how long would it take to mow half of it if 5 men refuse to labor ? 27. If 14J yd. of cloth cost $19 J, how much will 19| yd. cost? 28. If ^ of a ship costs 273 2s. 6d., what will -fo of her cost? 29. If 2J gal. of molasses cost 65 cents, what will 3J hhd. cost? 30. If a steeple 150 ft. high casts a shadow 210 ft., what is the length of the shadow cast, at the same time, by a staff 5ft. high? Suggestion : Let x represent the first term in the following problems. 31. If the interest of $600 for 6 mo. is $15, what principal will gain $64 in the same time ? 32. If 15 J yd. of silk that is f yd. wide will make a dress, how many yards of muslin that is 1 J yd. wide will be required to line it? 33. If I borrow $500 and keep it 1 yr. 4 mo., for how long a time should I lend $240 as an equivalent for the favor ? 292 PEACTICAL ARITHMETIC 34. A butcher in selling meat sells 14^ oz. for a pound. How much does he cheat a customer who buys of him to the amount of $30 ? 35. In what time can a man pump 64 hhd. of water if he can pump 12 hhd. in 2 hr. 15 min. ? 36. How many men can do in 24 days a piece of work which would employ 40 men 6 days? 37. If J of f of 6J bbl. of beef cost $78, how much will | of | of 3| bbl. cost? 38. If 450 tiles, each 1 2 in. square, will pave a cellar, how many tiles that are 9 in. by 8 in. will pave the same ? 39. If a distance of 48 miles is represented on a map by If in., what distance is represented on the same map by 7 J in. ? 40. Twenty-four men in 30 days can finish a piece of work. After 16 days 11 men quit work. In how many days can the rest finish the work ? COMPOUND PROPORTION. A Compound ratio indicates the product of two or more simple ratios ; for instance, -f- X -f is a compound ratio, being the product of the simple ratios ^ and -J, written \\\- A Compound proportion has one of its ratios compound. ILLUSTRATION. If 5 men build a wall 6 ft. high in 7 days of 8 hr., in how many days of 9 hr. can 10 men build a wall 11 ft. high? Process. The second ratio is simply 7 da. : x da. The first ratio is compound, and we construct it as follows : We write 10 : 5, for 10 men require less time than 5 men. We write 6 : 11, for 11 ft. require more time than 6 ft. We write 9 : 8, for 9 hr. per day require fewer days than 8 hr. COMPOUND PROPORTION 293 (10: 5^ Hence the proportion is : -[ 6 : 11 V : : 7 : x. ( 9: 8J 2 By formula A, a? = * " g^* = ? 3 r> A i 7 X 5 X 11 X 8 Rlq , By Analysis : ^-^rr T^ oM da. Since 5 men require 7 da., 10 men require a shorter time, i. e., T 5 months' credit and $600 worth on 3 months' credit. For what time should he give a note for the whole amount, $1800? 4. $1680 is to be paid in four equal instalments, in 1, 2, 3, and 4 mo. respectively. Equate the time. 5. $500 is due in 8 mo., $900 in 6 mo., $1000 in 3 mo., $1200 in cash [1200 X = 0]. Find the term of credit for a single payment of the whole indebtedness. 6. Equate the time for the payment of $5000, due Feb. 1 ; of $4000, due June 1 ; of $3000, due Aug. 1, and of $3000, due Oct. 1. Suggestion : Count time from Feb. 1. 7. A person owes a certain sum, of which J is payable in 8 mo., J in 9 mo., and the balance in 12 mo. Equate the time of payment. 8. Johnson & Co. sold a bill of lumber on the following terms: $1500 cash, $3000 payable in 30 days, and $2000 payable in 90 days. When may the whole debt be cancelled by one payment ? 9. If a person lends me $250 for 8 mo., for how long ought I to lend him $480 as an equivalent? 1 0. I bought on July 5th goods to the amount of $2400. $630 was to be paid at once, $820 in 8 mo., and $950 in 9 mo. What is the equated time for the payment of the whole? 11. A man owes $600, of which J is to be paid in 1 yr., and the remainder in 2 yr. Equate the time, and find the present value, money being worth 6%. 12. I bought bills of goods as follows: June 1, $250, on 3 mo. credit; July 5, $300, on 3 mo. credit; Aug. 6, $150, on 3 mo. credit ; Oct. 2., $400, on 2 mo. credit. Find the equated time of payment. AVERAGING OR EQUATING OF PAYMENTS 305 Process. Explanation. 250 X 0^00000 1. Add the terms of credit to their 300 X 34 = 10200 respective dates. 1 50 y 6f> - 9900 ^' -F 1 * 110 ^ the i n t erva l between the ~ earliest resultin s date and each of v Q9 the other dates. 1100 56900 3. Multiply the debts by their re- 5g900 -f- 1100 = 51-jSj- spective intervals, and proceed as Sept. 1 + 51 T 8 r = Oct. 23 13. Mr. B. bought goods as follows : April 15, $150 on 2 mo. credit ; May 10, $200 oil 3 mo. credit ; June 5, $250 on 4 mo. credit. Fiud the equated date of payment. 14. What is the average time at which the following bills become due : Feb. 1, 1898, $200 on 1 mo. credit ; March 10, 1898, $500 on 3 mo. credit; April 12, 1898, $275 on 2 mo. credit; and May 1, 1898, $400 on 4 mo. credit? 15. I owe Mr. Wilson $100, to be paid on the 15th of July; $200 on the 15th of August, and $300 on the 9th of September. What is the mean time of payment? 16. Find the equated time for the payment of $112.30 due July 6, $115.25 due July 30, $232.15 due Sept. 4, and $102.36*due Oct. 1. 17. A merchant bought goods as follows : Mar. 19th, $350 on 4 mo. ; Apr. 1st, $430 on 130 da. ; May 16th, $540 on 95 da. ; June 10th, $730 on 3 mo. ; what is the average time for the payment of the whole ? 18. $1200 worth of mdse., bought Nov. 5, and $1000 worth bought on the following Jan. 9, have a credit of 2 mo. When may both be paid at once ? 19. A man bought the following bills of goods: Jan. 15, $600 on 2 mo. credit; Feb. 1, $300 on 3 mo. credit; March 25, $550 on 30 da. credit ; and April 8, $400 on 60 da. credit. Find the equated time of payment. 20 306 PRACTICAL ARITHMETIC 20. Find the equated time of payment for the following obligations : 1. $400, due June 15 ; $375, due July 11 ; $195, due Sept. 4. 2. $1394.50, due Dec. 1, 1898 ; $129.80, due Dec. 10, 1898 ; $960, due Feb. 1, 1899. 21. A. owes $600, due in 8 mo. If he pays $160 in 3 mo. and $120 in 6 mo., when should he pay the balance? 8 mo. 3 mo. = 5 mo. 8 mo. 6 mo. = 2 mo. Therefore A. has to his credit : $160 for 5 mo. = $1 for 800 mo. ) fl1 (-ion.p o ai f nAf\ f $1 lor 1040 mo. $120 for 2 mo. = $1 for 240 mo. J But A. still owes $600 280 = $320. $1 for 1040 mo. = $320 for % 4 (t + u '2d step . . 4 . . . 1? 3d step . -v 4th step . |> 43) 133 . . 2t + u) 2t X u + w 2 (+ 4) 5th step . J 6th step. . 129 2t X u + u^ 7th step. . 461 \ 461 The above formula furnishes two figures of the root. Calling 23 the tens of the root, 2t = 46. SQUARE ROOT 315 EXERCISES. Find the square root of: 1. 100. 11. 2809. 21. 674,041. 2. 10,000. 12. 3969. 22. 784,996. 3. 625. 13. 4489. 23. 944,784. 4. 961. 14. 7056. 24. 998,001. 5. 2704. 15. 9216. 25. 5,875,776. 6. 6889. 16. 16,129. 26. 6,270,016. 7. 15,625. 17. 70,756. 27. 12,574,116. 8. 141,376. 18. 118,336. 28. 30,858,025. 9. 160,801. 19. 262,144. 29. 40,005,625. 10. 100,000,000. 20. 368,449. 30. 29,735,209. SQUARE ROOT OP COMMON AND DECIMAL FRACTIONS. INDUCTIVE STEPS. 1. What is the square root of J ? 2. What is the square root of $ ? = | ; f or ^ X | = 4. Hence. I/Fraction^ V Denominator 3. Find the square root of .25. 1/^25 = .5; for .5 X .5^.25. 4. The V^i= what? Not .2, for .2 X .2 = .04. But V~A = V~Ab = 1/.4000. 316 PRACTICAL ARITHMETIC By rule : .40'00 ( .632 36 123)400 369 Hence 1262) 3100 2524 V~A = .632+. In such cases annex ciphers and make periods from the point toward the right. PROBLEMS. 1. Find the square root of: 1. i 5. TTHT' 9- H*i- 13. mm- 2. A- 6. TTT- 10. HM- 14. T*WHHHHb 3. Tthr- 7. AT- 11. T^ft V 15. ifUMf 4. rMHhr- 8. fff 12. mn * is. im 7 ^9- 2. Find the square root of: 1. .09. 6. .12345. 11. .003969. 2. .9. 7. .763876. 12. 1.679616. 3. .0144. 8. .30858025. 13. 204.7761. 4. .144. 9. .093636. 14. .00009801. 5. .0100. 10. .099225. 15. .00010201. 3. Find the square root of: 1. f. 6. H 2. f 7. 3. T VW 8- f 4- tfff 9 - I- 5. jffy. 10. f. 11. f. 12. If. 13- I + | + f 1/1 a 9 15. 16. f 18. 19. 20. Suggestion : | = .875 ; i/.ST'SO = what? SQUARES 317 SQUARES. Since the area of a square lot whose side is 12 rods equals 12 X 12 or 144 square rods, a side of the lot = T/144. Hence the formula : Side of Square = v/Area. PROBLEMS. 1. What is the side of a square whose area is 1225 sq. It. ? 2. What is the side of a square whose area is 2025 sq. rd. ? 3. What is the side of a square farm containing 40 A. ? 4. A square plot of ground contains 320 A. How many feet long is each side ? 5. A circular pond has an area of 529 sq. rd. What is the side of a square of equal area ? 6. If an acre of land be laid out in a square farm, what will be the length of each side in rods ? 7. To arrange 7225 men in the form of a square, how many men must be put in each line ? 8. What would it cost to fence a square lot containing 640 A. at $4.00 per rod ? 9. If it cost $312 to enclose a field 216 rd. long and 24 rd. wide, what will it cost to enclose a square field of equal area with a like kind of fence ? 10. The attempt to form a square of 10,200 men excluded 200 of the men. How many men stood in each line of the square ? 11. If the faces of a cubical box measure 23,064 sq. in., how many linear inches in one of its edges ? 12. Which will cost the more to fence, a field measuring 40 by 80 rd. or a field of the same area in the form of a square? How much more at $1.33J per rod? 318 PRACTICAL ARITHMETIC TRIANGLES. A Triangle is a figure bounded by three straight lines. A Right Triangle has one right angle. h denotes the hypotenuse, the side opposite the right angle ; />, the perpendicular ; 6, the base. These three lines are so related that h 2 = b 2 + P 2 - Hence it follows that l.li = i/b 2 + p 2 2. b = i/h 2 p 2 3. p = l/h' b 2 Formulae. Right Triangle. 1. The base of a right triangle is 10 feet, its perpendicular 15 feet. Find its hypotenuse. h. = 1/b 2 + p 2 = 1/100 + 225 = 1/326 = 18 very closely. 2. Find the sides indicated by x in the table, using formulae 1, 2, and 3. No. 3. b = Vtf f = ~49~=: 1/15 = 3.87 +. 1 X 3 2 2 Q X 5 3 8 i \ * 4 10 q X 5 12 X 11 6 X 13 14 3. The perpendicular of a right triangle is 30 ft. and the hypotenuse is 50 ft. What is the base? 4. A square floor contains 400 sq. ft. Find the length of the longest straight line that can be drawn thereon. 5. A tree 150 ft. high stood on the bank of a stream. A part broken off 125 ft. from the top exactly measured the distance to the opposite banb. How wide was the stream ? 6. How far from a tower 40 ft. high must the foot of a ladder 50 ft. long be placed that it may exactly reach the top of the tower ? TRIANGLES 319 7. The inner dimensions of a box are 36, 24, and 12. Find the length of the longest straight rod that can be put therein. 8. A ladder 40 ft. long is so placed in a street that, with- out being moved at the foot, it will reach a window on one side 33 ft. and on the other side 21 ft. from the ground. What is the breadth of the street ? 9. x = 1/h 2 b 2 . Draw a figure for this equation, and write x upon the line to be found. 10. Make a ten-foot pole the hypotenuse, and find exact lengths for the base and perpendicular. Three Sides Given to Find the Area. 1. If the three sides of a triangle are 2, 5, and 6, what is its area? Process. (a.) *i* = J = 6.5. (6.) 6.5 2 = 4.5 ; 6.5 5 = 1.5 ; 6.5 6 = .5. (c.) Area = 1/6.5 X 4.5 X 1.5 X .5 = 1/21.9375 = 4.68. Brief directions are : 1. Find half the sum of the sides. 