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 THOMSON'S NEW GRADtD SERIES. 
 
 NEW 
 
 PRACTICAL 
 
 AEITHMETIC: 
 
 FOR 
 
 GRAMMAR DEPARTMENTS. 
 
 BY JAMKS B. THOMSON, LI, D., 
 
 AUTHOR OF DAT A THOMSON'S ARITHMETICAL 8ERIES ; EDITOR OP DAT'S SCHOOi 
 ALGEBRA, LEQENDRE'S GEOMETRY, ETC. 
 
 THIRTY-FIFTH EDITION. 
 
 NEW YORK: 
 
 CLARK & MAYNARD, PUBLISHERS, 
 
 5 BARCLAY STREET. 
 CHICAGO: 46 MADISOX STREET.
 
 TIJOIJSON'S NEW GRADED SERIES. 
 
 IN THREE BOOKS. 
 
 I. NEW MENTAL ARITHMETIC. 
 (For Primary Departments.) 
 
 I NEW RUDIMENTS OF ARITH- 
 METIC. (For Intermediate Departments.) 
 
 III. NEW PRACTICAL ARITHMETIC 
 (For Grammar Departments.) 
 
 KEY TO NEW PRACTICAL ARITH- 
 METIC. (For Teachers only.) 
 
 THOMSON'S SUPPLEMENTARY COURSE. 
 FOR HIGHER INSTITUTIONS. (In preparation.) 
 
 Entered according to Act of Congress, In the year 1872, bv 
 
 JAMES B. THOMSON, 
 In the Office of the Librarian of Congress, at Washington, D. C. 
 
 Electrotyped by SMVTH & McDouoAi. Ba Beekman Street, N. Y.
 
 03 
 
 1877 
 PREFACE. 
 
 r I ^HE New Practical Arithmetic now offered to the 
 _L public, is the third and last of the works which 
 constitute the author's " New Graded Series." 
 
 The old "Practical" was issued in 1845, and revised in 
 1853. Since that time important changes have taken 
 place in the commercial world. These changes necessarily 
 affect business calculations, and demand corresponding 
 modifications in text books. 
 
 To meet this demand, the "New Graded Series" was 
 undertaken. Each part of the series has been reinvesti- 
 gated and rewritten; the whole being readjusted upon 
 the graded plan, and brought down to the present wants. 
 
 Among the objects aimed at in the present work are 
 the following : 
 
 1. To make the definitions clear, concise, and com- 
 prehensive. 
 
 2. To present the principles of the science in a series 
 of distinct and consecutive propositions. 
 
 3. To lead the mind of the pupil, through the analysis 
 of the examples immediately following the respective 
 propositions, to discover the principles by winch all similar 
 examples are solved, and enable him to sum up the prin- 
 ciples thus developed, into a brief, comprehensive rule. 
 
 4. The "how" and the "why" are fully explained. 
 
 5. Great pains have been taken to ascertain the Standard 
 Weights and Measures authorized by the Government; and 
 to discard from the Tables such denominations as are 
 obsolete, or not used in this country.* 
 
 * Laws of Congress ; Profs. Hassler, Bnche and Egleston ; Reports of Supts. 
 Harrison and Calkins ; also of the Committee on Weights and Meat-urea of the 
 Fa: is Exposition, 1867. 
 
 409537
 
 PllEP A CE. 
 
 6. The Metric System is accompanied with brief and 
 appropriate explanations for reducing it to practice. Its 
 simplicity and comprehensiveness have secured its use in 
 the natural sciences and commerce to such an extent in 
 this and foreign countries, that no student can be said to 
 have a finished education, without a knowledge of it. 
 
 7. Particular attention has been paid to the develop- 
 ment of Analysis, the grand common sense rule which 
 business men intuitively adopt as they enter upon 
 practical life. 
 
 8. The examples are new and abundant ; being drawn 
 from the various industrial arts, commerce, science, etc. 
 
 9. The arrangement of the matter upon the page, and the 
 typography, have also received due attention. Teachers 
 who deal much with figures, will be pleased with the 
 adoption of the Franklin type. The ease with which 
 these figures are read, is sufficiently attested by their use 
 in all recent Mathematical Tables. 
 
 Finally, it has been the cardinal object to adapt the 
 science of numbers to the present wants of the farm, the 
 household, the workshop, and the counting-room ; in a 
 word, to incorporate as much information pertaining to 
 business forms, and matters of science, as the limits of the 
 book would permit. In this respect it is believed the 
 work is unrivaled. While it puts forth no claim to 
 mathematical paradoxes, it is believed teachers will find 
 that something worthy of their attention is gained, in 
 nearly every Article. 
 
 In conclusion, the author tenders his most cordial 
 thanks to teachers and the public for the very liberal 
 patronage bestowed upon his former Arithmetics, known 
 as "Day and Thomson's Series." It is hoped the "Ne\v 
 Graded Series" will be found worthy of continued favor. 
 
 JAMES B. THOMSON. 
 NEW YORK, July., 1872
 
 CONTENTS. 
 
 PAS* 
 
 Number, ....--..9 
 
 Notation, - - 10 
 
 Arabic Notation, - - 10 
 
 Roman Notation, - 15 
 
 To Express Numbers by Letters, - - 1 7 
 
 Numeration, - 17 
 
 French Numeration, - - 18 
 
 English Numeration, 20 
 
 Addition, - -21 
 
 When the Sum of each Column is less than 10, - 23 
 
 When the Sum of a Column is 10 or more, - - 24 
 
 Carrying Illustrated, - 24 
 
 Drill Columns, - - 29 
 
 Subtraction, - 31 
 When each Figure in the Subtrahend is less than 
 
 that above it,- - 32 
 When a Figure in the Subtrahend is greater 
 
 than that above it, - 34 
 
 Borrowing Illustrated, - - 34 
 
 Questions for Review, 38 
 
 Multiplication, - 40 
 
 When the Multiplier has but one Figure, - 43 
 
 When the Multiplier has more than one Figure, - 45 
 
 To find the Excess of 93, - 47 
 
 Contractions, - - - - 49
 
 VI CONTENTS. 
 
 FAGS 
 
 Division, - - - 53 
 
 The Two Problems of Division, - 55 
 
 Short Division, - '57 
 
 Long Division, - 6 1 
 
 Contractions, - 65 
 
 Questions for Review, 69 
 
 General Principles of Division, - - 7 1 
 Problems and Formulas in the Fundamental Rules, 7 2 
 
 Analysis, 77 
 
 Classification and Properties of Numbers, - - 80 
 
 The Complement of Numbers, - 82 
 
 Divisibility of Numbers, 83 
 
 Factoring, - 85 
 
 Prime Factors, - - 86 
 
 Cancellation, - 88 
 
 Greatest Common Divisor, - - 91 
 
 Least Common Multiple, - 96 
 
 Fractions, - - 99 
 
 To find a Fractional Part of a Number, - - 102 
 
 General Principles of Fractions, - - 103 
 
 Reduction of Fractions, - - 104 
 
 A Common Denominator, - - 1 1 1 
 
 The Least Common Denominator, - - 112 
 
 Addition of Fractions, - - 114 
 
 Subtraction of Fractions, - 117 
 
 Multiplication of Fractions, - -120 
 
 General Rule for multiplying Fractions, - - 125 
 
 Division of Fractions, - -126 
 
 General Rule for dividing Fractions, - i ;, i 
 
 Questions for Review, - - - 132 
 
 Fractional Relations of Numbers, - 134 
 
 Decimal Fractions, - 139 
 
 Reduction of Decimals, ... - 144 
 
 Addition of Decimals, - - - - - -146 
 
 Subtraction of Decimals, - - ... 148 
 Multiplication of Decimals, - ... - 149 
 
 Division of Decimals, - - - - 152
 
 CONTENTS. Vii 
 
 PAGE 
 
 United States Money, - 154 
 
 Addition of U. S. Money, - - - 158 
 
 Subtraction of U. S. Money, - - - - 159 
 
 Multiplication of U. S. Money, - - - 160 
 
 Division of U. S. Money, - - - - 1 6 1 
 
 Counting-room Exercises, - - - - 163 
 
 Making out Bills, - - - - - - -164 
 
 Business Methods, - 166 
 
 Compound Numbers, 171 
 
 Money, 171 
 
 Weights, 175 
 
 Measures of Extension, ... . 177 
 
 Measures of Capacity, - - 182 
 
 Circular Measure, - - - - - -184 
 
 Measurement of Time, - - - 186 
 
 Reduction, - 189 
 
 Application of "Weights and Measures, - - 194 
 
 Artificers' Work, ... - 196 
 
 Measurement of Lumber, - - 196 
 
 Denominate Fractions, - 202 
 
 Metric Weights and Measures, - - 207 
 
 Application of Metric Weights and Measures, - 2 1 .\ 
 
 Compound Addition, - -216 
 
 Compound Subtraction, - - -219 
 
 Compound Multiplication, - - 224 
 
 Compound Division, - - 226 
 
 Comparison of Time and Longitude, - - 227 
 
 Percentage^ - - 230 
 
 Notation of Per Cent, - - - - 230 
 
 Five Problems of Percentage, - - 233 
 
 Application* of Percentage, - 241 
 
 Commission and Brokerage, - - - - 241 
 
 Account of Sales, 2 4^ 
 
 Profit and Loss, 2 47 
 
 Interest, 
 
 Preliminary Principles, - 2 55 
 
 Six Per Cent Method, - - 256
 
 CONTENTS. 
 
 PAOl 
 
 Method by Aliquot Parts, - ... 259 
 
 Method by Days, ...... 2 6o 
 
 Partial Payments, - 267 
 
 Compound Interest, - - 273 
 
 Discount, - - ... . 276 
 
 Banks and Bank Discount, - 278 
 
 Stock Investments, - - 280 
 
 Government Bonds, - ... 2 8i 
 
 Exchange, - - 285 
 
 Insurance, - - - - - -292 
 
 Taxes, ------- - 295 
 
 Duties, --....-. 298 
 
 Internal Revenue, ... . 300 
 
 Equation of Payments, - - - 301 
 
 Averaging Accounts, ... - 304 
 
 Ratio, .... . 307 
 
 Proportion, .... . 309 
 
 Simple Proportion, - - - - -311 
 
 Simple Proportion by Analysis, - -313 
 
 Compound Proportion, - - 316 
 
 Partitive Proportion, ... - 319 
 
 Partnership, - - ... - 320 
 
 Bankruptcy,- - .... . 323 
 
 Alligation, - - - - - - -324 
 
 Tnvolution, - ...... 330 
 
 Formation of Squares, - -332 
 
 Evolution, - ..... 333 
 
 Extraction of the Square Root, ... 335 
 
 Applications of Square Root, - 339 
 
 Formation of Cubes, .... . 343 
 
 Extraction of the Cube Root, - 345 
 
 Applications of Cube Root, - ... 349 
 
 Arithmetical JProaression, - - - 350 
 
 (Geometrical Progression, - 353 
 
 Mensuration, ....... 355 
 
 Miscellaneous Examples, - ... 360
 
 ARITHMETIC. 
 
 Art. 1. Arithmetic is the science of numbers. 
 
 Arithmetic is sometimes said to be both a science and an art : a 
 science when it treats of the theory and properties of numbers ; an 
 art when it treats of their applications. 
 
 NOTES. i. The term arithmetic, is from the Greek arithmStike, 
 the art of reckoning. 
 
 2. The term science, from the Latin scientia, literally signifies 
 knowledge. In a more restricted sense, it denotes an orderly ar- 
 rangement of the facts and principles of a particular branch of 
 knowledge. 
 
 2. Nnnibcf is a imit,or a collection of units. 
 
 A Unit is any single thing, called one. One and 
 one more are called tu-o ; two and one more are called 
 three; three and one more, four, etc. The terms one, 
 two, three, four, are properly the names of numbers, but 
 are often used for numbers themselves. 
 
 NOTE. The term unit is from the Latin unus, signifying one. 
 
 3. The Unit Otie is the standard by which all num- 
 bers are measured. It may also be considered the base 
 or element of number. .For, all whole numbers greater 
 than one are composed of ones. Thus, two is composed of 
 one and one. Three is one more than two; but two, we 
 have seen, is composed of ones; hence, three is, and so on. 
 
 QUESTIONS. i. What is arithmetic? What else is it sometimes said to be? 
 When a science ? An art ? 2. Number ? A unit ? 3. The Lomdard 
 numbers are measured ? The base or element of number ?
 
 10 NOTATION. 
 
 4. Numbers are either abstract or concrete. 
 
 An Abstract Number is one that is not applied to 
 any object; as, three, five, ten. 
 
 A Concrete Number is one that is applied to some 
 object; as, five peaches, ten books. 
 
 NOTATION. 
 
 5. Notation is the art of expressing numbers by 
 figures, letters, or other numeral characters. 
 
 The two principal methods in use are the Arabic and 
 the Roman. 
 
 NOTE. Numbers are also expressed by words or common lan- 
 guage ; but this, strictly speaking, is not Notation. 
 
 ARABIC NOTATION. 
 
 6. The Arabic Notation is the method of express- 
 ing numbers by certain characters called figures. 
 
 It is so called, because it was introduced into Europe 
 from Arabia. 
 
 7. The Arabic figures are the following ten, viz : 
 
 1, 2, 3, 4, 5, 6, 7, 8, 9, O. 
 
 one, two, three, four, five, six, seven, eight, nine, naught. 
 
 The first nine are called significant figures, or digits ; 
 the last one, naught, zero, or cipher. 
 
 NOTES. i. The first nine are called significant figures, because 
 each always expresses a number. 
 
 2. The term digit is from the Latin digitus, a finger, and was 
 applied to these characters because they were employed as a substi 
 
 4 . What is an abstract number? Concrete? 5. What is notation? The 
 principal methods in use? 6. Arabic notation ? Why so called ? 7. TIow many 
 figures does it employ ? What are the first nine called ? The last one ? Note. 
 Why called significant figures ? Why digits? Meaning of digitu? ? Why is th 
 Jtt*t called naught ? Meaning of zero ? Of cipher ?
 
 NOTATION. 11 
 
 tute for the fingers upon which the ancients used to reckon. The 
 term originally included the cipher, but is now generally restricted 
 to the first nine. 
 
 3. The last one is called naught; because, when standing alone, it 
 has no value, and when connected with significant figures, it denotes 
 the absence of the order in whose place it stands. 
 
 4. Zero is an Italian word, signifying nothing. 
 
 The term cipher is from the Arabic sifr or sifreen, empty, vacant. 
 Subsequently the term was applied to all the- Arabic figures indis, 
 crimiuately ; hence, calculations by them were called ciphering. 
 
 8. Each of the first nine numbers is expressed by a 
 single figure, each figure denoting the number indicated 
 by its name. These numbers are called units of the first 
 order, or simply units. 
 
 8, a. ^Nine is the greatest number expressed by one figure. 
 Numbers larger than nine are expressed thus: 
 
 Ten (i more than 9) is expressed by an ingenious de- 
 vice, which groups ten single things or ones together, and 
 considers the collection a new or second order of units, 
 called ten. Hence, ten is expressed by writing the figure i 
 in the second place with a cipher on the right; as, 10. 
 
 The numbers from ten to nineteen inclusive are ex- 
 pressed by writing i in the second place, and the figure 
 denoting the units in the first ; as, 
 u, 12, 13, 14, 15, 16, 17, 18, 19. 
 
 eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen. 
 
 Twenty (2 tens) is expressed by writing the figure 2 in 
 the second place with a cipher on the right; as, 20. 
 
 Thirty (3 tens) by writing 3 in the second place with a 
 cipher on the right ; and so on to ninety inclusive ; as, 
 
 20, 30, 40, 50, 60, 70, 80, 90. 
 
 twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety. 
 
 8. How are the firat nine numbers expressed ? What are they called ? What 
 is the greatest number expressed by one figure ? How is ten expressed ? Twenty ? 
 Thirty, etc. ? The numbers between 10 and so? From 20 to 99 inclusive ?
 
 12 NOTATION. 
 
 The numbers from twenty to thirty, and so on to ninety^ 
 nine (99) inclusive, are expressed by writing the tens in the 
 second place, and the units in the first ; as, 
 
 21, 22, 23, 34, 35, 46, 57, 99. 
 
 twenty- twenty- twenty- thirty- thirty- forty- fifty- ninety- 
 one, two, three, four, five, six, seven, nine. 
 
 8, b. Ninety -nine is the greatest numbed that can be 
 expressed by two figures. 
 
 A hundred (i more than 99) is expressed by grouping 
 ten units of the second order together, and forming a new 
 or third order of units, called a hundred. Thus, a hun- 
 dred is expressed by writing i in the third place with two 
 ciphers on the right; as, 100. 
 
 In like manner, the numbers from one hundred to nine 
 hundred and ninety-nine inclusive, are expressed by writ- 
 ing the hundreds in the third place, the tens in the second, 
 and the units in the first. Thus, one hundred and thirty- 
 five (i hundred, 3 tens, and 5 units,) is expressed by 135. 
 
 8, c. Nine hundred and ninety-nine is the 
 
 greatest number that can be expressed by three figures. 
 
 Tliousands, and larger numbers, are expressed by form- 
 ing other new orders, called the fourth, fifth, etc., orders ; 
 as, tens of thousands, hundreds of thousands, millions, etc. 
 
 NOTES. i. The names of the first ten number?, one, two, three, 
 etc., are primitive words. The terms eleven and twelve are from the 
 Saxon endlffen and ticelif, meaning one and ten, two and ten. Thir- 
 teen is from thir and teen, which mean three and ten, and so on. 
 
 2. Twenty is from the Saxon tweentig, tween, two, and ty, tens ; 
 i. e., two tens. Thirty is from thir and ty, three tens, and so on. 
 
 3. The terms hundred, thousand, and million are primitive words, 
 having no perceptible analogy to the numbers they express. 
 
 From the foregoing illustrations we derive the following principle : 
 8. How is a hundred expressed T Thousands, and larger numbers t
 
 NOTATION. It 
 
 9. The Orders of Units increase by the scale of ten 
 That is, ten single units are one ten ; ten tens one hun- 
 dred; ten hundreds one thousand ; and, universally, 
 
 Ten of any lower order make a unit of the next 
 higher. 
 
 NOTES. i. If the term unit denotes one, how, it may be asked, can 
 ten things or ones be a unit, ten tens another unit, etc. And how- 
 can the figures i, 2, 3, etc., sometimes denote single things or ones ; 
 at others, tens of ones, and so on. 
 
 The answer is, units are of two kinds, simple and collective. 
 
 A simple unit is a single thing or one. 
 
 A collective unit denotes a group of ones, regarded as a whole. 
 
 2. To illustrate these units, suppose a basket of pebbles is before 
 us. Counting them out one by one, each pebble is a simple unit. 
 
 Again, counting out ten single pebbles and putting them togethei 
 in a group, this group forms a unit of the second order, called ten. 
 Counting out ten such groups, and putting them together in one 
 pile, this collection forms a unit of the third order, called hundred. 
 In like manner, a group of ten hundreds forms a unit of the fourth 
 order, called thousand, and so on. 
 
 These different groups of ten are called units on the same principle 
 that a group of ten cents forms a unit, called a dime ; or, a group of 
 ten dimes, a unit, called a dollar. Thus, it will be seen that the 
 figures i, 2, 3, 4, etc., always mean one, two, three, four units 
 as their name indicates ; but the value of these units depends upon 
 the place the figure occupies. 
 
 10. From the preceding illustrations, it will be seen 
 that the Arabic Notation is founded upon the following 
 principles : 
 
 ist. Numbers are divided into groups called units, of 
 the first, second, third, etc., orders. 
 
 zd. To express these different orders of units, a simple 
 and a local value are assigned to the significant figures, 
 according to the place they occupy. 
 
 3d. If any order is wanting, its place is supplied by a 
 cipher. 
 
 9. How do the orders of units increase? Units make a ten ? Tens a hundred' 
 to. Name the principles upon which the Arabic Notation in founded ?
 
 14 NOTATION. 
 
 11. The Simple Value of a figure is the number 
 of units it expresses when it stands alone, or in the right- 
 hand place. 
 
 The Local Value is the number it expresses when 
 connected with other figures, and is determined by the 
 place it occupies, counting from the right. 
 
 12. It is a general law of the Arabic Notation that the 
 value of a figure is increased tenfold for every place it is 
 moved from the right to the left ; and, conversely, 
 
 The value of each figure is diminished tenfold for every 
 place it is moved from the left to the right. Thus, 2 in 
 the first place denotes two simple units ; in the second 
 place, ten times two, or twenty ; in the third place, ten 
 times as much as in the second place, or two hundreds, 
 and so on. (Art. 8.) 
 
 13. The number denoting the scale by which the orders 
 of units increase is called the radix. 
 
 The radix of the Arabic Notation is ten ; hence it is 
 often called the decimal notation. 
 
 NOTES. i. The term radix, Latin, signifies root, or base. The 
 term decimal is from the Latin decent, ten. 
 
 2. The decimal radix was doubtless suggested by the number of 
 fingers (digiti) on both hands. (Art. 7, Note.) Hence, 
 
 14. To Express Numbers by Figures, 
 
 Begin at the left hand and write the figures of the given 
 orders in the successive places toward the right. 
 
 If any intermediate orders are omitted, supply their 
 places ivith ciphers. 
 
 ii. The simple value of a figure? Local? 12. What is the law as to mov'ig 
 a figure to the right or left ? 13. What is the radix of a system of notation ? Tho 
 radix of the Arabic ? What else is the Arabic system called ? Why ? Note. 
 Meaning of radix ? Decimal? What so"-p!>td the decimal radix? 14. Rule 
 for expressing numbers by figures ?
 
 NOTATION. 5 
 
 EXERCISES. 
 
 Express Uie following numbers by figures: 
 
 1. Three hundred and forty-five. 
 
 2. Four hundred and sixty. 
 
 3. Eight hundred and four. 
 
 4. Two thousand three hundred and ten. 
 
 5. Thirty thousand and nineteen. 
 
 6. Sixty-three thousand and two hundred. 
 
 7. One hundred and ten thousand two hundred and 
 twelve. 
 
 8. Four hundred and sixty thousand nine hundred 
 and thirty. 
 
 9. Six hundred and five thousand eight hundred and 
 forty-two. 
 
 10. Two millions sixty thousand and seventy-five. 
 
 ROMAN NOTATION. 
 
 15. The Roman Notation is the method of ex- 
 pressing numbers by certain letters. It is so called because 
 it was employed by the Romans. The letters used are 
 the following seven, viz. : I, V, X, L, C, D and M. The 
 letter I denotes one; V,ftve; X, ten; L, fifty; C, one 
 hundred ; D, five hundred ; M, one thousand. Interven- 
 ing and larger numbers are expressed by the repetition 
 and combination of these letters. 
 
 16. The Roman system is based upon the following 
 general principles : 
 
 ist. It proceeds according to the scale of ten as far 
 as a thousand, the unit of each order being denoted by 
 
 15. Roman notation? Why so called? Letters employed? The letter I de- 
 note? V? X? L? C? D? M? :6. Name the first principle upon which it 
 Is based. What is the effect of repeating n letter ? Of placing a letter of less 
 r:ilue before one of greater value? If placed after? If a line is placed over a 
 letter?
 
 16 
 
 NOTATION. 
 
 a single letter. Thus, I denotes one ; X, ten ; C, one hun- 
 dred ; M, one thousand. 
 
 2d. Repeating a letter, repeats its value. Thus, I denotes 
 one ; II, two ; III, three ; X, ten ; XX, twenty, etc. 
 
 3d. Placing a letter of less value before one of greater 
 value, diminishes the value of the greater by that of the 
 less ; placing the less after the greater, increases the value 
 of the greater by that of the less. Thus, V denotes five, 
 but IV denotes only four, and VI six. 
 
 4th. Placing a horizontal line over a letter increases its 
 value a thousand times. Thus, I denotes a thousand ; X, 
 ten thousand ; 0, a hundred thousand ; M, a million. 
 
 TABLE. 
 
 I, 
 
 denotes one. 
 
 XXX, denotes thirty. 
 
 n, 
 
 " two. 
 
 XL, " forty. 
 
 in, 
 
 " three. 
 
 L, " fifty. 
 
 IV, 
 
 " four. 
 
 LX, " sixty. 
 
 V, 
 
 " five. 
 
 LXX, " seventy. 
 
 VI, 
 
 " six. 
 
 LXXX, " eighty. 
 
 VII, 
 
 " seven. 
 
 XC, " ninety. 
 
 VIII, 
 
 " eight. 
 
 C, " one hundred. 
 
 IX, 
 
 " nine. 
 
 CC, " two hundred. 
 
 x, 
 
 " ten. 
 
 CCC, " three hundred. 
 
 XI, 
 
 " eleven. 
 
 CCCC, " four hundred. 
 
 XII, 
 
 " twelve. 
 
 D, " five hundred. 
 
 XIII, 
 
 " thirteen. 
 
 DC, " six hundred. 
 
 XIV, 
 
 " fourteen. 
 
 DCC, " seven hundred. 
 
 xv, 
 
 " fifteen. 
 
 DCCC, " eight hundred. 
 
 XVI, 
 
 " sixteen. 
 
 DCCCC, " nine hundred. 
 
 XVII, 
 
 " seventeen. 
 
 M, " one thousand. 
 
 XVIII, 
 
 " eighteen. 
 
 MM, " two thousand. 
 
 XIX, 
 
 " nineteen, 
 
 MDCCCLXXI, denotes one thous- 
 
 XX, 
 
 " twenty. 
 
 and eight hundred and seventy-one. 
 
 NOTES. i. Four was formerly denoted by IIII; nine by VIIII; 
 forty by XXXX ; ninety by LXXXX ; Jive hundred by 10 ; and a 
 thousand by CL>.
 
 NOTATION. 
 
 17 
 
 2. Annexing to 1,0 (five hundred) increases its value ten times. 
 Thus, 100 denotes five thousand ; 1000, fifty thousand. 
 
 Prefixing a C and annexing a to CIO (a thousand) increases its 
 value ten times. Thus, CC10J denotes ten thousand, etc. Hence, 
 
 17. To Express Numbers by Letters, 
 
 Begin at the left hand or highest order, and write the 
 letters denoting the given number of each order in succession. 
 
 NOTE. The Roman Notation is seldom used, except in denoting 
 chapters, sections, heads of discourses, etc. 
 
 EXERCISES. 
 Express the following numbers by letters : 
 
 i. Fourteen, 
 4. Sixty-six, 
 7. Eighty-eight, 
 
 10. 107, 
 
 13- 613, 
 
 1 6. 724, 
 
 19. 1004, 
 
 22. 1417, 
 25. 1748, 
 
 2. Twenty-nine, 
 
 5. Forty-nine, 
 
 8. Ninety-four, 
 
 II. 212, 
 
 14- S ?, 
 
 17. 829, 
 
 20. 1209, 
 
 23. 1614, 
 
 26. 1803, 
 
 3. Thirty-four, 
 6. Seventy-three, 
 9. Ninety-nine, 
 
 12. 498, 
 
 15. 608. 
 
 18. 928, 
 
 21. 1363, 
 
 24. 1671, 
 
 27. 1876. 
 
 NUMERATION. 
 
 18. Numeration is the art of reading numbers ex- 
 pressed by figures, letters, or other numeral characters. 
 
 NOTE. The learner should be careful not to confound Numeration 
 with Notation. The distinction between them is the same as that 
 between reading and vsritiny. 
 
 19. There are two methods oi' reading numbers, the 
 French and the English. 
 
 17. Role for expressing numbers byletterc? 18. Numeration? Note. Dis- 
 tinction between Numeration and Notation? ig. How many methods of readiujj 
 numbers V
 
 18 NUMERATION. 
 
 FRENCH NUMERATION. 
 
 20. The French divide numbers into periods of three 
 figures each, and then subdivide each period into units, 
 tens, and hundreds, as in the following 
 
 TABLE. 
 
 r3 GO S 
 
 5 .2 1 . 
 
 3 ^3to rf. So5 2 P 
 
 <| j s s ?i 
 
 e^ ^< t ti^ ' 
 
 tj C? __ r __ 
 
 H * " 
 
 O ~ O ^ O = 
 
 'S < 2 i> ^ -S^oj -S w . iS " ^ 
 
 e 
 
 S<"^ aJv-S 3t*H S'wS V ^ S 
 
 Sog So=a !-o~ ^og ^os C 
 
 11 li Is! lli lal 11 
 
 Sa !^oJ> So^ 3 ol 3 opS =Ot5 
 
 WEHO>. W ft w WHH^ WEH^ KH^ WH^ 
 
 823 561 729 452 789 384 
 
 6th period. 5th period. 4th period. 3d period. 2d period. 1st period. 
 
 The first period on the right is called units period ; the 
 second, thousands; the third, millions, etc. 
 
 The periods in the table are thus read : 823 quadrillions, 
 561 trillions, 729 billions, 452 millions, 789 thousand, three 
 hundred and eighty-four. 
 