2. From the half sum subtract each side separately. 3. Find the square root of the product of the half sum and the three remainders. 2. What is the area of a triangle whose sides are respect- ively 4 in., 5 in., and 6 in.? 3. Find the area of a triangular lot whose sides are respect- ively 20, 25, and 28 rods. 4. Find the area of a triangular farm whose sides are 400 yd., 500 yd., and 600 yd. 5. What is the area of a triangle whose sides are 6, 8, and 12 ft.? 320 PRACTICAL ARITHMETIC CUBE ROOT. 1. The Cube Root of a number is one of its three equal factors. 216 = 6 X 6 X 6; 216 is therefore a cube, and 6 is its cube root. 2. The numbers represented by the digits and their cubes are: Numbers, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Cubes, 0, 1, 8, 27, 64, 125, 216, 343, 512, 729. NOTE. This table should be memorized. 3. The numbers are the cube roots of their cubes. 1 is the cube root of 1 ; 2 is the cube root of 8 ; 3 is the cube root of 27, and so on. 4. The cube root of 216 may be thus expressed : ^216 or Cube Root Found by Factoring". The prime factors of 64 are 2X2X2X2X2X2 = 4X4X4; therefore 1^64"= 4. 216 = 2X2X2 X 3^X~3~X~3 = 6X6X6; therefore In like manner find the cube root of 27, 125, 343, 512, 729, 4096, 42,875, 166,375, 185,193. Periods and Roots Compared. 1. Separating the following numbers into three-figure periods as far as possible, we have : One Period. Two Periods. Three Periods. f]7= i. 1^1 'ooo = 10. fVooo'ooo == 100. ^729"' = 9. f980'001 = 99. 1^998'000'001 = 999. 2. Obviously one period in the cube gives but one figure CUBE EOOT 321 in the root ; two periods in the cube, two figures in the root ; three periods in the cube, three figures in the root. Hence the principle : The number of figures in the cube root equals the num- ber of three-figure periods into which the number can be pointed off, beginning at units. NOTE The period on the extreme left may contain only one or two figures. Extraction of the Cube Boot. General Method. Find the cube root of 74,088. Cube. Root. 1. The number, pointed off into periods, is '4 Ooo ( 41i 2. The greatest cube in 74 is 64 3. The cube root of 64 is 4, the first figure of the root. 4. Subtracting and annexing the second period, we have 10088 5. 300 times the root-figure 4* = 4800, the trial divisor. 6. 10088 -=- 4800 gives 2 for the second figure of the root. 7. The complete divisor consists of: (a. ) The trial divisor, 4800 (b.) 30 times 4 X 2, or 240 (c ) 2x2 = 2 2 , or _4 Sum = 5044 8. Multiplying the sum, 5044, by 2, we have .... 10088 9. Subtracting, we have Therefore the cube root of 74,088 is 42. The process, freed from explanation, stands thus : 74'088 ( 42 64 4800 10088 240 4 5044 10088 21 322 PRACTICAL ARITHMETIC The algebraic discussion is as follows : We have seen that the cube of any number consisting of tens and units = the tens 3 -f 3 times the tens 2 X the units -f 3 times the tens X the units 2 -f the units 3 . For example, 35 3 = (30 + 5) 3 = 30 3 + 3 X 30 2 X 5 + 3 X 30 X 5 2 + 5 3 . That is, the cube of a two-digit number consists of four parts, which may be presented thus : X u. c. 3 X 30 X 5 2 = 3 X t X u?. d. 5 3 =u*. By regarding these four parts we may readily see how the cube root of a number may be obtained. 1. What is the cube root of 42,875? Process. 42'875 ( 35 Pointing off, we have : 42'875 9700 97 Finding Part a, rf 3 , we have : 27 ' 1^27 = 3, the tens of the root. A KA -i K 40U 1 Subtracting 27, we have remaining parts6 l c,d= 15875 Assume 3 X V X u = 15875. Dividing by the factor 3 X t 2 , we shall obtain the other factor, the units. 3 x P = 2700 ; 15,875 -5- 2700 = 5, the units of the root. Having thus found by trial the units, we must now form the parts b, c, d, and subtract their sum. X 5 = 3175 X 5 = 15875 Hence, the cube root of 42,875 is 3 tens -f 5 units = 35. The rule, briefly stated, is : 1. Point off the number into three-figure periods. 2. Find in the first period the greatest cube and its root CUBE BOOT 323 3. Subtract and annex the second period. 4. To find the second figure of the root, divide the re- mainder by 3OO times the square of the first root-figure. 5. To this divisor add SO times the product of the two root-figures; also, the square of the second figure. 6. Multiply the sum by the second root-figure. 7. Then apply again 3, 4, 5, 6, and 7, if necessary. To apply the rule : 2. Find the cube root of 79,507. Process. (1.) 79'507(43 (2.) 64 _ (3.) 15507 (6.) 15 507 4800 (4.) 360 (5.) 9 (5.) 5169(6.) 3. Find the cube root of 2,048,383. 300 (4.) 60 (5.) 4(5.) 364 (6.) proceed to find the units by (4). 144 300 43200 (4.) 2520 (5.) 49 (5.) 45769 (6.) Process. 2'048'383(127, Ans. 1048 (3.) 728 (g.) 320383 (7.) Now call the root 12 tens, and 320383, Rem, 320383 0(6.) 324 PKACTICAL ARITHMETIC EXERCISES. Find the cube root of : 1. 614,125. 8. 2,000,376. 15. 592,704. 2. 74,088. 9. 153,990,656. 16. 1,860,867. 3. 15,625. 10. 41,063,625. 17. 34,328,125. 4. 32,768. 11. 12,167. 18. 145,531,576. 5. 103,823. 12. 32,768. 19. 264,609,288. 6. 1,953,125. 13. 79,507. 20. 1,879,080,904. 7. 5,545,233. 14. 59,319. 21. 12,895,213,625. CUBE ROOT OP COMMON AND DECIMAL FRACTIONS. 1. What is the cube root of ^? *^ = i; foriX JX J = (J) 3 = 3sV. FORMULA. ? Fraction = ^Numerator __ v Denominator 2. What is the cube root of .8 ? Not .2, for .2 X .2 X .2 = .008. && = 1^.800000. By rule : .SOO'OOO ( .92 + 729 24300 71000 540 49688 4 21312, Rem. 24844 Hence ^8~= .92 +. In such cases annex ciphers, and make periods from the point toward the right. CUBE ROOT OF COMMON AND DECIMAL FRACTIONS 325 3. Find the cube root of: *. 3*. 5! m- ! IPfT 3- TJft. 6. #&. 9. I- 4. Find the cube root of: 1. .008. 6. 2.197. 11. 7. 2. .08. 7. 9.261. 12. 34.965783. 3. .8. 8. 185.193. 13. 41.063625. 4. .125. 9. .1. 14. .000001. 5. .25. 10. 6. 15. .0000001. Suggestion : 1^.08 = 1^.080 = .4, etc. Find two more places. 5. Find the cube root of: 1. f. 3. f 5. f. 7. lff|. 9. 2. -fy. 4. f. 6. f. 8. ifff. 10. Suggestion : f = .75 ; ^^750 = what? VOLUME. Volume of a cube = side 3 . Therefore : ^Volume = side of cube. PROBLEMS. 1. A cubical cistern contains 1331 solid feet. What is the length of one side of the cistern ? Volume = 1331. 1^1331 = 11, length of one side. 2. A cubical pedestal contains 373,248 cu. in. What is the length of one of its sides ? 3. A cubical box contains 474,552 cu. in. What is the area of one of the surfaces of the box ? 4. How much paper will cover the six surfaces of a cubical box whose volume is - cu. ft. ? 326 PRACTICAL ARITHMETIC 5. What is the depth of a cubical box that will hold a bushel ? 6. A wagon-box holds 100 bu. The length is twice the width and the width and depth are equal. Find the dimensions. 7. Find the cost, at 83 cts. per square yard, of lining the inside of a cubical box holding 900 gal. of water. 8. Find the height of a cubical pile of wood containing 179 cords. SIMILAR FIGURES. 1. Similar figures have the same shape, but differ in size. 2. Figures are either surfaces or solids. 3. A surface has dimensions and area. 4. A solid has dimensions and volume. 5. The relation of similar figures is in accordance with the following general principles : Similar Surfaces. 1. The areas of similar surfaces are to each other as the squares of their like dimensions. 2. The like dimensions of similar surfaces are to each other as the square roots of their areas. PROBLEMS. 1. Two surfaces having the same shape are to each other as 114 to 36. What is the ratio of their lengths ? Process. L. : 1. = 1/144 : 1/36. L. : 1. = 12: 6. L. : 1. == 2 : 1. Hence the ratio of their lengths is 2 : 1. CUBE HOOT OF COMMON AND DECIMAL FRACTIONS 327 2. The radius of a certain circle is 5 ft. What is the radius of another circle containing twice the area of the first? Suggestion : 5 : K. = V\ : 1/2. 3. The surfaces of two bodies having the same shape are as 100 : 25. What is the ratio of their widths? 4. If the area of a circle, whose diameter is 2 ft., is 6.2832 sq. ft., what is the diameter of a circle whose area is 25.1328 sq. ft? Suggestion : 25.1328 = 4 times 6.2832. 5. A farmer has a field 50 rd. wide by 80 rd. long, which contains 25 A. Find the dimensions of a similar field con- taining 16.81 A. 6. If a horse tied to a stake by a rope 8.79 rd. long can graze upon 1J A. of land, how long must the rope be that he may graze upon 6 A . ? 7. If a pipe whose diameter is 1.5 in. fills a cistern in 5 hours, in what time will a pipe whose diameter is 3 in. fill the same cistern ? 8. A half-inch pipe discharges a barrel of water in a cer- tain time. How much will a 2-in. pipe discharge in the same time? 9. If a 1-in. pipe discharges 1 gal. in 45 seconds, how much will a 2-in. pipe discharge in 60 seconds ? 10. A rectangular piece of land has a width of 160 ft. and is valued at $1200. What is the value of a similar piece of land having twice the length and breadth ? Similar Solids. 1. The volumes of similar solids are to each other as the cubes of their like dimensions. 2. The like dimensions of similar solids are to each other as the cube roots of their volumes. 328 PRACTICAL ARITHMETIC PROBLEMS. 1. Of two spheres, one is 1000 times the size of the other. If the diameter of the smaller is 6 inches, how many feet are in the diameter of the larger ? Process. ri : 1^1000 = 6 in. : x. I : 10 = 6 in. : x. x = 10 X 6 = 60 in. = 5 feet. 2. The diameter of a ball weighing 32 Ib. is 6 in. What is the diameter of a ball weighing 4 Ib. ? 3. The diameters of two spheres are respectively 4 and 1 2 in. The larger sphere is how many times the smaller? 4. If a 2-in. globe of gold is worth $500, what is the value of a 6-in. globe of gold ? 5. If the diameter of the sun is 112 times as long as that of the earth, how much greater is the mass of the sun than that of the earth ? 6. If the diameter of the moon is 2000 mi. and that of the earth is 8000 mi., what is the ratio of their volumes? 7. The weights of two cylinders of the same shape are as 27 to 64. What is the ratio of their lengths? Process. 1. : L. = 1^27 : ^64. 1. : L. = 3 : 4. 8. If a log 1J ft. in diameter contains 35 cu. ft., what is the diameter of a log of the same length that contains 105 cubic feet? 9. If a pyramid of hay 12 ft. high contains 8 tons, how high is a similar pyramid that contains 60 tons ? MENSURATION 329 MENSURATION. Mensuration treats of the measurement of lines, surfaces, and volumes. Important Suggestion. Experience has shown that much, if not all, of the difficulty in mensuration results from the pupil's failure fully to under- stand the terms used in describing surfaces and solids, and from the conse- quent failure to get a clear conception of the objects themselves. There- fore it is suggested that pupils be required to learn all definitions. This can best be done by a careful study of the figures in connection with the definitions. Concrete illustration should be used whenever possible, and pupils should be permitted to handle objects. In the absence of geomet- rical forms, pupils should draw correct and neat figures to represent the conditions of each problem. Time thus spent will produce good results. DEFINITIONS. 1. A Line has length, but no width. 2. A Straight Line is one which has the same direction throughout its whole length. It is the shortest distance be- tween two points. 3. A Curved Line is one which changes its direction at every point in its length. 4. Parallel Lines are equidistant throughout their whole length. 5. A Horizontal Line is a line paral- lel to the horizon. The line A B is hori- zontal. 6. When two straight lines meet or intersect in such manner as to form right angles, they are said to be Perpendicular, the one to the other. 330 PRACTICAL ARITHMETIC 7. A Vertical Line is one that is c perpendicular to the horizon. C is a vertical line. 8. An Angle is the amount of diver- gence of two lines which meet at a point. The point is called the Vertex. In the angle A C, is the vertex. The size of an angle is not dependent upon the length of the lines which form the angle. 9. There are three kinds of angles : 1. Right Angle. 2. Acute Angle, less than a right angle. 3. Obtuse Angle, greater than a right angle. Draw an angle of each kind. 10. A Diagonal is a straight line joining opposite angles. 11. The Perimeter measures the bounding line of a sur- face. 12. An Inscribed Figure is the largest figure of a given kind that can be drawn within another. (See page 343.) 13. A Circumscribed Figure is the smallest figure of a given kind that can be drawn about another. (See page 343.) 14. Concentric Circles are those having the same centre. The space between two concentric circles is called a Ring. Draw two concentric circles. SURFACES. 1. Surface is the outside of anything. Every surface has two dimensions, length and breadth. 2. Area is the extent of a surface, and is estimated in square units ; as, square inches, square feet, square yards, etc. MENSURATION 331 3. A Plane Surface is flat, like the walls and the floor of the school-room. Name some plane surfaces. 4. A Curved Surface is like that of a ball. Name some curved surfaces. 5. Surfaces are bounded by straight or curved lines ; hence the terms rectilinear and curvilinear as applied to surfaces. TRIANGLES. 1. A Triangle is a plane surface having three angles and three sides. Every triangle has two dimensions, altitude and base. base 2. Triangles, classified according to their angles, are of three kinds : 1. Right Triangle, having one right angle. 2. Obtuse- Angled Triangle, having one obtuse angle. 3. Acute-Angled Triangle, having three acute angles. Draw a triangle of each kind. 3. Triangles classified according to their sides are of three kinds : 1 . Equilateral Triangle, all sides equal. 2. Isosceles Triangle, two sides equal. 3. Scalene Triangle, no two sides equal. Draw a triangle of each kind. 4. We have learned (page 164) that the area of a rectangle is the product of the length and breadth (base and altitude). Every triangle is regarded as one-half of a rectangle having the same base and altitude ; hence the formula for the area of a triangle is : Area of triangle = base x altitude 332 PRACTICAL ARITHMETIC By drawing figures and by cutting paper let pupils prove the foregoing. PROBLEMS. 1. The base of a triangle is 150 yd. and its altitude is 75 yd. What is its area? 2. Required the area of a triangle whose base is 40 rd. and altitude 30 rd. 3. What is the area of an equilateral triangle whose sides are each 10 chains? 4. A board 5 ft. long has the shape of an isosceles triangle and measures at its base 15 inches. Find the number of square feet it contains. 5. Find the area of a right triangle, base 23.1 ft., altitude 32.1 ft. PARALLELOGRAM. alt. Rhomboid. 1. A Parallelogram is a plane surface whose opposite sides are parallel. 2. There are four paral- lelograms : 1. Square Sides parallel and equal; four right angles. 2. Oblong Sides parallel ; opposite sides equal, ad- jacent sides unequal ; four right angles. 3. Rhombus Sides parallel and equal ; two angles obtuse and two acute. 4. Rhomboid Sides parallel ; opposite sides equal ; two angles obtuse and two acute. 3. The altitude of the Rhombus and the Rhomboid is the perpendicular distance between the parallel sides. 4. Make correct forms of the parallelograms. Draw the diagonal and mark the altitude. MENSURATION 333 5. The formula for the area of a parallelogram is : Area = base X altitude. PROBLEMS. 1. A field in the form of a square is 64 rd. long. Find its area in acres. 2. How many square feet in an oblong board 90 in. long and 14 in. wide? 3. A pane has the form of a rhombus, measures 16 in. on each side, and the perpendicular distance between its sides is one-half the length of a side. Find its area. 4. Find the area in acres of a rhomboidal field which meas- ures 10 ch. in length and 8 ch. in breadth. TRAPEZOID. 1. A Trapezoid is a four- sided plane figure having two / ,. sides parallel. 2. The altitude of a trape- zoid is the perpendicular distance between the parallel sides. 3. The formula for the area of a trapezoid is : Area = gum^Qf_parallel sides x altitude . What do you get when you divide the sum of the parallel sides by 2 ? PROBLEMS. 1 . Find the area of a trapezoid with parallel sides of 50 rd. and 78 rd., and with a distance between them of 39J rd. 2. A trapezoidal field contains 12J A. Its parallel sides are 220 rd. and 180 rd. How far apart are the parallel sides? 334 PRACTICAL ARITHMETIC THE TRAPEZIUM. 1. A Trapezium is a four- sided plane surface having no two of its sides parallel. 2 . The diagonal of a trape- zium is a straight line con- necting opposite angles. The diagonal divides the trape- zium into two triangles. 3. The altitude of each triangle is the perpendicular dis- tance between the diagonal and the opposite angle. 4. To find the area of a trapezium, the diagonal and the altitude of each triangle being given, first find the area of each triangle, then add the areas. 5. The following is the formula : Area = diagonal x * um of Attitudes. 2 PROBLEMS. 1. A field has the form of a trapezium with a diagonal length of 1000 ft., and with perpendicular distances of 450 and 350 ft. Find the area. 2. Require the area of a trapezium whose diagonal meas- ures 145 ft. and the altitudes of the two triangles are 34 and 44 ft. respectively. THE REGULAR POLYGON. 1. Every plane surface bounded by straight lines has as many angles as it has sides. Naming plane figures according to the number of angles each contains, we have the following : MENSURATION 335 Triangle, three angles; Quadrangle, four angles; Pen- tangle, or Pentagon, five angles; Hexagon, six angles; Heptagon, seven angles; Octagon, eight angles; Nonagon, nine angles; Decagon, ten angles, etc. 2. Polygon is a general term, and is applicable to any figure having three or more angles. 3. A Regular Polygon is one having all its angles and sides equal. 4. Any regular polygon may be divided Hexagon, into as many equal triangles as the polygon has sides. If the base and the altitude of the triangles be known, the area of the polygon may be found by multiplying the area of one triangle by the number of triangles. 5. The formula for the area of a regular polygon is : Area = perimeter X Perpendicular 2 NOTE. The word "perpendicular'' is here used to denote the altitude of one triangle. PROBLEMS. 1. Find the art-a of a hexagon whose sides are each 12 in. and the perpendicular distance from the centre to a side is 8 in. 2. What is the area of a regular pentagon whose side is 15 ft. and the altitude of the triangles into which it may be divided is 8.602 ft. ? THE CIRCLE. 1. A Circle is a plane surface bounded by a curved line every point of which is equally distant from a point within the circle called the centre. The point where the straight lines in the figure meet is the Centre of the circle. 336 PRACTICAL ARITHMETIC 2. The Circumference of a circle is the bounding line. 3. The Diameter is the distance across the circle measured through the centre. 4. The Radius is one-half of the diameter. By a geometrical process, it has been found that if the di- ameter of a circle is 1, the circumference is 3.1416. Hence, if we know the diameter of a circle, we may find the circum- ference by multiplying the diameter by 3.1416 ; and knowing the circumference, we may find the diameter by dividing the circumference by 3.1416. The number 3.1416 is called the ratio of a circumference to its diameter. Pupils should remember this number, as it is of much use in measuring circular surfaces, etc. 5. The following formulae apply to the circle : 1. Circumference = diameter X 3.1416. 2. Diameter = gkg|niference 3. Area = circumference X I ^ l ~ f *~. 4. Area = Radius 2 X 3.1416. 5. Area = diameter 2 X .7854. . Observe that .7854 is one-fourth of 3.1416. MISCELLANEOUS PROBLEMS. 1. What is the circumference of a circle having a diameter of 21 ft. ? 2. What is the diameter of a circle 33 yd. in circum- ference ? 3. What is the circumference of a circle whose radius is 16yd.? 4. What is the area of a circle whose circumference is 18 in.? 5. Find the perimeter of a triangle whose sides are re- spectively 3J ft., 4f ft., and 5{ ft. MENSURATION 337 6. A horse is tied by a rope 7 rd. long, and can reach 2 ft. beyond the end of the rope. How much surface can he graze over? 7. Find the circumference of a circle whose diameter is 14 ft. 8. Find the diameter of a circle whose circumference is 1 ft, 9. Find the area of a circle whose radius is 7 yd. 10. The radius of a grass plot is 42 ft. Find the area of a walk 4 ft. wide running around the grass plot. 11. Find the area of a triangle whose base is 10 ft. and altitude 2| ft. 12. What is the area of a trapezium the diagonal of which is 1 1 ft., and the perpendiculars to the diagonal are 40 ft. and 60 ft. respectively ? 13. If a horse is tethered by a rope 20 rd. long, over how much surface can he graze ? 14. The base of a triangle is 300 yd. and its altitude is 150 yd. Find the area. 15. Two opposite sides of a quadrangular field are parallel, and are 140 yd. and 170 yd. long. The shortest measure across the field is 90 yd. What is the area? 16. A rectangular tank is 12 ft. long, 4 ft. wide, and 3 ft. high. How many square feet of sheet lead will be required to line it? 17. A diagonal of a field in the form of a trapezium is 17 chains 56 links ; the perpendiculars to that diagonal from the opposite angles are 8 chains 82 links, and 7 chains 73 links. What is the area? 18. Find the diameter of a circle whose circumference is 316 ft. 19. What is the circumference of a circular pond whose diameter is 45 rods ? 22 338 PRACTICAL ARITHMETIC 20. What is the area in acres of a circular island whose circumference is 2 miles ? 21. A farm in the form of a trapezoid has its parallel sides 72 ch. and 84 ch. in length, and the perpendicular distance between them is 40 ch. How large is the farm ? 22. How many rods of fence will be needed to go round a circular park containing 120 A.? Suggestion : Draw figures to illustrate the following problems. 23. A circular yard 200 feet in diameter has a walk 6 feet wide bordering on the circumference and extending entirely around the yard. What is the area of the walk? 24. If within a circle 10 feet in diameter a circle 6 feet in diameter be drawn so that the two circles shall meet at one point, what will be the area of the crescent thus formed ? 25. The side of the largest regular hexagon that can be in- scribed within a circle 6 ft. in diameter is equal to the radius of the circle. How much waste will there be in cutting such hexagon from the circle? 