 NOTE. The terms billion, trillion, quadrillion, etc., are derived 
 from the Italian milione and the Latin bis, tres, guatuor, etc. Thus, 
 bis, united with million, becomes billion, etc. 
 
 21. To read Numbers according to the French Numeration. 
 
 Divide them into periods of three figures each, counting 
 from the right. 
 
 Beginning at the left hand, read the periods in succes- 
 sion, and add the name to each, except the last. 
 
 20. The French method ? Repeat the Table, bcgiuning at the right. What is 
 the i Bt period called? The ad? 3d? 4th? sth? 6th? 21. The rule for reading 
 numbers by the French method ? Note Why omit the name of the right hand 
 period ? The difference between orders and periods ?
 
 NUMERATION. 
 
 19 
 
 NOTES. i. The name of the right-hand period is omitted for the 
 sake of conciseness ; and since this period always denotes units, the 
 omission occasions no obscurity. 
 
 2. The learner should observe the difference between the orders of 
 units and the periods into which they are divided. The former 
 increase by tens ; the latter by thousands. 
 
 3. This method of reading numbers is commonly ascribed to the 
 French, and is thence called FrencJi numeration. Others ascribe it to 
 the Italians, and thence call it the Italian method. 
 
 Read the following numbers by the French numeration : 
 
 21. 3006017 
 
 22. 2460317239 
 
 23. 5100024000 
 
 24. 70300510 
 
 25. 203019060 
 
 26. 7800580019 
 
 27. 86020005200 
 
 28. 51036040 
 
 29. 621000031 
 
 30. 93000275320 
 
 I. 
 
 270 
 
 ii. 
 
 840230 
 
 2. 
 
 39 
 
 12. 
 
 4603400 
 
 3- 
 
 1270 
 
 J 3- 
 
 35040026 
 
 4- 
 
 2036 
 
 14. 
 
 8o6oOOOO 
 
 5- 
 
 8605 
 
 15- 
 
 9001307 
 
 6. 
 
 40300 
 
 1 6. 
 
 65023009 
 
 7- 
 
 85017 
 
 17- 
 
 SlOOOO 
 
 8. 
 
 160401 
 
 1 8. 
 
 753O6O2O 
 
 9- 
 
 405869 
 
 19. 
 
 165380254 
 
 10. 
 
 1365406 
 
 20. 
 
 310400270 
 
 31. 216327250516 
 
 32. 4260300210109 
 
 33. 300073004000 
 
 34. 41295000400649 
 
 35. 264300439000200 
 
 36. 
 37. 
 
 38. 
 
 39- 
 
 40. 
 
 Express the following numbers by figures 
 
 1. Ten millions, five thousand, and two hundred. 
 
 2. Sixty-one millions, three hundred and forty. 
 
 3. Three hundred and ten millions, and five hundred. 
 
 4. Twenty-six billions, seventy millions, three hundred. 
 
 5. One hundred billions, four hundred and twenty-five. 
 
 6. Sixty-eight trillions, seven hundred and twenty-five. 
 
 7. Eight hundred and twenty millions, five hundred 
 and twenty- three.
 
 20 NUMERATION. 
 
 8. Sixty-seven quadrillions, ninety-seven billions. 
 
 9. Four hundred and sixty quadrillions, and eighty, 
 seven millions. 
 
 10. Seven hundred and sixty-one quadrillions, seventy- 
 one trillions, two hundred billions, eighteen millions, five 
 thousand, and thirty-six. 
 
 ENGLISH NUMERATION. 
 
 22. The English divide numbers into periods of six 
 figures each, and then subdivide each period into units, 
 lens, hundreds, thousands, tens of . thousands, and hundreds 
 of thousands, as in the following 
 
 TABLE. 
 
 <n .rt 
 II 
 
 00 
 
 a o 
 
 K ^3 
 
 
 ^-33.2 s|35 . i| 
 
 .Sc5f3a5 'ra^^d - 
 
 tj_ C 3ti c -;9 * a P ^! S $<"> 
 
 oS^OK ot<o t,ogj^- ^ 
 
 s M I i ijii^i ^1^1 1 2 
 
 407692 958604 413056 
 
 3d period. id period. ist period. 
 
 The periods in the Table are thus read : 407692 billions, 958604 
 millions, 413 thousand, and fifty-six. 
 
 NOTE. This method is called English numeration, because it was 
 invented by the English. 
 
 For other numbers to read by this method, the pupil is referred 
 to those in Art. 21. 
 
 as. What is the English method of numeration ? Note. Why BO called T
 
 ADDITION. 
 
 23. Addition is uniting two or more numbers in one 
 The Sum or Amount is the number found by addi- 
 tion. Thus, 5 added to 7 are 12; twelve, the number 
 obtained, is the sum or amount. 
 
 NOTES. i. The sum or amount contains as many units as the 
 numbers added. For, the numbers added are composed of units ; 
 and the whole is equal to the sum or all its parts. (Art. 3.) 
 
 2. When the numbers added are the same denomination, the 
 operation is called Simple Addition. 
 
 SIGNS. 
 
 24. Signs are characters used to indicate the relation 
 of numbers, and operations to be performed. 
 
 25. The Sign of Addition is a perpendicular 
 cross called plus ( + ), placed before the number to be 
 added. Thus 7+5, means that 5 is to be added to 7, and 
 is read " 7 plus 5." 
 
 NOTE. The term plus, Latin, signifies more, or added to. 
 
 26. The Sign of Equality is two short parallel 
 lines ( = ), placed between the numbers compared. Thus 
 7 + 5 = 12, means that 7 and 5 are equal to 12, and is read, 
 " 7 plus 5 equal 12," or the sum of 7 plus 5 equals 12. 
 
 Head the following numbers : 
 
 i. 8 + 4 + 2=6 + 8 
 
 2 - 7+3 + 5 = 
 
 3. 19 + 1+0=6 + 5+ 9 
 
 4. 23+ 7 = 19 + 11 
 
 5. 37+ 8=30 + 15 
 
 6. 58+ 10=40 + 28 
 
 23. What is addition ? The result called ? Note. When the numbers added 
 are the same denomination, what is the operation called ? 24. What are Picons ? 
 25. The sifjn of addition ? Note. The meaning of plus? 26. Sign of equality? 
 How is 7 +5 =12 read?
 
 22 
 
 ADDITION. 
 
 27. The Sign of Dollars is a capital S with two 
 perpendicular marks across it ($), prefixed to the number 
 of dollars to be expressed. Thus, $245 means 245 dollars. 
 
 NOTE. The term prefix, from the Latin prefigo, signifies to place 
 before. 
 
 Read the following expressions : 
 
 1. $17+ 
 
 2. $i3 + 
 
 3. $21+ 
 
 =$i5+$io 
 
 4- 
 5- 
 6. $105 - 
 
 ADDITION TABLE. 
 
 + $8 = 
 
 $36 -$96 + $45 
 
 i and 
 
 2 and 
 
 3 and 
 
 4 and 
 
 5 and 
 
 i are 2 
 
 i are 3 
 
 i are 4 
 
 i are 5 
 
 i are 6 
 
 2 3 
 
 2 " 4 
 
 2 5 
 
 2 " 6 
 
 2 7 
 
 3 " 4 
 
 3 " 5 
 
 3 " 6 
 
 3 " 7 
 
 3 " 8 
 
 4 " 5 
 
 4 " 6 
 
 4 " 7 
 
 4 " 8 
 
 4 9 
 
 5 " 
 
 5 " 7 
 
 5 " 8 
 
 5 " 9 
 
 5 " 10 
 
 6 " 7 
 
 6 8 
 
 6 " 9 
 
 6 " 10 
 
 6 ii 
 
 7 " 8 
 
 7 " 9 
 
 7 " 10 
 
 7 " ii 
 
 7 12 
 
 8 9 
 
 8 " 10 
 
 8 " ii 
 
 8 " 12 
 
 8 13 
 
 9 " 10 
 
 9 " ii 
 
 9 " 12 
 
 9 " 13 
 
 9 14 
 
 10 II 
 
 10 " 12 
 
 10 I 3 
 
 10 I 4 
 
 10 " 15 
 
 6 and 
 
 7 and 
 
 8 and 
 
 9 and 
 
 10 and 
 
 i are 7 
 
 i are 8 
 
 i are 9 
 
 i are 10 
 
 i are n 
 
 2 " 8 
 
 2 " 9 
 
 2 " 10 
 
 2 " II 
 
 2 " 12 
 
 3 " 9 
 
 3 " 10 
 
 3 " ii 
 
 3 " 12 
 
 3 " 13 
 
 4 10 
 
 4 " ii 
 
 4 " 12 
 
 4 " 13 
 
 4 " 14 
 
 5 " ii 
 
 5 " 12 
 
 5 " 13 
 
 5 " i4 
 
 5 " 15 
 
 6 " 12 
 
 6 13 
 
 6 14 
 
 6 15 
 
 6 " 16 
 
 7 " 13 
 
 7 " 14 
 
 7 15 
 
 7 " 16 
 
 7 " 17 
 
 8 " 14 
 
 8 15 
 
 8 " 16 
 
 8 " 17 
 
 8 18 
 
 9 " 15 
 
 9 " 16 
 
 9 " 17 
 
 9 " 18 
 
 9 " 19 
 
 10 " 16 
 
 10 " 17 
 
 10 " 18 
 
 10 " 19 
 
 10 ". 20 
 
 More mistakes are made in adding than in any other arith- 
 metical operation. The fiist five digits are easily combined; the 
 results of adding 9, being i less than if 10 were added, are also easy. 
 The othera 6, 7, 8, are more difficult, and therefore should receive 
 special attention. 
 
 -^ 7. What Is the Blgn of dollars T
 
 ADDITION. 23 
 
 CASE I. 
 
 28. To find the Amount of two OP more numbers, when 
 the Sum of each column is Less than 10. 
 
 Ex. i. A man owns 3 farms; one contains 223 acres, 
 another 51 acres, and the other 312 acres: how many 
 acres has he ? 
 
 ANALYSIS. Let the numbers be set down as in 
 
 _. . . OPERATION. 
 
 the margin. Beginning at tlie right, we proceed tt - 
 
 thus : 2 units and i unit are 3 units, and 3 are 6 "H g's 
 units ; the sum being less than ten units, we set it 
 
 under the column of units, because it is units. Next, 2 2 3 
 
 i ten and 5 tens are 6 tens, and 2 are 8 tens ; the 5 1 
 
 sum being less than 10 tens, we set it under the ? 1 2 
 
 column of tens, because it is tens. Finally, 3 hun- 
 
 dreds and 2 hundreds are 5 hundreds; the sum A.ns. 586 
 being less than 10 hundreds, we set it under the 
 
 column of hundreds, for the same reason. Therefore, he lias 586 
 acres. All similar examples are solved in liko manner. 
 
 By inspecting the preceding illustration, the learner 
 will discover the following principle : 
 
 Units of the same order are added together, and the 
 sum is placed under the column added. (Art. 9.) 
 
 NOTES. i. The same orders are placed under each other for the 
 sake of eon cenience and rapidity in adding. 
 
 2. We add the same orders together, units to units, tens to tens, 
 etc., because different orders express units of different values, and 
 therefore cannot be added to each other. Thus, 5 units and 5 tens 
 neither make 10 units nor 10 tens, any more than 5 cents and 5 
 dimes will make 10 cents or 10 dimes. 
 
 3. We add the columns separately, because it is easier to add one 
 order at a time than several. 
 
 4. The sum of each column is set under the column added, becaiir e 
 being less than 10, it is the name order as that column. 
 
 (2.) (3-) (4.) (5-) (6.) 
 
 2IO3 3O24 I2II 2IO2 2IO32 
 
 4022 1230 2002 I2 53 52010 
 
 1674 4603 5340 4604 24603 
 
 What is the sum of $2321 +$123 + 13245?
 
 24 ADDITION. 
 
 8. What is the sum of 3210 pounds+2023 pounds-f 
 4601 pounds? 
 
 9. What is the sum of 130230 + 201321-1-402126 ? 
 10. What is the sum of 24106324-1034246 + 320120 ? 
 
 CASE II. 
 
 29. To find the Amount of two or more numbers, when 
 
 the Sum of any column is 10, or more. 
 
 i. What is the sum of $436, 8324, and $645 ? 
 
 ANALYSIS. Let the numbers be set down as in the 
 margin. Adding as before, the sum of the first column 
 is 15 units, or i ten and 5 units. We set the 5 units 
 under the column added, and add the i ten to the next 
 column because it is the same order as that columa. 
 Now, i added to 4 tens makes 5 tens, and 2 are 7 tens, and 
 3 are 10 tens, or I hundred and o tens. We set the o, or $ 1 405 
 right hand figure, under the column added, and add the 
 i hundred io the next column, as before. The sum of the next 
 column, with the i added, is 14 hundreds ; or I thousand and 4 hun- 
 dreds. This being the last column, we set down the wJiote sum. 
 The answer is $1405. All similar examples may be solved in like 
 manner. 
 
 By inspecting this illustration, it will be seen, 
 When the sum of a column is i o or more, we write the 
 units' figure under the column, and add the tens' figure to 
 the next column. 
 
 NOTES. i. We set the units' figure under the column added, and 
 add the tens to the next column, because they are the same orders 
 as these columns. 
 
 2. We begin to add at the right Juind, in order to carry the tens as 
 we proceed. We set down the whole sum of the last column, because 
 there are no figures of the same order to which its left hand figure 
 can be added. 
 
 30. Adding the tens or left hand figure to the next 
 column, is called carrying the tens. The process of carry- 
 ing the tens, it will be observed, is simply taking a certain 
 number of units from a lower order, and adding their 
 equal to the next higher ; therefore, it can neither increase 
 nor diminish the amount.
 
 m 
 
 ADDITION. 25 
 
 NOTE. We carry for ten instead of seven, nine, eleven, etc., 
 because in the Arabic notation the orders increase by the scale of 
 ten. If they increased by the scale of eight, twelve, etc., we should 
 carry for that number. (Art. 13.) 
 
 (*-} (3-) (4-) (5-) (6.) 
 
 5689 6898 7585 8456 97504 
 
 3792 33 6 5 3748 5078 38786 
 
 435 8 79 8 7 8667 6904 75979 
 
 31. The preceding principles may be summed up 
 the following 
 
 GENERAL RULE. 
 
 I. Place the numbers one under another, units under 
 units, etc. ; and beginning at the right, add each column 
 separately. 
 
 II. If the sum of a column does not exceed NINE, write 
 it under the column added. 
 
 If the sum exceeds NINE, write the units' figure under 
 the column, and add the tens to the next higher order. 
 Finally, set down the whole sum of the last column. 
 
 NOTES. i. As soon as the pupil understands the principle of 
 adding, he should learn to abbreviate the process by simply pro- 
 nouncing the successive results, as he points to each figure added. 
 Thus, instead of saying 7 units and 9 units are 16 units, and 8 are 
 24 units, and 7 are 31 units, he should say, nine, sixteen, twenty-fair, 
 thirty-one, etc. 
 
 Again, if two or more numbers together make 10, as 6 and 4, 
 7 and 3 ; or 2, 3, and 5, etc., it is shorter, and therefore better, to add 
 10 at once. 
 
 31. How write numbers to be added ? The next step ? If the sum of a column 
 does not exceed nine, what do yon do with it ? If it exceeds nine ? The sum 
 of the last column ? 28. Note. Why write units under units, etc. ? Why add the 
 columns separately ? Why not add different orders together promiscuously ? Is 
 the sum of 3 units and 4 tens, 7 units or 7 tens ? When the sum of a column does 
 not exceed 9, why set it under the column 1 29. Note. If the sum of a column is 
 10 or more, why set the units' figure under the column added, and carry the tens 
 to the next column ? 30. What is meant by carrying the tens ? Why does not 
 carrying change the amount f Why carry for 10 instead of 6, 8, 12, etc.
 
 26 ADDITION. 
 
 2. Accountants sometimes set the figure carried under the right 
 hand figure in a line below the answer. In this way the sum of 
 each column is preserved, and any part of the work can be reviewed, 
 if desired ; or if interrupted, can be resumed at pleasure. 
 
 32. PEOOF. Begin at the top and add each column 
 downward. If the two results agree, the work is right. 
 
 NOTE. This proof depends upon the supposition that reversing 
 the order of the figures, will detect any error that may have occurred 
 in the operation. 
 
 EXAMPLES. 
 
 i. Find the sum of 864, 741, 375, 284, and 542, and 
 prove the operation. 
 
 (2.) (3-) (4-) (5-) (6.) 
 
 Dollars. Pounds. Yards. Rods. Feet. 
 
 263 4780 2896 23721 845235 
 
 425 7642 8342 70253 476234 
 
 846 5036 257 4621 6897 
 
 407 7827 3261 342 7 2 3468 
 
 7. What is the sum of 675 acres + 842 acres + 904 acres 
 4-39 acres? 
 
 8. What is the sum of 8423 + 286 + 7932 + 28 + 6790? 
 
 9. What is the sum of 82431 + 376 + 19 + 62328 + 4521 
 
 + 35787? 
 10. What is the sum of 63428 + 78 + 4236 + 628 + 93 + 
 
 8413? 
 
 (n.) (12.) ( I3 .) .(14.) (15-) 
 
 2685 89243 7 20 94 825276 93 I2 53 
 
 6543 8284 96308 704394 43 2 5 6 7 
 
 8479 34567 763 37783 65414 
 
 6503 865 4292 1697 9236 
 
 1762 3952 23648 349435 843 
 
 7395 42678 75965 697678 68 
 
 i6. If a man pays $2358 for his farm, $1950 for stock, 
 and $360 for tools, how much does he pay for all ? 
 
 32. How prove addition ? Note. Upon what is this proof based ?
 
 ADDITION. 27 
 
 17. A merchant bought 371 yards of silk, 287 yards of 
 calico, 643 yards of muslin, and 75 yards of broadcloth: 
 how many yards did he buy in all ? 
 
 18. Eequired the sum of $2404 + $100 + 11965 +$1863. 
 
 19. Eequired the sum of 968 pounds + 81 pounds + 7 
 pounds + 639 pounds. 
 
 20. Required the sum of 1565 gals. + 870 gals. +31 
 gals. + 1 60 gals. + 42 gals. 
 
 21. Add 2368, 1764, 942, 87, 6, and 5271. 
 
 22. Add 281, 6240, 37, 9, 1923, 101, and 45. 
 
 23. Add 888, 9061, 75, 300, 99, 6, and 243. 
 
 24. 243 + 765+980 + 759 + 127 how many? 
 
 25. 9423 + 100 + 1600 + ii9 + 4oo4=how many? 
 
 26. 81263 + 16319 + 805 +25oo + 93=how many? 
 
 27. 236517 +460075 + 2353oo + 275i6i = how many? 
 
 28. A savings bank loaned to one customer $1560, to 
 another $1973, to a third $2500, and to a fourth $3160: 
 how much did it loan to all ? 
 
 29. A young farmer raised 763 bushels of wheat the 
 first year, 849 bushels the second, ion bushels the third, 
 and 1375 bushels the fourth : how many bushels did he 
 raise in 4 years ? 
 
 30. A man bequeathed the Soldiers' Home $8545 ; the 
 Blind Asylum $7538; the Deaf and Dumb Asylum 
 $6280; the Orphan Asylum $19260; and to his wife the 
 remainder, which was $65978. What was the value of 
 his estate ? 
 
 31. A merchant owns a store valued at $17265, his 
 goods on hand cost him $19230, and he has $1563 in 
 bank : how much is he worth ? 
 
 32. "What is the sum of thirteen hundred and sixty- 
 three, eighty-seven, one thousand and ninety-four, and 
 three hundred ? 
 
 33. "What is the sum of three thousand two hundred 
 and forty, fifteen hundred and sixty, and nine thousand?
 
 28 ADDITION. 
 
 34. Add ninety thousand three hundred and two, sixty- 
 five thousand and thirty, forty-four hundred and twenty- 
 three. 
 
 35. Add eight hundred thousand and eight hundred, 
 forty thousand and forty, seven thousand and seven, nine 
 hundred and nine. 
 
 36. Add 30006, 301, 55000, 2030, 67, and 95000. 
 
 37. Add 65139, 100100, 39, iniii, 763002, and 317. 
 
 38. Add 8 1, 907, 311, 685, 9235, 7, and 259. 
 
 39. Add 4895, 352, 68, 7, 95, 645, and 3867. 
 
 40. Add 631, 17, i, 45, 9268, 196, and 3562. 
 
 41. Add 77777, 33333> 88888, 22222, and nui. 
 
 42. Add 236578, 125, 687256, and 4455- 
 
 43. Add 23246, 8200461, 5017, 8264, and 39. 
 
 44. Add 317, 21, 9, 4500, 219, 3001, 17036, and 45. 
 
 45. Add 10000000, joooooo, 100000, 10000, 1000, 100, 
 und 10. 
 
 46. Add 22000000, 22000, 202000, 22oo, and 2 20. 
 
 47. If a young man lays up $365 in i year, how much 
 will he lay up in 4 years ? 
 
 48. In what year will a man who was born in 1850, be 75 
 years old ? 
 
 49. A man deposits $1365 in a bank per day for 6 days 
 in succession: what amount did he deposit during the 
 week? 
 
 50. A planter raised 1739 pounds of cotton on ono 
 section of his estate, 703 pounds on another, .20 15 pounds 
 on another, and 2530 pounds on another: how many 
 pounds did he raise in all ? 
 
 51. A man was 29 years old when his eldest son was 
 born ; that son died aged 47, and the father died 1 7 years 
 later : how old was the man at his death ? 
 
 52. A speculator bought 3 city lots for $21213, an ^ sold 
 so as to gain $375 on each lot: for what sum did he sell 
 them?
 
 ADDITION. 29 
 
 53. A's income-tax for 1865 was $4369, B's $3978; C's 
 was $135 more than A's and B's together, and D's was 
 equal to all the others: what was D's tax? What the 
 tax of all ? 
 
 54. How many strokes does a clock strike in 24 hours? 
 
 55. A raised 3245 bushels of corn, and B raised 723 
 bushels more than A : how many bushels did both raise ? 
 
 56. In a leap year 7 months have 31 days each, 4 months 
 30 days each, and i month has 29 days: how many days 
 constitute a leap year ? 
 
 57. The entire property of a bankrupt is $2648, which 
 is only half of what he owes : how much does he owe ? 
 
 58. What number is 19256 more than 31273 ? 
 
 59. A has 860 acres of land, B 117 acres more than A, 
 and G as many as both : how many acres have all ? 
 
 DRILL COLUMNS. 
 
 (60.) 
 
 (61.) 
 
 (62.) 
 
 (630 
 
 (640 
 
 (650 
 
 Pols. cts. 
 
 Dols. cts. 
 
 Dols. cts. 
 
 Dols. cts. 
 
 Dols. cts. 
 
 Dols. cts. 
 
 14 65 
 
 25 76 
 
 37 38 
 
 24 9 1 
 
 34 45 
 
 49 68 
 
 21 43 
 
 3 2 37 
 
 3 25 
 
 42 73 
 
 32 61 
 
 26 52 
 
 64 61 
 
 54 61 
 
 46 72 
 
 9 61 
 
 64 12 
 
 38 43 
 
 37 28 
 
 16 45 
 
 28 41 
 
 3 2 44 
 
 7 33 
 
 27 59 
 
 43 2 4 
 
 67 38 
 
 7 5 
 
 51 62 
 
 24 54 
 
 12 38 
 
 46 03 
 
 34 9 2 
 
 63 04 
 
 9 28 
 
 32 67 
 
 45 63 
 
 19 41 
 
 76 41 
 
 16 28 
 
 7 3 6 
 
 48 32 
 
 5 7i 
 
 3 2 34 
 
 47 69 
 
 7 39 
 
 84 52 
 
 25 6 7 
 
 26 83 
 
 34 32 
 
 3i 04 
 
 82 01 
 
 19 24 
 
 13 09 
 
 34 26 
 
 15 73 
 
 83 26 
 
 7 63 
 
 32 41 
 
 58 32 
 
 72 61 
 
 62 64 
 
 2 7 13 
 
 24 07 
 
 42 35 
 
 24 63 
 
 23 45 
 
 The ability to add with rapidity and correctness is con- 
 fessedly a most valuable attainment ; yet it is notorious that few 
 pupils ever acquire it in school. The cause is the neglect of the 
 Table, and the want of drilling. Practice is the price of skill.
 
 30 ADDITION. 
 
 (66.) 
 
 (67.) 
 
 (68.) 
 
 (6 9 .) 
 
 (7.) 
 
 Dols. cts. 
 
 Dols. cts. 
 
 Dols. cts. 
 
 Dols. cts. 
 
 Dols. cts. 
 
 25 6 3 
 
 346 25 
 
 273 42 
 
 3424 27 
 
 4386 48 
 
 46 74 
 
 405 3 1 
 
 534 97 
 
 6213 39 
 
 3275 26 
 
 82 31 
 
 830 62 
 
 286 31 
 
 5382 13 
 
 5327 64 
 
 60 46 
 
 642 43 
 
 347 34 
 
 45 61 6 3 
 
 4613 04 
 
 75 38 
 
 761 38 
 
 721 35 
 
 8324 29 
 
 1729 56 
 
 22 76 
 
 482 71 
 
 635 26 
 
 5 2 7 6 34 
 
 3537 63 
 
 64 28 
 
 395 65 
 
 453 94 
 
 6594 S 2 
 
 8274 31 
 
 37 3 1 
 
 762 34 
 
 587 63 
 
 1723 45 
 
 5026 73 
 
 25 16 
 
 25 25 
 
 658 92 
 
 2674 56 
 
 i5 86 37 
 
 32 10 
 
 874 5 
 
 894 28 
 
 3295 3 1 
 
 2345 6 7 
 
 75 87 
 
 328 25 
 
 946 25 
 
 4463 79 
 
 4326 43 
 
 62 75 
 
 432 12 
 
 973 22 
 
 5324 28 
 
 5437 26 
 
 48 12 
 
 5i9 75 
 
 5 6 4 33 
 
 6543 20 
 
 6325 4i 
 
 23 15 
 
 438 29 
 
 283 12 
 
 7035 28 
 
 7396 27 
 
 18 ii 
 
 533 25 
 
 597 3i 
 
 8546 37 
 
 8325 34 
 
 5 25 
 
 453 62 
 
 296 54 
 
 9 6 34 25 
 
 9274 S 2 
 
 78 26 
 
 374 S 2 
 
 458 73 
 
 8346 52 
 
 3465 23 
 
 39 44 
 
 25 37 
 
 605 42 
 
 6275 35 
 
 4289 67 
 
 72 5 1 
 
 862 75 
 
 486 54 
 
 4235 34 
 
 5365 72 
 
 33 4i 
 
 953 25 
 
 193 12 
 
 3271 05 
 
 6582 39 
 
 58 75 
 
 846 62 
 
 586 32 
 
 6137 94 
 
 1205 32 
 
 29 31 
 
 553 12 
 
 101 53 
 
 6283 59 
 
 4396 84 
 
 54 62 
 
 228 51 
 
 253 72 
 
 4346 3 2 
 
 5724 33 
 
 27 3i 
 
 312 52 
 
 X S4 53 
 
 7294 5 8 
 
 1065 43 
 
 29 50 
 
 455 63 
 
 276 32 
 
 6275 63 
 
 4953 2 7 
 
 68 71 
 
 729 3i 
 
 586 34 
 
 3284 32 
 
 2586 54 
 
 97 53 
 
 426 76 
 
 235 20 
 
 1635 34 
 
 4234 62 
 
 82 43 
 
 623 25 
 
 463 52 
 
 2586 89 
 
 i73 6 44 
 
 64 25 
 
 321 35 
 
 958 76 
 
 7434 26 
 
 539 s 29 
 
 18 12 
 
 238 17 
 
 386 29 
 
 5869 73 
 
 1234 5 6 
 
 J 9 5 
 
 125 51 
 
 3*5 46 
 
 3276 42 
 
 7891 01 
 
 62 25 
 
 436 25 
 
 434 57 
 
 1635 38 
 
 1234 16 
 
 6 4 37 
 
 536 63 
 
 372 46 
 
 59*3 84 
 
 6843 75 
 
 53 63 
 
 257 47 
 
 657 32 
 
 6284 35 
 
 7616 24
 
 STJBTEAOTION. 
 
 33. Subtraction is taking one number from another. 
 ^The Subtrahend is the number to be subtracted. 
 
 The Minuend is the number from which the sub- 
 traction is. made. 
 
 The Difference or Remainder is the number found 
 by subtraction. Thus, when it is said, 5 taken from 12 
 leaves 7, 12 is the minuend; 5 the subtrahend; and 7 the 
 difference or remainder. 
 
 NOTES. i. The term subtraction, is from the Latin subtraho, to 
 draw from under, or take away. 
 
 The term subtrahend, from the same root, signifies that which is to 
 be subtracted. 
 
 Minuend, from the Latin minuo, to diminisJi, signifies that which 
 is to be diminished ; the termination nd in each caso having the force 
 of to be. 
 
 2. Subtraction is the opposite of Addition. One unites, the other 
 separates numbers. 
 
 3 When both numbers are the same denomination, the operation 
 is called Simple Subtraction. 
 
 34. The Sign of Subtraction is a short horizontal 
 line called minus ( }, placed before the number to be 
 subtracted. Thus, 64 means that 4 is to be taken from 
 6, and is read, " 6 minus 4." 
 
 NOTE. The term minus, Latin, signifies less, or diminished by. 
 Head the following expressions : 
 
 i. 21 3 = 2+ 3 + 13- 2. 31 10= 6+ 4+n. 
 3. 45 15 = 8 + 12 + 10. 4. 10030 = 55 + 10+ 5. 
 
 33. What is subtraction ? What is the number to be subtracted called ? The 
 number from which the subtraction is made? The result? Note. Meaning of 
 term Subtraction ? Subtrahend ? Minuend ? Difference between Subtraction 
 and Addition ? When the Nos. are the same denomination, what ia the operation 
 called 1 34. Sign of Subtraction ? How read 6 -4 ? Meaning of the term minus ;
 
 SUBTRACTION". 
 
 SUBTRACTION TABLE. 
 
 I from 
 
 2 from 
 
 3 from 
 
 4 from 
 
 5 from 
 
 i leaves o 
 
 2 leaves o 
 
 3 leaves o 
 
 4 leaves o 
 
 5 leaves o 
 
 2 I 
 
 3 " i 
 
 4 " i 
 
 5 " I 
 
 6 i 
 
 3 " 2 
 
 4 " 2 
 
 5 " 2 
 
 6 2 
 
 7 2 
 
 4 " 3 
 
 5 3 
 
 6 3 
 
 7 " 3 
 
 8 " 3 
 
 5 " 4 
 
 6 " 4 
 
 7 " 4 
 
 8 4 
 
 9 " 4 
 
 6 " 5 
 
 7 " 5 
 
 8 5 
 
 9 " 5 
 
 10 5 
 
 7 " 6 
 
 8 " 6 
 
 9 " 6 
 
 10 " 6 
 
 ii 6 
 
 8 " 7 
 
 9 " 7 
 
 10 " 7 
 
 ii 7 
 
 12 " 7 
 
 9 " 8 
 
 10 8 
 
 ii 8 
 
 12 8 
 
 13 " 8 
 
 10 " 9 
 
 ii " 9 
 
 12 9 
 
 13 9 
 
 14 " 9 
 
 II " 10 
 
 12 " 10 
 
 13 10 
 
 14 10 
 
 15 " 10 
 
 6 from 
 
 7 from 
 
 8 from 
 
 9 from 
 
 10 from 
 
 6 leaves o 
 
 7 leaves o 
 
 8 leaves o 
 
 9 leaves o 
 
 i o leaves o 
 
 7 " i 
 
 8 " I 
 
 9 i 
 
 10 " I 
 
 ii " i 
 
 8 " 2 
 
 9 " 2 
 
 10 " 2 
 
 II " 2 
 
 12 " 2 
 
 9 " 3 
 
 10 3 
 
 ii 3 
 
 12 " 3 
 
 i3 " 3 
 
 10 " 4 
 
 ii " 4 
 
 12 " 4 
 
 13 " 4 
 
 14 '* 4 
 
 ii 5 
 
 12 " 5 
 
 13 " 5 
 
 14 " 5 
 
 15 5 
 
 12 " 6 
 
 13 " 6 
 
 14 " 6 
 
 15 " 6 
 
 16 6 
 
 13 " 7 
 
 14 " 7 
 
 15 " 7 
 
 16 7 
 
 17 " 7 
 
 14 " 8 
 
 15 " 8 
 
 16 " 8 
 
 17 " 8 
 
 18 8 
 
 15 " 9 
 
 16 " 9 
 
 17 9 
 
 18 " 9 
 
 19 9 
 
 16 10 
 
 17 10 
 
 18 " 10 
 
 19 " 10 
 
 20 " 10 
 
 It will aid the pupil in learning the Subtraction Table to 
 observe that it is the reverse of Addition. Tims, he has learned 
 that 3 and 2 are 5. Reversing this, he will see that 2 from 5 leaves 
 3, etc. Exercises combining the two tables will be found useful, as a 
 review. 
 
 CASE I. 
 
 35. To find the Difference, when each figure of the Subtra- 
 hend is less than the corresponding figure of the Minuend. 
 
 Ex. i. From 568 dollars, take 365 dollars. 
 
 ANALYSIS Let the numbers be set down, the less OPBBATIOK. 
 tinder the greater, as in the margin. Beginning at $568 Mln. 
 the right hand, we proceed thus : 5 units from 8 units 365 g nl)t> 
 
 leave 3 units; set the 3 in units' place under the 
 
 figure subtracted, because it is units. Next, 6 tens $203 Rem. 
 
 from 6 tens leave o tens. Set the o in tens' place 
 
 under the figure subtracted, because there are no tens. Finally, 3
 
 SUBTRACTION. 33 
 
 hundreds from 5 hundreds leave 2 hundreds. Set the 2 in hundreds' 
 place, because it is hundreds. The remainder is $203. All similar 
 examples are solved in like manner. 
 
 By inspecting this illustration, the learner will discover 
 the following principle : 
 
 Units of the same order are subtracted one from 
 another, and the remainder is placed under the figure suo- 
 tr acted. (Arts. 9, 28.) 
 
 NOTES. 1. The less number is placed under the greater, with 
 units under units, etc., for the sake of convenience and rapidity 
 in subtracting. 
 
 2. Units of the same order are subtracted from each other, for 
 the same reason that they are added to each other. (Art. 28, n.) 
 
 3. We subtract the figures of the subtrahend separately, because it 
 is easier to subtract one figure at a time than several. 
 
 4. The remainder is set under the figure subtracted, because it is 
 the same order as that figure. 
 
 From 
 
 () 
 
 648 
 
 (30 (4-) (50 (60 
 726 8652 7230 9621 
 
 Take 
 
 415 
 
 516 3440 412 
 
 o 8510 
 
 
 (7-) 
 
 (8.) 
 
 (90 (10.). (ii.) 
 
 From 
 
 534 
 
 6546 
 
 7852 8462 9991 
 
 Take 
 
 IIO2 
 
 3 2I 4 
 
 5220 413 8880 
 
 
 (12.) 
 
 (130 
 
 (MO 
 
 (isO 
 
 From 
 
 $6948 
 
 43iS ft- 
 
 $5346 
 
 9657 yds. 
 
 Take 
 
 $2416 
 
 3212 ft. 
 
 $3106 
 
 5021 yds. 
 
 
 (16.) 
 
 (170 
 
 (18.) 
 
 (190 
 
 From 
 
 645921 
 
 8256072 
 
 72567803 
 
 965235804 
 
 Take 
 
 435 010 
 
 6i35 5 2 
 
 42103201 
 
 604132402 
 
 20. What is the difference between 3275 and 2132? 
 
 21. "What is the difference between 5384 and 3264? 
 
 22. A father gave his son $5750 and his daughter 42505 
 how much more did he give his son than his daughter?
 
 34 SUBTRACTION. 
 
 CASE II. 
 
 36. To find the Difference, when a figure in the Subtra- 
 hend is greater than the corresponding figure of the Minuend. 
 
 Ex. i. What is the difference between 8074 and 4869 ? 
 
 ist ANALYSIS. Let the numbers be set down as in ist METHOD. 
 the margin. Beginning at the right, we proceed 8074 Min. 
 thus: 9 units cannot be taken from 4 units; we A 860 Sub 
 therefore take i ten from the 7 tens in the upper 
 number, and add it to 4 units, making 14. Sub- -120$ Rem. 
 tracting 9 units from 14 units leaves 5 units, which 
 we place under the figure subtracted. Since we have taken i ten 
 from 7 tens, there are but 6 tens left, and 6 tens from 6 tens leaven 
 
 tens. Next, 8 hundreds cannot be taken from o hundreds; hence, 
 we take i thousand from the 8 thousand, and adding it to the o, 
 we have 10 hundreds. Taking 8 hundreds from 10 hundreds leaves 
 2 hundreds. Finally, 4 thousand from 7 thousand (8 i) leave 3 
 thousand. The remainder is 3205. 
 
 2d ANALYSIS. Adding 10 to 4, the upper figure, zd METHOD. 
 
 makes 14, and 9 from 14 leaves 5. Now, to balance 8074 Min. 
 
 the ten added to the upper figure, we add i to 4860 Sub 
 the next higher order in the lower number. Adding 
 
 1 to 6 makes 7, and 7 from 7 leaves o. Next, since -j 2 o"; Rem 
 we cannot take 8 from o, we add 10 to the o, and 8 
 
 from 10 leaves 2. Finally, adding i to 4 makes 5, and 5 from 8 
 leaves 3. The remainder is 3205, the same as before. 
 
 NOTE. The chief difference between the two methods is this : In 
 the first, we subtract i from the next figure in the upper number ; in 
 the second, we add i to the next figure in the lower number. The 
 former is perhaps the more philosophical ; but the latter is more 
 convenient, and therefore generally practiced. 
 
 By inspecting this illustration, it will be seen, 
 
 If a figure in the lower number is larger than that above 
 
 it, we add 10 to the upper figure, then subtract, and add i 
 
 to the next figure in the lower number. 
 
 Hem Instead of adding 10 to the upper figure, many prefer to 
 take the lower figure directly from 10, and to the remainder add the 
 tipper figure. Thus, 9 from 10 leaves i, and 4 make 5, etc. 
 
 37. Adding 10 to the upper figure is called borrowing ; 
 and adding i to the next figure in the lower number 
 pays it.
 
 SUBTRACTION. 35 
 
 NOTES. i. The first method of borrowing depends upon the obvi- 
 ous principle that the value of a number is not altered by transfer- 
 ring a unit from a higher order to the next lower. 
 
 2. The reason that the second method of borrowing does not affect 
 the difference between the two numbers, is because they are equally 
 increased ; and when two numbers are equally increased, their dif- 
 ference is not altered. 
 
 3. The reasor for borrowing 10, instead of 5, 8, 12, or any other 
 number, is because ten of a lower order are equal to one of the nest 
 higher. If numbers increased by the scale of 5, we should add 5 to 
 the upper figure ; if by the scale of 8, we should add 8, etc. (Art. 9.) 
 
 4. We begin to subtract at the right, because when we borrow we 
 must pay before subtracting the next figure. 
 
 (2.) (3-) (4-) (5-) (6.) 
 
 From 784 842 6704 8042 9147 
 Take 438 695 3598 5439 8638 
 
 38. The preceding principles may be summed up in the 
 following 
 
 GENERAL RULE. 
 
 I. Place the less number under the greater, units under 
 units, etc. 
 
 II. Begin at the right, and subtract each figure in the 
 lower number from the one above it, setting the remainder 
 under the figure subtracted. (Art. 35.) 
 
 III. If a figure in the lower number is larger than the 
 one above it, add 10 to the upper figure ; then subtract, and 
 add i to the next figure in the lower number. (Art. 36.) 
 
 39. PROOF. Add the remainder to the subtrahend ; if 
 the sum is equal to the minuend, the work is right. 
 
 38. How write numbers for subtraction ? How proceed when a figure in the 
 lower number is greater than the one above it? 35. Note. Why write the lesa 
 number under the greater, etc. ? Why subtract the figures separately ? Why 
 set the remainder under the figure subtracted ? 37. What is meant by borrow- 
 ing 1 What is meant by paving or carrying ? Note. Upon what principle does 
 the first method of borrowing depend ? Why does not the second method of 
 borrowing affect the difference between the two numbers? Why borrow 10 
 Instead of 5, 8, 12, etc.? Why begin to subtract at the right hand? 39. How 
 prove subtraction ? Note. Upon what does this proof depend?
 
 30 SUBTRACTION. 
 
 NOTES. i. This proof depends upon the Axiom, that the wJtole it 
 equal to the sum of all its parts. 
 
 2. As soon as the Class understand the rule, they should learn to 
 omit all words but the results. Thus, ; n Ex. 2, instead of saying 9 
 from 4 you can't, 9 from 14 leaves 5, etc., sayjice, naught, three, etc. 
 
 EXAMPLES. 
 i. Find the difference between 84065 and 30428. 
 
 (2.) (3-) (4-) (50 
 
 From 824 rods, 4523 pounds, 6841 years, 735 acres i 
 Take 519 rods. 445^ pounds, 3062 years. 5^3* acre8k 
 
 (6.) 
 
 (7-) 
 
 (8.) 
 
 (9-) 
 
 From 23941 
 
 46083 
 
 52300 
 
 483672 
 
 Take 12367 
 
 23724 
 
 25121 
 
 216030 
 
 (10.) 
 
 (II.) 
 
 (12.) 
 
 (130 
 
 From 638024 
 
 7423614 
 
 8605240 
 
 9042849 
 
 Take 403015 
 
 2414605 
 
 3062431 
 
 6304120 
 
 14. From 85269 pounds, take 33280 pounds. 
 
 15. From 412685 tons, take 103068 tons. 
 
 ' 1 6. From 840005 acres, take 630651 acres. 
 
 17. What is the difference between 95301 and 60358? 
 
 18. What is the difference between 1675236 and 439243 ? 
 
 19. Subtract 2036573 from 5670378. 
 
 20. Subtract 35000384 from 68230460. 
 
 21. Subtract 250600325 from 600230021. 
 
 22. A man bought a house and lot for 636250, and 
 paid $17260 down : how much does he still owe ? 
 
 23. A man bought a farm for $19200, and sold it for 
 $17285 : what was his loss? 
 
 24. A merchant bought a cargo of tea for $1265235, and 
 Bold it for $1680261 what was his gain? 
 
 25. A's income is 645275, and B's $845280: what is 
 the difference in their incomes ?
 
 SUBTRACTION. 3? 
 
 26. A bankrupt's assets are $569257, and his debts 
 $849236 : how much will his creditors lose ? 
 
 27. Subtract 9999999 from iiiiino. 
 
 28. Subtract 666666666 from 7000000000. 
 
 29. Subtract 8888888888 from 10000000000. 
 
 30. Take 200350043010 from 490103060756. 
 
 31. Take 53100060573604 from 80130645002120. 
 
 32. Take 675000000364906 from 901638000241036. 
 
 33. From two millions and five, take ten thousand. 
 
 34. From one million, take forty-five hundred. 
 
 35. From sixty-five thousand and sixty-five, take five 
 hundred and one. 
 
 36. From one billion and one thousand, take one million. 
 
 37. Our national independence was declared in 1776: 
 how many years since ? 
 
 38. Our forefathers landed at Plymouth in 1620: how 
 many years since ? 
 
 39. A father having 3265 acres of land, gave his son 
 1030 acres: how many acres has he left? 
 
 40. The Earth is 95300000* miles from the sun, and 
 Venus 68770000 miles: required the difference. 
 
 41. Washington died in 1799, at the age of 67 years: in 
 what year was he born ? 
 
 42. Dr. Franklin was born in 1706, and died in 1790: 
 at what age did he die ? 
 
 43. Sir Isaac Newton died in 1727, at the age of 85 
 years : in what year was he born ? 
 
 44. Jupiter is 496000000 miles from the sun, and 
 Saturn 909000000 miles : what is the difference ? 
 
 45. In 1865, the sales of A. T. Stewart & Co., by official 
 returns, were $39391688; those of H. B. Claflin & Co., 
 $42506715 : what was the difference in their sales? 
 
 46. The population of the United States in 1860 was 
 31445080; in 1850 it was 23191876: what was the in crease? 
 
 * Kiddle's Astronomy. 
 
 409537
 
 SUBTRACTION. 
 
 QUESTIONS FOR REVIEW. 
 
 1. A man's gross income was $1565 a month for two 
 successive months, and his outgoes for the same time 
 $965 : what was his net profit ? 
 
 2. A man paid $2635 for his farm, and $758 for stock; 
 he then sold them for $4500 : what was his gain ? 
 
 3. A flour dealer has 3560 barrels of flour; after selling 
 1380 barrels to one customer, and 985 to another, how 
 many barrels will he have left ? 
 
 4. If I deposit in bank $6530, and give three checks of 
 $733 each, how much shall I have left ? 
 
 5. What is the difference between 3658 + 256 + 4236 and 
 2430 + 1249? 
 
 6. What is the difference between 6035+560 + 75 and 
 5003 + 360 + 28? 
 
 7. What is the difference between 891+306+5007 and 
 40 + 601 + 1703 + 89? 
 
 8. What is the difference between 900130 + 23040 and 
 19004 + 100607 ? 
 
 9. A man having $16250, gained $3245 by specula- 
 tion, and spent $5203 in traveling: how much had he 
 left? 
 
 10. A farmer bought 3 horses, for which he gave $275, 
 $320, and $418 respectively, and paid $50 down: how 
 much does he still owe for them ? 
 
 11. A young man received three legacies of $3263, 
 $5490, and $7205 respectively; he lost $4795 minus $1360 
 by gambling : how much was he then worth ? 
 
 12. What is the difference between 6286 + 850 and 
 6286-850? 
 
 13. What is the difference between 11325 2361 and 
 8030-3500?- 
 
 14. What number added to 103256 will make 215378? 
 
 15. What number added to 573020 will make 700700?
 
 SUBTBACTIOST. 39 
 
 1 6. What number subtracted from 230375 will leave 
 121487 ? 
 
 17. What number subtracted from 317250 will leave 
 190300? 
 
 1 8. If the greater of two numbers is 59253, and the 
 difference is 21231, what is the less number? 
 
 19. If the difference between two numbers is 1363, and 
 the greater is 45261, what is the less number? 
 
 20. The sum of two numbers is 63270, and one of them 
 is 29385 : what is the other? 
 
 21. What number increased by 2343 131, will become 
 3265291 ? 
 
 22. What number increased by 3520+1060, will become 
 
 6539-279? 
 
 23. What number subtracted from 5009, will become 
 2340 + 471? 
 
 24. Agreed to pay a carpenter $5260 for building a 
 house; $3520 for the masonry, and $1950 for painting: 
 how much shall I owe him after paying $6000 ? 
 
 25. If a man earns $150 a month, and it costs him $63 
 a month to support his family, how much will he 
 accumulate in 6 months ? 
 
 26. A's income-tax is $1165, B's is $163 less than A's; 
 and C's is as much as A and B's together, minus $365 : 
 what is C's tax ? 
 
 27. A is worth $15230, B is worth $1260 less than A; 
 and C is worth as much as both, wanting $1760: what is 
 B worth, and what C ? 
 
 28. The sum of 4 numbers is 45260; the first is 21000, 
 the second 8200 less than the first, the third 7013 less 
 than the second : what is the fourth? 
 
 29. A sailor boastingly said; If I could save $26.3 
 more, I should have $1000: how much had he ? 
 
 30. The difference between A and B's estates is 61525 ; 
 B, who has the least, is worth $17250: what is A worth ?
 
 MULTIPLICATION. 
 
 40. Multiplication is finding the amount of a 
 
 number taken or added to itself, a given number of times. 
 
 The Multiplicand is the number to be multiplied. 
 
 The Multiplier is the number by which we mul- 
 tiply. 
 
 The Product is the number found by multiplication. 
 Thus, when it is said that 3 times 6 are 18, 6 is the mul- 
 tiplicand, 3 the multiplier, and 18 the product. 
 
 41. The multiplier shows how many times the multi- 
 plicand is to be taken. Thus, 
 
 Multiplying by i is taking the multiplicand once ; 
 Multiplying by 2 is taking the multiplicand twice ; and 
 Multiplying by any whole number is taking the multi- 
 plicand as many times as there are units in the multiplier. 
 
 NOTES. i. The term multiplication, from the Latin 
 multus, many, and plico, to fold, primarily signifies to increase by 
 regular accessions. 
 
 2. Multiplicand, from the same root, signifies that which is to be mid- 
 tiplied ; the termination nd, having the force of to be. (Art, 33. n.) 
 
 42. The multiplier and multiplicand are also called 
 Factors ; because they make or produce the product' 
 Thus, 2 and 7 are the factors of 14. 
 
 NOTES. i. The term factor, is from the Latin facio, to produce. 
 2. When the multiplicand contains only one denomination, the 
 operation is called Simple Multiplication. 
 
 40. What is multiplication? What is the number multiplied called? The 
 number to multiply by? The result? When it is said, 3 times 4 are 12, which is 
 the multiplicand? The multiplier? The product? 41. What does the multi- 
 plier show ? What is it to multiply by i ? By 2 ? 42. What else aro (he nntnbers 
 to be multiplied together called? Why? Not'. Meaning of the term factor? 
 What is the operation called when the multiplicand contains only one denomi- 
 nation ? 43. The sign of multiplication.
 
 MULTIPLICATION. 
 
 43. The Sign of Multiplication is an oblique 
 cross ( x ), placed between the factors. Thus, 7x5 means 
 that 7 and 5 are to be multiplied together, and is read 
 " 7 times 5," " 7 multiplied by 5," or " 7 into 5." 
 
 44. Numbers subject to the same operation are placed 
 within & parenthesis ( ), or under a line called a vincuhim 
 
 ( ). Thus (4 + 5) x 3, or 4 + 5 x 3, shows that the sum 
 
 of 4 and 5 is to be multiplied by 3. 
 
 MULTIPLICATION TABLE. 
 
 2 times 
 
 3 times 
 
 4 times 
 
 5 times 
 
 6 times 
 
 7 times 
 
 i are 2 
 
 i are 3 
 
 i are 4 
 
 i are 5 
 
 i are 6 
 
 i are 7 
 
 2 " 4 
 
 2 " 6 
 
 2 " 8 
 
 2 " 10 
 
 2 " 12 
 
 2 " 14 
 
 3 " 6 
 
 3 " 9 
 
 3 12 
 
 3 " 15 
 
 3 " 18 
 
 3 27 
 
 4 8 
 
 4 " 12 
 
 4 " 16 
 
 4 " 20 
 
 4 " 24 
 
 4 " 28 
 
 5 " 10 
 
 5 " 15 
 
 5 " 20 
 
 5 " 25 
 
 5 " 30 
 
 5 " 35 
 
 6 " 12 
 
 6 18 
 
 6 " 24 
 
 6 30 
 
 6 36 
 
 6 42 
 
 7 " i4 
 
 7 " 21 
 
 7 " 28 
 
 7 " 35 
 
 7 42 
 
 7 " 49 
 
 8 16 
 
 8 " 24 
 
 8 " 32 
 
 8 " 40 
 
 8 48 
 
 8 " 56 
 
 9 " 18 
 
 9 " 2 7 
 
 9 " 36 
 
 9 " 45 
 
 9 " 54 
 
 9 " 63 
 
 10 " 20 
 
 10 3 
 
 10 " 40 
 
 10 " 50 
 
 10 60 
 
 10 " 70 
 
 II " 22 
 
 ii " 33 
 
 ii " 44 
 
 ii 55 
 
 ii 66 
 
 n 77 
 
 12 " 24 
 
 12 36 
 
 12 48 
 
 12 " 60 12 " 72 
 
 12 " 84 
 
 8 times 
 
 9 times 
 
 10 times 
 
 ii times 
 
 12 times 
 
 i are 8 
 
 i are 9 
 
 i are 10 
 
 i are n 
 
 i are 12 
 
 2 16 
 
 2 " l8 
 
 2 " 20 
 
 2 " 22 
 
 2 2 4 
 
 3 * 24 
 
 3 " 27 
 
 3 " 30 
 
 3 " 33 
 
 3 " 36 
 
 4 " 32 
 
 4 " 36 
 
 4 " 40 
 
 4 " 44 
 
 4 " 48 
 
 5 " 40 
 
 5 " 45 
 
 5 " 50 
 
 5 " 55 
 
 5 " 60 
 
 6 " 48 
 
 6 " 54 
 
 6 " 60 
 
 6 66 
 
 6 " 72 
 
 7 " 56 
 
 7 63 
 
 7 " 7o 
 
 7 " 77 
 
 7 " 84 
 
 8 64 
 
 8 " 72 
 
 8 80 
 
 8 " 88 
 
 8 96 
 
 9 72 
 
 9 " 81 
 
 9 90 
 
 9 99 
 
 9 108 
 
 10 " 80 
 
 10 " 90 
 
 IO " IOO 
 
 10 " IIO 
 
 10 " 120 
 
 ii " 88 
 
 ii " 99 
 
 II "lIO 
 
 II " 121 
 
 II " 132 
 
 12 " 96 
 
 12 108 
 
 12 " 120 
 
 12 " 132 
 
 12 " 144 
 
 Tlie pupil will observe that the products by 5, terminate in 
 5 and o, alternately. Thus, 5 times i are 5 ; 5 times 2 are 10.
 
 42 MULTIPLICATION. 
 
 The products by 10 consist of the figure multiplied and a ciplwr. 
 Thus, 10 times i are 10 ; ten times 2 are 20, and so on. 
 
 The first nine products by n are formed by repeating the figure 
 multiplied. Thus, n times I are n ; n times 2 are 22, and so on. 
 
 The first figure of the first nine products by 9 increases, and the 
 second decreases regularly by I ; while the sum of the digits in each 
 product is 9. Thus, 9 times 2 are 18, 9 times 3 are 27, and so on. 
 
 45. When the factors are abstract numbers, 8 at & S 
 it is immaterial in what order they are multi- & $ $ a 
 plied. Thus, the product of 3 times 4 is equal gj $ & 
 to 4 times 3. For, taking 4 units 3 times, is 
 
 the same as taking 3 units 4 times ; that is, 4 x 3=3 x 4. 
 
 NOTES. i. It is more convenient and therefore customary, to 
 take the larger of the two given numbers for the multiplicand. 
 Thus, it is better to multiply 5276 by 8, than 8 by 5276. 
 
 2. Multiplication is the same in principle as Addition; and is 
 sometimes said to be a short method of adding a number to itself a 
 given number of times. Thus: 4 stars + 4 stars + 4 stars=3 times 4 
 or 12 stars. 
 
 46. The product is the same name or kind as the multi- 
 plicand. For, repeating a number does not change its 
 nature. Thus, if we repeat dollars, they are still dollars, etc. 
 
 47. TJhb multiplier must be an abstract number, or con- 
 sidered as such for the time being. For, the multiplier 
 shows how many times the multiplicand is to be taken ; and 
 to say that one quantity is taken as many times as another 
 \s heavy is obi urd. 
 
 NOTE. When it is asked what 25 cts. multiplied by 25 cts., or 
 23. 6d. by 2s. 6d., will produce, the questions, if taken literally, 
 are nonsense. For, 2s. 6d. cannot be repeated 2s. 6d. times, nor 25 cts. 
 25 cts. times ; but we can multiply 25 cts. by the number 25, and 
 23. 6d. by 2^. In like manner we can multiply the price of i yard by 
 the number of yards in an article, and the product will be the cost. 
 
 45. When the factors are abstract, docs it make any difference with the pro- 
 duct which is taken for the multiplicand f Note. Which is commonly taken f 
 Why ? What is Multiplication the same as ? 46. What denomination is the pr 
 duct ? Why ? 47. What must the multiplier be ? Why ?
 
 MULTIPLICATION. 43 
 
 CASE I. 
 
 48. To find the Product of two numbers, when the 
 
 Multiplier has but one figure. 
 
 i. What is the cost of 3 horses, at $286 apiece ? 
 
 ANALYSIS. 3 horses will cost 3 times as much as OPEBATION. 
 
 1 horse. Let the numbers be set down as in the $286 Multd. 
 margin. Beginning at the right, we proceed thus : ^ Mult. 
 
 3 times 6 units are 18 units. We set the 8 in units' 
 
 place because it is units, and carry the i ten to the $858 Prod, 
 product of the tens, as in Addition. (Art. 29.) Next, 
 
 3 times 8 tens are 24 tens and i (to carry) make 25 tens, or 2 hun- 
 dred and 5 tens. We set the 5 in tens' place because it is tens, and 
 carry the 2 hundred to the product of hundreds. Finally, 3 times 
 
 2 hundred are 6 hundred, and 2 (to carry) are 8 hundred. We set 
 the 8 in hundreds' place because it is hundreds. Therefore the 3 
 horses cost $858. All similar examples are solved in like manner. 
 
 By inspecting the preceding analysis, the learner will 
 discover the following principle : 
 
 Each figure of the multiplicand is multiplied ly the 
 multiplier, beginning at the right, and the result is set 
 down as in Addition. (Art. 29.) 
 
 NOTES. i. The reason for setting the multiplier under the multi- 
 plicand is simply for convenience in multiplying. 
 
 2. The reasons for beginning to multiply at the right hand, as 
 well as for setting down the units and carrying the tens, are the 
 same as those in Addition. (Art. 29, TO.) 
 
 3. In reciting the following examples, the pupil should care- 
 fully analyze each ; then give the results of the operations re- 
 quired. 
 
 49. Units multiplied by units, it should be observed, 
 produce units ; tens by units, produce tens ; hundreds by 
 units, produce hundreds ; and, universally, 
 
 If the multiplying figure is units, the product will be 
 the same order as the figure multiplied. 
 Again, if the figure multiplied is units, the product will 
 
 49. What do units multiplied by units produce? Tens by units? Hundredi 
 V units ? When the multiplying figure is units, what is the product?
 