26. After making the hexagon in problem 25, suppose you should decide to make from the hexagon as large a circle as possible, what would be the diameter of the circle ? VOLUMES. 1. A Solid has three dimensions, length, breadth, and thick- ness. 2. The Volume of a solid is the number of cubic units which it contains ; it may be cubic inches, cubic feet, etc. 3. The Lateral Surface of a solid is the area of its sides or faces. This is also called Convex Surface. 4. To find the volume of a solid three dimensions or their equivalent must be given ; and to find any one of the dimen- sions of a solid the volume and two dimensions or their equivalent must be given. MEJXSUKATION 339 THE PRISM AND CYLINDER. 1. Prism is a solid whose ends are equal parallel polygons, and whose sides are rectangles. The ends are called bases and the sides are called lateral faces. 2. The form of the base gives a prism its distinguishing name. If the base be a triangle, the prism is called a triangular prism; if the base be a square, the prism is called a square prism ; if the base be a pentagon, the prism is called a pent- angular prism, etc. 3. A Cylinder is a solid with circular ends and uniform diameter. The ends are called the bases, and the curved surface is called the lateral surface, or convex surface. 4. The following formula apply to prisms and cylinders : 1. L. S. = Perimeter of Base X Altitude. 2. Vol. = Area of Base X Altitude. Base a circle PROBLEMS. 1. Find the lateral surface of a pentangular prism, the side of the base being 8 in. and the height 35 in. 2. Find the lateral surface of a cylinder whose height is 25 in. and diameter of the base 15 in. 3. Find the lateral surface of a triangular prism 24 ft. high, the sides of the base being 3 ft., 4 ft., and 5 ft. 4. Find the entire surface of a cylinder 9 ft. high and 3 ft. in diameter. 5. Find the entire surface of a prism 18 in. square and 7 ft. high. 340 PRACTICAL ARITHMETIC 6. Find the entire surface of a prism 18 in. high, the base being a triangle whose sides are 3 in., 4 in., and 4J in. 7. Estimate the volumes of the solids described in prob- lems 2, 3, 4, 5, and 6. 8. What must be the diameter of a cylindrical tank 10 ft. deep to contain 8460.288 gal. ? 9. A rectangular bin 5 ft 4 in. long and 3 ft. 2 in. wide contains 64 bu. What is the depth ? 10. If you cut a cylinder as large as can be made from a prism 6 in. square and 18 in. long, how much of the prism will be wasted ? THE PYRAMID AND CONE. 1. A Pyramid is a solid having a regular polygon for a base and ending in a point at the top. Draw a triangular pyramid. A square pyramid. 2. A Cone is a solid having a circular base and tapering to a point. 3. The point of a pyramid and of a cone is called the Vertex. 4. The Altitude of a cone and of a pyra- mid is a straight line drawn from the vertex perpendicular to the base, 5. The Slant Height of a pyramid is a straight line drawn from the vertex perpen- dicular to one side of the base, as A B. 6. The Slant Height of a cone is a straight line drawn from the vertex to any point on the circumference of the base. Slant Height Base a circle 1. L. S. = Perimeter of Base X 2. Vol. = Area of Base X Altitude 3 MENSURATION 341 PROBLEMS. 1 . Find the lateral surface of a hexagonal pyramid whose slant height is 20 ft. and each side of the base 5 ft. 2. What is the extent of the lateral surface of a cone the base of which is 27 in. in diameter and the slant height 5 ft. ? 3. If wheat be piled in a corner of a rectangular room in such manner as to form a portion of a cone, how many bushels are in the pile if the top of the pile is 8 ft. from the floor and the outer edge 5 ft. from the angle formed by the walls ? 4. A conical glass is 7 in. deep and 5J in. in diameter. What part of a gallon will it hold? 5. A quadrangular pyramid is 16 in. square at the base and 3 ft. high. In making from this pyramid the largest possible cone, how much must be cut off? FRUSTUM OF PYRAMID AND OF CONE. 1. A Frustum of a Pyramid is that part of a pyramid which remains when the top is cut off by a plane parallel to the base. 2. A Frustum of a Cone is that part of a cone which remains when the top is cut off by a plane parallel to the base. 3. The Altitude and the Slant Height of frustums are found in the same manner as in the case of the pyramid and the cone. 4. The following formulae are applicable to pyramids and cones : 1. L. S = Sum of Perimeters of the 2 Bases 2 2. Vol. = [Sum of Bases + V Product of Bases] X X Slant Height. Altitude 342 PEACTICAL ARITHMETIC PROBLEMS. 1. Find the lateral surface of a frustum of a pentangular pyramid if the side of the lower and upper bases be 3 ft. and 2 ft., respectively, and the slant height 9 ft. 2. What is the entire surface of a frustum of a cone, the bases being 16 in. and 10 in. in diameter and the altitude -30 in. Suggestion : First find the slant height. 3. What is the volume of the frustum described in the second problem? 4. Find the volume of a frustum of a pyramid 4 J ft. square at the lower base, 2J ft square at upper base, and 6J ft. high. 5. At $1.25 a square foot what will be the cost of lining with copper a vat in the shape of an inverted frustum of a cone if the upper diameter is 7 ft., the lower diameter 5 ft., and the depth 6 ft. ? THE SPHERE. 1. A Sphere is a solid bounded by a curved surface of which every point is equally distant from a point called the centre. 2. The following formulae are for the surface and volume of a sphere : 1. Sur. = Diameter X Circumference. 2. Sur. = Diameter 2 X 3.1416. 3. Vol. = Sur. X |p 4. Vol. = Diameter 3 X .5236. NOTE. .5236 is one-sixth of 3.1416. PROBLEMS. 1. What is the surface of a sphere 18 in. in diameter? Its volume? 2. The diameter of a sphere is 12 in., the circumference is 37.6992 in. What is the surface? MENSURATION 343 3. What is the volume of a sphere the surface of which is 78.54 sq. in. and the radius is 2.5 in. ? 4. If the diameter of a cannon-ball is 15 in., what is the volume ? What is the surface ? 5. A hemispherical bowl 12 in. in diameter is filled with water. An iron ball put into the water is just large enough to extend from the bottom of the bowl to the surface of the water. Find the amount of water that remains in the bowl after the sinking of the ball. CIRCLE AND LARGEST SQUARE. h is obviously both diameter of the circle and hypotenuse of a right triangle ; b and p are base and perpendicular, and also sides of the square. Since 6 = - p, h 2 = 26 2 . Let h = 10 ; then 26 2 = 100, and 6 2 = 1M-. Taking the square root, we have b = Hence the formula : Side of square = When the diameter = 1, the side of the square = v\ or .5 = .7071 4~? and the formula becomes : Side of square = diameter X .7O71. PROBLEMS. 1. If 6 and p each equal 1 (see figure), what is the length of A? 2. If p or b equal 1, what is the length of the circum- ference ? 3. When the diameter of a circle equals 5, what is the side of the inscribed square? 4. Find the area of the inscribed square and of the circum- scribed circle, when the diameter equals 5. 344 PRACTICAL ARITHMETIC SPHERE AND LARGEST CUBE. h f is obviously both diameter of the sphere and hypotenuse of the erect right triangle, ///, 7i, p- } h is the hypotenuse of the horizontal tri- angle, h, b, b. h 2 = 2b 2 . (hj = h 2 -f p 2 . Hence (hj == 2b 2 + p 2 . But b = p; therefore (h') 2 = 36 2 . Let h' = 10 ; then 36 2 = 100, and b 2 = ^. Taking the square root, we have 6 = Hence the formula : Side of Cube = /diameter' When the diameter = 1, the side of the cube = V \ or .3333, etc. .57735 -f-, and the formula becomes : Side of cube = diameter x .57735. PROBLEMS. 1. What is the volume of a pyramid whose base is a rec- tangle 13 by 14 feet, and whose height is 18 feet? 2. What is the volume of a cylinder 108 in. in diameter and 10 ft. long? 3. What is the lateral surface of a cone whose base is 10 ft. in diameter and slant height 20 ft. ? Find also the entire surface. 4. Find the surface of a sphere whose radius is 12 inches. 5. How many gallons will a hollow globe contain whose inside diameter is 20 inches ? 6. What is the lateral surface of a triangular prism whose sides are each 6 feet and whose altitude is 8 feet ? 7. What is the lateral surface of a quadrangular pyramid whose base is 15 feet square and the slant height 18 feet? MENSURATION 345 8. What is the lateral surface of a cone whose base is 10 ft. in diameter and whose slant height is 10 ft. ? 9. Find the volumes in problems No. 6, 7, and 8. 10. Required the surface of the frustum of a cone whose slant height is 12 feet, diameter of lower base 10 ft. and upper base 6 feet. What is the volume? 11. Find the entire surface of the frustum of a triangular pyramid whose slant height is 40 in., and the sides of the upper base 4 in. and the lower base 10 in. 12. Required the contents of a cannon ball whose diameter is 9 inches. What is the surface ? 13. At 45 cents a square foot, how much will it cost to gild a ball 25 inches in diameter? 14. Find how many cubic inches of iron there are in a hollow sphere, the diameter being 15 inches long and the shell 3 inches thick ? 15. A cylindrical can is 6 inches deep and 4 inches in diameter. If a cone of the same height and diameter be placed in the can, how much water will be required to fill the remaining space? 16. In the above problem, what is the ratio of the volume of the cone and cylinder? Does this show why 3 is used in the formula for the volume of a cone? 17. Find the side of the greatest square that can be in- scribed in a circle whose diameter is 10 feet? 18. Find the edge of the greatest cube that can be cut-from a wooden ball whose diameter is 5.5 inches. 19. I have a cubical box whose faces each contain 64 square inches. Find the diameter of the sphere that will exactly contain the box. 20. I have a circular garden whose circumference is 31.416 rods. I wish to reduce it within the circumference to the largest possible square form. Find the area of the square. 346 PEACTICAL AKITHMETIC GENERAL REVIEW. The following problems have been selected from the ex- amination papers of the University of the State of New York. They are introduced here for the purpose of affording a complete review of the principles and methods set forth in the previous pages of the book. It is suggested that the best efforts of both teacher and pupil be applied to these problems, and that the science and art of arithmetic, as already illustrated, be faithfully recalled, studied afresh, and securely fixed in mind. Let every solution, therefore, proceed systematically, and every principle involved be distinctly stated. 1. Define sum, and illustrate your definition by a practical example. 2. A man deposits in bank $986.46. At different times he has drawn the following amounts : $314.18, $49.25, $57.62, $39.84, $25.13. Find the amount remaining in the bank. 3. Find the least number of bushels of grain that can be exactly measured either by a 3-quart, a peck, a 20-quart, or a bushel measure. 4. Reduce M-JI4 to its lowest terms. & t y o / 5. Simplify * _ . and express the result both as a common and as a decimal fraction. 6. Define composite number and give an example. 7. Make a receipted bill for the following : Harold Kirby bought of Pliny Hall, 10 Ib. sugar at 5 cts., J Ib. tea at 60 cts., 3 Ib. coffee at 40 cts., 1 sack flour at $1.50. 8. If the shadow of a post 6 ft. high is 4 ft. 6 in. long, what is the height of a tree whose shadow at the same time is 125 ft. long? (Solve by analysis.) GENERAL KEVIEW 347 9. What would it cost to dig a cellar 80 ft. X 35 ft. X 8 ft. at $.84 per cubic yard ? 10. A railway train runs f of a mile in f of a minute. Find its velocity per hour ? (Solve by analysis.) 11. Define quotient, and give an illustration. 12. Find the prime factors of 1001 and 1309, and from these factors form the G. C. D., and the L. C. Dd. (least common multiple) of the two numbers. 13. A field 10 chains 50 links long and 8 chains 40 links wide produces 40 bushels of oats per acre ; what is the value of the crop at 35 cents a bushel ? 14. Find the sum of 9f, 8J, 5f, and ^-. Express the re- sult both as a fraction in lowest terms and as a decimal. 15. What part of an ounce (apothecaries' weight) is 5 drachms and 2 scruples ? 16. Find the cost of a stick of timber 40 ft. long, 12 in. wide, 9 in. thick, at $12.50 per M., board measure. 17. A roll of wall paper 8 yd. long and 18 in. wide costs 25 cts. What will be the cost of paper for the four walls of a room 30 ft. X 27 ft. X 9 ft., no allowance being made for openings ? 18. I bought 240 barrels of apples at $1.75 a barrel ; lost 40 barrels through frost ; at what price a barrel must I sell the remainder to gain 25% on the money invested? 19. If 2 men plough 15 acres in 5 days, working 10 hours a day, how many acres will 3 men plough in 4 days, working 8 hours a day ? 20. Define greatest common divisor and least common divi- dend (multiple). Illustrate. 348 PRACTICAL ARITHMETIC 21. What is meant by cancellation f 22. Simplify J of -^- of 2J X 14. 23. What part of a bushel is contained in a rectangular box 3 in. deep and 4 in. square? [A bushel 2150.42 cu. in.] 24. From sixty subtract forty-seven and sixteen ten-mil- lionths and express the decimal as a common fraction. 25. Find the cost of carpeting a room 18 ft. long, 15 ft. wide, with carpet 27 in. wide, at 75 cts. a yard. 26. Define divisor, root, proportion, fraction. 27. I retail oranges at 3 cts. each, gaining 150% on the purchase price. What did the oranges cost a dozen? 28. I sell an article at an advance of 25% on the cost and then discount the bill 5% for cash payment. My net gain is $63.75. Find the cost, 29. A cubic foot of water weighs 62J Ib. Find the weight of a barrel of water. JJ0T On a bill of goods amounting to $485.50 I receive commercial discounts of 15%, 10%, and 5%. Find the net cost of the goods. 31. What principal loaned for 1 yr. and 3 mo. at 6% simple interest will amount to $1000? 32. A 30-day note discounted at a New York bank yields $358.02. What was the face of the note? 33. A note for $500 at 90 days, with interest at 6%, is discounted at a bank 30 days after it is dated. Find the proceeds. 34. A certain stock pays annual dividends of 4%. At what rate must it be bought to pay 5% on the investment? 35. Find the square root of 4,004,231 to two places of decimals. GENERAL REVIEW 349 36. If I buy 10 shares of railway stock at 80 and sell them at 90, how many dollars do I gain and what is the rate per cent, of profit ? 37. Find the smallest number that will exactly contain 15, 18, 21, 24, and 30. 38. Two men hire a pasture for $30. A. puts in 8 horses for 10 weeks and B. 6 horses for 12 weeks. How much should each pay? 39. A house valued at $6000 is insured for f of its value at the rate of J of 1% a year. How much is the annual premium? 40. Find the prime factors of 1226, 1938, and 2346. In- dicate which of these factors must be combined to produce (a) the greatest common divisor, (b) the least common dividend. 41. Make a receipted bill of the following: Sold this day to Anson White, 3 bbl. flour, at $3.75 ; 75 Ib. sugar, at 5 cts. ; 10 Ib. coffee, at 35 cts. ; 2 Ib. tea, at 60 cts. 42. Find the amount at simple interest of $865.35 for 1 yr. 5 mo. 17 da. at 4J%. 43. In a certain school district the assessed valuation of property is $136,395, and the amount to be raised by local tax is $785.72. Find the amount of A.'s tax, whose property is assessed at $8500. 44. A bar of iron in the form of a cylinder, 6 feet long and 2 inches in diameter, is forged into a square bar whose cross- section is 2J square inches. Find the length of the new bar. 45. A man plants corn on of his land, potatoes on 2J times as much, and sows the remainder with wheat. He sells the wheat at 60 cts. a bushel, and receives for it $180. If the yield of wheat was 20 bushels an acre, how much land had he ? 350 PRACTICAL ARITHMETIC "46. Simplify the following : a 8 * * 2 V ? X 9 X 3 47. A note for $624 is dated August 26, 1893; July 15, 1894, there was paid on it $62.50. Find the amount now due. 48. Find the amount of $685 at 4J% simple interest from July 1, 1894, to the present time. 49. Define and illustrate dividend, power, ratio, factor. 50. I buy hats at $18 a dozen and sell them at $2.50 apiece. Find the gain per cent. 51. I sell goods at a discount of 10% from the marked price and still make a profit of 8%. How many per cent, above cost was the marked price ? 52. What single discount is equal to a commercial discount of 10%, 10%, and 5% ? 53. Find the square root of 1,080,234 to two decimal places. 54. Find the least possible cost of carpeting a room 15 feet long, 12 feet wide, with carpet f yd. wide, at 75 cts. a running yard. 55. Write the table of avoirdupois weight. For what is this weight used ? 56. Two men start from the same point on a level plain and travel, one due north at the rate of 3 miles an hour, the other due east at the rate of 5 miles an hour. How far apart will they be at the end of 10 hours ? 57. Divide one millionth by eight ten-thousandths, and express the result in words. 58. Find the prime factors of 2964, and all the different composite factors into which the prime factors may be com- bined. 59. Define minuend, multiplication, prime factor, common divisor, ratio. GENERAL REVIEW 351 60. Find the amount at simple interest, at 5%, of $860 from Sept. 1, 1894, to the present time. 61. Show that if four quantities are in proportion the pro- duct of the means equals the product of the extremes. 62. How much is due Aug. 15, 1893, on an interest-bearing promissory note for $250, dated Buffalo, June 1, 1886, on which $50 was paid Dec. 24, 1886, and $10 Jan. 5, 1888? 63. Find the cost, at $7 per 100 sq. ft., of slating a trape- zoid of which the parallel sides are 64 ft. and 32 ft., and the perpendicular distance between them is 20 ft. I Ql ^ >J2 64. Simplify and express decimally 8 J*_ 61 65. Find the square root of 8.5849. 66. Find the cost of shingles required to cover a roof 40 ft. long, 20 ft. wide at $5.00 a thousand, if it requires 36 shingles to cover 5 sq. ft. 67. Find the amount due this day on a note given in New York May 10, 1890, for $500, with interest, a payment of $35 having been made July 5, 1891. 68. Reduce to its lowest terms , (1 J + lf) 4 X 3 .- -g x -% ^ f 69. Define least common dividend, factor, numerator, divisor, root, proportion, fraction. 70. A cistern is 6 ft. square. How deep must it be to hold 30bbl. of water? 71. Find the least common dividend (multiple) and the greatest common divisor of 45, 70, and 105. 72. How many times will a wheel 4 ft. in diameter revolve in going one mile ? 352 PEACTICAL ARITHMETIC 73. Find the diagonal of a rectangle whose sides are 15 ft. and 20 ft, 74. I invest $6000 in 6% bonds at 125. What rate per cent, do I receive on the investment and what is the income from it ? 75. A field is 42 rd. long and 35 rd. wide. Find its value ut $37.50 an acre. 76. A man 6 ft. high casts a shadow 42 in. long. Find the height of a flagstaff which at the same time casts a shadow 28 ft. long. 77. Multiply 2 thousand 9 ten-millionths by 30 thousand 2 and 7 tenths, and divide the product by 3 ten-thousandths. 78. An agent remits to me $247.38, after retaining a com- mission of 5% for collection. What sum did he collect? What was the amount of his commission? 79. Three men engage in partnership. A. puts in $1200, B. $1550, C. $1900. They gain $350. What is each man's share of the profits? 80. The owner of -fj- of a mine sold -f$ of his share for $40,500. What should he who owns ^ of the mine get for f of his share? 81. If 18 men can dig 128 yards of ditch in 32 days, how many yards can 1 2 men dig in 64 days ? 82. If a square field contains 10 acres, what is the length of the diagonal ? 83. At what price must 6% bonds be bought to yield 4% on the investment ? 84. If 8 men reap 36 acres of grain in 9 days, working 9 hours a day, how many men will reap 48 acres in 12 days, working 12 hours a day? 85. Find the cost, at 35 cts. per cubic yard, of excavating a trench 6 rods long, 1 \ yards wide, 1 foot 6 inches deep. GENERAL EEVIEW 353 86. A note for $560, payable in 90 days, is discounted at a bank 30 days after it is dated. Find the proceeds. 87. Find the amount of $945.15 from December 15, 1891, to November 22, 1892, at 4J% simple interest. 88. Divide $720 among A., B., and C., so that the number of dollars they receive shall be as the numbers 5, 6, and 7. 89. A merchant marks an article $2.80, but in selling it takes off 5% for cash. If the rate of his profit is 33%, what was the cost of the article ? 90. What part of an ounce is 53 %3 ? 91. Find the amount of $375 for 11 mo. 17 da., at simple interest. 92. Find the cost, at 25 cts. a rod, of building a fence round a square 10-acre field. 93. How many gold rings, each weighing 5 pwt. 18 gr., can be made from 2 oz. 6 pwt. of gold ? 94. Find the face of a 60-day note which, when discounted at a New York bank, will yield $250. 95. If it costs $80 to plough a field 40 rods by 80 rods when we pay $5 a day for man and team, how much will it cost to plough a field 30 rods by 60 rods if we pay $4 a day ? Suggestion : Solve by proportion and by analysis. 96. What number divided by the sum of and 2^ will give a quotient of 2^- ? 97. If rain-drops are falling directly downward, how much more ground surface would be protected from the rain by a board 20 feet long and 18 inches wide when in a horizontal position than when one end of it is elevated 9 feet higher than the other? 23 354 PRACTICAL ARITHMETIC 98. A certain town raised a tax of $4607.50. The real estate was valued at $420,000, the personal property at $189,000, and 1250 persons paid a poll-tax of $1.25 each. Find the tax on $1.00 of the property. 99. How high must be a pile of wood 10 feet long and 2J feet wide to contain one cord ? 100. How much should be paid for 40 shares of railroad stock at 3J% discount and \\% interest after deducting 35 cts. from every $12 of the income ? 155. David Palmer borrows this day of Samuel Hill $350, and gives his note for this amount for 4 months at 6%. Make out the promissory note in proper form. 156. When it is 3 P.M. at Rome, longitude 12 27' east, it is 8.20 A.M. at Chicago ; find the longitude of Chicago. 157. A and B run a mile in opposite directions : A's run- ning is to B's as 6-J- : 5 ; B gets 4 seconds start, during which time he runs 12^ yards. Find when he will pass A. ANSWERS. Page 14. 1.1. Fifteen. 2. Four. 3. Fourteen. 4. Twenty-four. 5. Nineteen. 6. Thirty-nine. 7. Thirty-three. 8. Twenty-nine. 9. Forty-nine. 10. Forty-five. 11. Ninety -nine. 12. Sixty-five. 13. One hundred nine. 14. One hundred eleven. 15. Ninety-one. 16. Six hundred ninety. 17. Three hundred 39. 