 44 MULTIPLICATION. 
 
 be the same order as the multiplying figure; for the 
 product is the same, whichever factor is taken for the 
 multiplier. (Art. 45.) 
 
 (') (3-) (4-) (50 
 
 Multiply 2563 13278 2648203 48033265 
 
 By 4 5 6 
 
 6. "What will 8 carriages cost, at $750 apiece? 
 
 7. What cost 9 village lots, at $1375 a lot? 
 
 8. At $3500 apiece, what will 10 houses cost? 
 
 9. At $865 a hundred, how much will u hundred- 
 weight of opium come to ? 
 
 10. If a steamship sail 358 miles per day, how far will 
 she sail in 1 2 days ? 
 
 11. Multiply 86504 by 5. 12. Multiply 803127 by 7. 
 13. Multiply 440210 by 6. 14. Multiply 920032 by 8. 
 15. Multiply 603050 by 9. 16. Multiply 810305 by 10. 
 17. Multiply 753825 by ii. 18. Multiply 954635 by 12. 
 
 19. What cost 4236 barrels of apples, at $7 a barrel ? 
 
 20. What cost 5167 melons, at n cents apiece? 
 
 21. At 6 shillings apiece, what will 1595 arithmetics 
 cost? 
 
 22. At $12 a barrel, what will be the cost of 1350 bar- 
 rels of flour ? 
 
 23. What will 1735 boxes of soap cost, at $9 a box ? 
 
 24. When peaches are 12 shillings a basket, what will 
 2363 baskets cost ? 
 
 25. If a man travels 8 miles an hour, how far will he 
 travel in 3260 hours ? 
 
 26. A builder sold 10 houses for $4560 apiece: how 
 much did he receive for them all ? 
 
 27. What will it cost to construct ii miles of railroad, 
 at $12250 per mile? 
 
 28. What will be the expense of building 12 ferry-boats, 
 at $23250 apiece ?
 
 MULTIPLICATION. 45 
 
 CASE II. 
 
 50. To find the Product of two numbers, when the 
 Multiplier has two or more figures. 
 
 i. A manufacturer bought 204 bales of cotton, at $176 
 a bale : what did he pay for the cotton ? 
 
 ANALYSIS. Write the numbers as in the mar- $176 Multd. 
 gin. Beginning at the right : 4 times 6 units are 2Q . Mult 
 
 24 units, or 2 tens and 4 units. Set the 4 in units' 
 
 place, and carry the 2 to the next product. 4 74 
 
 times 7 tens are 28 tens and 2 are 30 tens, or 3 352 
 
 hundred and o tens. Set the o in tens' place and 
 
 carry the 3 to the next product. 4 times i hun- $35904 Am. 
 are 4 hun. and 3 are 7 hun. Set the 7 jn hundreds' 
 place . The product by o tens is o ; we therefore omit it. Again, 
 2 hundred times 6 units are 12 hun. units, equal to i thousand and 
 2 hundred. Set the 2 in hundreds' place and carry the I to the 
 next product. 2 hundred times 7 tens are 14 hundreds of tens, equal 
 to 14 thousand, and i are 15 thousand. Set the 5 in thousands' 
 place, and carry the i to the next product, and so on. Adding 
 these results, the sum is the cost. 
 
 REMAKK. The results which arise from multiplying the multipli- 
 cand by the separate figures of the multiplier, are called partial pro- 
 ducts ; because they are parts of the whole product. 
 
 By inspecting this analysis, the learner will discover, 
 
 1. The multiplicand is multiplied by each figure of the 
 multiplier, beginning at the right, and the result set down 
 as in Addition. 
 
 2. The first figure of each partial product is placed 
 under the multiplying figure, and the sum of the partial 
 products is the true answer. 
 
 NOTES. i. When there are ciphers between the significant 
 figures of the multiplier, omit them, and multiply by the next sig- 
 nificant figure. 
 
 2. We multiply by each figure of the multiplier separately, when it 
 exceeds 12, for the obvious reason, that it is not convenient to multi- 
 ply by the whole of a large number at once. 
 
 3. The first figure of each partial product is placed under the 
 multiplying figure; because it is the same order as that figure. 
 
 4. The several partial products are added together, because tha 
 whole product is equal to the sain of all its parts.
 
 46 MULTIPLICATION. 
 
 (') (30 (4-) (50 
 
 Multiply 426 563 1248 2506 
 
 By 24 35 52 304 
 
 51. The preceding principles may be summed up in 
 the following 
 
 GENERAL RULE 
 
 I. Place the multiplier under the multiplicand) units 
 under units, etc. 
 
 II. When the multiplier has but one figure, beginning at 
 the right, multiply each figure of the multiplicand by it, 
 and set down the result as in Addition. 
 
 III. If the multiplier has two or more figures, multiply 
 the multiplicand by each figure of the multiplier separately, 
 and set the first figure of each partial product under the 
 multiplying figure. 
 
 Finally, the sum of the partial products will be the 
 answer required. 
 
 NOTE. The pupil should early learn to abbreviate the several 
 steps in multiplying as in Addition. (Art. 31, n.) 
 
 PROOF. 
 
 52. By Multiplication. Multiply the multiplier by the 
 multiplicand ; if this result agrees with the first, the work 
 is right. 
 
 NOTE. This proof is based upon the principle that the result 
 will be the same, whichever number is taken as the multiplicand. 
 
 53. By excess of 95. Find the excess of 9.*? in each factor 
 separately ; then multiply these excesses together, and reject 
 the gs from the result ; if this excess, agrees with the excess 
 of 93 in the answer, the work is right. 
 
 51. How write numbers for multiplication? Wlien the multiplier has but one 
 figure, how proceed ? When it has two or more ? What is finally done with the 
 partial products ? 4*?. Note. Why write the multiplier under the multiplicand, 
 units under units ? Why begin at the right hand ? 50. What are partial products ? 
 Note. Why multiply by each figure separately ? Why set the first figure of each 
 under the multiplying figure f Why add the several partial products together T 
 52. How prove multiplication by multiplication T Note. Upon what is this proof 
 based ? 53. How prove it by excess of 98 f
 
 MULT IPLICATIOX. 
 
 47 
 
 NOTE. This method of proof, if deemed advisable, may be omitted 
 till review. It is placed here for convenience of reference. Though 
 depending on a peculiar property of numbers, it is easily applied, 
 and is confessedly the most expediitous method of proving multipli- 
 cation yet devised. 
 
 54. To find the Excess of 9s in a number. 
 
 Beginning at the left hand, add the figures together, and as soon 
 as the sum is 9 or more, reject 9 and add the remainder to the next 
 figure, and so on. 
 
 Let it be required to find the excess of 93 in 7548467. 
 
 Adding 7 to 5, the sum is 12. Rejecting 9 from 12, leaves 3 ; and 
 3 added to 4 are 7, and 8 are 15. Rejecting 9 from 15, leaves 6 ; and 
 <> added to 4 are 10. Rejecting 9 from 10, leaves i ; and i added to 
 6 are 7, and 7 are 14. Finally, rejecting 9 from 14 leaves 5, the 
 excess required. 
 
 NOTES. i. It will be observed that the excess of 93 in any two 
 digits is always equal to the sum, or the excess in the sum, of those 
 digits. Thus, in 15 the excess is 6, and 1 + 5=6; so in 51 it is 6, 
 and 5 + i=T>. In 56 the sum is n, the excess 2. 
 
 2. The operation of finding the excess of 93 in a number is called 
 casting out the gs. 
 
 EXAMPLES. 
 
 i. What is the product of 746 multiplied by 475 ? 
 
 OPERATION. Proof by Excess of 9*. 
 
 746 Excess of 93 in multd. is 8 
 475 9 s in mult. " 7 
 
 Proof by Mult. 
 
 475 
 746 
 
 Now 8x756 
 
 
 -222 The excess of 93 in 56 is 2 
 2984 The excess of 93 in prod, is 2 
 
 2850 
 1900 
 3325 
 
 dns. 35435 
 (2.) (30 (4-) 
 
 Multiply 5645 18934 48367 
 By 43 65 75 
 
 Ans. 35435 
 
 (50 
 23I45 6 
 87 
 
 
 
 54. How find the excess of 98 ?
 
 48 MULTIPLICATION. 
 
 (6.) (7.) (8.) (9.) 
 
 Multiply 1421673 2342678 4392460 5230648 
 
 By 234 402 347 526 
 
 10. Mult. 640231 by 205. ii. Mult. 520608 by 675. 
 
 12. Mult. 431220 by 1234. 13. Mult 623075 by 2650. 
 
 14. Mult. 730650 by 2167. 15. Mult. 593287 by 6007. 
 
 16. Mult. 843700 by 3465. 17. Mult. 748643 by 2100. 
 
 18. Mult. 9000401 by 5000 1. 19. Mult. 82030405 by 23456. 
 
 20. How many pounds in 1375 chests of tea, each chest 
 containing 63 pounds ? 
 
 21. What cost 738 carts, at $75 apiece? 
 
 22. At 43 bushels per acre, how many bushels of wheat 
 will 520 acres produce? 
 
 23. At $163 apiece, what will be the cost of 1368 covered 
 buggies ? 
 
 24. If a man travel 215 miles per day, how far can he 
 travel in 365 days ? 
 
 25. There are 5280 feet in a mile: how many feet in 
 256 miles? 
 
 26. What cost 2115 revolvers, at $23 apiece? 
 
 27. Bought 1978 barrels of pickles, at $17 ; how much 
 did they come to ? 
 
 28. If railroad cars are $4735 apiece, what will be the 
 expense of 500? 
 
 29. How far will 2163 spools of thread extend, each 
 containing 25 yards ? 
 
 30. Bought 15265 ambulances, at. $117 apiece: what 
 was the amount of the bill ? 
 
 31. What will 3563 tons of railroad iron cost, at $68 
 per ton ? 
 
 32. How far will a man skate in 6 days, allowing he 
 skates 8 hours a day, and goes ) miles an hour ?
 
 MULTIPLICATION. 4S 
 
 CONTRACTIONS. 
 
 55. A Composite Number is the product of two 
 or more factors, each of which is greater than i. Thus 
 15 = 3x5, and 42 = 2 x 3 x 7, are composite numbers. 
 
 NOTES. i. The product of a number multiplied into Uself is 
 called a power. Thus, 9=3 x 3, is a power. 
 
 2. Composite is from the Latin compono, to place together. 
 
 3. The terms factors &n& parts must not be confounded with each 
 other. The former are multiplied together to produce a number ; 
 rl:e latter are added. Thus, 3 and 5 are the factors of 15 ; but 5 and 
 10, 6 and 9, 7 and 8, etc., are the parts of 15. 
 
 1. What are the factors of 45 ? Ans. 5 and 9. 
 
 2. What are the factors of 24 ? Of 2 7 ? Of 28 ? Of 30 ? 
 
 3. What are the factors of 32? Of 35 ? Of 42? Of 48? 
 
 4. What are the factors of 54 ? Of 63? Of 72? Of 84? 
 
 CASE I. 
 56. To multiply by a Composite Number. 
 
 i. A farmer sold i5boxes of butter, each weighing 20 
 pounds: how many pounds did he sell? 
 
 ANALYSIS. 15 = 5 times 3; hence, 15 boxes 
 
 OPERATION. 
 
 will weigh 5 times as much as 3 boxes. Now, 
 
 if i box weighs 20 pounds, 3 boxes will weigh 
 
 3 times 20, or 60 pounds. Again, if 3 boxes 
 
 weigh 60 pounds, 5 times 3 boxes will weigh 5 
 
 times 60, or 300 pounds. He therefore sold 300 
 
 rounds. In the operation, we first multiply by 5 
 
 the factor 3, and the product thus arising by the 
 
 other factor 5. Hence, the Ans. 300 pounds, 
 
 EULE. Multiply the multiplicand by one of (he facers 
 of the multiplier, then this product by another, and so on, 
 till all the factors have been used. 
 
 The last product will be the answer. 
 
 35. A composite number ? Note. The difference between factors and pane t 
 ?6. How multiply by a composite number ? 
 
 3
 
 50 MULTIPLICATION. 
 
 NOTES. i. This rule is based upon the principle that it is 
 immaterial in what order two factors are multiplied. (Art. 45.) 
 
 The sanifc, illustration may be extended to three or more numbers. 
 For, the product of two of the factors may be considered as one 
 Dumber, and this may be used before or after a third factor, etc. 
 
 2. The process of multiplying three or more factors together, ia 
 called Continued Multiplication; the result, the continued product. 
 
 2. Multiply $568 by 35. Ans. $19880. 
 
 3. Multiply 2604 by 25. 4. Multiply 6052 by 48. 
 5. Multiply 8091 by 63. 6. Multiply 45321 by 72. 
 
 7. In i cubic foot there are 1728 cubic inches: how 
 many cubic inches are there in 84 cubic feet ? 
 
 8. If a ton of copper ore is worth $5268, what is the 
 worth of 56 tons? 
 
 9. What cost 125 houses, at $1580 apiece? 
 
 CASE II. 
 
 57. To multiply by 1O, 1OO, 1OOO, etc. 
 
 10. What will 100 cows cost, at $31 apiece? 
 
 ANALYSIS. Annexing a cipher to a number moves each figure one 
 place to the left ; but moving a figure orie place to the left increases 
 vis value ten times ; therefore annexing a cipher to a number 
 multiplies it by 10. In like manner, annexing two ciphers, multi- 
 plies it by a hundred, etc. (Art. 12.) Therefore, $31 x ioo=$3ioo. 
 Hence, the 
 
 RULE. Annex as many ciphers to the multiplicand as 
 there are ciphers in the multiplier. 
 
 NOTE. The term annex, xrom the Latin ad and necto, to join to, 
 signifies to place after. 
 
 11. What cost 1000 horses, at $356 apiece? 
 
 12. Multiply 40530 by 1000. 
 
 13. Multiply 9850685 by 10000. 
 
 14. Multiply 84050071 by 100000. 
 
 15. Multiply 360753429 by 1000000. 
 
 57. How multiply by 10, 100, 1000. etc. ?
 
 <2^L 
 
 MULTIPLICATION. 51 
 
 CASE III. 
 
 58. To multiply, when there are Ciphers on the right of 
 either or both Factors. 
 
 20. What is the product of 87000 multiplied by 230 ? 
 
 ANALYSIS. The factors of the multiplicand are 
 87 and 1000; the factors of the multiplier are 23 
 and 10. We first multiply the factors consisting 
 of significant figures ; then multiply this product 
 by the other two factors (1000 x 10), or 10000, 
 by annexing 4 ciphers to it. Hence, the Ans. 20010000 
 
 RULE. Multiply the significant figures together ; and 
 to the result annex as many -ciphers as are found on the 
 right of both factors. 
 
 NOTE. This rule is based upon the two preceding cases; for, by 
 supposition, one or both the given numbers are composite ; and one 
 of the factors of this composite number is 10, 100, etc. 
 
 (21.) (22.) (23.) 
 Multiply 2130 64000 83046 
 
 By 700 52 2000 
 
 24. In i barrel of pork there are 200 pounds: how many 
 pounds in 3700 barrels? 
 
 25. What will 2300 head of cattle cost, at $80 per head? 
 
 26. In $i there are 100 cts. : how many cts. in $26000? 
 
 27. The salary of the President is $50000 a year: how 
 much will it amount to in 21 years ? 
 
 28. If a clock ticks 86400 times in i day, how many 
 times will it tick in 7000 days? 
 
 29. If one ship costs $150000, what will 49 cost? 
 
 30. 670103700 x 60030040? 
 
 31. 800021000x80002100? 
 3 2 - 570305000x40000620? 
 
 58. When one or both factors have ciphers on the right? Note, Upon what la 
 this rule based ?
 
 52 MULTIPLICATION. 
 
 33. 467234630X27000000? 
 
 34. 890000000x350741237? 
 
 35. 9400000027 x 28000000 ? 
 
 36. Multiply 39 millions and 200 thousand by 530 
 thousand. 
 
 37. Multiply 102 times 700 thousand and i hundred by 
 60 1 thousand and twenty. 
 
 38. Multiply 74 millions and 21 thousand by 5 millions 
 and 5 thousand. 
 
 39. Multiply 31 millions 31 thousand and 31 by 21 
 thousand and twenty-one. 
 
 40. Multiply 2 billions, 2 millions, 2 thousand and 2 
 hundred by 200 thousand and 2 hundred. 
 
 CASE IV. 
 
 59. To multiply 13, 14, 15, OP I with a Significant Figure 
 annexed. 
 
 41. If one city lot costs $3245, what will 17 lots cost? 
 
 ANALYSIS. 17 lots will cost 17 times as much as 3245 x 17 
 
 I lot. Placing the multiplier on the right, we 22715 
 
 multiply the multiplicand by the 7 unite, set each . 
 
 figure one place to the right of the figure multi- $55165 An. 
 plied, and add the partial product to the multi- 
 plicand. The result is $55165. Hence, the 
 
 EULE. I. Multiply the multiplicand by the unit*? figure 
 of the multiplier, and set each figure of the partial product 
 one place to the right of the figure multiplied. 
 
 II. Add tliis partial product to the multiplicand, and the 
 result will be the true product. 
 
 NOTE. This contraction depends upon the principle that as the 
 tens' figure of the multiplier is i, the multiplicand is the second par- 
 tial product ; hence its first figure must stand in tens' place. 
 
 42. Multiply 1368 by 13. 43. Multiply 2106 by 14. 
 44. Multiply 3065 by 15. 45. Multiply 6742 by 16. 
 46. Multiply 25269 by 18. 47. Multiply 83467 by 19. 
 
 59. How multiply by 13, 14, 15, or i with a significant figure annexed ? Note. 
 Upon what is this contraction based ?
 
 
 PIVISION. 
 
 60. Division is finding how many times one numb** 1 
 is contained in another. 
 
 The Dividend is the number to be divided. 
 
 The Divisor is the number by which we divide. 
 
 The Quotient is the number found by division, and 
 shows how many times the divisor is contained in the 
 dividend. 
 
 The Remainder is a part of the dividend left after 
 division. Thus, when it is said, 5 is contained in 17,3 
 times and 2 over, 17 is the dividend, 5 the divisor, 3 the 
 quotient, and 2 the remainder. 
 
 NOTES. i. The term division, is from the Latin divido, to part, 
 or divide. 
 
 The term dividend, from the same root, signifies that which is to be 
 divided ; the termination nd, having the force of to be. (Art. 33, n.) 
 
 Quotient is from the Latin quoties, signifying how often or how 
 many times. 
 
 2. The remainder is always the same denomination as the divi- 
 dend ; for it is an undivided part of it. 
 
 3. A proper remainder is always less than the divisor. 
 
 61. An obvious way to find how many times the divisor 
 is contained in the dividend, is to subtract the divisor 
 from the dividend continually, till the latter is exhausted, 
 or till the remainder is less than the divisor ; the number 
 of subtractions will be the quotient. Thus, taking any 
 two numbers, as 4 and 12, we have 124=8; 84=4; 
 and 44=0. Here are three subtractions; therefore, 4 i8 
 contained in 12, 3 times. Hence, 
 
 Division is sometimes said to be a sliort method of con- 
 tinued subtraction. 
 
 60. What is division ? The number to be divided called ? To divide by ? Tho 
 result? Tlip r>art left ? Note. Mr-anine- of the term division 1 Dividend? Quo- 
 tient ? What denomination is the remainder ? Why ?
 
 54 
 
 DIVISION. 
 
 But there is a shorter and more direct way of obtaining 
 the quotient. For we know by the multiplication table, 
 that 3 times 4, or 4 taken 3 times, are 1 2 ; hence, 4 is 
 contained in 1 2, 3 times. 
 
 62. Division is the reverse of multiplication. In mul- 
 tiplication loth factors are given, and it is required to find 
 the product ; in division, one factor and the product (which 
 answers to the dividend) are given, and it is required to 
 find the other factor, which answers to the quotient. 
 Hence, division may be said to be finding a quotient which, 
 multiplied into the divisor, will produce the dividend. 
 
 NOTE. When the dividend contains only one denomination, the 
 operation is called Simple Division. 
 
 DIVISION TABLE. 
 
 i is in 
 
 2 is in 
 
 3 is in 
 
 4 is in 5 is in 6 is in 
 
 i, once. 
 
 2, once. 
 
 3, once. 
 
 4, once. 5, once. 6, once. 
 
 2, 2 
 
 4, 2 
 
 6, 2 
 
 8, 2 10, 2 12, 2 
 
 3, 3 
 
 6, 3 
 
 9, 3 
 
 12, 3 15, 3 18, 3 
 
 4, 4 
 
 8, 4 
 
 12, 4 
 
 16, 4 20, 4 24, 4 
 
 5, 5 
 
 io, 5 
 
 15, 5 
 
 20, 5 25, 5 30, 5 
 
 6, 6 
 
 12, 6 
 
 18, 6 
 
 24, 630, 636, 6 
 
 7, 1 
 
 14, 7 
 
 21, 7 
 
 28, 7l35, 742, 7 
 
 8, 8 
 
 16, 8 
 
 24, 8 
 
 32, 840, 848, 8 
 
 9, 9 
 
 18, 9 
 
 27, 9 
 
 3 6 , 9 45, 9 54, 9 
 
 10, 10 
 
 20, 10 
 
 30, 10 
 
 40, 1050, 1060, 10 
 
 7 i* in 
 
 8 is in 
 
 9 is in 
 
 10 is in 
 
 ii is in 
 
 12 is in 
 
 7, once. 
 
 8, once. 
 
 9, once. 
 
 10, once. 
 
 1 1, once. 
 
 12 once. 
 
 14, 2 
 
 16, 2 
 
 18, 2 
 
 20, 2 
 
 22, 2 
 
 24, 2 
 
 21, 3 
 
 24, 3 
 
 27, 3 
 
 3, 3 
 
 33, 3 
 
 36, 3 
 
 28, 4 
 
 32, 4 
 
 36, 4 
 
 40, 4 
 
 44, 4 
 
 48, 4 
 
 35, 5 
 
 40, 5 
 
 45, 5 
 
 5, 5 
 
 55, 5 
 
 60, 5 
 
 42, 6 
 
 48, 6 
 
 54, 6 
 
 60, 61 66, 6 
 
 72, 6 
 
 49, 7 
 
 56, 7 
 
 63, 7 
 
 7, 7 
 
 77, ' 7 
 
 84, 7 
 
 56, 8 
 
 64, 8 
 
 72, 8 
 
 80, 8 
 
 88, 8 
 
 96, 8 
 
 63> 9 
 
 72, 9 
 
 81, 9 
 
 90, 9 
 
 99, 9 
 
 1 08, 9 
 
 70, 1080, 10 
 
 90, 10 
 
 IOO, IO 
 
 IIO, 10 
 
 1 2O, IO 
 
 61. What is an obvious way to find how many times the divisor IB contained 
 In the dividend ? What is division sometimes called t
 
 DIVISION. 05 
 
 OBJECTS OF DIVISION. 
 63. The object or office of Division is twofold: 
 ist, To find how many times one number is contained in 
 another. 
 
 zd, To divide a number into equal parts. 
 
 63, <( To find how many times one number is contained 
 
 in another. 
 
 i. A man has 15 dollars to lay out in books, which are 
 3 dollars apiece : how many can he buy ? 
 
 ANALYSIS. In this problem the object is to find Jww many times 
 3 dols. are contained in 15 dols. 
 
 Let the 15 dols. be represented by 15 counters, or unit marks. 
 Separating these into groups of 3 each, there are 5 groups. There- 
 fore, he can buy 5 books. 
 
 63, & To divide a number into equal parts. 
 
 2. If a man divides 15 dollars equally among 3 persons, 
 how many dollars will each receive ? 
 
 ANALYSIS. The object here is to divide 15 dols. into 3 equal parts. 
 
 Let the 15 dols. be represented by 15 counters. If we form 3 groups, 
 first putting I counter in each, then another, till the counters are ex- 
 hausted, each group will have 5 counters. Therefore, each person 
 will receive 5 dollars. 
 
 ^oo^^lo^^^ol^^-^^^ 
 
 REMARK. The preceding are representative examples of the two 
 classes of problems to which Division is applied. In the first, the 
 divisor and dividend are the same denomination, and the quotient is 
 times, or an abstract number. 
 
 In the second, the divisor and dividend are different denominations, 
 and the quotient is the same denomination as the dividend. Hence, 
 
 64. When the divisor and dividend are the same denom- 
 ination, the quotient is always an abstract number. 
 
 62. Of what is division the reverse ? What is given in multiplication ? What 
 required? What is given in division? What required? Note. When the divi- 
 dend contains but one denomination, what is the operation called ? 63. What is 
 the ohject or office of division ? 63, a. What is the object in the first problem f 
 63, b. What in the second ? Bern. What is said of first two problems I
 
 DIVISION. 
 
 When the divisor and dividend are different denomina- 
 tions, the quotient is always the same denomination as the 
 dividend. 
 
 NOTES. i. The process of separating a number into equal parts, as 
 required in the second class of problems, gave rise to the name 
 " Division." It is also the origin of Fractions. (Art. 134.) 
 
 2. The mode of reasoning in the solution of these two classes of 
 examples is somewhat different ; but the practical operation is the 
 same, viz. : to find 7iow many times one number is contained in 
 another, which accords with the definition of Division. 
 
 65. When a number or thing is divided into two equal 
 parts, the parts are called halves; into three, the parts are 
 called thirds ; into four, they are called fourths ; etc. 
 
 The number of parts is indicated by their name. 
 
 66. A number is divided into two, three, four, five, etc., 
 equal parts by dividing it by 2, 3, 4, 5, etc., respectively. 
 
 3. What is a half of 12 ? A third of 15 ? A fourth of 
 20 ? A fifth of 35 ? A seventh of 28 ? 
 
 4. What is a sixth of 42 ? A seventh of 56 ? A ninth 
 of 63 ? An eighth of 72 ? " A twelfth of 108 ? 
 
 67. The Siffn of Division is a short horizontal 
 line between two dote (-T-), placed before the divisor. 
 Thus, the expression 28-7-7, shows that 28 is to be divided 
 by 7, and is read, "28 divided by 7." 
 
 68. Division is also denoted by writing the divisor 
 under the dividend, with a short line between them. 
 Thus, - 2 ,- is equivalent to 28-5-7. I* is rea ^> " 2 & divided 
 by 7," or "28 sevenths." 
 
 69. Division is commonly distinguished as Short Divis~ 
 ion and Long Division. 
 
 64. When the divisor and dividend are the same denomination, what is the 
 quotient ? When different denominations, what ? 65. When a number is divided 
 into two equal parts, what are the parts called ? Into three ? Four ? 66. How 
 divide a number into 2, 3, 4, etc., equal parts? 67. What is the sitrn of division? 
 What does the expression 35-1-7 show? 68. How else is division denoted?
 
 SHOET DIVISION. 
 
 70. Short Division is the method of dividing, 
 when the results of the several steps are carried in the 
 mind, and the quotient only is set down. 
 
 71. To divide by Sliort Division. 
 
 Ex. i. If apples are $3 a barrel, how many barrels can 
 you buy for $693 ? 
 
 ANALYSIS. Since $3 will buy i barrel, $693 OPERATIC*. 
 will buy as many barrels as $3 are contained ^$>6ot 
 times in $693. Let the numbers be set down as 
 in the margin. Beginning at the left, we proceed ** uot - 2 3* Kt " 
 thus : 3 is contained in 6 hundred, 2 hundred times. 
 Set the 2 in hundreds' place, under the figure divided, became it is 
 hundreds. Next, 3 is contained in 9 tens, 3 tens times, feet the 3 
 in tens' place under the figure divided. Finally, 3 is contained in 
 3 units, i time. Set the i in units' place. Ans. 231 barrels. 
 
 REM. This problem belongs to the ist class, the object being to 
 find how many times one number is contained in another. (Art. 63, a.) 
 
 Solve the following examples in the same manner : 
 
 (2.) (3-) (4-) (5-) 
 
 2)4468 3)3696 4)4848 5)5555 
 
 6. A man having 27543 pounds of grapes, packed them 
 for market in boxes containing 5 pounds each : how many 
 boxes did he fill, and how many pounds over? 
 
 ANALYSIS. He used as many boxes as there OPERATION. 
 are times 5 pounds in 27543 pounds. Let the 5)27543 pounds, 
 numbers be set down as in the margin. Since o uo t. cco8~boxes, 
 the divisor 5 is not contained in the first figure and p over 
 
 of the dividend, we find how many times it is 
 contained in the first two figures, which is 5 times and 2 over. We 
 set the quotient figure 5 under the right hand figure divided, 
 because it is the same order as that figure, and prefix the remainder 
 2, mentally to the next figure of the dividend, making 25. Now 5 
 is in 25, 5 times, and no remainder. Again, 5 is not contained in 4, 
 the next figure of the dividend ; we therefore place a cipher in the
 
 58 DIVISION. 
 
 quotient, and prefix the 4 mentally to tlie next figure of the dividend, 
 as if it were a remainder, making 43. Finally, 5 is in 43, 8 times, 
 and 3 over. He therefore filled 5508 boxes, and had 3 pounds 
 remainder. Hence, the 
 
 EULE. I. Place the divisor on the left of the dividend, 
 and beginning at the left, divide each figure by it, setting 
 the result under the figure divided. 
 
 II. If the divisor is not contained in a figure of the div- 
 idend, put a cipher in the quotient, and find hoiv many 
 times it is contained in this and the next figure, setting the 
 result under the right hand figure divided. 
 
 III. If a remainder arise from any figure before the last, 
 prefix it mentally to the next figure, and divide as before. 
 
 If from the last figure, place it over the divisor, and 
 annex it to the quotient. 
 
 NOTES. i. In the operation, the divisor is placed on the left of 
 the dividend, and the quotient under it, as a matter of convenience. 
 When division is simply represented, the divisor is either 
 placed under the dividend, or on the right, with the sign (-J-) be- 
 fore it. 
 
 2 The reason for beginning to divide at the left hand is, that in 
 dividing a higher order there may be a remainder, which must be 
 prefixed to the next lower order, as we proceed in the operation. 
 
 3. We place the quotient figure under the figure divided; because 
 the former is the same order as the latter. (Art. 71.) 
 
 4. When the divisor is not contained in a figure of the dividend, 
 we place a cipher in the quotient, to show that the quotient has no 
 units corresponding with the order of this figure. It also preserves 
 the local value of the subsequent figures of the quotient. 
 
 5. The final remainder shows that a part of the dividend is not 
 divided. It is placed over the divisor and annexed to the quotient 
 to complete the division. 
 
 70. What is Short Division? 71 How write nnmters for short division? 
 The next step ? When the divisor is not contained in a figure of the dividend, 
 how proceed ? When there is a remainder after dividing a figure, how ? If there is 
 a remainder after dividing the last figure, what ? Note. Why place the divisor on 
 the left of the dividend ? Why begin to divide at the left hand ? Why place each 
 quotient figure under the figure divided? Why place a cipher in the quotient, 
 when tho divisor is not contained tn a figure of the dividend ? Whnt doos the final 
 remainder show ? Why place it over the divisor and annex it to the Quotient?
 
 DIVISION. 59 
 
 53^" The pupil should early learn to abbreviate the language used 
 In the process of dividing. Thus, in the next example, instead of 
 eaying 5 is contained in 7 once, and 2 over, let him pronounce the 
 quotient figures only; as, one, four, five, seven. 
 
 i. A man divided 7285 acres of land equally among his 
 5 sons : what part, and how much, did each receive ? 
 
 SOLUTION. i is i fifth of 5 ; hence each had i fifth part. (Art. 65.) 
 Again, 7285 A -5-5 = 1457 A; hence each received 1457 A. (Art. 66.) 
 
 (2.) 
 
 (3-) (40 (50 
 3)560346 4)689034 5)748239 
 
 (6.) 
 
 6)3972647 
 
 (7-) (8.) ( 9 .) 
 7)4806108 8)7394 6 4 9)8306729 
 
 (10.) 
 
 10)57623140 
 
 (II.) (12.) 
 11)66730145! 12)8160252397 
 
 13. At $2 apiece, how many hats can be bought for 
 $16486 ? (Art 48, Note 3.) 
 
 14. At $4 a head, how many sheep can be bought for 
 $844? 
 
 15. How many times are 3 rods contained in 26936 
 rods? 
 
 1 6. A man having $42684, divided it equally among 
 his 4 children : how much did each receive ? 
 
 17. If a quantity of muslin containing 366 yards is di- 
 vided into 3 equal parts, how many yards will each part 
 contain ? 
 
 1 8. If 84844 pounds of bread are divided equally among 
 6 regiments, how many pounds will each regiment re- 
 ceive ? 
 
 19. Eight men found a purse containing $64968, which 
 they shared equally: how much did each receive?
 
 60 DIVISION. 
 
 20. Divide 4268410 by 4. 21. Divide 5601234 by 6. 
 
 22. Divide 6403021 by 5. 23. Divide 7008134 by 7. 
 
 24. Divide 8210042 by n. 25. Divide 9603048 by 8. 
 
 26. Divide 23468420 by 10. 27. Divide 32064258 by 9. 
 
 28. Divide 46785142 by 8. 29. Divide 59130628 by 7. 
 
 30. Divide 653000638 by n. 31. Divide 774230029 by 12 
 
 32. In 7 days there is i week: how many weeks in 
 26563 days? 
 
 33. If $38472 are divided equally among 6 persons, how 
 much will each receive ? 
 
 34. How many tons of coal, at $7 a ton, can be pur- 
 chased for ,$63456 ? 
 
 35. At $9 a barrel, how much flour can be bought for 
 $47239? 
 
 36. In 12 months there is i year: how many years in 
 41260 months? 
 
 37. A merchant laid out $45285 in cloths, at $7 a yard: 
 how many yards did he buy ? 
 
 38. At 8 shillings to a dollar, how many dollars are 
 there in 75240 shillings? 
 
 39. In 9 square feet there is i square yard : how many 
 square yards are there in 52308 square feet? 
 
 40. If a person travel 10 miles an hour, how long will 
 it take him to travel 25000 miles ? 
 
 41. A market woman having 845280 eggs, wished to 
 pack them in baskets holding i dozen each : how many 
 baskets did it take ? 
 
 42. If a prize of $116248 is divided equally among 8 
 men, what will be each one's share ? 
 
 43. If in 7 townships there are 2346281 acres, how many 
 acres are there to a township ? 
 
 44. At $8 a barrel, how many barrels of sugar can be 
 bought for $i 1 1364 ? 
 
 45. At $11 each, how many cows can be had for $88990 ?
 
 " ' 
 
 LOKG Division*. 
 
 72. Long Division is the method of dividing, when 
 the results of the several steps and the quotient are both 
 set down. 
 
 73. To divide by Long Division. 
 
 i. A speculator paid $31097 for 15 city lots: what did 
 the lots cost apiece ? 
 
 ANALYSIS. Since 15 lots cost $31097, I OPERATION. 
 
 lot will cost as many dollars as 15 is con- 
 tained times in 31097. Let the divisor and 3 
 
 dividend be set down as in the margin. 
 
 Beginning at the left, the first step is to 109 
 
 find how many times the divisor 15, is IQC 
 
 contained in 31 (which is 2 times), and set _ 
 
 the 2 on the right of the dividend. x 7 
 
 Second, multiply the divisor by the ** 
 
 quotient figure, and set the product 30, _ 
 
 under the figures divided. TJiird, sub- 2 
 
 tract this product from the figures 
 
 divided. FourtJi, bring down the next figure of the dividend, 
 and place it on the right of the remainder, making 10 for tho 
 next partial dividend, and proceed as before. But 15 is not con- 
 tained in 10 ; we therefore place a cipher in the quotient, and 
 bring down the next figure of the dividend, making 109. Now 
 15 is in 109, 7 times. Set the quotient figure 7 on the right, mul- 
 tiply the divisor by it, subtract the product from the partial divi- 
 dend, and to the right of the remainder, bring down the succeeding 
 figure for the next partial dividend, precisely as before. Now 15 is 
 in 47, 3 times. Setting the 3 in the quotient, multiplying, and sub- 
 tracting, as above, the final remainder is 2. We place this remainder 
 
 72. What is Long Division? 73. How write the numbers? What is the first 
 Btep? The second? The third? The fourth? If the divisor is not contained 
 in a partial dividend, how proceed? What is to be done with the last remainder? 
 Note. What is the difference between short and long division ? Of what order itt 
 the quotient figure ?
 
 62 DIVISION. 
 
 over the divisor and annex it to the quotient. The divisor and divi- 
 dend being different denominations, the quotient is the same as the 
 dividend (Art. 64). Therefore the lots cost $2073 -& apiece. Hence, the 
 
 EULE. I. Find how many times the divisor is contained 
 in the fewest figures on the left of the dividend, that will 
 contain it, and set the quotient on the right. 
 
 II. Multiply the divisor by this quotient figure, and sub- 
 tract the product from the figures divided. 
 
 III. To the right of the remainder, bring down the next 
 figure of the dividend, and divide as before. 
 
 IV. If the divisor is not contained in a partial dividend, 
 place a cipher in the quotient, bring down another figure, 
 and continue the operation till all the figures are divided. 
 
 If there is a remainder after dividing the last figure, set 
 it over the divisor, and annex it to the quotient. 
 
 NOTES. i. The parts into which the dividend is separated in 
 finding the quotient figure, are called partial dividends, because 
 they are parts of the whole dividend. 
 
 2. Short and Long Division, it will be seen, are the same in prin- 
 ciple. The only difference is, that in one the results of the several 
 steps are carried in the mind, in the other they are set down. 
 
 Short Division is the more expeditious, and should be employed 
 when the divisor does not exceed 12. 
 
 The reasons for the arrangement of the parts, and for beginning 
 to divide at the left hand, are the same as in Short Division. 
 
 3. The quotient figure in Long as well as in Short Division, is 
 always of the same order as that of the right hand figure of the par- 
 tial dividend. 
 
 4. To prevent mistakes, it is customary to place a mark under 
 the several figures of the dividend as they are brought down. 
 
 5. After the first quotient figure is obtained, for each succeeding 
 figure of the dividend, either a significant figure or a cipher must be 
 put in the quotient. 
 
 6. If the product of the divisor into the figure placed in the quo- 
 tient is greater than the partial dividend, it is plain the quotient 
 figure is too large, and therefore must be diminished. 
 
 If the remainder is equal to or greater than the divisor, the quo- 
 tient figure is too small, and must be incrcawL
 
 DIVISION. 63 
 
 PROOF. 
 
 74. By Multiplication. Multiply the divisor and quo- 
 tient together, and to the product add the remainder. If 
 the result is equal to the dividend, the work is right. 
 
 NOTE. This proof depends upon the principle, that Division 
 is the reverse of Multiplication ; the dividend answering to the pro- 
 duct, the divisor to one of the factors, and the quotient to the other. 
 (Art. 62.) 
 
 75. By excess of 98. Multiply the excess of gs in 
 the divisor by that in the quotient, and to the product add 
 the remainder. If the excess of gs in this sum is equal to 
 that in the dividend, the ivork is right. 
 
 2. Divide 181403 by 67, and prove the operation. 
 
 Ans. 270714. 
 
 Proof. By Multiplication. 2707x67=181369, and 181369 + 34 
 the rem. = 181403 the dividend. 
 
 By excess of gs. The excess of gs in the divisor is 4, and the 
 excess in the quotient is 7. Now 4x7=28, and 28 + 34=62; the 
 excess of gs in 62 is 8. The excess of 93 in the dividend is also 8. 
 
 3. Divide 34685 by 15. 4. Divide 65456 by 16. 
 
 5. Divide 41534 by 20. 6. Divide 52663 by 25. 
 
 7. Divide 420345 by 39. 8. Divide 506394 by 47. 
 
 9. Divide 673406 by 69. 10. Divide 789408 by 77. 
 ii. Divide 4375023 by 86. 12. Divide 5700429 by 93. 
 13. Divide 6004531 by 59. 14. Divide 8430905 by 78. 
 15. Divide 7895432 by 89. 16. Divide 9307108 by 98. 
 
 17. How many acres of land at $75 per acre, can I buy 
 for $18246 ? 
 
 1 8. At $83 apiece, how many ambulances can be bought 
 for $37682 ? 
 
 73. Note. If the product of the divisor into the figure placed in the quotient, is 
 greater than the partial dividend, what does it show ? If the remainder is equal 
 to or greater than the divisor, what f How is Division proved ?
 
 64 DIVISION. 
 
 76. To find the Quotient Figure, when the Divisor is large. 
 
 Take t\\e first figure of the divisor for a trial divisor, and 
 find how many times it is contained in the first or first 
 two figures of the dividend, making due allowance for 
 carrying the tens of the product of the second figure of the 
 divisor into the quotient figure. 
 
 19. Divide 18046 by 673. 
 
 ANALYSIS. Taking 6 for a trial divisor, it is 673)18046(26 
 
 contained in 18, 3 times. But in multiplying 7 by 1346 
 3, there are 2 to carry, and 2 added to 3 times 6, 
 
 make 20. But 20 is larger than the partial divi 4586 
 
 dend 18 ; therefore, 3 is too large for the quotient 4038 
 figure. Hence, we place 2 in the quotient, and 
 
 proceed as before. (Art. 73, n.) c^g 
 
 20. Divide 3784123 by 127. 21. Divide 436 17 29 by 219. 
 22. Divide 8953046 by 378. 23. Divide 9073219 by 738. 
 
 24. How many shawls at $95, can be bought for $42750 ? 
 
 25. In 144 square inches there is i square foot: how 
 many square feet are there in 59264 square inches ? 
 
 26. A quartermaster paid $29328 for 312 cavalry horses: 
 how much was that apiece ? 
 
 27. If 128 cubic feet of wood make i cord, how many 
 cords are in 69240 cubic feet? 
 
 28. If a purse of $150648 is divided equally among 250 
 sailors, how much will each receive ? 
 
 29. In 1728 cubic inches there is i cubic foot: how 
 many cubic feet are there in 250342 cubic inches ? 
 
 30. If 560245 pounds of bread are divided equally 
 among 11200 soldiers, how much will each receive? 
 
 31. Div. 36942536 by 4204. 32. Div. 57300652 by 5129. 
 33. Div. 629348206 by 52312. 34. Div. 730500429 by 61073, 
 
 35. Divide 7300400029 by 236421. 
 
 36. Divide 8230124037 by 463205. 
 
 37. Divide 843000329058 by 203963428.
 
 DIVISION. 65 
 
 38. A stock company having $5000000, was divided 
 into 1250 shares: what was the value of each share? 
 
 39. A railroad 478 miles in length cost $18120000 : what 
 was the cost per mile ? 
 
 40. A company of 942 men purchased a tract of land 
 containing 272090 acres, which they shared equally: what 
 was each man's share ? 
 
 41. A tax of $42368200 was assessed equally upon 5263 
 towns : what sum did each town pay ? 
 
 42. The government distributed $9900000 bounty equally 
 among 36000 volunteers: how much did each receive? 
 
 43. Since there is i year in 525600 minutes, how many 
 years are there in 105192000 minutes. 
 
 CONTRACTIONS. 
 CASE I. 
 
 77. To divide by a Composite Number. 
 
 Ex. i. A farmer having 300 pounds of butter, packed it 
 in boxes of 15 pounds each : it is required to find how 
 many boxes he had, using the factors of the divisor. 
 
 ANALYSIS. 15=5 times 3. Now if he puts 
 5 pounds into a box, it is plain he would use 
 as many boxes as there are 53 in 300 or 60 OI 
 
 fine-pound boxes. But 3 five-pound boxes 5/3 lba - 
 make i fifteen-pound box ; hence, 60 fife- 3)60 5 lb. boxes. 
 
 pound boxes will make as many 15 pound _Ans. 20 boxes. 
 boxes, as 3 is contained times in 60. which is 
 20. In the operation, we first divide by one 
 of the factors of 15, and the quotient by the other. Hence, the 
 
 EULE. Divide the dividend ~by one of the factors of the 
 divisor, and the quotient thence arising by another factor, 
 and so on, until all the factors have been used. TJie last 
 quotient will be the one required. 
 
 77. When the divisor Is a composite number, how procoed T
 
 66 DIVISION. 
 
 NOTES. I. This contraction is the reverse of multiplying by a 
 composite number. Hence, dividing the dividend (which answers to 
 the product) by the several factors of one of the numbers which pro- 
 duced it, will evidently give the quotient, the other factor of which 
 the dividend is composed. (Art. 56.) 
 
 2. When the divisor can be resolved into different sets of factors, 
 the result will be the same, whichever set is taken, and whatever 
 the order in which they are employed. The pupil is therefore at 
 liberty to select the set and the order most convenient. 
 
 2. Divide 357 by 21, using the factors. 
 
 3. If 532 oranges are divided equally among 28 boys, 
 what part, and how many will each receive ? 
 
 4. A dairyman packed 805 pounds of butter in 35 jars : 
 how many pounds did he put in a jar ? 
 
 5. How many companies in a regiment containing 756 
 soldiers, allowing 63 soldiers to a company ? 
 
 6. Divide 204 by 12, using different sets of factors. 
 
 7. Divide 368 by 16, using different sets of factors. 
 
 8. Divide 780 by 30, using different sets of factors. 
 
 78. To find the True Remainder, when Factors of the 
 divisor are used. 
 
 Ex. 9. A lad picked 2425 pints of chestnuts, which he 
 wished to put into bags containing 64 pints each: it is 
 required to find the number of bags he could fill, and the 
 number of pints over, or the true remainder. 
 
 ANALYSIS. The divisor 64 
 equals the factors 2x8x4. Di- 
 
 .,. . , , ,, OPERATION. 
 
 vidmg 2425 pints by 2, the quo- 2 ^2A2C 
 
 tient is 1212 and i remainder. J- 
 
 But the units of the quotient 1212 8 ) I2I2 ~ * J P*- I8t r - 
 
 are 2 times as large as pints, 4)151 45 4x2= 8pt. 2dr. 
 
 which, for the sake of distinction, -77 ^ i x 8 X 2 48 pt. *d r 
 
 we will call quarts; and the re- 
 mainder i, is a pint, the same as An ^ ^ b and s ; t 
 the dividend. (Art. 60, n.) 
 
 Next, dividing 1212 quarts by 
 8, the quotient is 151, and 4 remainder. But the units of the 
 
 78. How is the true remainder fonnd 1 Note. To what is It eqnnl ? \Vht 
 should he done with it? The oh.iect In multiplying each partial remainder by all 
 the preceding divisors except its own ?
 
 DIVISION. 67 
 
 quotient 151 are 8 times as large as quarts; call them pecks ; the 
 remainder 4, which denotes quarts, must be multiplied by the 
 preceding divisor 2, to reduce it back to pints. Finally, dividing 151 
 pecks by 4, the quotient is 37, and 3 remainder. But the units of 
 the quotient 37 have four times the value of pecks ; call them bags ; 
 and the remainder 3, which denotes pecks, must be multiplied by 
 the last divisor 8, to reduce them back to quarts, and be multiplied 
 by 2 to reduce the quarts to pints. Having filled 37 bags, we have 
 three partial remainders, I pint, 4 quarts, and 3 pecks. 
 
 The next step is to find the true remainder. We have seen that 
 a unit of the 2d remainder is twice as large as those of the given 
 dividend, which are pints ; hence, 4 quarts = 8 pints. Again, each 
 unit of tlve 3d remainder is 8 times as large as those of the preceding 
 dividend, which are quarts , therefore, 3 pecks = 3 x 8 or 24 quarts ; 
 and 24 qU. 24 x 2 or 48 pts. The sum of these partial remainders, 
 i pt. + 8 pt, + 48 pt. = 57 pts., is the true remainder. Hence, the 
 
 KULE. Multiply each partial remainder by all the 
 divisors preceding its own ; the sum of these results added 
 to the first, will be the true remainder. 
 
 NOTES. i. Multiplying each remainder by all the preceding di- 
 visors except its own, reduces them to units of the same denomina- 
 tion as the given dividend. Hence, the true remainder is equal to 
 the sum of the partial remainders reduced to the same denomina- 
 tion as the dividend. 
 
 2. When found, it should be placed over the given divisor, and be 
 annexed to the quotient. 
 
 10. Divide 43271 by 45. n. Divide 502378 by 63. 
 12. Divide 710302 by 72. 13. Divide 3005263 by 84. 
 14. Divide 63400511 by 96. 15. Divide 216300265 by 144. 
 
 CASE II. 
 
 79. To divide by 10, 100, 1000, etc. 
 
 Ex. 1 6. Divide 2615 by 100. 
 
 ANALYSIS. Annexing a cipher to a figure, we 
 have seen, multiplies it by 10 ; conversely, re- OTKHATIOB. 
 
 moving a cipher from the right of a number i|Qo)26 15 
 must diminish its value 10 times, or divide it Ans. 26 15 Rem. 
 by 10; for, each figure in the number is re- 
 moved one plas.e to the right. (Art. 12.) In like manner, cutting
 
 68 DIVISION. 
 
 off two figures from the right, divides it by 100 ; cutting off three, 
 divides it by 1000, etc. 
 
 In the operation, as the divisor is 100, we simply cut off two 
 figures on the right of the dividend ; the number left, viz., 26, is the 
 quotient; and the 15 cut off, the remainder. Hence, the 
 
 EULE. From the right of the dividend, cut off as many 
 figures as there are ciphers in the divisor. The figures left 
 tvill be the quotient ; those cut off, the remainder. 
 
 17. Divide 75236 by 100. 20. 9820341 by 100000. 
 
 18. Divide 245065 by 1000. 21. 9526401 by 1000000. 
 79. Divide 805211 by 10000. 22. 80043264 by 10000000. 
 
 CASE III. 
 
 80. To divide, when the Divisor has Ciphers on the right. 
 
 23. At $30 a barrel, how many barrels of beef can be 
 bought for $4273? 
 
 ANALYSIS. The divisor 30 is composed of 
 the factors 3 and 10. Hence, in the operation, 
 we first divide by 10, by cutting off the right- 3,|4__7 [3 
 hand figure of the dividend ; then dividing the Quot. 142, I Eem. 
 remaining figures of the dividend by 3, the Ans. I4 2 vo tls- 
 quotient is 142, and i remainder. Prefixing 
 the I remainder to the 3 which was cut off, we have 13 for the true 
 remainder, which being placed over the given divisor 30, and an- 
 nexed to the quotient, gives 142^ bis. for the answer. Hence, the 
 
 EULE. I. Cut off the ciphers on the right of the divisor 
 and as many figures on the right of the dividend. 
 
 II. Divide the remaining part of the dividend ly the 
 remaining part of the divisor for the quotient. 
 
 III. Annex the figures cut off to the remainder, and the 
 result will be the true remainder. (Art. 78.) 
 
 NOTE. This contraction is based upon the last two Cases. The 
 1rue remainder should be placed over the whole divisor, and be an- 
 nexed to the quotient. 
 
 79. How proceed when the divisor is 10, 100, etc. ? 80. How when there art 
 ciphers on the right of the divisor? Note. Upon what is this contraction based'
 
 DIVISION. 69 
 
 4. Divide 45678 by 20. 25. 81386 by 200. 
 
 26. Divide 603245 by 3400. 27. 740321 by 6500. 
 
 28. Divide 7341264 by 87000. 29. 8004367 by 93000. 
 30. Divide 61273203 by 125000. 31. 416043271 by 670000. 
 
 32. In 100 cents there is i dollar: how many dollars in 
 37300 cents ? 
 
 33. At $200 apiece, how many horses will $45800 buy? 
 
 34. If $75360 were equally distributed among 1000 men, 
 
 how much would each receive ? 
 
 35. At $4800 a lot, how many can be had for $25200? 
 
 36. How many bales, weighing 450 pounds each, can be 
 made of 27000 pounds of cotton ? 
 
 QUESTIONS FOR REVIEW, INVOLVING THE 
 PRECEDING RULES. 
 
 1. William, who has 219 marbles, has 73 more than 
 James: how many has James ? How many have both ? 
 
 2. A farmer having 368 sheep, wishes to increase his 
 flock to 775 : how many must he buy? 
 
 3. The difference of two persons' ages is 19 years, and 
 the younger is 57 years: what is the age of the elder? 
 
 4. What number must be added to 1368 to make 3147 ? 
 
 5. What number subtracted from 4118 leaves 1025 ? 
 
 6. What number multiplied by 95 will produce 7905 ? 
 
 7. The product of the length into the breadth of a field 
 is 2967 rods, and the length is 69 rods: what is the breadth? 
 
 8. A man having 5263 bushels of grain, sold all but 
 145 bushels: how much did he sell? 
 
 9. What number must be divided by 87, that the 
 quotient may be 99 ? 
 
 10. If the quotient is 217, and the dividend 7595, what 
 must be the divisor ? 
 
 11. If the divisor is 341 and the quotient 589, what 
 must be the dividend ?
 
 70 DIVISION. 
 
 12. A merchant bought 516 barrels of flour at $9 a bar- 
 rel, and sold it for $5275 : how much did he gain or lose ? 
 
 13. How long can 250 men subsist on a quantity of 
 food sufficient to last i man 7550 days? 
 
 14. How many pounds of sugar, at n cents, must be 
 given for 629 pounds of coffee, at 17 cents? 
 
 15. At $13 a barrel, how many barrels of flour must be 
 given for 530 barrels of potatoes worth $3 a barrel ? 
 
 1 6. A man having $15260, deducted $4500 for personal 
 use, and divided the balance equally among his 7 sons : 
 how much did each son receive ? 
 
 17. A man earns 12 shillings a day, and his son 8 shil- 
 lings: how long will it take both to earn 1200 shillings? 
 
 1 8. If a man earns $19 a week, and pays $2 a week fo* 
 boarding each of his 3 sons at school, how much will he 
 lay up in 12 weeks ? 
 
 19. A farmer sold 6 cows at $23, 150 bushels of wheat 
 at $2, and 75 barrels of apples at $4, and laid out his money 
 in cloth at $7 a yard: how many yards did he have? 
 
 20. If I buy 1361 barrels of flour at $7, and sell the 
 whole for $12249, how much shall I make per barrel ? 
 
 21. The earnings of a man and his two sons amount to 
 $3560 a year; their expenses are $754. If the balance is 
 divided equally, what will each have ? 
 
 22. A man having $23268, owed $1733, and divided the 
 rest among four charities : how much did each receive ? 
 
 23. How many sheep at $5 a head, must be given for 30 
 cows at $42 apiece ? 
 
 24. A father bought a suit of clothes for each of his 
 3 sons, at $123 a suit, and agreed to pay 17 tons of hay at 
 $12 a ton, and the rest in potatoes at $43 barrel: how 
 many barrels of potatoes did it take ? 
 
 25. If you add $435 to $567, divide the sum by $334, 
 multiply the quotient by 217, and divide the product by 
 59, what will be the result?
 
 DIVISION. . 71 
 
 26. If from 1530 you take 319, add 793 to the remainder, 
 multiply the sum by 44, and divide the product by 37, 
 what will be the result ? 
 
 27. A man's annual income is $4250 ; if he spends $1365 
 for house rent, $1439 for other expenses, and the balance 
 in books, at $3 apiece, how many books can he buy ? 
 
 28. A farmer having $3038, bought 15 tons of hay at 
 81 1, 3 yoke of oxen at $155, 375 sheep at $5, and spent the 
 rest for cows at $41 a head : how many cows did he buy ? 
 
 GENERAL PRINCIPLES OF DIVISION. 
 
 81. From the nature of Division, the absolute value of the 
 quotient depends both upon the divisor and the dividend. 
 
 The relative value of the quotient ; that is, its value com- 
 pared with the dividend, depends upon the divisor. Thus, 
 
 1. If the divisor is equal to the dividend, the quotient is i. 
 
 2. If the divisor is greater than the dividend, the 
 quotient is less than i. 
 
 3. If the divisor is less than the dividend, the quotient 
 is greater than i. * 
 
 4. If the divisor is i, the quotient is equal to the dividend. 
 
 5. If the divisor is greater than i, the quotient is less 
 than the dividend. 
 
 6. If the divisor is less than i, the quotient is greater 
 than the dividend. 
 
 82. The relation of the divisor, dividend, and quotient 
 is such that the divisor remaining the same, 
 
 Multiplying the dividend by any number, multiplies the 
 quotient by that number. Thus, 12-7-2=6; and (12x2) 
 -7-2 = 6x2. Conversely, 
 
 81. Upon what does the absolute value of the quotient dapead? Its relative 
 value 1 If the divisor is eqnal to the dividend, what is true ok the quotient ? If 
 greater? If less? If the divisor is i, what is the quotient? If preater tl.nn i T 
 If less than 1 1 82. Wliat is the effect of multiplying the dividend ? 83. Of divid- 
 ing it?
 
 Tt . DIVISION. 
 
 83. Dividing the dividend by any number, divides the 
 quotient by that number. Thus, (12-^- 2)-+ 2 = 6+ 2. 
 
 84. Multiplying the divisor by any number, divides the 
 quotient by that number. Thus, 24-7-6=4; and 24-;- 
 (6 x 2) =4-7- 2. Conversely, 
 
 85. Dividing the divisor by any number, multiplies the 
 quotient by that number. Thus, 24-=- (6 4-2) =4 x 2. 
 
 86. Mulfyilying or dividing both the divisor and rfwn- 
 dfewc? by the same number, does not ^er the quotient 
 Thus, 48-7-8-6; so (48x2)-7-(8x2) = 6; and (48^-2)^- 
 
 (8-r-2) = 6. 
 
 NOTE. Multiplying or dividing the dividend, produces a like 
 effect on the quotient ; but multiplying or dividing the divisor, pro- 
 duces the opposite effect on the quotient. 
 