18. Seven hundred 34. 19. 790. 20. 1029. 21. 5555. 22. 550600. 23. 210506. 24. 8000. 25. 200090. 26. 149. 27. 2500. 28. 70899. 29. 1595864. Page 15. 2. 1. XV. 2. XXXVI. 3. LXXXVII. 4. LVI. 5. XLIX. 6. XCIX. 7. ML. 8. MMMMMX = VX. 9. DCCLXXXIX. 10. MDCCCXCVIII. 11. XVIII. 12. XLII. 13. LXVI. 14. LXXXVI. 15. LXIII. 16. C. 17. IIIDC. or MMMDO. 18. DLXXXVII. 19. CCVII. 20. VIIIIV. 21. XXVII. 22. LXXXI. 23. XCV. 24. XL. 25. XLV. 26. DXXXIV. 27. V. 28. CDXXXVI. 29. CMXCIX. 30. LXXVTCMLIX, 359 360 ANSWERS ADDITION. Page 23. Page 18. 21. 1. 66892. 5. 67573. 1. 1. 599. 6. $6.95. 2. 58434. 6. 46997. 2. 676. 7. $9.55. 3. 508785. 7. 51871. 3. 1026. 8. $92.79. 4. $9622942. 8. 49845. 4. 794. 9. $983.9Q 5. 748. Page 24. Page 19. 9. 134083.44. 11. 2356.9657. 10. 16193. 12. 10615. 10. 108349.98. 11. 10333 13, 7720. 2. 8649. SUBTRACTION. 3. 1. 11429. 4. 77230. Page 26. 2. 20681. 5. 235308. 1. 411. 3. 254. 5 352. 3. 101391. 2. 324. 4. 213. 6. 5533 4. 1. 4164. 1. 2519. 2. 1461. 2. 3046. Page 27. 3. 2867. 3. 1965. 7. $25.62. 11. 32,154. 4. 3285. 4. 2690. 8. $35.09. 12. 27,312. 5. 2791. 5. 3332. 9. $11.13. 13. 422.641. 6. 1453. 6. 2469. 10. $21.40. 14. 145.325. 5. 104367. 7. 83619. 2. 1520. 6. $1447.845. 8. $132.90. Page 28. Page 2O. 4. $3264. 8. 127,420. 5. 1212. 9. 289. 9. 7076. 10. 31164. 6. 6,550,216. 10. 9 yrs. 2. 95 acres. 4. 20694. 7. $4,820,411. 3. $604.20. 5. 1065. Page SO. Page 21. 1. 1. 305. 12. 2131. 6. 528408. 10. 402399. 2. 228. 13. $3.07. 7. 13587. 11. $2454.63. 3. 292. 14. $2.17. 8. $2275.00. 12. $8513.75. 4. 272. 15. $16.17. 9. 58639. 5. 1879. 16. $24.96. Page 22. 6. 61. 17. $33.66. 13. 1646619. 18. 61. 7. 1919. 18. $.995. 14. 73941. 19. LXII. 8. 388. 19. $88.996. 15. 365. 20. $1906.50, 9. 1299. 20. 1,410,273. 16. 72 days. $6140.66, 10. 40. 21. 3120473. 17. $171800. $8047.16. 11. 6828. 22. 998.78. ANSWERS 361 2. 1. 27,747. 6. 132,890. MULTIPLICATION. 2. 45,860. d. 430,875. 3. 493,879. 7. 5,741,182. Page 39. 4. 382,717. 8. 1,987,588. 2. 1. 730. 6. $25.45. 2. 2696. 7. $63.60. Page 31. 3. 2268. 8. $89.25. 4. 1962. 9. $495 54. 2. $2125. 8. 7795. 5. 2040. 10. $523.20. 4. 45,558,897. 9. 69,191,517. 5. 45yrs. 10. $3149. 6. 67yrs. 11. $925,985. 7. 7600 ft. 4. 1. 69536. 6. 41C88. 2. 37296. 7. 261045. 3. 51590. 8. 478709. 4. 65601. 9. 318352. Page 32. 5. 69380. 10. 827847. 12. $1046. 17. 4908 ft. 5. 1. 3780. 5. 162108. 13. $4.365. 18. 1437. 2. 18118. 6. 1526190. 14. $19.81. 19. 1706. 3. 234177. 7. 243582. 15. $27,404. 20. LXlVII. 4. 12533346. 8. 7282896 16. 14,162ft. 6. 1. 302. 4. 687. 7. 2484. 1. 23,527. 2. 33,958. 2. 54. 5. 8537. 3. 4160. 6. 1553. Page 33. 3. 13,181. 5. $207.61. Page 4O. 4 120,091. 2. $11.25. 7. 672. 1. 224,980 3. 919. 3. 1227.45. 8. 2,400,000. 2. 19,553068. 4. 55. 4. 15840. 9. 82287UOOO. 5. $29316. 10. $2499.96. Page 34. 6. $6655. 11. $42,592. 2, 165. 6. 91,145; 58,905. 8. $1365. 7. 3561. Page 41. 4. 1155. 8. $4484. 12. .03. 14. Lost $10. 5. 2070. 9. D. 697. 13. Cows, 20. 15. 9050. Page 35. Page 42. 10. 447. 15. 796. 2. 1. 4755. 6. $440.55. 11. 245. 16. 96. 2 7728. 7. $767.55. 12. 115. 17. 168.89. 3. 19481. 8. $2176.56. 13. 273. 18. 82. 4. 17082. 9. $3477.33. 14. 2410. 19. 11,220. 6. 12691. 10. $1614.14. 362 ANSWERS 3. 1. 7258. 11. 59424. Page 45 2. 13440. 12. 66822. 1. 4860, 48600, 194400. 3. 21465. 13. 47320. 4. 19758. 14. 45384. 2. 382400, 764800, 9560000. 3. 1722000, 2296000, 2583000. 5. 47085. 15. 78027. 6. 45522. 16. 21909. 4. 747000, 7470000, 10956000. 5. 21492000,256710000,453720000. 7. 42182. 17. 88445. 8. 66822. 18. 90159. 6. 13536000, 156510000, 2411100000. 9. 53963. 19. 229554. 7. 1315170000000, 1480784000000. 10. 47974. 20. 307395. 8. $3139972.00, $39249650.00. 4. 31806. 9. 3604200000, 2405202800000. 5. 138104.50. 10. 440000000, 25,960,000. 6. 119239. 12. 3168000. 7. 350090. 13. 126000000. 8. 46529640. 14. 48000. 9. $16808.61. 15. 1610000. 10. 1. $6141.720. 3. 27154202. 16. 1140000. 2. 5107212. 4. 96332187. 17. 44000000. Page 43. Page 46. 11. 35843685. 12. 214007086881. 18. $650. 20. 86,400. 1 Q $80 000 13. 764,819,895,290,424. At7 ipOvjUV/v. 14. 2,324,334,767,296. 15. 99253.80. Page 47. 16. 152323.35. 1. 333641. 5. 3306564. 17. $69520.33. 2. 27421443. 6. 401193. 18. CMXII. 3. 14889792. 7. 2153232. 19. CLXXXVIICDLVI. 4. 4382415. 8. 49308. 20. 5859385041295896. 21. 21,842,100. 1. 500. 3. $72. 5. 9328. 2. $503.50. 4. 913,920. 1. $20604. 4. 984072. 2. 11025. 5. $494. ',} $2533 50. 6. 137664. Page 48. 6. 320,000. Page 44. 7. 213,192. 7. 95040. 8. $1,377. 8. (65 57) X 54 == 432. 9. 103,615 9. (17 + 2fi) X $42.50 (17 X 10. $2,583. 38.75) -f (26 X 40.25) = 11. 89,232 $122.25. 12. $31.80. ANSWERS 363 13. 5,865,696,000,000. 17. 93716. 22. $.92. 14. 13,176. 18. 209758. 23. $108.50. 15. 968710. 19. 189572. 24. 32793. 16. $17,979,365. 20. 26485. 25. 11750. 21. $6.07. 26. 63362. Page 49. 2. 406. 3. 432. 17. 276. 18. 693. Page 56. 19. 63,210,541,205,000. 4. 167. 13. 32. 1. (16 11 -f 2) X 6 = 35. 5. 172, with4rem. 14. 123. 2. (4-f 15)X(15 4)X 6 = 1254. 6. 230, with 7 rem. 15. 182. 3. 63915 + 936085 = 1000000. 7. 403. 16. 92. 4. 3149 -f- 4872 = 8021. 8. 286, with 8 rem. 17. 88. 5. 5301 _ 1046 = 4255. 9. 315, with 14 rem. 18. 217. 6. 300,003, 300,003. 10. 312. 19. 136. 7. MMMIII, CI. 11. 1899, with 5 rem. 20. 72.' 8. 397,056. 12. 439, with 38. rem. 21. 35. 9. 12,343,200. 2. 95. 4. 144. 6. 108. 10. 674. 3. 8. 6. 13178. DIVISION. Page 57. Page 54. 7. 81. 15. 365. 1. 283. 9. 5263. 17. 209758. 8. 104. 16. 5280. 2. 188. 10. 6238. 18. 189572. 9. 37. 17. 17443, with 16 rem. 3. 71. 11. 4812. 19. $264.85. 10. 72. 18. 175. 4. 124. 12. 4809. 20. $924.67 11. 66. 19. 327. 5. 834. 13. 247. 21. $128.21. 12. 162. 20. 7328. 6. 4169. 14. 4138. 22. $222.22. 13. 2640. 21. $89. 7. 9451. 15. 6559. 14. 153. 8. 9485. 16. 93716. Page 58. 1. 283. 4. 248. 7. 9451 1. 1. 11,572, with 110 rem. 2. 198. 5. 1668. 8. 9485. 2. 6284. 3. 142. 6. 4169. 3. 1938. 4. 664. 5. 736. Page 55. 6. 893. 9. 5263. 13. 27680. 7. 969, with 344 rem. 10. 6238. 14. 2470. 8. 1064. 11. 4812. 15. 8276. 9. 985. 12. 4809. 16. 6559. 10. 692, with 533 rem. 364 ANSWERS 3. 1. 527, with 380 rem. 1. 576,544. 9. 319,099. 2. 692, with 533 rem. 2. 103,075. 10. 801,587. 3. 5205, with 38 rem. 3. 213,789. 11. 117,554. 4. 814, with 167 rem. 4. 4. 12. 388,129. 5. 1259, with 581 rem. 5. 9,042,049. 13. 8,886,859. ( ; . 645, with 312 rem. 6. 8161. 14. 253, with 21, 700 7. 283, with 736 rem. 7. 1162. rem. 8. 3241. 8. 28. 99401 OTzOX. 10. 876, with 110 rem. Page 62. 11. 474,536, with 523 rem. 15. 17,115,520. 12. 4567. 16. . 13. 4207. 17. 67, with 999 rem. 14. 10,110, with 9 rem. 18. 25. 19. 240. 4. 1. 1,672,940, with 165,534 rem. 2. 206,008,604, with 24 rem. 20. 300, with 9999 rem. 3. 100,000,000, with 102,345,678 2. 18. 3. 276. 4. 40. rem. 4. 100,000,000. Page 63. 5. 48,100,720,009. 5. $10.00. 6. 3000 too much in 2d member. Page 59. 7. 41. 8. 132699. 1. 96. 6. 329. 10. 58. 9. 90. 2. 1760. 7. 85 +. 11. 7 +. 10. 2122. 3. 60. 8. 22 -}-. 12. 3579. 11. 19,868. 4 19. 9. 36. 13. 25. 12. 139,806. 5. 425. 13. $384.25. Page 6O. 14. Lost $952. H. 491 sec. 21. 548,501. 15. 5475 hr. 15. 15. 22. 9238. 16. 12,295. 16. 500. 23. 5. Page 64. 17. 605. 24. 28. 18. $137 nearly. 25. 308. 19. 357 -f. 26. 0. 17. 5 yr. 19. 2.83. 18. 105. 20. $12.40. 20. 13. 27. IX. Page 65. Page 61. 2. $3.60. 5. $.66. 8. $2552 28. CLXXX. 30. 437. 3. $6.10. 6. $4.50. 9. 116. 29. MCCLXXX 4. $3125. 7. $28.00. 10. 6480. ANSWERS 365 Page 66. PROPERTIES OF NUM 2. 93. 4. 12 5. 47. BERS. 1. 20. 3. 1785. 5. 5. Page 74. 2. 57, 656. 4. 210. 1. 9 ) 2, 2O O , 6, 6. 2. 5 7. Page 67. 3. 2, 2, 2, 2, 2, 2. 7 36. 11. 10 14. $60. 4. A 23 8 10 hr. 12. $.25. 15. $3010. 5. 2, 2, 2, 2, 3, 7. 9 114. 13. $1 50. 16. 40. 6. 3, 37 10 108. 7. 5 7, 11. Page 69. 8. 3, 11 , 13. Q. Rem. Q- Rem. 9. 5, 5, 37. 2. 1. 632, 7. 9. 55, 33. 10. 2, 2, 3,41. 2. 532, 7. 10. 12, 34. 11. 2, 2, 2, 3, 5, 11. 3. 973, 2. 11. 6, 173. 12. 2, 2, 2, 3, 3, 3, 3, 18. 4. 926, 7. 12. 5, 432. 13. 2, 3, 1283. 5. 256, 7. 13. 8, 650. 14 743, prime number. 6. 32, 67. 14. 3, 000. 15. 3, 5, 5, 7, 7. 7. 53, 27. 15. 5, 678 16. 2, 2, 2, 3, 3, 3, 3, 7. 8. 92, 73. 17. 2, 2, 2, 3, 7. Q. Rem. Q. Rem. 18. 3, 7, 11. 2. 1 33, 13. 5. 13, 34. 19. 2, 89. 2. 31, 27. 6. 12, 16. 20. 2, 2, 3, 3, 5. 3. 17, 6. 7. 28, 136. 21. 2, 2, 2, 2, 3, 3. 4. 15, 40. 8. 24, 100. 22. 3, 3, 5,7. 23. 2, 2, 3, 5, 7. Page 70. 24. 2, 2, 3, 5, 11. Q. Rem. Q. Rem. 25. 2, 2, 5,37. 3. 1. 1, 273 7. 2, 2432. 26. 3, 3, 3, 5, 7. 2. 1, 3o2. 8. 2, 37. 27. 2, 2, 2, 2, 2, 2, 3, 3, 3. 3. 1, 295 9. 1, 3396. 28. 2, 2, 3, 3, 7, 17. 4. 1, 173. 10. 1, 2116. 29. 2, 29, 29. 5. 2, 1327. 11. 1, 2370. 30. 997, prime number. 6 2, 2645. 12. 1, 1573. 31. 2, 2, 3, 5, 7, 11. Q. Rem. Q. Rein. 32. 2, 3, 5, 5, 5, 7. 2. 1. 29, 1958. 7. 45, 5896. 33. 2, 3, 7,19. 2. 12, 4425. 8. 20, 17432. 34. 2, 2, 11, 11. 3. 14, 4495. 9. 10, 1959. 35. 2, 2, 2, 2, 2, 2, 2, 2, 5. 4. 20, 1765. 10. 38, 9938. 36. 2, 13, 73. 5. 10, 4543 11. 23, 25548. 37. 2, 2, 3, 6, 7, 13. 6. 2725, 250. 12. 14, 1337. 38. 2 2, 3, 3, 5, 19. 366 ANSWERS 39. 2, 3, 5, 7, 7. 2. 1. 563. 7. 89. 40. 2, 2, 373. 2. 324. 8. 16. 41. 2, 3, 5, 7, 11. 3. 728. 9. 38. 42. 2, 2, 3, 3, 7, 11. 4. 18. 10. 864. 43. 2, 2, 2, 2, 2, 2, 5, 5. 5. 53. 11. 892. 44. 2, 2, 3, 17, 41. 6. 430. 12. 2735. 45. 6, 11, 47. 46. 1997, prime number. Page 77. 47. 3, 3, 7, 7, 11. 48. 3, 3, 7, 11, 11. Q. R. Q. R. 1. 304, 8. 5. 44, 28. Page 75. 2. 91, 9. 6. 119, 16. 1 3. 121, 9. 7. 49, 20. {2 X 3 = 6. 2 X 17 = 34. 4. 58, 27. 8. 23, 11. 3X17 = 51. r 3 X 6 15 Page 78. 105. | 3 X 7 = 21. 3. 1. 45. 3. Y- 5. 46. 7. 20. U X 7 = 35. 2. 45. 4. 21. 6. 4. 2x2 = 4. 2X3 = 6. 2. 39. Page 79. 108 ^ 2 X 3 X 3 = 18. j 2 X 3 X 3 X 3 = 54. 3. 20. 9. 20. 1 2 X 2 X 3 = 12. 4. 1.50. 10. 9. 12x2x3x3 = 36. 5. 2745. 11. 8. 221 has prime factors only. 6. 43. 80 very nearly. 12. 21. r 5 X U = 55. 7. 6. 13. 26. 715. | 5X13 = 65. 8. 6. Ill X 13 = 143. Page 81. 845 I 5 X 13 = 65 ' 118X 13 = 169. 2. 1. 7. 3. 7. 6. 6. 7. 6. The answers to the remaining 2. 6. 4. 9- 6. 6. 8. 35. eleven examples are omitted. 3. 1. 14. 3. 15. 5. 16. 7. 120. 2. 1. $1855. 3. $334.80. 2. 42. 4. 20. 6. 42. 8. 22. 2. $227.70. Page 76. Page 82. 4. $215.82. 10. $67.155. 1. 9. 3. ' 15. 13. 21. 3. 5. 694.95. 11. $17424. 10. 12. 16. 14. 22. 37. 6. $6375. 12. $12600. 11. 9. 17. 60. 23. 101. 7. $61.25. 13. $87.04. 12. 6. 18. 72. 24. 2. 8. $118.125. 14. $76.95 13. 75. 19. 29. 9. $1165.50. 14. 144. 20. 1. ANSWERS 367 2. 1. 11. 8. 37. 15. 37. 7. 4284. 2. 23. 9. 283. 16. 47. 8. 160,121. 3. 31. 10. 2. 17. 41. 9. 441.000. 4. 41. 11. 3. 18. 53. 10. 7770. 5. 47. 12. 17. 19. 267. 11. 290,177. 6. 53. 13. 48. 20. 396. 12. 1,639,872. 7. 61. 14. 11. 13. 314,259. 14. 86,394. ' Page 83. 15. 1,009,091. 3. 1. 12. 5. 43. 8. 126. 16. 1,038,007. 2. 8. 6. 1. 9. 42. 17. 240,463. 3. 4. 7. 3. 10. 37. 18. 179,655. 4. 15. 19. 50,552. 1. 4. 4. 12. 2. 6. 5. 14. 3fq 6. 16. 7. 23. 20. 473,989. 21. 23,760. 22. 71,842,008. * DO. Page 84. 23. 31,154,994,649. 8. 2. 9. 5. 10. 940. 24. 260,117. 25. 329,616. 26. 4340. Page 86. 27. 42,149,000. 1. 84. 7. 1080. 13. 330. 28. 3,268,080. 2. 720. 8. 840. 14. 720. 3. 448. 9. 1200. 4. 144. 10. 1440. 15. 1200. 16. 225. Page 89. 5. 180. 11. 2016. 17. 576. 1. 840. 5. 210. 9. 720. 6. 360. 12. 360. 18. 900. 2. 180. 6. 720. 10. 876. 3. 5040. 7. 60. 11. 82,063,340. Page 87. 4. 120. 8. 460. 1. 300. 7. $10,800. 12. 216. 2. 120. 8. 630. 3. 280. 9. 156. 13. 5040. 14. 510. Page 9O. 4. 54. 10. 72. 15. 10,920. 1. 490. 5. 3. 8. 509. 5. 540. 11. 72. 16. 6300. 2. 