 86, ' If a number is both multiplied and divided by the 
 same number, its value is not altered. Thus, (7 x 6) ~- 6 is 
 equal to 7. 
 
 PROBLEMS AND FORMULAS PERTAINING TO THE 
 FUNDAMENTAL RULES. 
 
 87. A Problem is something to be done, or a question 
 to be solved. 
 
 88. A Formula, is a specific rule by which problems 
 are solved, and may be expressed by common language, or 
 by ,si^r5. 
 
 89. The/owr #raz Problems of Arithmetic have already 
 been illustrated. They are 
 
 ist. When two or more numbers are given, to find their 
 sum, or amount. (Art 29.) 
 
 2d. When two numbers are given, to find their difference. 
 
 84. What is the effect of multiplying the divisor? Of dividing it? 86. WTiat 
 is the effect of multiplying or dividing both the divisor and dividend ? 87. What 
 ia a problem? S3. A formula? 89. The four great problems ot arithmetic!'
 
 PROBLEMS AND FORMULAS. 73 
 
 3d. When two factors are given, to find their product. 
 (Art. 50.) 
 
 4tb. When two numbers are given, to find how many 
 times one is contained in the other. (Art. 73.) 
 
 90. These problems constitute the four fundamental 
 rules of Arithmetic, called Addition, Subtraction, Hulti~ 
 plication, and Division. 
 
 NOTES. i. These rules are called fundamental, because upon 
 them are based all arithmetical operations. 
 
 2. As multiplication is an abbrevated form of addition, and division, 
 of subtraction, it follows that every change made upon the value of a 
 number, must increase or diminish it. Hence, strictly speaking, 
 there are but two fundamental operations, viz. : aggregation and 
 diminution, or increase and decrease. 
 
 3. The following problems, though subordinate, are so closely con- 
 nected with the preceding, that a passing notice of them may not be 
 improper, in this connection. 
 
 91. To find the greater of two numbers, the less and their 
 difference being given. 
 
 1. A planter raised two successive crops of cotton, the 
 smaller of which amounted to 4168 bales, and the dif- 
 ference between them was 1123 bales: what was the 
 greater crop ? 
 
 ANALYSIS. If the difference between two numbers be added to 
 the less, it is obvious the sum must be equal to the greater. There- 
 fore, 4168 bales + 1123 bales or 5291 bales must be the greater crop. 
 Hence, the 
 
 RULE. To the less add the difference, and the sum will 
 be the greater. (Art. 39.) 
 
 2. One of the two candidates at a certain election, re- 
 ceived 746 votes, and was defeated by a majority of 411: 
 how many votes did the successful candidate receive ? 
 
 90. What do these constitute? Note. Why so called? What is given and 
 what required in addition ? In subtraction ? In multiplication ? In division ? 
 Where begin the operation in addition, subtraction, and multiplication ? Where 
 In division ? What is the difference between addition and subtraction ? Between 
 addition and multiplication? Between subtraction and division? Between 
 multiplication and division ?
 
 74 PROBLEMS AND FOBMULAS. 
 
 92. To find the less of two numbers, the greater and 
 their difference being given. 
 
 3. The greater of two cargoes of flour is 5267 barrels, 
 and their difference is 1348 barrels: how many barrels does 
 the smaller contain ? 
 
 ANALYSIS. The difference added to the less number equals the 
 greater ; therefore, the greater diminished by the difference, must be 
 equal to the less ; and 5267 barrels minus 1348 barrels leaves 3919 
 barrels, the smaller cargo. Hence, the 
 
 RULE. From the greater subtract the difference, and the 
 result will be the less. (Art. 38.) 
 
 4. At a certain election one of the two candidates re- 
 ceived 1366 votes, and was elected by a majority of 219 
 yotes : how many votes did the other candidate receive ? 
 
 93. The Product and one Factor being given, to find the 
 
 other Factor. 
 
 5. A drover being asked how many animals he had, 
 replied that he had 67 oxen, and if his oxen were multi- 
 plied by his number of sheep, the product would be 
 37520: how many sheep had he? 
 
 ANALYSIS. Since 37520 is & product, and 67 one of its factors, the 
 other factor must be as many as there are 673 in 37520 ; and 
 37520-7-67=560. (Art. 93.) Therefore, he had 560 sheep. Hence, the 
 
 EULE. Divide the product by the given factor, and the 
 quotient will be the factor required. (Art. 62.) 
 
 6. The length of a certain park is 320 rods, and the 
 product of its length and breadth is 5 1 200 rods : what is 
 its breadth ? 
 
 94. The Product of three or more Factors and all the 
 
 Factors but one being given, to find that Factor. 
 
 7. The product of the length, breadth, and height of 
 a certain mound is 62730 feet; its length is 45 feet, and 
 its breadth 41 feet: what is its height? 
 
 91. How find the greater of two numbers, the less and difference being given ? 
 93. How find the less, the greater and difference being given ?
 
 PROBLEMS AND FORMULAS. 75 
 
 ANALYSIS. The contents of solid bodies are found by multiplying 
 their length, breadth, and thickness together. Now as the length is 
 45 feet, and the breadth 41 feet, the product of which is 1845, the 
 height must be 62730 feet divided by 1845, or 34 feet. Hence, the 
 
 RULE. Divide the given product by the product of 
 the given factors, and the quotient will be the factor re- 
 quired. (Art. 93.) 
 
 8. The continued product of the distances wK ;h 4 men 
 traveled is 1944630 miles; one traveled 45, .nother 41, 
 and another 34 miles: how far did the fourth travel? 
 
 95. To find the Dividend, the Divisor and Quotient being 
 
 given. 
 
 9. If the quotient is 7071, and the divisor 556, what is 
 the dividend ? 
 
 ANALYSIS. Since the quotient shows how many times the cficisor 
 is contained in the dividend, it follows, that the product of the 
 divisor and quotient must be equal to the dividend. Now 7071 x 556 
 =3931476 the dividend. Hence, the 
 
 RULE. Multiply the divisor by the quotient, and the 
 result will be the dividend. (Art. 74.) 
 
 10. "What number of dollars divided among 135 persons 
 will give them $168 apiece? 
 
 96. To find the Divisor, the Dividend and Quotient 
 
 being given. 
 
 11. What must 8640 be divided by that the quotient 
 may be 144? 
 
 ANALYSIS. Since the quotient shows how many times the divisor 
 is contained in the dividend, it follows that if the dividend is divided 
 by the quotient the result must be the divisor, and 8640-*- 144=60. 
 Therefore, the divisor is 60. Hence, the 
 
 RULE. Divide the dividend by tb^ quotient, and the 
 result will be the divisor. (Arts. 62, 93.) 
 
 93. The product and one fiactor being given, how find the other factor T
 
 76 PROBLEMS AND FOBMULAS. 
 
 12. If the dividend is 7620, and the quotient 127, what 
 must be the divisor ? 
 
 97. The Sum and Difference of two numbers being given, 
 to find the Numbers. 
 
 13. The sum of two numbers is 65, and their difference 
 15 : what are the numbers ? 
 
 ANALYSIS. The sum 65 is equal to the greater number increased 
 by the less; and the greater diminished by the difference 15, is 
 equal to the less (Art. 38). Hence, if 15 is taken from 65, the re- 
 mainder 50, must be twice the less. But 50-5-2=25 the less num- 
 ber ; and 25 + 15 =40 the greater number. 
 
 Proof. 40 + 25=65 the given sum. Hence, the 
 
 KULE. From the sum take the difference, and half the 
 remainder will be the less number.' 
 
 To the less add the difference, and the result will be the 
 greater. (Art. 39.) 
 
 14. A merchant made $5368 in two years, and the dif- 
 ference in his annual gain was 6976 : what was his profit 
 each year ? 
 
 15. The whole number of votes cast for the two candi- 
 dates at a certain election was 5564, and the successful 
 candidate was elected by a majority 708 : how many votes 
 did each receive ? 
 
 16. A lady paid $250 for her watch and chain; the 
 former being valued $42 higher than the latter : what was 
 the price of each ? 
 
 17. Two pupils A and B, solved 75 examples; B solving 
 15 less than A: how many did each solve? 
 
 1 3. A and B found a pocket-book, and returning it to 
 the owner, received a reward of $500, of which A took 
 $38 more than B : what was the share of each ? 
 
 94. Wlien the product of three or more factors, and all but one are given, bow 
 find that one ? 95. How find the dividend, the divisor and quotient being given? 
 96. How find the divisor, the dividend and quotient being given ?
 
 ANALYSIS. 
 
 98. Analysis primarily denotes the separation of an 
 object into its elements. 
 
 99. Analysis, in Arithmetic, is fas process of tracing 
 the relation of the conditions of a problem to each other, 
 and thence deducing the successive steps necessary for its 
 solution. 
 
 NOTES. i. The application of Analysis to arithmetic is of recent 
 date, and to this source the late improvements in the mode of 
 teaching the subject are chiefly due. Previous to this, Arithmetic 
 to most pupils, was a hidden mystery, regarded as beyond the reach 
 of all but the favored few. 
 
 2. The pupil has already learned to analyze particular examples, 
 and from them to deduce specific rules by which similar examples 
 may be solved ; but Arithmetical Analysis has a much wider range. 
 It is applied with advantage to those classes of examples commonly 
 placed under the heads of Barter, Percentage, Profit and Loss, 
 Simple and Compound Proportion, Partnership, etc. In a word, it 
 is the grand Common-Sense Rule by which business men perform 
 the great majority of commercial calculations. 
 
 100. No specific directions can be prescribed for analyti- 
 cal solutions. The following suggestions may, however 
 be serviceable to beginners : 
 
 ist. In general, we reason from the given value of one, 
 to the value of two or more of the same kind. Or, 
 
 2d. From the given value of two or more, to that of one. 
 
 In the first instance we reason from a part to the 
 wJiole ; in the second, from the whole to a part. 
 
 3d. Sometimes the result of certain combinations ia 
 given, to find the original number or base. 
 
 98. What is the primary meaning of analysis ? 99. What is arithmetical 
 analysis? Note. What ia said as to its utility in arithmetical and business 
 calculations ? 100. Can specific rules be given for analytical solutions ? What 
 general directions can you mention ?
 
 78 ANALYSIS. 
 
 In such cases, it is generally best to begin with the 
 result, and reverse each operation in succession, till the 
 original number is reached. That is, to reason from the 
 result to its origin, or from effect to cause. 
 
 1. A farmer bought 150 sheep at $2 a head, and paid 
 for them in cows at $20 a head : how many cows did it 
 take to pay for the sheep ? 
 
 2. How much tea, at 85 cents a pound, must be given 
 for 425 pounds of rice, at 10 cents a pound ? 
 
 3. How much sugar, at 12 cents a pound, must be given 
 for 288 pounds of raisins, at 18 cents a pound? 
 
 4. How much corn, at 80 cents a bushel, must be given 
 for 1 60 pounds of tobacco, at 30 cents a pound? 
 
 5. How much butter, at 40 cents a pound, must be 
 given for 62 yards of calico, at 20 cents a yard ? 
 
 6. Bought 189 yards of linen, at 84 cents a yard, and 
 paid for it in oats, at 42 cents a bushel : how many bushels 
 did it take ? 
 
 7. Paid 1 8 barrels of flour for 30 yards of cloth, worth 
 $6 a yard : what was the flour a barrel ? 
 
 8. A farmer gave 15 loads of hay for 45 tons of coal, 
 worth $6 a ton : what did he receive a load for his hay ? 
 
 9. A sold B 35 hundred pounds of hops, at $27 a hun- 
 dred, and took 45 sacks of coffee, at $14 a sack, and the 
 balance in money : how much money did he receive ? 
 
 10. James bought 96 apples, at the rate of 4 for 3 cents, 
 and exchanged them for pears at 4 cents apiece: how 
 many pears did he receive ? 
 
 11. A farmer being asked how many acres he had, 
 replied, if you subtract 20 from the number, divide the 
 remainder by 8, add 15 to the quotient, and multiply by 5, 
 the product will be 125 : how many had he ? 
 
 ANALYSIS. Taking 125 as the base, and reversing the several 
 operations, beginning with the last, we have 125 -f- 5 = 25, the 
 number before the multiplication. Again, subtracting 15 from 25., 
 we have 25 15 = 10, the number before the addition.
 
 ANALYSIS. 79 
 
 Next, multiplying 10 by 8, we have 10 x 8=80, the number before 
 the division. Finally, adding 20 to 80, we have 80 + 20=100, the 
 number required. 
 
 12. What number is that, to which, if 25 be added, and 
 the sum multiplied by 9, the product will be 504 ? 
 
 13. What number, if diminished by 40, and the remain- 
 der divided by 8, the quotient will be 58 ? 
 
 14. A man being asked how many children he had, 
 answered, if you multiply the number by 1 1, add 23 to 
 the product, and divide the sum by 9, the quotient will 
 be 16: how many children had he? 
 
 15. The greater of two numbers is 3 times the less, and 
 the sum of the numbers is 36 : what are the numbers ? 
 
 ANALYSIS. The smaller number is i part, and the larger 3 parts ; 
 hence, the sum of the two is 4 parts, which by the conditions is 36. 
 Now, if 4 parts of a number are 36, i part is equal to as many units 
 as there are 43 in 36, or 9. Therefore 9 is the smaller number, and 
 3 times 9 or 27, the greater. 
 
 16. The sum of two numbers is 72, and the greater is 
 5 times the less* what are the numbers? 
 
 17. Divide 472 into three such parts, that the second 
 shall be twice the first, and the third 3 times the second 
 plus 13. 
 
 ANALYSIS. Calling the first i part, the second will be 2 parts, 
 and the third 6 parts plus 13 ; hence the sum of the three, in the 
 terms of the first, is 9 parts plus 13, which by the conditions is 472. 
 Taking 13 from 472 leaves 459, and we have 9 parts equal to 459. 
 Now if 9 parts equal 459, I part is equal to as many units as 9 is 
 contained times in 459, or 51. Therefore the first is 51, the second 
 2 times 51 or 102, the third 3 times 102 plus 13 or 319. 
 
 1 8. A and B counting their money, found that both 
 had $473, and that A bad 3 times as much as B plus $25 : 
 how much had each ? 
 
 19. The sum of two numbers is 243, the second is 
 three times the first minus 25 : what are the numbers? 
 
 20. What number is that to which if 315 be added, the 
 sum will be 250 less than 2683 ? 
 
 (For further applications of Analysis, see subsequent pages.)
 
 CLASSIFICATION 
 
 AND PROPERTIES OF NUMBERS. 
 
 101. Numbers are divided into abstract and concrete, 
 simple and compound, prime and composite, odd and even, 
 integral, fractional, and mixed, known and unknown, 
 similar and dissimilar, commensurable and incommen- 
 surable, rational and irrational or surds. 
 
 DEP. i. Abstract Numbers are those which are not applied to 
 things ; as, one, two, three. 
 
 2. Concrete Numbers are those which are applied to things ; as, 
 two caps, three pencils, six yards. 
 
 3. Simple Numbers are those which contain only one denomination, 
 and may be either abstract or concrete; as, 13, n pounds. 
 
 4. A Compound Number is one containing two or more denomina- 
 tions, which have the same base or nature ; as, 3 shillings and 6 
 pence ; 4 yards 2 feet and 6 inches. 
 
 5. A Prime Number is one which cannot be produced by multi* 
 plication of any two or more numbers, except a unit and itself. 
 
 All prime numbers except 2 and 5, end in i, 3, 7, or 9. 
 
 6. A Composite Number is the product of two or more factors, each 
 of which is greater than I ; as, 15 (5 x 3), 24 (2 x 3 x 4), etc. (Art. 55.) 
 
 A prime number differs from a composite number in two respects : First, In 
 Its origin ; Second, In its divisibility ; the former being divisible only by a unit 
 and itself; the latter by each of the factors which produce it. 
 
 7. Two numbers are prime to each other or relatively prime, when 
 the only number by which both can be divided without a remainder, 
 is a unit or i ; as, 5 and 6. 
 
 8. An Even Number is one which can be divided by 2 without a 
 remainder ; as, 4, 6, 10. 
 
 9. An Odd Number is one which cannot be divided by 2 without a 
 remainder ; as, 3, 5, 7, 9, n. 
 
 Name the kind of each of the following numbers, and why : 3, 10, 
 17, 21, 28, 31, 56, 63, 72, 81, 44, 39, 91, 67, 51, 84, 99, 100. 
 
 10. An Integer is a number which contains one or more entire 
 units only; as, i. 3, 7, 10, 50, 100. 
 
 101. How are numbers divided ? An abstract number ? Concrete ? Simple ? 
 Compound? Prime? Composite? When are two numbers prime to each other ! 
 Even? Odd? An integer? A fraction? A mixed number?
 
 PROPERTIES OF NUMBERS. 81 
 
 n. A Fraction is one or more of the equal parts into which a unit 
 is divided ; as, i-half, 2-thirds, 3-fourths, etc. 
 
 12. A Mixed Number is an integer and a fraction expressed 
 together ; as, 5^, nf, etc. 
 
 13. Like or similar numbers are those which express units of the 
 Bame kind or denomination ; as, 3 shillings and 5 shillings, four and 
 Beven, etc. 
 
 14. Unlike Numbers are those which express units of different 
 kinds or denominations ; as, 2 apples and 3 oranges. 
 
 15. Commensurable Numbers are those which can be divided by 
 the same number, without a remainder ; as, 9 and 12, each of which 
 can be divided by 3. 
 
 1 6. Incommensurable Numbers are those which cannot be divided 
 by the same number without a remainder. Thus, 3 and 7 are incom- 
 mensurable. 
 
 17. A given number is one whose value is expressed. 
 
 A number is also said to be given when it can be easily inferred 
 from something else which is given. Thus, if two numbers are 
 given, their sum and difference are given. 
 
 18. An Unknown Number is one whose value is not given. 
 
 102. A Factor of a number is one of the numbers, 
 which multiplied together, produce that number. 
 
 103. An Exact Divisor of a number is one which 
 will divide it without a remainder. Thus 2 is an exact 
 divisor of 6,, 3 of 15, etc. 
 
 NOTES. i. An exact divisor of a nutnbor is always & factor of that 
 number ; and, conversely, a factor of a number is always an exact 
 (Ucisor of it. For, the dividend is the product of the duisor and 
 quotient, and therefore is divisible by each of the numbers that pro- 
 duce it. (Art. 62.) 
 
 2. The terms dicisor and factor aro here restricted to integral 
 numbers. 
 
 104. A Pleasure of a number is an exact divisor of 
 that number. It is so called because the comparative 
 magnitude of the number divided, is determined by tl.'is 
 standard. 
 
 Like numbers? Unlike? Commensurable? Incommensurable? What is a 
 given number? An unknown? A tactor? An exact divisor ? A measure?
 
 
 PROPERTIES OF NUMBERS. 
 
 105. An aliquot part of a number is a factor or an 
 exact divisor of that number. 
 
 NOTE. The terms factors, measures, exact divisors, and aliquot 
 parts, are different names of the same thing, and are often used na 
 synonymous. Thus, 3 and 5 are respectively the factors, measures, 
 exact divisors, and aliquot parts of 15. 
 
 106. The reciprocal of a number is i divided by 
 that number. Thus, the reciprocal of 4 is 1-7-4, or {. 
 
 COMPLEMENT OF NUMBERS. 
 
 107. The Complement of a Number is the dif- 
 ference between the number and a unit of the next higher 
 order. Thus, the complement of 7 is 3; for 10 7=3; 
 the complement of 85 is 15 ; for 100 85 15. 
 
 108. To find the Complement of a Number. 
 
 Subtract the given number from i ivith as many ciphers 
 annexed, as there are integral figures in the given number. 
 
 Or, begin at the left hand, and subtract each figure of the 
 given number from 9, except the last significant figure on 
 th-e right, which must be taken from 10. 
 
 NOTE. The second method is based npon the principle that when 
 we borrow, the next upper figure must be considered i less than it is. 
 (Art. 37, ?z.) 
 
 Find the complement of the following numbers : 
 
 1. 328. 
 
 6. 6072. 
 
 ii. 56239. 
 
 16. 73245. 
 
 2. 567. 
 
 7. 8256. 
 
 12. 64123. 
 
 17. 1234567. 
 
 3. 604. 
 
 8. 9061. 
 
 13. 102345. 
 
 18. 2301206. 
 
 4. 891. 
 
 9. 13926. 
 
 14. 261436. 
 
 19. 3021238. 
 
 5 . 4638. 
 
 10. 23184. 
 
 15. 40061. 
 
 20. 7830426. 
 
 toe,. An aliquot part? Note. What is said as to the use of these four term*T 
 106. The reciprocal of a number? 107. The complement? 108. How find the 
 complement of a number T
 
 ^ 
 
 PROPERTIES OF LUMBERS. 83 
 
 / / 
 
 DIVISIBILITY OF NUMBERS. 
 
 109. One number is said to be divisible by another, I .... 
 when there is no remainder. The division is then com- 
 plete. 
 
 When there is a remainder, the division is incomplete; 
 and the dividend is said to be indivisible by the divisor. 
 
 NOTES. i. In treating of the divisibility of numbers, the term 
 divisor is commonly used for exact divisor. 
 
 2. Every integral number is divisible by the unit i, and by itself. 
 It is not customary, however, to consider the unit i, or the number 
 itself, as a factor. (Art. 102.) 
 
 110. In determining the divisibility of numbers the 
 following properties or facts are useful: 
 
 PROP. i. Any number is divisible by 2, which ends with o, 2, 4, 
 6, or 8. 
 
 2. Any number is divisible by 3, if the sum of its digits is divisible 
 by 3. Thus, 147 the sum of whose digits is 1+4 + 7=12, is divisible 
 by 3- 
 
 3. Any number is divisible by 4, if its too right hand figures are 
 divisible by 4 ; as, 2256, 15368. 19384. 
 
 4. Any number is divisible by 5, whicli ends with o or 5 ; as, 
 130, 675. 
 
 5. Any even number is divisible by 6, which is divisible by 3. 
 Taus, 1344-1-3=448; and 1344-^-6=224. 
 
 6. Any number is divisible by 8, if its three right Jutnd figures are 
 divisible by 8 ; as, 1840, 1688, 25320. 
 
 7. Any number is divisible by 10, which ends with o ; by too, if it 
 ends with oo ; by 1000, if it ends with ooo, etc. 
 
 8. Any number is divisible by 12 which is divisible by 3 and 4. 
 
 9. Any number divided by 9 will leave the same remainder as the 
 turn of its digits divided by 9. Hence, 
 
 10. Anv number is divisible by 9 if the sum of its digits is divisible 
 by 9. Thus, the sum of the digits of 54378 is 5 + 4 + 3 + 7 + 8, or 27. 
 Now, as 27 is divisible by 9, we may infer that 54378 is divisible 
 
 ic.. WT.en is one number divisible by another? Wben indivisible ? MO. Whe 
 Is a number divisible by 2 By 3? By<? By 5? By 6? By 8? By 10?
 
 84 PROPERTIES OF NUMBERS. 
 
 n. As 9 is a multiple of 3, any number divisible by g, is also 
 divisible by 3. 
 
 12. Any number divided by n will leave the same remainder as 
 the sum of its digits in the even places, taken from the*w?re of those 
 in the odd places, counting from the right, the latter being increased 
 or diminished by n or a multiple of 11. Thus, the sum of the 
 digits in the even places of 314567, viz., 6 + 4 + 3, or 13. is equal to 
 the sum of the digits in the odd places, viz., 7 + 5 + 1, or 13 ; there- 
 fore, 314567 is divisible by u. Heu.ce, 
 
 13. Any number is divisible by n when the sum of its digits in 
 the even places is equal to the sum of those in the odd places, or 
 when their difference is divisible by n. (For a demonstration of the 
 properties of 9 and n, see Higher Arithmetic.) 
 
 14. If one number is a divisor of another, the former is also a 
 divisor of any multiple of the latter. (Art. 103, n.) Thus, 2 is a divisor 
 of 6 ; it is also a divisor of the product of 3 times 6, of 5 x 6, and of any 
 whole number of times 6. 
 
 15. If a number is an exact divisor of each of two numbers, it is 
 nlsa an exact divisor of their sum, their difference, and their product. 
 Thus, 3 is a divisor of 9 and 15 respectively ; it is also a divisor ot 
 9 + 15, or 24; of 159, or 6; of 15 xg, or 135. 
 
 16. A composite number is divisible by each of its prime factors, by 
 the product of any two or more of them, and by no other number. 
 Thus, the prime factors of 30 are 2x3x5. Now 30 is divisible by 
 by 2, 3 and 5, by 2 x 3, by 2 x 5, by 3 x 5, and by 2 x 3 x 5, and by no 
 other number. Hance, 
 
 17. The least divisor of a composite number, is a prime number. 
 
 18. An odd number cannot be divided by an even number .vithout 
 a remainder. 
 
 19. If an odd number divides an even number, it will also divide 
 half of it. 
 
 20. If an even number is divisible by an odd number, it will also 
 be divisible by double that number. 
 
 If a number is divided by 9, to what is the remainder equal 1 When is a 
 number divisible by 9? If a number is divided by u, to what is the remainder 
 equal ? When is a number divisible by 1 1 V If a number is a divisor of another, 
 what is true of it in regard to any multiple of that number? If a number is a 
 divisor of each of two numbers, what is true of it in regard to their sum, dif- 
 ference, and product? By what is a composite number divisible? Note. What 
 is true of the least divisor of every composite number? Is an odd number 
 dirisible by an even ? If an odd divides an even number, what is true of half 
 of it? If an even number is divisible by an odd, what is true of it in regard to 
 double that number ?
 
 FACTORING. 
 
 111. Factors, we have seen, are numbers which mul- 
 tiplied together, produce & product. (Art. 42.) Hence, 
 
 112. Factoring a number is finding two or more fac- 
 tors, which multiplied together, produce the number. Thus, 
 the factors of 15 are 3 and 5 ; for, 3 x 5 = 15. (Art. 56, n.) 
 
 NOTE. Every number is divisible by itself and by i ; hence, if 
 multiplied by i, the product will be the number itself. (Art. 41.) 
 
 But it cannot properly be said that i is & factor, nor that a number 
 is a factor of itself. If so, all numbers are composite. (Art. 109, n.) 
 
 113. To resolve a Composite Number into two Factors. 
 
 Divide the number by any exact divisor ; the divisor and 
 quotient will be factors. (Art. 62.) 
 
 NOTE. Every composite number must have two factors at least ; 
 some have three or more ; and others may be resolved into several 
 different pairs of factors. (Art. 101, Del'. 6.) 
 
 1. What are the two factors of 35 ? Of 49 ? Of 12 1 ? 
 
 2. Name two factors of 45 ? Of 56 ? Of 72 ? Of 108 ? 
 
 114. To find the Different Pairs cf factors of a 
 
 Composite Number. 
 
 3. What are the different pairs of factors of 36 ? 
 
 ANALYSTS. Dividing 36 by 2, we have 36-5-2=18. 36 2-= 18 
 
 Again, 36-1-3=12; 36-4-4=9; and 36-5-6=6. That 36 3=12 
 
 is, the different pairs of factors of 36 are 2x18; 36 4= o 
 
 jx 12; 4x9; and 6x6. Hence, the 36 6 5 
 
 EULE. Divide the given number continually by each of 
 its exact divisors, beginning ivith the least, until the quo* 
 tient obtained is less than the divisor employed. 
 
 The divisors and corresponding quotients will be th& 
 different pairs of factors required. 
 
 m. What arc factors? Is i a factor? Is a number a factor of itself ? Why 
 not ? What is meant by Factoring? Note. How resolve a number into two fkc. 
 tore ? 1 13. How find the different pairs of factors T
 
 86 FACTORING. 
 
 NOTE. The division is stopped as soon as the quotient is lest than 
 the divisor ; for, the subsequent factors will be similar to those 
 already found. 
 
 Find the different pairs of factors of the following 
 numbers : 
 
 4- 20, 8. 38, 12. 96, 16. 256, 
 
 5- 2 7> 9- 45> 13- 1 10, 17. 475, 
 6. 30, 10. 56, 14. 144, 18. 600, 
 7- 32, " 75, !5- 225, 19. 1240. 
 
 PRIME FACTORS. 
 
 115. The Prime Factors of a number are the 
 Prime Numbers which, multiplied together, produce the 
 number. 
 
 116. Every Composite Number can be resolved 
 into prime factors. For, the factors of a composite num- 
 ber are divisors of it, and these divisors are either prime 
 or composite. Ifjyrime, they accord with the proposition. 
 If composite, they can be resolved into other factors, and 
 
 on, until all the factors are prime. (Art. 1 14.) 
 
 117. To find the Prime factors of a Composite Number. 
 Ex. i. What are the prime factors of 210 ? 
 
 ANALYSIS. Dividing 210 OPERATION. 
 
 by any prime number that ist divisor, 2 | 2 1 o, given. 
 
 will exactly divide it, aa 2, 2d " 3 
 
 we resolve it into the factors 3d " 5 
 
 2 and 105. Again, dividing 4th " 7 
 the ist quotient 105 by any 
 
 105, ist quot. 
 
 35, 2d " 
 
 7,3d 
 
 i, 4 th 
 
 prime number, as 3, we re- Hence, 210 = 2x3x5x7. 
 solve it into the factors 3 and 
 
 35. In like manner, dividing the 2d and 3d quotients by the prime 
 numbers 5 and 7, the 4th quotient is i. The divisors 2, 3, 5 and 7 
 are the prime factors required. For, each of these divisors ia a 
 prime number, and the division is continued until the quotient is a 
 
 115. What are prime factors? 116. Wliat is said of composite numbers? 
 How find the prime factors of a number?
 
 FACTORING. 87 
 
 unit ; therefore, the several divisors must be the prime factor* 
 required. Hence, the 
 
 RULE. Divide the given number by any prime number 
 that will divide it without a remainder. Again, divide 
 this quotient by a prime number, and so on, until the quo- 
 tient obtained is i. The several divisors are the prime 
 factors required. 
 
 NOTE. As the least divisor of every number is prime, beginners 
 may be less liable to mistakes by taking for the divisor the smallest 
 number that will divide the several dividends without a remainder 
 
 Find the prime factors of the following numbers : 
 
 2. 
 
 72. 
 
 7- 
 
 184. 
 
 12. 
 
 IOOO. 
 
 17- 
 
 2348. 
 
 3- 
 
 96. 
 
 8. 
 
 215- 
 
 13- 
 
 1208. 
 
 1 8. 
 
 10376. 
 
 4- 
 
 121. 
 
 9- 
 
 320. 
 
 14. 
 
 1560. 
 
 19. 
 
 25600. 
 
 5- 
 
 I 3 2. 
 
 10. 
 
 468. 
 
 15- 
 
 1776. 
 
 20. 
 
 64384. 
 
 6. 
 
 144. 
 
 ii. 
 
 576. 
 
 1 6. 
 
 1868. 
 
 21. 
 
 98816. 
 
 118. To find the Prime Factors common to two OP more 
 Numbers. 
 
 22. What are the prime factors common to 42, 168, and 
 210 ? 
 
 ANALYSIS. Dividing the given numbers by the 
 
 prime factor 2, the quotients are 21, 84, and 105. OPERATION. 
 
 Again, dividing these quotients by the prime 2)42, 168, 210 
 
 factor 3, the quotients are 7, 28, and 35. Finally, ~\^ 84 IO5 
 
 dividing by 7, the quotients are i, 4, and 5, no two r-- 
 
 of which can be divided by any number greater > 7> 2 "> 35 
 
 than i. Therefore, the divisors 2, 3, and 7 aro ij 4> 5 
 the common prime factors required. Hence, the 
 
 RULE. I. Write the given numbers in a horizontal line, 
 and divide them by any prime number which will divide 
 each of them without a remainder. 
 
 II. Divide the quotients thence arising in the same 
 manner, as long as the quotients have a common factor ; 
 
 the divisors ii'ill be the prime factors required. 
 
 ~* 
 
 118. How find the prime factors common to two or more numbers?
 
 88 CANCELLATION. 
 
 Find the prime factors common to the 
 numbers : 
 
 23. 12, 1 8, 30. 29. 75, 90, 135, 150. 
 
 24. 48, 66, 72. 30. 132, 144, 196, 240. 
 
 25. 64, 108, 132. 31. 168, 256, 320, 500. 
 
 26. 71, 113, 149. 32. 200, 325, 540, 625. 
 
 27. 48, 72, 88, 120. 33. 316, 396, 484, 936. 
 
 28. 60, 84, 108, 144. 34. 462, 786, 924, 858, 972. 
 
 119. To find the Prime Numbers as far as 250. 
 
 I. Write the odd numbers in a series, including 2. 
 II. After 3, erase all that are divisible by 3 ; after 5, erase 
 all that are divisible by 5 ; after 7, all that are divisible by 7 ; 
 after 1 1 and 1 3, all that are divisible by them : those left 
 are primes. 
 
 35. Find what numbers less than 100 are prime. 
 
 36. Find what numbers less than 200 are prime. 
 
 CANCELLATION. 
 
 120. Cancellation is the method of abbreviating opera- 
 tions by rejecting equal factors from the divisor and dividend. 
 
 The Sign of Cancellation is an oblique mark 
 drawn across the face of a figure ; as, S, 5, 1, etc. 
 NOTE. Tlie term cancellation is from the Latin cancello, to erase. 
 
 121. Cancelling a Factor of a number divides 
 the number by that factor. For, multiplying and divid- 
 ing a number by the same factor does not alter its value. 
 (Art. 86, a.} Thus, let 7 x 5 be a dividend. Now cancelling 
 the 7, divides the dividend by 7, and leaves 5 for the 
 quotient ; cancelling the 5, divides it by 5, and leaves 7 
 for the quotient. 
 
 120. What is cancellation T 121. Effect of cancelling a factor? WhyT
 
 CANCELLATION". 89 
 
 122. To divide one Composite Number by another. 
 
 1. What is the quotient of 72 divided by 18 ? 
 
 ANALYSIS. The dividend 72=9 x 4 x 2, and OPERATION. 
 
 the divisor 18=9x2. Cancelling the com- 72 SI X 4 X 2 
 
 mon factors 9 and 2, we have 4 for the Y8~ Six 2 = ^ 
 quotient. 
 
 2. Divide the product of 45 x 17, by the product of 9 x 5, 
 
 ANALYSIS. In this example the product of the two 9 45 
 
 factors composing the divisor, viz. : 9x5, equals one 
 
 ' 
 
 factor of the dividend. We therefore cancel these equals ~\ 
 at once, and the factor 17 is the quotient. 
 
 Remark. In this operation we place the numbers composing the divisor on 
 the left of a perpendicular line, and those composing the dividend on the right. 
 The result is the same as if the divisor were placed under the dividend as before. 
 Each method has its advantages, and may be used at pleasure. Hence, the 
 
 RULE. Cancel all the factors common to the divisor and 
 dividend, and divide the product of those remaining in the 
 dividend by the product of thosz remaining in the divisor. 
 (Art 86.) 
 
 NOTES. i. When either the divisor, dividend, or both, consist of 
 two or more factors connected by the sign x , they may be regarded 
 as composite numbers, and the common factors be cancelled before 
 the multiplication is performed. 
 
 2. This rule is founded upon the principle that, if the divisor and 
 dividend are both divided by the same number, the quotient is not 
 altered. (Art. 86.) Its utility is most apparent in those operations 
 which involve both multiplication and division; as Fractions, Pro- 
 portion, etc. 
 
 3. When the factor or factors cancelled equal the number itself, 
 the unit i, is always left ; for, dividing a number by itself, the 
 quotient is i. When the i stands in the dividend, it must be 
 retained. But when in the divisor, it is disregarded. (Art. 81.) 
 
 3. Divide the product 5 x 6 x 8 by 30 x 48. 
 
 SOLUTION. Cancelling the common 5x6x8 5 x 6 x "8. i i 
 factors, we have I in the dividend ~ Q = , jTiT? 6 = 6 
 
 and 6 in the divisor. (Art. 112, n.) 
 
 .13. Rule for cancelling in division ? Note. On what is the rule founded J
 
 90 CANCELLATION. 
 
 4. Divide 24 x 6 by 12 x 7. 7. Divide 18 x 15 by 5 x 6. 
 
 5. Divide 42 x 8 by 21 x 2. 8. Divide 56 x 16 by 8 x 4. 
 
 6. 18 X3 X4by 12 x6. 9. 28 x6 x 12 by 14 x 8. 
 
 10. Divide 21 x 5 x 6 by 7 x 3 x 2. 
 
 1 1. Divide 32 x 7 x 9 by 8 x 5 x 4. 
 
 12. Divide 27 x 35 x i4by 9 x 7 x 21. 
 
 13. Divide 33 x 42 x 25 by 7 x 5 x 1 1. 
 
 14. Divide 36 x 48 x 56 x 5 by 96 x 8 x 4 x 2. 
 
 15. Divide 63 x 24 x 33 x 2 by 9 x 12 x n. 
 
 16. Divide 175 x 28 x 72 by 25 x 14 x 12. 
 
 17. Divide 220 x 60 x 48x69 by 13 x nox 12 x8. 
 
 18. Divide 350 x 63 x 144 by 50 x 72 x 24. 
 
 19. Divide 500 x 128 x 42 x 108 by 256 x 250 x 12. 
 
 20. How many barrels of flour, at $12 a barrel, must be 
 given for 40 tons of coal, at $9 per ton ? 
 
 21. How many tubs of butter of 56 pounds each, at 
 28 cents a pound, must begiven for 8 pieces of shirting, 
 containing 45 yards each, at 26 cents a yard ? 
 
 22. How many bags of coffee 42 pounds each, worth 
 4 shillings a pound, must be given for 18 chests of tea, 
 each containing 72 pounds, at 6 shillings a pound? 
 
 23. Bought 24 barrels of sugar, each containing 168 
 pounds, at 20 cents a pound, and paid for it in cheeses, 
 each weighing 28 pounds, worth 16 cents a pound: how 
 many cheeses did it take ? 
 
 24. A grocer sold 10 hogheads of molasses, each con- 
 taining 63 gallons, at 7 shillings per gallon, and took his 
 pay in wheat, worth 14 shillings a bushel: how much 
 wheat did he receive ? 
 
 25. A young lady being asked her age, replied: If you 
 divide the product of 64 into 14, by the product of 8 into 
 4, you will have my age : what was her age ? 
 
 26. A has 60 acres, worth $15 an acre, and B 100 acres, 
 worth $75 an acre: B's property is how many times the 
 value of A's ?
 
 COMMON DIVISORS. 
 
 123. A Common Divisor is any number that will 
 divide two or more numbers without a remainder. 
 
 124. To find a Common Divisor of two or more Numbers. 
 
 1. Required a common divisor of 10, 12, and 14. 
 
 AHALYSIS. By inspection, we perceive that 10= 10 = 2x5 
 
 2x5; 12=2x6, and 14=2x7. The factor 2, is com- 12 = 2x6 
 
 mon to each of the given numbers, and is therefore 14 = 2 X 7 
 a common divisor of them. (Art. 123.) Hence, the Ans. 2. 
 
 RULE. Resolve each of the given numbers into two 
 factors, one of which is common to them all. 
 
 Find common divisors of the following numbers : 
 
 2. 8, 1 6, 20, and 24. Ans. 2 and 4. 
 
 3. 12, 15, 1 8, and 30. 6. 21, 28, 35, 49, 63. 
 
 4. 36, 48, 96, and 108. 7. 20, 30, 70, and 100. 
 
 5. 42, 54, 66, and 132. 8. 60, 75, 120, and 240. 
 9. 16, 24, 40, 64, 116, 120, 144, 168, 264, 1728. 
 
 GREATEST COMMON DIVISOR. 
 
 125. The Greatest Common Divisor of two or 
 
 more numbers, is the greatest number that will divide 
 each of them without a remainder. Thus, 6 is the greatest 
 common divisor of 18, 24, and 30. 
 
 NOTEa I. A common divisor of two or more numbers is the same 
 as a common factor of those numbers ; and the greatest common 
 divisor of them is their greatest common factor. Hence, 
 
 2. If any two numbers have not a common factor, they cannot 
 have a common divisor greater than i. (Art. 112, n.) 
 
 3. A common divisor is often called a common measure, and the 
 greatest common divisor, the greatest common measure. 
 
 123. A common divisor? 125. Greatest common divisor? Note. A common 
 divisor is the same as what ? Often called what ?
 
 COMMON DIVISOBS. 
 
 FIRST METHOD. 
 
 126. To find the Greatest Common Divisor by continued 
 Divisions. 
 
 i. What is the greatest common divisor of 28 and 40 ? 
 
 ANALYSIS. If we divide the greater number 
 by the less, the quotient is i, and 12 remainder. OPERATION. 
 Next, dividing the first divisor 28, by the first 28)40(1 
 remainder 12, the quotient is 2, and 4 remainder. 28 
 
 Again, dividing the second divisor by the second ~\ n/ 
 
 remainder 4, the quotient is 3, and o rem. The 
 last divisor 4, is the greatest common divisor. 
 
 Demonstration. We wish to prove two points : 4) 1 2 (3 
 
 ist, That 4 is a common divisor of the given num- - 1 2 
 
 bers. 2d, That it is their greatest common divisor. 
 
 First. We are to prove that 4 is a common divisor of 28 and 40. 
 By the last division, 4 is contained in 12, 3 times. Now, as 4 is a 
 divisor of 12 ; it is also a divisor of the product of 12 into 2, or 24. 
 (Art. no, Prop. 14.) Next, since 4 is a divisor of itself and 24, it 
 must be a divisor of the sum of 4 + 24, or 28, which is the smaller 
 number. (Prop. 15.) For the same reason, since 4 is a divisor of 12 
 and 28, it must also be a divisor of the sum of 12 + 28, or 40, which 
 is the larger number. Hence, 4 is a common divisor of 28 and 40. 
 
 Second. We are now to prove that 4 is the greatest common divisor 
 of 28 and 40. If the greatest common divisor of these numbers is 
 not 4, it must be either greater or less thau 4. But we have shown 
 that 4 is a common divisor of the given numbers; therefore, no 
 number less than 4 can be the greatest common divisor of them. 
 The assumed number must therefore be greater than 4. By supposi- 
 tion, this assumed number is a divisor of 28 and 40; hence, it must 
 be a divisor of their difference 4028 or 12. And as it is a divisor of 
 12, it must also divide the product of 12 into 2 or 24. Again, since 
 the assumed number is a divisor of 28 and 24, it must also be a di- 
 visor of their difference, which is 4 ; that is, a greater number will 
 divide a less without a remainder, which is impossible. (Art. 81, 
 Prin. 2.) Therefore, 4 must be the greatest common divisor of 28 
 and 40. Hence, the 
 
 126. Explain Ex. i. Prove that 4 is the greatest common divisor of 28 and 40. 
 Enle i Note. If there are more thau two numbers, how proceed ? If tb-e Uet 
 divisor is i, then what? What is the greatest common divisor of two or 
 prime numbers, or numbers prime to each other ?
 
 COMMON DIVISOES. 93 
 
 RULE. Divide the greater number by the less; then 
 divide the first divisor by the first remainder, the second 
 divisor by the second remainder, and so on, until nothing 
 remains; the last divisor will be the greatest common 
 divisor. 
 
 NOTES. i. If there are more than two numbers, begin with the 
 smaller, and find the greatest common divisor of two of them ; then 
 of this divisor and a third number, and so on, until all the numbers 
 have been taken. 
 
 2. The greatest common divisor of two or more prime numbers, 
 or numbers prime to each other, is i. (Art. 112, n.) 
 
 3. If the last divisor is i, the numbers are prime, or pri-me to each 
 other ; therefore their greatest common divisor is i. Such numbers 
 are said to be incommensurable. (Art. 101, Def. 16.) 
 
 2. What is the greatest com, divisor of 48, 72, and 108 ? 
 
 3. What is the greatest common divisor of 72 and 120? 
 
 SECOND METHOD. 
 
 127. To find the greatest Common Divisor by Prime 
 Factors. 
 
 4. What is the greatest common divisor of 28 and 40 ? 
 ANALYSIS. Resolving the given numbers into 
 
 OPERATION. 
 
 factors common to both, we have 28=2x2x7, 
 
 XT *i. j f ^i. 2o = 2 X 2 X 7 
 
 and 40=2x2x10. Now the product of these 
 
 factors, viz., 2 into 2, gives 4 for the greatest 4 O = 2 x 2 x IO 
 
 common divisor, the same as by the first method. Ans. 2x2 = 4. 
 
 5. What is the greatest com. div. of 30, 45, and 105 ? 
 
 ANALYSIS. Setting the numbers in a hori- OEEKATIOH. 
 
 zontal line, we divide by any prime number, as 3)30? 45? l5 
 3, that will divide each without a remainder, e^io T< T? 
 
 and set the quotients under the corresponding 
 numbers. Again, -dividing each of these quo- ' * 
 
 tieuts by the prime number 5, the new quotients 3 * 5 ~ *5> 
 2, 3, and 7, are prime, and have no common 
 factor. Therefore, the product of the common divisors 3 into 5, or 
 15, is the gxaatest common divisor. (An. no, Prop. 16) Hence, the 
 
 127. What is the rule for the second method? Note. What is the object of 
 placing the numbers in a horizontal line ?
 
 COMMON DIVISORS. 
 
 RULE. I. Write the numbers in a horizontal line, and 
 divide by any prime number that will divide each without 
 a remainder, setting the quotients in a line below. 
 
 II. Divide these quotients as before, and thus proceed till 
 *io number can be found that will divide all the quotients 
 without a remainder. The product of all the divisors will 
 be the greatest common divisor. 
 
 NOTES. i. This rule is the same as resolving the given numbers 
 into prime factors, and multiplying together all that are common. 
 
 2. The advantage of placing the numbers in a horizontal line, is 
 that the prime factors that are common, may be seen at a glance. 
 
 3. When required to find the greatest Common divisor of three or 
 more numbers, the second method is generally more expeditious, and 
 therefore preferable. 
 
 4. When the given numbers have only one common factor, that 
 factor is their greatest common divisor. 
 
 5. Two or more numbers may have several common divisors ; but 
 they can have only one greatest common divisor. 
 
 Find the greatest common divisor of the following: 
 
 6. 63 and 42. 14. 10, 28, 40, 64, 90, 32. 
 
 7. 135 and 105. 15. 12, 36, 60, 108, 132. 
 
 8. 24, 36, 72. 16. 16, 28, 64, 56, 160, 250. 
 
 9. 60, 75, 12. 17. 576 and 960. 
 
 10. 75, 125, 250. 18. 1225 and 592. 
 
 11. 42, 54, 60, 84. 19. 703 and 1369. 
 
 12. 72, ioo, 168, 136. 20. 1492 and 1866. 
 
 13. 60, 84, 132, 108. 21. 2040 and 4080. 
 
 22. A merchant had 180 yards of silk, and 234 yards 
 of poplin, which he wished to cut into equal dress pat- 
 terns, each containing the greatest possible number of 
 yards : how many yards would each contain ? 
 
 23. Two lads had 42 and 63 apples respectively : how 
 many can they put in a pile, that the piles shall be equal, 
 and each pile have the greatest possible number ? 
 
 24. A man had farms of 56, 72, and 88 acres respectively, 
 which he fenced into the largest possible fields of the same 
 number of acres : how many acres did he put in each ?
 
 MULTIPLES. 
 
 Multiple is a number which can be dividec 
 by another number, without a remainder. Thus, 12 is a 
 multiple of 4. 
 
 REMAKK. The Term Multiple is also used in the sense 
 of product : as when it is said, " if one number is a di- 
 visor of another, the former is also a divisor of any mul- 
 tiple (product) of the latter." Thus, 3 is a divisor of 6 ; 
 it is also a divisor of 7 times 6, or 42. 
 
 NOTE. Multiple is from the Latin multiplex, having many folds, 
 or taken many times ; hence, a product, 
 
 But every product is divisible by its factors; hence, the terra came 
 to denote a dividend. The former signification is derived from the 
 formation of the number ; the latter, from its divisibility. 
 
 129. A Common Multiple is any number that 
 can be divided by two or more numbers without a re- 
 mainder. Thus, 1 8 is a common multiple of 2, 3, 6, and 9. 
 
 130. A common multiple of two or more numbers may 
 be found by multiplying them together. That is, the pro- 
 duct of two numbers, or any entire number of times their 
 product, is a common multiple of them. (Art. 128, n.) 
 
 NOTES. i. The factors or divisors of a multiple are sometimes 
 called sub-multiples. 
 
 2. A number may have an unlimited number of multiples. For, 
 according to the second definition, every number is a multiple of 
 itself; and if multiplied by 2, the product will be a second multiple ; 
 if multiplied by 3, the product will be a third multiple ; and univer- 
 sally, its product into any ichole number will be a multiple of that 
 number. (Art. 128.) 
 
 Find a common multiple of the following numbers: 
 
 1. 2, 3 and 5. 4. 2, 8 and 10. 7. 13, 7 and 22. 
 
 2. 3, 7 and n. 5. 7, 6 and 17. 8. 19, 2 and 40. 
 
 3. 5, 7 and 13. 6. n, 21 and 31. 9. 17, 10 and 34. 
 
 128. Meaning of the term multiple? 129. What is a common multiple? 
 130. How found? Not*. \Vhat are sub-multiples ? How many multiples has a 
 number T
 
 96 MULTIPLES. 
 
 LEAST COMMON MULTIPLE. 
 
 131. The Least Comm-on Multiple of two or 
 
 more numbers, is the least number that can be divided 
 by each of them without a remainder. Thus, 15 13 the 
 least common multiple of 3 and 5. 
 
 132. A Multiple, according to the first definition, is a 
 composite number. But a composite number contains all 
 the prime factors of each of the numbers which produce 
 it. Hence, we derive the following Principles : 
 
 . i. That a multiple of a number must contain all the prime 
 factors of that number. 
 
 2. A common multiple of two or more numbers must contain all 
 the prime factors of each of the given numbers. 
 
 3. The least common multiple of two or more numbers is the least 
 number which contains all their prime factors, each factor being 
 taken as many times only, as it occurs in either of the given numbers, 
 and no more. 
 
 133. To find the Least Common IMLultiple of two OP 
 
 more numbers.. % 
 
 i. What is the least common multiple of 1 8, 21 and 66 ? 
 
 ist METHOD. We write the num- , 6 t OPERATION. 
 
 bera in a horizontal line, with a comma 2) 18 21 66 
 
 between them. Since 2 is a prime fac- 
 tor of one or more of the given num- 
 
 bers, it must be a factor of the least 3> 7> * 
 
 common multiple. (Art. 132, Prin. i.) 2x3x3x7x11 = 13801 
 We therefore divide by it, setting the 
 
 quotients and undivided numbers in a line below. In like manner, we 
 divide these quotients and undivided numbers by the prime number 
 3, and set the results in another line, as before. Now, as the numbers 
 in the third line are prime, we can carry the division no further ; 
 for, they have no common divisor greater than i. (Art. 101.) 
 Hence, the divisors 2 and 3, with the numbers in the last line, 
 3, 7 and n, are all the prime factors contained in the given numbers, 
 and each is taken as many times as it occurs in either of them. 
 
 131. What !s the least common multiple ? 132. What principles are derived?
 
 MULTIPLES. 97 
 
 Therefore, the continued product of these factors, 2x3x3x7x11, 
 or 1386, is the least common multiple required. 
 
 2d METHOD. Resolving each num- ad OPKRATIOK. 
 
 ber into its prime factors, we have 1 8 = 2 X 3 X 3 
 
 18=2x3x3, 21=3x7, and 66=2x3 21 = 3x7 
 
 xii. Now, as the least common mul- 66 =2x3x11 
 
 tiple must contain all the prime fac- 
 
 tors of the given number, it must con- 2X3XIIX7X 3 = 1386 
 tain those of 66, viz., 2x3x11; we 
 
 therefore retain these factors. Again, it must contain the prime 
 factors of 21, which are 3x7, and of 18, which are 2x3x3, each 
 being taken as many times as it is found in either of the given 
 numbers. But 2 is already retained, and 3 has been taken once; we 
 therefore cancel the 2 and one of the 33, and retain the other 3 and 
 the 7. The continued product of these factors, viz., 2x3x11x7x3, 
 is 1386, the same as before. Hence, the 
 
 RULE. I. Write the numbers in a horizontal line, and 
 divide by any prime number that will divide two or more 
 of them without a remainder, placing the quotients and 
 numbers undivided in a line below. 
 
 II. Divide this line as before, and thus proceed till no 
 two numbers are divisible by any number greater than i. 
 The continued product of the divisors and numbers in the 
 last line will be the anszver. 
 
 Or, resolve the numbers into their prime factors ; mid- 
 tiply these factors together, taking each the greatest number 
 of times it occurs in either of the given numbers. TJie 
 product will be the answer. 
 
 NOTES. i. These two methods are based upon the same principle, 
 viz. : that the least common multiple of two or more numbers is the 
 least number which contains all their prime factors, each factor 
 being taken as many times only, as it occurs in either of the given 
 numbers. (Art. 132, Prin. 3.) 
 
 2. The reason we employ prime numbers as divisors is because the 
 given numbers are to be resolved into prime factors, or factors prime 
 
 133. How find the least common multiple ? Note. Upon what principle are these 
 two methods based ? Whr employ prime numbers for divisors in the first 
 method? Why write the numbers in a horizontal line? What is the second 
 method ? Advantage of it ? How may the operation be shortened ? When the 
 given numbers are prime, or prime to each other, what U to be done ?
 
 98 MULTIPLES. 
 
 to each other. If the divisors were composite numbers, they would 
 be liable to contain factors common to some of the quotients, or 
 numbers in the last line ; and if so, their continued product would 
 not be the least common multiple. 
 
 3. The object of placing the numbers in a horizontal line, is to 
 resolve all tho numbers into prime factors at the same lime. 
 
 4. The chief advantage of the second method lies in the distinct- 
 ness with which the prime factors are presented. The former is the 
 more expeditious and less liable to mistakes. 
 
 5. The operation may often be shortened by cancelling any num- 
 ber which is a factor of another number in the same line. (Ex. 2.) 
 When the given numbers are prime, or prime to each other, they 
 have no common factors to be rejected ; consequently, their con- 
 tinued product will be the least common multiple. Thus, the least 
 common multiple of 3 and 5 is 15 ; of 4 and 9 is 36. 
 
 2. Find the least common multiple of 5, 6, 9, 10, 21. 
 
 ANALYSIS. In the first line 5 2)5, 6, 9, 10, 21 
 
 is a factor of 10, and is therefore \ ~ ~~ 
 
 cancelled ; in the second, 3 is a 
 iactor of 9, and is also cancelled. 3> 5> 7 
 
 The product of 2x3x3x5x 7=630, the answer required. 
 
 Find the least common multiple of the following : 
 
 3. 8, 12, 16, 24. 10. 36, 48, 72, 96. 
 
 4. 14, 28, 21, 42. ii. 42, 68, 84, 108. 
 
 5 36, 24, 48, 60. 12. 120, 144, 1 68, 216. 
 
 6. 25, 40, 75, 100. 13. 96, 108, 60, 204. 
 
 7. 1 6, 24, 32, 40. 14. 126, 154, 280, 560. 
 
 6 22, 33, 55, 66. 15. 144, 256, 72, 300. 
 9. 30, 40, 60, 80. 16. 250, 500, 1000. 
 
 17. Investing the same amount in each, what is the 
 smallest sum with which I can buy a whole number of 
 pears at 4 cents, lemons at 6, and oranges at 10 cents ? 
 
 1 8. A can hoe 16 rows of corn in a day, B 18, C 20, and 
 D 24 rows : what is the smallest number of rows that will 
 keep each employed an exact number of days ? 
 
 19. A grocer has a 4 pound, 5 pound, 6 pound, and a 
 1 2 pound weight : what is the smallest tub of butter that 
 can be weighed by each without a remainder ?
 
 FRACTIONS. 
 
 134. A Fraction, is one or more of the equal parts 
 into which a unit is divided. 
 
 The number of these parts indicates their name. Thus, 
 when a unit is divided into two equal parts, the parts are 
 called halves; when into three, they are called thirds; when 
 into fou r, fourths, etc. Hence, 
 
 135. A half is one of the two equal parts of a unit ; a 
 third is one of the three equal parts of a unit ; a fourth, 
 one of the four equal parts, etc. 
 
 NOTE. The term fraction is from the Latin frango, to break. 
 Hence, Fractions are often called Broken Numbers. 
 
 136. The value of these equal parts depends, 
 First. Upon the magnitude of the unit divided. 
 Second. The number of parts into which it is divided. 
 
 ILLUSTRATION. ist. If a large and & small apple are each divided 
 into two, three, four, etc., equal parts, it is plain that the parts of the 
 former will be larger than the corresponding parts of the latter. 
 
 2d. If one of two equal apples is divided into two equal parts, and 
 the other into four, the parts of the first will be twice as large as 
 those of the second ; if one is divided into two equal parts, the other 
 into six, one part of the first will be equal to three of the second, etc. 
 Hence, 
 
 NOTE. A half is twice as large as a fourth, three times as large as 
 a sixth, four times as large as an eighth, etc. ; and generally, 
 
 The greater the number of equal parts into which the unit is di- 
 vided, the less will be the value of each part. Conversely, 
 
 The less the number of equal parts, the greater will be the value 
 of each part.' 
 
 134. What is a fraction ? What does the number of parts indicate ? 135. What 
 Is a half? A third? A fonrth? A tenth? A. hundredth ? 136. Upon what doei 
 the size of these parts depend ? 99
 
 100 FRACTIONS. 
 
 137. Fractions are divided into two classes, common and 
 decimal. 
 
 A Common Fraction is one in which the unit is 
 divided into any number of equal parts. 
 