3. 6. 40,170. 9. $10. 6. 1512. 3. 283. 7. 119. 10. 720. Page 88. 4. 1044. 1. 2871. 2. 13,889. FRACTIONS. 3. 10013. 4. 819. Page 94. 5. 2160. 2. 1. ff. 3. ft. 6- Jft. 6. 2873. 2. ft- 4.yflh,. 6-Tttr- 368 ANSWERS Page 95. Page 98. 7. ff- 12. / . 17. ff. 3. 1. " 8 ^- 5 2 ~i 15. ^F. 8. TrV 13. ft. 18. T V 2. ^F- 9. lifi. 16. 6 i- 9. ff 14. ff. 19. i. 3. 10. iif 4 . 17. ^|f. 10. H 15. |f. 20. -V/- 4. 1 f-. 11. iy* 18. - 8 T 7 7-. 11 if- 16. |f. 5. -af^. 12. -\% 8 -. 19. W- 2.. 1. if 5. f. 9. f 6. ^F- 13. ifp. 20. -97-. 2. f 6. T %. 10. Iff- 7. 1 -l 1 ' 14. ifp. 3. TV 7- T 9 _, if 2. ff if, !i- 4. f f, ^t ^t- Page 97. Page 1O1 31. Ti- 35. if. 39. \. 5. T ' TcF ' T^ f iff- 32. ff- 36. f. 40- rVs- 6- iff, Ml, iff. 33. H- 37. ift. 41- yVf- 7. Mf, if. 34. f 38. fi. 8 - t lili I, Iff- 9. T 8 o ? -_7_ >_ } -6,^. 2. 1. -? 5 -- 8. *ff. 15. f|. 10 ] f , 2^( [ 2. f 9. l|p. 16. ^ffl. 11. -^ -> II > -W-' 3. 10. iffi. 17. - 2 ^ OJL . 12. M* ff. 4. 11. - 7 /-. 18. J- 2 T 3 /i. 13. f Ml, ft. 5. -\ 5 - 12. -V 9 - 19. 31U8.. 14. \ V -, if , W f f 6. 101. 13. - 2 r Y-. 20. -^ff A. 15. - 3 3 V, ff , If' ff 7. 401 14. - 4 /. 16. Jg j-, Vcf-, W ANSWERS 369 17. W, W, H, if 11- 1- if 5. T ^L 9. 1211. is. f, -I, , M, *g, if 2- yVV 6- ^ 3 2-- 10. 18ff. 19 - ihro> IPf, TjVi&i tfM, lfj> 3. i. 7. 1048if. 11. 12|i. MM- 4. ft. 8. 52111 12. 41 T W OA 7 125 18 140 12 30 2 1 . - 3 - 6 2 -$ 8 -, - 3 - 5 - 1 - 1 -, - 2 - 8 - 8 - -- , JL3L&A.QL Page 1O6. ' -AMMFtfHttv 12. 1. 30ft. 8. 25ft. 15. 8^,. 22. j 3 ^, AW, roV^J TeVV, fil 2. 29ft. 9. 6 5 2 T . 16. 15if^. iVsV 3. 4f, 10. 17. 17. 15. 4. Oft. 11. 89f. 18. If ADDITION. 5. 261. 12. 45|. 19. llff Page 102. 1. 2||. 5. lif. 9. lOff 2. 17ft. 6. 84ft. 10. Uf- 3. 31. 7. 12ft. 11. 26H- 6. 79ft. 13. 14if, 20. 27f^. 7. 9. 14. 34f|5. 21. 35ff. 13. 1. If. 6. 10^. 11. 4ff. 2. f. 7. f^. 12. 1251. 3Q77 Q O 3 1 1Q QQ23 4. 4ii|. 8. lOJff 12- 2H- T8^* ^^^"* AO ^O-K^TT. 4. 4|f. 9. 19 f. 14. 40ft. Page 1O3. 6. 6 ? |. 10. 23f|. 15. ^L. 13. f. is. 3ft. 19. 1426|f. 1. 9f 2. if 14. 2 T 3 r . 17. 19|. 20. Iffi. 15. 21 T 3 7 . 18. 33|f Page 1O7. 3. 300. 1. 2|. 5. |f. 9. 191ifi. 4- is 1 = Iff, 1 =F || 5 , f - Iff. 2. 2ft. 6. 49i. 10. SftV^r- 5. Hi. 3. 91. 7. 1130|f. 6- Iff 4. 3101. 8. 9632if. 8. 1. 373ft. 4. 7811. 6. 429ft. 1. If- 5. 31ff. 9. 34f 2. 6ft. 5. 26913. 7. 234|f. 2. I 7 4ft- 6. 25|. 10. 63 |f. 3. 35f|. 3. 4- 7. 29 V 4. 2909||. 8. 18^. 1. 271ft. 2. 37ff. 3. 241if MULTIPLICATION. Page 108. 3. 1. f. 12. ft. Page 1O4. 2- f 13. 4 2 i|. 4. 141ft. 5. 81J. 3. y =6. 14. 5ft. 4. |. 15. 8ff. SUBTRACTION. 5. f = If. 16. 5|. 6. 2i. 17. 17|. Page 1O5. 7. 51. 18. f. 4- i- I 8. H, tt- 8. f. 19. 2ff. 5 - T 4 ^, 8. 7. 15. IXT 3. 1. 2|. 5. i. 9 1296 O 4 2. 3. 3 64. 6if 6. 7. 2 T* t 10 11 8f 5 if 1. 6}. 2. 26f 3. 4. 18. 5. f 27. 4. 4 *V 8. 2f 12 . 4f Page 111. Page 115. 4. 1. H- 5. ^01^ 8. /6- 6. if 8. 98ff 10. 6|f 2 6. *r 9. sAV 7. 10 T V 9. 1689|. 3. rift (-I) 4 = -0001, (.02) 3 == .000008. 1. * = 625. 2. a: = 1225. 3. x = 7744. 4. x = 10,201. 5. a? = 169. 6. x = 729. 7. a? = 4913. 8. a; = 1000. 9. x = 9261. 10. x --= ft. 11. (.001 ) 3 =:. 000000001. 12. 15 3 = . 003375. 13. .04 3 = .000064. 14. .001953125. 15. .000729. 16. .00000625 17. .005 = .000000125. 18. 4.2025. 19. (25t)" ANSWERS 393 20. (4.6 tO I ' Qi CO ^.7 ^ O *- \%. 27. A life annuity costs a person 44 yrs. old $5933.35. Find the amount of the annuity, interest at 3?%. 28. Find the amount of $365 at compound interest for 20 yrs. at 5$. 29. Find the amount of $1728 at compound interest for 25 yrs. 420 PRACTICAL ARITHMETIC CIRCULATING DECIMALS. | = ~ = .4, quotient exact, f = s -~ = .4285 -f, quotient inexact. To reduce a common fraction to a decimal, we annex ciphers to the numerator, perform the operation indicated, and point off in the quotient as many places for decimals as there are ciphers an- nexed. Annexing a cipher to the numerator multiplies it by the factors 2 and 5 ; hence, when the denominator contains no other factors than 2 and 5, the quotient will be exact ; when it contains other factors than 2 and 5, the quotient will be inexact. T \ reduced to a decimal becomes .384615, and so on without end ; but in this instance a very noticeable peculiarity is that the 384,615 will be constantly repeated, however far the reduction be carried. Such a decimal fraction is called a circulating or repeating decimal. To show the fact of repetition we place a point over the first and the last digit, thus : 384615. T 3 T = .272727, etc. = .27. The constantly repeating figures are called a repetend. A mixed repetend begins with one or more non-repeating figures, as .5243. The practical advantage of recognizing circulates will appear in the following illustrations : 1. Reduce .72 to a common fraction. 100 times .72 = 72.7272 1 time .72 = .7272 99 times .72 = 72 1 time .72 = f = T 8 T . 2. Reduce .1172 to a common fraction. .1172 = .ll|f = .11 T 8 T = ^ = T oV Hence the formula : _Digits_of_repetend^__ = value of repe tencl. As many 9's as digits 421 EXAMPLES. Reduce the following repetends to common fractions : 1. .3. 7. .753. 13. .852. 19. .0003. 25. .2297. 2. .4. 8. .216. 14. .144. 20. .246789. 26. 2.1873. 3. .6. 9. .531. 15. .527. 21. .2564. 27. .4306. 4. .36, 10. .0234. 16. .0009. 22. .8716. 28. 5.0415. 5. .21. 11. .8232. 17. .048. 23. .35135. 29. .84234. 6. .018. 12. .81. 18 .08199. 24. 3.04i2. 30. 4.2674. THE GREATEST COMMON DIVISOR OF FRACTIONS. Find the G. CD. of *|, f, f Each of the given fractions divided by the G. C. D. must give an integer for quotient. In dividing by a fraction, we invert the divisor and then multiply. Hence the G. C. D. sought must have for its numerator the G. C. D. of the given numerators and the L. C. Dd. of the given denominators. The G. C. D. of 2, 4, and 6 = 2. The L. C. Dd. of 3, 5, and 7 = 105. Hence the G. C. D. of f , f , and f = r f 5 . GK C. D. = FORMULA. G-. C. D. of Numerators L. O. Dd. of Denominators' THE LEAST COMMON DIVIDEND OF FRACTIONS. Find the L. C. Dd. of f , f , and T V TL L. C. Dd. sought, when divided by each of the given frac- tions, must give an integral quotient. In dividing by a fraction, we invert the divisor and then multiply. Hence the L. C. Dd. 422 PRACTICAL ARITHMETIC required must have for its numerator the L. C. Dd. of the given numerators and the G. C. D. of the denominators. The L. C. Dd. of 2, 4, and 8 = 8. The G. C. D. of 3, 9, and 15 = 3. Hence the L. C. Dd. of f , f , and T 8 T = f . FORMULA. L C Dd L - G - Pd - of Numerators ~ G. C. D. of Denominators' EXERCISES. 1. Find the G. C. D. of : 1. f, f , f . 3. 12i, 3|, 17}. 5. 1A, 1&, 1&. * i 7 *, A, A, A- 4- 10, 2j, f . . 6. if, H, If 2. Find the L. C. Dd. of: 1. f, T V, f . 3. 6f , 3f , 2f. 5. A, rV, 2i 5, 6}. 2. I, if, if. 4. ff, A, &. 6. H f f If THE THERMOMETER. The Thermometer is an instrument for measuring change of temperature by means of the expansion of liquid substances. Mercury is the pre-eminently suitable substance. Thermometers are of three principal kinds : The Fahrenheit, the Centigrade, and the Reaumur. The Centigrade is used largely for scientific purposes. It is sometimes necessary to transform readings from one scale to another. Freezing Point. Boiling Point. Interspace. Fahrenheit 32 212 180 Centigrade 100 100 Reaumur 80 80 Hence the number of degrees F. 32 = \\% or f C. = Vo- or |B. From which we deduce the formulae : 1. (F. 32) X f = C. 3. O. X | = F. 32. 2. (F. 32) X $ = R. 4. R. X f = F. 32. APPENDIX 423 1. Change 40 F. to C. 40 32 = 8. 8 X f = -V- = 4f C. 2. Change 40 C. to F. 40 X | = *P = 72. 72 + 32 = 104 F. The minus sign ( ) prefixed to a reading signifies below zero. 3. Change 10 C. to F. 10 X f = =nr = 18 We now have 18 + 32. Difference of sign implies subtrac- tion. 18 -f 32 = 14 F. above zero. 4. Change 30 C. to F. - 30 X f = =^ = 54. 54 -f 32 = 22 F.; that is, 22 below zero. 5. Change 32 F. to C. and B. 6. Change 50 C. to F. and B. 7. Change 20 C. to F. and B. THE CLOCK. The hour and minute hands are together at 12 o'clock, and the minute hand may be regarded as setting out at that point to rejoin the hour hand. In 60 minutes the minute hand will have re- turned to 12, but the hour hand will have passed on to 1. In 60 minutes, therefore, the minute hand has gained on the hour hand 11 spaces. Hence, a single space was gained in ^ of 60 minutes = 5 T 5 T minutes. 1. At what time between 4 and 5 o'clock are the hands of a clock together? At 4 o'clock the hands are 4 spaces apart. Since the minute hand gains 1 space in 5 T 5 r minutes, it will gain 4 spaces in 4 times 5^ r minutes = 21 T 9 T minutes. Hence, the hands will be together at 21 T 9 T minutes past 4. 2. When will the hands of a clock be together between : 1. 6 and 7 o'clock ? 4. 3 and 4 o'clock ? 2. 8 and 9 o'clock? 5. 9 and 10 o'clock. 3. 1 and 2 o'clock? 6. 5 and 6 o'clock. 424 PRACTICAL ARITHMETIC 3. When will the hands of a clock be opposite each other between : 1. 12 and 1 o'clock? 3. 9 and 10 o'clock? 2. 3 and 4 o'clock? 4. 11 and 12 o'clock? 4. When will the hands of a clock be at right angles between : 1. 2 and 3 o'clock ? 3. 9 and 10 o'clock ? 2. 4 and 5 o'clock? 4. 11 and 12 o'clock? 5. When between those hours will they make with each other an angle of 30 ? Of 60 ? Of 120 ? Of 150 ? WORK. 1. A. can do a certain piece of work in 6 days, B. in 8 days, and C. in 9 days. How long will it take them to do it together? A. can do of the work in 1 day. B. can do | of the work in 1 day. C. can do ^ of the work in 1 day. All can do i + i -f = f f in 1 day. All can do -fa of the work in ^ of 1 day. All can do ff of the work in |f = 2f days. 2. A. and B. together can do a piece of work in 2} days, A. and C. in 3J days, B. and C. in 3| days. How long will it take the three working together to do the work, and how long will it take each alone ? A. and B. in one day can do ~ f the work. A. and C. in one day can do of the work. B. and C. in one day can do ~ of the work. 2A. + 2B. + 2C. in one day can do ~ + -^ + ~ of the work. A. + B. -f C. in one day can do + ~- + -J- of the work = I -f "73 1/2 & + & = ^-^ = f* of the work. In one day A. can do | f ~ = f f $$ = ft of the work. Hence A. can do f in ff or 4 T \ days. Find time required by B. and C. 3. If it takes A., working alone, 4 days, B. 3 days, and C. 4 days to do a piece of work, how long will it take them to do the work if all three work together? APPENDIX 425 4. A. can do a piece of work in 10 days, A. and C. can do it in 7 days, and A. and B. can do it in 6 days. How long will it take them all to do it? 5. One pipe can fill a cistern half full in f of an hour, and an- other can fill it three-quarters full in J an hour. How long will it take both pipes together to till the cistern ? 6. Pipes A. and B. can fill a cistern in 3 minutes and 5 minutes respectively, and C. can empty it in 7J min. In what time will the cistern be filled when A., B., and C. are all open ? 7. A., B., and C. together can do a piece of work in 10 days, A. and B. together in 12 days, B. and C. together in 20 days. How long will it take each alone to do the work? 8. A. does T 4 T of a piece of work in 6 days, when B. comes along and helps him, and they finish it in 5 days. How long would it take B. alone to do the work ? 9. A. and B. can do a piece of work in 4 days, A. and C. in 6 days, and A., B., and C. in 3 days. In how many days can each do the work alone? 10. A reservoir has two sluices, one of which alone would drain it in 7 hours and the other in 13 hours. How soon would it be emptied if both were opened together? AVERAGING OP ACCOUNTS. 1. Find the average term of credit of the following account : DR. JACOB HART. CR. 1898 1 1898 Jan. 1 To Mdse 448 00 Jan. 20 Amt. br. forwd 560 00 Feb. 4 " Cash 364 00 Feb. 11 By 1 Carriage 264 00 " 20 u u 232 00 " 25 " Cash 900 00 Process. DR. Due. Da. Items. Prod. Jan. 1, 00 Feb. 4, 34 " 20, 50 448 00000 364 12376 232 11600 Due. Jan. 20, Feb. 16, " 25, 1044 23976 CR. Da. Items. Prod. 19 560 10640 46 264 12144 55 900 49500 1724 72284 1044 23974 oces, 680 '48308 48308 -=- 680 = 71 da. Jan. 1 + 71 da. = March 13. 426 PRACTICAL ARITHMETIC Explanation. Jan. 1, the earliest date, was assumed to be the starting point or focal date. From Jan. 1 to Jan. 1 there are days ; from Jan. 1 to Feb. 4 there are 34 days ; from Jan. 1 to Feb. 20 there are 50 days. On the credit side we proceed in a similar way, saying from Jan. 1 to Jan. 20 are 19 days, and so on. The balance of the products divided by the balance of the items gives the average term of credit, 71 da., which, added to Jan. 1, gives us March 13 as the day of payment. 2. When should interest begin on the following account : DR. JACOB JOHNSON. CR. 1898 1898 Jan. 1 To Mdse, 3 mo. 145 86 May 11 By Cash 11 00 " 12 u u 5 ii 37 48 July 12 u u 15 00 June 3 11 It U 12 25 Oct. 12 u u 82 00 Aug. 4 II II O II 66 48 Process. DR. CR. Due. Da. Items. Prod. April 1, 00 145.86 0000.00 June 12, 72 37.48 2698.56 Sept. 3, 155 12.25 1898.75 Oct. 4, 186 66.48 12365.28 262.07 16962.59 108.00 Bal. of Items, 154.07 Due. Da. Items. Prod. May 11, 40 11.00 440.00 July 12, 102 15.00 1530.00 Oct. 12, 194 82.00 15908.00 108.00 17878.00 16962.55 Bal. of Prod., 915.41 915 + 154.07 = 6 da. April 1 6 da. = March 26, 1898. Jan. 1 + 3 mo. April 1 Jan. 12 + 5 mo. = June 12 June 3 + 3 mo. = Sept. 3 Aug. 4 + 2 mo. = Oct. 4 which shows April 1 to be the focal date. We now proceed as before. It must be observed that when the balances are both on the same side of the account, the term of credit must be added to the focal date ; other- wise subtracted. APPENDIX 427 3. Find the average term of credit of the following account : DR. RICHARD STEVENS, in Acct. with HENRY BECK. CR. 1898 Apr. 10 To Mdse 150 00 1898 Apr. 30 By Cash 250 00 "' 30 u u 400 00 May 1 u u 200 00 May 16 u 100 00 Jun. 27 (( U 400 00 Jun. 24 <( 500 00 4. What will be the cash balance of the following account Jan. 1, 1899, interest at 6f ? DR. ENOCH HOBSON. CR. 1898 July 10 To Mdse, 2 mo. 500 00 1898 July 20 By Cash 400 00 Aug. 1 " " 3 " 700 00 Aug.20 u 1000 00 Sept. 9 (( U ^ (( 800 00 " 20 <( It O H 600 00 REVIEW. Simplify the following : I X | X 3. 5 *f + 5* 26.7 11.80 + 6.45 f X 3H X .72 6 3f - f X 4.2 7. 4f X A + 1.8 .25 X n.it=ii 4fX5f 12 3j X 2J If 4 21 + 1J' 13 f X 1.25 ' 5f 4.25' 14 8$ X 1 T V + 4rV 3ft 6* - 7f -3- 28 A + i ' 15. 7 ^ X T7 H-i 16. 4J of 1 A- of A + 31 + 2 J of If 1 7 3f Of 1-jrV JL / ! of 6f 18. 19. 2| of WT + yV of 10. + I X H + 3 + 2| of IT> T 20 ^ + 5 ^ of f ' t of 5f + f 428 PRACTICAL ARITHMETIC 21. '4 , 2| , 2 1 , _2f 23. f of f of 5 T V-4M 7if-6i 24. tt=|* 25. SjjV + 2f 3 1 + 6| + lOf . 28. (2* + (2J 11). If of 2f of 3f - 4 of -s-4*. 33. % * 34.|^|f 35. JL* -4* 36. .04478256 ^- 5.48 .036X2.043 6? 37. If of If + 38. 2|. 41. 3.01 5.314 + 2.4. 42. |- + 2^1 + ! & + ? 43 15! 4f of If ' | of 231 + 2if 44 -02048 ' .00003125' 45 TT -*- 12 f + A ()f 9|. 5i-4| -f- 1| Of If 46. 48. .111 + . 6666| +.222222$. so. 4 + _j, 51. 53. 1 HX 2^ + 21x5^ ~l| 2H 3* of 41 (2J i) of (3i - 54. -321 X. 321 -. 179 X. 179 .321 .179 .562 X. 562 .188 X. 188 , 55 - - .562 -.188 - f$75 57 (-OQ056542) 2 (12.534) 2 * 40. 3J + 2| of 5J of 1 ^ _ 2 T V X f - APPENDIX 429 , If of If 62 T % of 1.82 + | of .35 , _ . - 4 ^ 65. of 2 yd. 2 ft. 11J in. 6i 2 of 4 66. (f of f + f of 2i) X - 3 + X 3i 70 }> 7i. . _J_- T.5-f ~ 10 79 . of . J> FTTTT+l+l 2 |_3 2 TT7 I + -S .0038425 .00183 of 2.179 - | of .8684 i Q , 86> 87. -. X of 12s. 9fd. 2.1742 2.78 .203 X .0003 X 16 .008 X .0029 430 PRACTICAL ARITHMETIC 93 > X X * X * X ''** 2*3*5*4 94. .016 + 4.0808 .0008 + 50.1 .1966. 5 of 2 5 r + j of 1?) . AX If XI}- rV ' I of 3-f 7 X 96. A of **~~k of (3f + i - 2^). 97. i of (i + i +i) + 7 X GV + A) - A * 98. If of 5 T 4 T 1/3 of 5f -f Iff of 2 T V 99. | (! X i + | -f- f) - || of ^ + f of ^. 100. l/ 7.4538 6.8 -5- 8.5 2.03 X 1.17. MISCELLANEOUS PROBLEMS. 1. Prove the product will be the same in whatever order the factors be taken. 2. Prove that in division of fractions, multiplying the divi- dend by the divisor inverted will produce the quotient. 3. Show how you determine whether a given common frac- tion can be exactly expressed as a decimal, and give reasons. 4. What factors of two or more numbers must be combined to produce their greatest common divisor, and what ones to produce the least common dividend. 5. State a method of multiplying a fraction by a fraction, and demonstrate the correctness of the method. 6. On what theory was the length of the metre originally de- termined ? 7. Explain the process of finding the greatest common divisor by division. 8. Prove that any common divisor of two numbers is a divisor of their sum and of their difference. 9. Explain a method of finding the greatest common divisor of two fractions. 1O. Given interest, principal, and time, how may the rate be found ? APPENDIX r 431 11. Explain and illustrate a method of finding the least com- mon dividend of fractions. 12. Indicate the following operations by signs in one connected expression : The sum of 3 and 4 multiplied by the difference be- tween 9 and 5, and the product divided by 2 times 7. Perform the operations indicated. 13. Write a complex fraction. State the reasons for regarding it as complex. Reduce it to a simple fraction, and this result to a decimal. 14. Distinguish between a compound and a denominate num- ber ; also, interest and discount. Illustrate by examples. 15. Write a number which shall be at the same time simple, composite, abstract, and even. State why it fills each of these requirements. 16. Name the principal unit of length, of surface, of capacity, and of weight in the metric system, and show the relation among these units. 17. The sum of two numbers is 260 and their difference is 12 ; find the numbers and demonstrate the principle involved. 18. A. and B. can do a piece of work in three days ; B. and C. can do it in four days ; A. and C. can do it in six days ; if all work together for the same length of time, what part of the sum paid to all should each receive? 19. Find the fourth term of the following proportion, and demonstrate the principle on which it is based : 8 : 12 = 10 : x. 20. A cylindric vessel is 8 ft. in diameter ; how deep must it be to contain 75 bbl. of water? 21. Find the square root of 104976, and give a reason for each step in the process. 22. Deduce a rule for finding the sum of an arithmetic series, and illustrate its use by finding the sum of ten terms of the series whose first term is 2 and whose common difference is 4. 23. Find the sixth root of 191102976, and show why you believe your method to be correct. 24. Indicate the following by signs : The difference of 9 and 5 is multiplied by 8, this product is divided by 10 and the quotient increased by 1, the sum is squared, increased by 2, and the cube root of the result taken. 25. A franc is worth 9.5d. ; a mark is worth 11.7d. ; a pound sterling is worth $4.86. Find the value of $100 in each of the three other currencies, 3 432 PKACTICAL ARITHMETIC 26. Explain and illustrate a method of finding the least com- mon denominator of fractions. 27. Insert four geometric means between 3 and 96. Insert two arithmetic means between 3 and 96. 28. Assuming that iron is 7.8 times as heavy as water, find the weight in kilogrammes of a round bar of iron .60 centimetres in diameter and 3 metres long. 29. Find the cost in United States money of a bill of exchange on London for 12 15s. 9d., exchange being at $4.86. 30. A room 5 m long, 4 m wide, and 3 m high has opening from it one door 2 m high, l^ m wide, and two windows, each 2J[ m high, l m wide. Find the cost of plastering the walls and ceiling at 15 cts. a square metre, deducting half the openings. 31. In an arithmetic progression of 8 terms, the first term is 3 and the last is 31 ; find the remaining terms. 32. Show the exact value of the decimal .666 .... to infinity. 33. The length of a tank which holds 100 bbl. of water is twice its height, and its height is twice its width ; find its dimensions to the nearest inch. 34. Loaned $6000 to be paid, with interest at 6%, in six equal annual instalments ; what is the amount of each payment? 35. Explain a method of finding difference of longitude from difference of time, and show its application in finding the longi- tude of a place. 36. A bar of aluminum 2 cm thick and 2 cm wide weighs l k ; find its length, assuming that aluminum is 2 times as heavy as water. 37. State the process of finding the cost in U. S. money of a time draft on a foreign country, giving the reasons for each step. 38. Reduce the repetend .16213 to a common fraction in its lowest terms. 39. The diameter of a cylindric vessel is 42 cm and its depth is 6 dm ; how many litres of water will it hold and how many kilos will this water weigh ? 40. What factors of two or more numbers must be combined to produce their greatest common divisor, and what ones to pro- duce the least common dividend? Give reasons. 41. Find the G. C. D. and the L- C. Dd. of |, f, f. Explain the process fully. 42. When it is Monday, 7 A.M., at San Francisco, longitude 122 24' 15" W., what day and time of day is it at Berlin, longitude 13 23' 55" E.? APPENDIX 433 43. A gallon contains 231 cu. in. ; a cubic foot of water weighs 62.5 Ib. ; mercury is 13.5 times as heavy as water. How many gallons of mercury will weigh a ton ? 44. Find the sum of 23.3, 42.61, 78.3452. \ 2 *\ 45. Find the value of \ 2 *\2 + .8. 46. At how many minutes after 3 o'clock will the hour and minute hands of a watch be opposite each other? 47. A general formed his army into a solid square, and had 200 men left over ; he then received a reinforcement of 1000 men, and, increasing each side of the square by 5 men, lacked 25 men to complete the square ; how many men were there in the original army? 48. Find the cubic inches in a pail 12 in. deep, 16 in. wide at top and 12 in. at bottom. 49. A life annuity costs a person 44 yrs. old $5933.35. Find the amount of the annuity, interest at 3%%. 50. A person investing in a 4% stock receives 4f % for his money. What is the price of the stock ? 5 1. A piece of work is to be completed in 30 days, and 15 men are employed upon it ; at the end of 24 days the work is only half done. How many more men must be employed to fulfil the con- tract? 52. A man buys eggs at a certain price per score, and sells them at half that price per dozen. What is his gain or loss per cent.? 53. A.'s present age is to B.'s as 9 is to 5 ; three years ago the proportion was 10 to 3. Find the present age of each. 54. Simplify If of || of ||| 55. Simplify 54 434 PRACTICAL ARITHMETIC Some Commercial Laws Tabulated. STATES AND TERRITORIES. GRACE ALLOWED. LEGAL RATE OF INTEREST. P "l sg PENALTY FOE USURY. ARREST FOR DEBT. i Alabama . . Yes 8553 UNIVERSITY OF CALIFORNIA LIBRARY