 138. Common fractions are expressed by figures writ- 
 ten above and below a line, called the numerator and 
 denominator; as \, f, -j%. 
 
 139. The Denominator is written lelow the line, 
 and shows into how many equal parts the unit is divided. 
 It is so called because it names the parts; as halves, thirds, 
 fourths, tenths, etc. 
 
 The Numerator is written above the line, and 
 shows how many parts are expressed by the fraction. It 
 is so called because it numbers the parts taken. Thus, iu 
 the fraction f, four is the denominator, and shows that 
 the unit is divided into four equal parts ; three is the nu- 
 merator, and shows that three of the parts are taken. 
 
 140. The Terms of a fraction are the numerator and 
 denominator^ 
 
 NOTE. Fractions primarily arise from dividing a single unit into 
 equal parts They are also used to express a part of a collection of 
 units, and apart of & fraction itself; as i half of 6 pears, i third of 
 3 fourths, etc. . But that from which they arise, is always regarded 
 as a whole, and is called the Unit or Base of the fraction. 
 
 141. Common fractions are usually divided into proper, 
 improper, simple, compound, complex, and mixed numbers. 
 
 1. A Proper Fraction is one whose numerator is 
 less than- the denominator; as -, , f. 
 
 2. An Improper Fraction is one whose numerator 
 equals or exceeds the denominator ; as , f . 
 
 3. A Simple Fraction is one having but one 
 
 137. Into how many classes arc fractions divided? A common fraction! 
 > 38. How expressed ? m. What does the denominator show ? Why so called? 
 The numerator ? Why so called ? 140. What are the terms of a fraction ?
 
 FRACTION'S. 107 
 
 numerator and one denominator, each of which is a whole 
 number, and may be proper or improper ; as f , |. 
 
 4. A Compound Fraction is a fraction of a frac- 
 tion ; as | of f . 
 
 5. A Complex Fraction is one which has a frac' 
 
 tional numerator, and an integral denominator; as, -, . 
 
 ( 4 *> 
 
 REMARK. Those abnormal expressions,^&v\ngfnictionaljdenQm-^'' 
 
 inators, commonly called complex fractions, do not strictly come under 
 the definition of a fraction. For, a fraction is one or more of the 
 equal parts into which a unit is divided. But it cannot properly be 
 said, that a unit is divided into s^ths, 4|dths, etc. ; that is, into 3^ 
 equal parts, 4! equal parts, etc. They are expressions denoting divi- 
 sion, having & fractional divisor, and_should be treated as such. 
 
 6. A Mixed Nutnbe^\s> a whole number and a frac- 
 tion expressed together/fas, 5 f , 344-j . 
 
 NOTE. The primary idea of a fraction is a part of a unit. Hence, 
 a fraction of less value than a unit, is called a proper fraction. 
 
 All other fractions are called improper, because, being equal to 01 
 greater than a unit, they cannot be said to be a part of a unit. 
 
 Express the following fractions by figures: 
 
 1. Two fifths. 6. Thirteen twenty-firsts. 
 
 2. Four sevenths. 7. Fifteen ninths. 
 
 3. Three eighths. 8. Twenty-three tenths. 
 
 4. Five twelfths. 9. Thirty-one forty-fifths. 
 
 5. Eleven fifteenths. 10. Sixty-nine huudredths. 
 
 11. 112 two hundred and fourths. 
 
 12. 256 five hundred and twenty-seconds. 
 
 13. Explain the fraction . 
 
 ANALYSIS. \ denotes \ of i ; that is, one such part as is obtained 
 by dividing a unit into 4 equal parts. The denominator names the 
 parts, and the numerator numbers the parts taken. It is common 
 because the unit is divided into any number of parts taken at ran- 
 dom ; proper, because its value is less than i ; and simple, because it 
 has but one numerator and one denominator, each of which is a 
 whole number. 
 
 141. Into what are common fractions divided ? A proper fraction ? Improper? 
 Simple ? Compound ? Complex ? A mixed number ?
 
 102 FEACTIONS. 
 
 14. Explain the fraction . 
 
 ANALYSIS. J denotes J of i, or \ of 3. For, if 3 equal lines are 
 each divided into 4 equal parts, 3 of these parts will be equal to J of 
 i line, or \ of the 3 lines. It is common, etc. (Ex. 13.) 
 
 Read and explain the following fractions : 
 
 15. f. 20. 4^-. 25. 1 6. 
 
 16. $off. 21. |f 26. zSfJ. 
 
 17. -J-J-. 22. ff. 27. | Of | Of f 
 
 1 8. fofif 23. If. 28. I off of 2f 
 
 ii M_ IX ^ 
 
 NOTE. The 2ist and 22d sliould be read, "68 twenty -first*," 
 " 85 thirty -seconds," and not 68 twenty-on.es, 85 thirty-twos. 
 
 142. Fractions, we have seen, arise from division, the 
 numerator being the dividend, and the denominator the 
 divisor. (Art. 64, n.) Hence, 
 
 The value of a fraction is the quotient of the numerator 
 divided by the denominator. Thus, the value of i fourth 
 is i ~-4 f or ; of 6 halves is 6H-2, or 3; of 3 thirds is 
 3-^3>or i. 
 
 143. To find a Fractional Part of a number. 
 
 i. "What is i half of 12 dimes? 
 
 ANALYSIS. If 12 dimes are divided into 2 equal parts, i of these 
 parts will contain 6 dimes. Therefore, J ^ of 12 dimes is 6 dimes. 
 
 NOTE. The solution of this and similar examples is an illustra- 
 tion of the second object or office of Division. (Art. 63, b.) 
 
 4. What is ^ of 36 ? | of 45 ? of 60 ? | of 63 ? 
 
 5. What is | of 48 ? of63? ^ of 190? ^ of 132? 
 
 6. What is f of 12 dimes? 
 
 ANALYSIS. 2 thirds are 2 times as much as j. But i third of 12 
 is 4. Therefore, of 12 dimes are 2 times 4, or 8 dimes. Hence, the 
 
 142. From what do fractions arise ? Which part is the dividend ? Which the 
 ot'isor ? What is the value of a fraction >
 
 FRACTIONS. 103 
 
 RULE. Divide the given number by the denominator, and 
 multiply the quotient by the numerator 
 
 To find , divide the number by 2. 
 To find |, divide the number by 3. 
 To find f, divide by 3, and multiply by 2, etc. 
 
 7. What is | of 56 ? Ans. 
 
 8. What is f. of 30 apples ? Of 45 ? Of 60 ? 
 ^9. What is of 42 ? } of 56 ? f of 72 ? 
 
 foTWhat is f of 45 ? -^ of 50 ? ^ of .66-?^ T 7 7 of 108 ? 
 
 GENERAL PRINCIPLES OF FRACTIONS. 
 
 144. Since the numerator and denominator have the 
 tame relation to each other as the dividend and divisor, it 
 follows that the general principles established in division 
 are applicable to fractions. (Art. 81.) That is, 
 
 1. If the numerator is equal to the denominator, the 
 value of the fraction is i. 
 
 2. If the denominator is greater than the numerator, the 
 value is less than i. 
 
 3. If the denominator is less than the numerator, the 
 ralue is greater than i. 
 
 4. If the denominator is i, the value is the numerator. 
 
 5. Multiplying the numerator, multiplies the fraction. 
 
 6. Dividing the numerator, divides the fraction. 
 
 7. Multiplying the denominator, divides the fraction. 
 
 8. Dividing the denominator, multiplies the fraction. 
 
 9. Multiplying or dividing both the numerator and de- 
 nominator by the same number does not alter the value of 
 the fraction. 
 
 144. If the numerator Is equal to the denominator, what Is the value of th 
 fraction ? If the denominator Is the greater, what ? If less, what ? If the de- 
 nominator Is i, what ? What Is the effect of multiplying the numerator ? Divid- 
 ing the numerator f Multiplying the denominator ? Dividing the denominator ?
 
 104 FRACTIONS. 
 
 REDUCTION OF FRACTIONS. 
 
 145. Reduction of fractions is changing their terms, 
 without altering the value of the fractions. (Art. 142.) 
 
 CASE I. 
 
 146. To Reduce a Fraction to its Lowest Terms. 
 
 DBF. The Lowest Terms of a fraction are the 
 smallest numbers in which its numerator and denominator 
 can be expressed. (Art. 101, Def. 7.) 
 
 Ex. i. Reduce ff to its lowest terms. 
 
 ANALYSIS. Dividing both terms of a fraction ist METHOD. 
 
 by the same number does not alter its value. ,24 12 
 
 (Art. 144, Prin. 9.) Hence, we divide the given 2 ' ^2 7(5 ~ 
 
 terms by 2, and the terms of the new fraction I2 ? 
 
 by 4 ; the result is 1. But the terms of the frac- 4)~~7 == ~ Ans. 
 tion J are prime to each other ; therefore, they 
 are the lowest terms in which f -J can be expressed. 
 
 Or, if we divide both terms by their greatest *L XETHOD. 
 common divisor, which is 8, we shall obtain the 24 3 
 same result. (Art. 126.) Hence, the 3^ = 4 ' 
 
 RULE. Divide the numerator and denominator con- 
 tinually by any number that will divide both without a re- 
 mainder, until no number greater than i will divide them. 
 
 Or, divide both terms of the fraction by their greatest 
 common divisor. (Art. 126.) 
 
 NOTES. i. It follows, conversely, that a fraction is reduced to 
 higher terms, by multiplying the numerator and denominator by a 
 common multiplier. Thus, =H> both terms being multiplied by 8. 
 
 2. These rules depend upon the principle, that dividing both terms 
 of a fraction by the same number does not alter its value. When the 
 terms are large, the second method is preferable. 
 
 145. What is Reduction of Fractions* 146. What are the lowest terms of a 
 fraction How reduce a fraction to its lowest terms f
 
 SEDUCTION OF FRACTIONS. 105 
 
 Eeduce the following fractions to their lowest terms : 
 
 2. ||. 9. $f 1 6. iff. 23. 
 
 3- M- I0 - iH- J 7- fj- 24. 
 
 4- if- ". %H. 1 8. f^f 25. |f. 
 5. ff. 12. f|f. 19. ,flfr. 26. #ft. 
 
 6- H- 13- -Hi- 20. #$. 27. Htf. 
 
 7- ft- 14- T 7 ^- 21. -fl&. 28. i^. 
 
 8- & 15- if- 22. -#&. 29. -ftV 
 
 CASE II. 
 
 147. To reduce an Improper Fraction to a Whole or 
 Mixed Number. 
 
 i. Eeduce ^ 5 - to a whole or mixed number. 
 
 ANALYSIS. Since 4 fourths make a unit or i, 45 4)45 
 fourths will make as many units as 4 is contained T A 
 
 times in 45, which is n^. Hence, the 
 
 ETJLE. Divide the numerator ~by the denominator. 
 
 NOTES. i. This rule, in effect, divides both terms of the fraction 
 by the same number ; for, removing the denominator cancels it, and 
 cancelling the denominator divides it by itself. (Art. 121.) 
 
 2. If fractions occur in the answer, they should be reduced to the 
 lowest terms. 
 
 Eeduce the following to whole or mixed numbers : 
 2 . i-Jl. 8. -ip-. 14. -^V-- 20. 
 
 3- *$* 9- - 4 i 4 2 3 "- J 5- "Wr- 
 
 4. 1 1 -2-. 10. -^-. l6. T ggg. 22. -3J 
 
 5- /- ii. W- 17- -ViW 23. ^ 
 
 6. -!. 12. ^ 8 /. 1 8. -^fl^. 24 
 
 7- W^ J 3- Hf J 9- Hfl^- 25, 
 
 26. Iix ^f fir 5 - of a pound, how many pounds ? 
 
 27. In VzVW- f a dollar, how many dollars ? 
 
 28. In 25 -^p^- of ?. y^ar, how many years? 
 
 147, Bow reduce an improper fraction to a whole or mixed number T
 
 106 DEDUCTION OF FRACTIONS. 
 
 CASE III. 
 
 148. To reduce a Mixed Number to an Improper Fraction. 
 
 i. Reduce 8f to an improper fraction. 
 
 ANALYSIS. Since there are 7 sevenths in a unit, there ga 
 must be 7 times as many sevenths in a number as there _ 
 are units, and 7 times 8 are 56 and 3 sevenths make ty. ^ 
 Therefore, 8*=^. *f Am. 
 
 Or thus: Since i=$, 8 = 8 times \ or &P-, and * make V. In the 
 operation we multiply the integer by the given denominator, and to 
 the product add the numerator. Hence, the 
 
 RULE. Multiply the whole number by the given denomi- 
 nator ; to the product add the numerator, and place the 
 sum over the denominator. 
 
 REMAKK. A whole number may be reduced to an improper 
 fraction by making I its denominator. Thus, 4=f . (Art. 81.) 
 
 Reduce the following to improper fractions : 
 
 2. isf 6. i 4 5ff 10. i573f 14- 
 
 3. i8f 7- i7H- " 2 56if 15- 
 4- 35f 8 - 2 95iff- 12. 3 6 4 of. 1 6. 
 5. 81^. 9. 806^. 13. 8624^- 17- 
 
 1 8. In 263-^5- pounds how many sixteenths ? 
 
 19. Change 641^ mile to fortieths of a mile. 
 
 CASE IV. 
 
 149. To reduce a Compound Fraction to a Simple one. 
 
 i. Reduce | of $ to a simple fraction. 
 
 ANALYSIS. 3 fourths of f = 3 times i fourth of f. iBt METHOD. 
 
 Now i fourth of J is ft ; for multiplying the de- 326 
 
 nominator divides the fraction: and 3 fourths of 4 X 3~~72 
 | = 3 times -ft or & ; for multiplying the numerator 
 
 multiplies the fraction. (Art. 144.) Reduced to its A -_* 
 
 lowest terms i 6 2 =. ' 12 2 
 
 148. How reduce a mixed number fQ an Improper fraction T Bern, A wholq 
 pnnrt>er to an Improper ft-action T
 
 REDUCTION OF FRACTIONS. 107 
 
 Or, since numerators are dividends and *d METHOD. 
 
 denominators divisors, the factors 2 and 3, i, $ 2 i 
 
 which are common to both, may be cancelled. 2 7 x ~~ Ans. 
 (Art. 144, Prin. 9.) The result is . Heuce, the 
 
 RULE. Cancel the common factors, and place the product 
 of the factors remaining in the numerators over the product 
 of those remaining in the denominators. 
 
 NOTES. i. Whole and mixed numbers must be reduced to improper 
 fractions, before multiplying the terms, or cancelling the factors. 
 
 2. Cancelling the common factors reduces the result to the lowest 
 terms, and therefore should be employed, whenever practicable. 
 
 3. The numerators being dividends, may be placed on the right 
 of a perpendicular line, and the denominators on the left, if pre- 
 ferred. (Art. 122, Bern.) 
 
 4. The reason of the rule is this: multiplying the numerator of 
 one fraction by the numerator of another, multiplies the value of the 
 former fraction by as many units as are contained in the numerator 
 of the latter ; consequently the result is as many times too large as 
 there are units in the denominator of the latter. (Art. 144, Prin. 5.) 
 This error is corrected by multiplying the two denominators to- 
 gether. (Art. 144, Prin. 7.) 
 
 2. Reduce f of of ~fa of 2 J to a simple fraction. Ans. $. 
 Reduce the following to simple fractions: 
 
 S-fof-ft- 9. I of A of 60. 15- A of ft of ft. 
 
 4. i of f . 10. | of H of ffr. 1 6. | of f of | of f 
 
 5. 4 of ^. ii. & of H of IJ. 17. H of 1J of 65}. 
 
 6. | of -H. 12. Jf of | of |f. 1 8. fj- of ff of 8 4 J 
 
 7. ^ of . 13. || of A of 45- 19- I of 8f of 
 
 8. f of tV. 14. i of f of ||. 20. of | of 
 
 21. Reduce | of 4$ of $ of 9! to a simple frac- 
 tion. 
 
 22. To what is | of fj of 3^ bushels equal? 
 
 23. To what is f of 4^ of -fo of a yard equal ? 
 
 149. How reduce a compound fraction to a simple one T Note. What must be 
 done with wbole aod mjjed oumberi T What i the advantage of cancelling r
 
 108 BEDUCTION OF FEACTIONS. 
 
 CASE V. 
 150. To reduce a Fraction to any required Denominator. 
 
 1. Keduce f to twenty-fourths. 
 
 IST ANALYSIS. 8 is contained in 24, 3 times ; there- 24-1-8:= 3 
 fore, multiplying both terms by 3, the fraction be- 6x3 18 
 comes if, and its value is not altered. (Art. 144.) 8 X V = ~ 
 
 REM. If required to reduce to fourths, we should divide both 
 terms by 2, and it becomes J. (Art. 144.) 
 
 20 ANALYSIS. Multiplying both terms of f 6x24=144 
 
 by 24, the required denominator, we have |Jf. g ^ , . _ ~7T 
 (Art. 144, Prin. 9.) Again, dividing both terms of } - * 
 
 by 8, the given denominator, we have i, the same 
 
 as before. Hence, the 192-7-8 124 
 
 EULE. Multiply or divide loth terms of the fraction 
 by such a number as will make the given denominator 
 equal to the required denominator. 
 
 Or, Multiply both terms of the given fraction by the 
 required denominator ; then divide both terms of the re- 
 sult by the given denominator. (Art. 144, Prin. 9.) 
 
 NOTES. i. The multiplier required by the ist method is found 
 by dividing the proposed denominator by the given denominator. 
 
 2. When the required denominator is neither a multiple, nor an 
 exact divisor of the given denominator, the result will be a complex 
 
 3V 4* 
 
 fraction. Thus, ^ reduced to tenths= ; reduced to sevenths= 
 
 Change the following to the denominator indicated : 
 
 2. -^ to fifty-seconds. 7. f to 246ths. 
 
 3. -fa to sixtieths. 8. %\ to 288ths. 
 
 4. -jUj- to eighty-eighths. 9. f& to 36oths. 
 
 5. - to i44ths. 10. T^ to loooths. 
 
 6. to 2O4ths. ii. ^ looooths. 
 
 12. Keduce f to fourths. Ans. . 
 
 1 3. Reduce ^ to sixths. 15. Reduce f to twenty-sevenths. 
 14. Reduce to thirds. 16. Reduce -f to fourths. 
 
 ijo. How reduce a fraction to any required den.pminticm ?
 
 REDUCTION OF FRACTIONS. 109 
 
 151. To reduce a Whole Number to a Fraction having a 
 given Denominator. 
 
 1. Eeduce 7 to fifths. 
 
 ANALYSIS. ist. Since there are 5 fifths in a unit, an> number 
 must contain 5 times as many fifths as units ; and 5 time& 7 are 35. 
 Therefore 7= a . 
 
 Or, 2d. Since i=f, 7 must equal 7 times , 
 or 3 s a . In the operation 7 is both multiplied OFBKATIOH. 
 
 and divided by 5; hence, its value is not 7=^(7 x 5) -1-5 -5*. 
 altered. (Art. 86, a.) Hence, the 
 
 RULE. Multiply the whole number by the given denomi* 
 nator, and place the product over it. 
 
 2. Change 7 to a fraction having 9 for its denominator. 
 
 3. Change 63 to 5ths. 9. 468 to 76ths. 
 
 4. Reduce 79 to 7ths. 10. 500 to 87ths. 
 
 5. Reduce 83 to gths. n. 1560 to hundredths. 
 
 6. Reduce 105 to i6ths. 12. 2004 to thousandths. 
 
 7. Reduce 217 to 2oths. 13. 500 to ten-thousandths. 
 
 8. Reduce 321 to 49ths. 14. 25 to millionths. 
 
 CASE VI. 
 152. To reduce a Complex Fraction to a Simple One, 
 
 i. Reduce the complex fraction to a simple one. 
 
 
 
 ANALYSIS. The denominator of a fraction, we OPBKATIOH. 
 have seen, is a divisor. Hence, the given complex 3! 
 fraction is equivalent to 3^-2-5. (Art. 142.) Now ~ 3T~^~5> 
 3i=J; therefore, 3i--5=-5-5. (Art. 148.) But ? ,l = ip . 
 ^^-5= $ ; for, dividing the numerator by any num- 
 ber, divides the fraction by that number. (Art. '7~ r 5~3r- 
 144, Prin. 6.) Therefore, f is the simple fraction Ans. f. 
 required. 
 
 Or, multiplying the denominator by 5, divides the fraction, and 
 we have !=$, the same as before. (Prin. 7.) 
 
 150. Upon what principle IB this mle founded ? 151. How redoce ft wholp 
 number to & fraction having a given denominator ?
 
 '10 RRDUCTIOK OP FRACTIONS. 
 
 A 
 
 2. Reduce the complex fraction - to a simple one. 
 
 ANALYSIS.' Reasoning as before, the given fraction OPIBJ non. 
 
 ts equivalent to f -r-y. But we cannot divide the 4 
 
 numerator of the fraction 4 by 7 without a remainder, ~ = i~^~ 7 5 
 we therefore multiply the denominator by it ; for, 
 
 multiplying the denominator, divides the fraction; *~7 Ts- 
 and ^-T-7=2\, the simple fraction required. (Art. 144, Ans. ^ 8 . 
 Prin. 7.) Hence, the 
 
 KULE. I. Reduce the numerator of the complex fraction 
 to a simple one. 
 
 II. Divide its numerator by the given denominator. 
 Or, Multiply its denominator by the given denominator. 
 
 NOTES. i. When the numerator can be divided without a remain- 
 the former method is preferable. When this cannot be dona 
 tbe latter should be employed. 
 
 s. After the numerator of the complex fraction is reduced to a 
 simple one, the factors common to it and the given denominator, 
 should be cancelled. 
 
 If this case is deemed too difficult for beginners, it may be 
 fitted till review. 
 
 (For the method of treating those expressions which have fractional 
 'mominatora, see Division of Fractions, p. 130.) 
 
 c-s 
 2. Reduce ~ to a simple fraction. 
 
 24 
 SOLUTION. 5$=^ ; and ^-H 24 =->*&, or , Ant. 
 
 Reduce the following complex fractions to simple ones. 
 
 , f . 
 
 8. -Z-. 12. 
 
 6. ^-. 10. 
 
 2 2 
 
 152. How reduce a complex fraction to a simple one? Not*. What should bq 
 done with common factors ?
 
 BEDUCTION OF FBACTIONS. Ill 
 
 CASE VII. 
 153. To reduce Fractions to a Common Denominator. 
 
 DBF. A Common Denominator is one that 
 belongs equally to two or more fractions ; as, f, f, f. 
 
 i. Keduce ^, f , f to a common denominator. 
 
 REMARK. In reducing fractions to a common denominator, it 
 hould be observed that the value of the fractions is not to be altered. 
 Hence, whatever change is made in any denominator, a corresponding 
 change must be made in its numerator. 
 
 ANALYSIS. If each denominator is mul- OPERATION. 
 
 tiplied by all the other denominators, the i 1x3x4 12 
 
 fractions will have a common denominator. ~ = 2 X. 1 X d. == 24* 
 and if each numerator is multiplied into all 
 
 the denominators except its own, the terms __ 2 x 2 x 4 _ J 
 
 of each fraction will be multiplied by the 3 3x2x4 24 
 
 same numbers ; consequently their value , 7 x 2 x ^ 18 
 
 will not be altered. (Art. 144.) The frac- - = - = . 
 
 tions thus obtained are iJ, if, and if. 4 4x2x3 24 
 Hence, the 
 
 RULE. Multiply the terms of each fraction by all the 
 denominators except its own. 
 
 NOTES. i. Mixed numbers must first be reduced to improper 
 fractions, compound and complex fractions to simple ones. 
 
 2; This rule is founded upon the principle, that if both terms of a 
 fraction are multiplied by the same numbers, its value is not altered. 
 
 3. A common denominator, it will be seen, is the product of all the 
 denominators ; hence it is a common multiple of them. (Art. 129.) 
 
 Reduce the following to a common denominator : 
 
 2. | and f 5. $, |, and -ft. 8. f, f, tf, #. 
 
 3. i, |, and f . 6. -f, T 6 T , and A- 9- H> ft, 
 
 4. |, f , and -A, 7- I i, A and ^ 10. -fi ||, 
 
 153. What la a common denominator? How reduce fractions to a common 
 denominator ? Note. What is this rnle based upon ? What ia to be done wlttj 
 mixed numbers, compound and complex fractions ?
 
 112 REDUCTION OF FRACTIONS. 
 
 ii. Find a common denominator of 4, 3^, and of | 
 
 ANALYSIS. 4 = } ; 3$ = ; and $ of = \. Reducing t, 3 fe 
 to a common denominator, they become 3 8 % ^ and f . 
 
 12. Keduce 3^, i|, 2-$-. 15. Reduce 5-$-, \ of 8, {. 
 
 13. Reduce 6, 7-fJ, -,%-, |. 16. Reduce f of \ of 9, i i-J, /p 
 
 14. Reduce f of }, 5$-, f. 17. Reduce 13^, 17, f of f. 
 
 1 8. Find a common denominator of 2^, 5-^ and 2f. 
 
 19. Reduce -^, f, and , to the common denominator 
 100. 
 
 SOLUTIOK. The fraction -&=-&&', =-Mr; and M=\ t a- 
 
 20. Change |, f, and f to 72ds. 
 ^I. Change f, T 7 ^, and { to 96ths. 
 
 22. Reduce f of f and to poths. 
 
 23. Reduce ^ of lof and ^, to 75ths. 
 
 24. Reduce f, -j^, and ^| to i68ths. 
 
 25. Reduce -^5-, f, and fj$, to icooths. 
 
 CASE VIII. 
 
 154. To Reduce Fractions to the Least Common 
 Denominator. 
 
 DEF. A Common Denominator is a common 
 multiple of all the denominators. (Art. 153, n.) Hence, 
 
 The Least Common Denominator is the least 
 common multiple of all the denominators. 
 
 i. Reduce f, ^, and -f| to the least common denomi- 
 nator. 
 
 ANALYSIS. Here are two steps : 1st. To find OPERATION. 
 
 the least common multiple of the denominators ; 3)3> I2 > J 5 
 
 2d. To reduce the given fractions to this denomi- ~^ T r 
 
 nator. The least common multiple of the given 5 x 4 x C 60 
 denominators is 60. (Art. 133.) 
 
 154. What is the least common denominator? How reduce fractions to the 
 least common denominator ? Note. What is to be done with mixed numbers, 
 >und and complex fractions ?
 
 REDUCTION OF .FRACTIONS. 113 
 
 To reduce the given fractions to 6oths, we multiply both terms of 
 | by 20, and it becomes $g. In like manner, if both terms of T 6 ^ are 
 multiplied by 5, it becomes ; and multiplying both terms of |by 
 4, it becomes $$. Therefore, f , ^f, and {% are equal to $g, f $ and 
 |^. Hence, the 
 
 RULE. Find the least common multiple of all the denom- 
 inators, and multiply both terms of each fraction by such a 
 number as will reduce it to this denominator. (Art. 150, n.} 
 
 NOTES. i. Mixed numbers must be reduced to improper frac- 
 tions ; compound and complex fractions to simple ones ; and all frac- 
 tions to the lowest terms, before applying the rule. If not reduced 
 to the lowest terms, the least common multiple of the denominators 
 is liable not to be the least common denominator. (See Ex. 2, 18.) 
 
 2. This rule, like the preceding, is based upon the principle that 
 multiplying both terms of a fraction by the same number does not 
 alter its value. (Art. 144, Prin. 9.) 
 
 2. Reduce 15, 2^, f of f, and & ^ the least common 
 denominator. 
 
 15 7 2 ,$ 2 , 6 i 
 
 ANALYSIS. 1 1 = -, 2-J- = -. - of - = -, and - 
 
 i 3*55 12 2 
 
 Now the least common multiple of the denominators of 
 
 15 7 2 I . . 450 70 12 15 
 
 - - -, is 30. A.ns. . 
 
 I 3 5 2 30 30 30 30 
 
 Reduce the following to least common denominator : 
 
 ? 2 3 5 TT 8 6. 14 C 2 
 
 6" T> 3> 7' TfTS> TT5> 55> J^' 
 
 6. y, , 4^. 14. ^ of 13, 
 
 8. A> 1? f of to. 16. 
 
 10. ^ of 8|, i of 40. 18. ifj, $ffi, H||.
 
 114 ADDITION OF FBACTIONS. 
 
 ADDITION OF FRACTIONS. 
 
 155. Addition of Fractions embraces two classes 
 of examples, viz. : those which have a common denominator 
 and those which have different denominators. 
 
 156. When two or more fractions have a common de- 
 nominator, and refer to the same kind of unit or base, 
 their numerators are like parts of that unit or base, and 
 therefore are like numbers. Hence, they may be added, 
 subtracted, and divided in the same manner as whole num- 
 bers. Thus, f and are 12 eighths, just as 5 yards and 
 7 yards are 12 yards. (Art. 101, Def. 13.) 
 
 157. "When two or more fractions have different denom- 
 inators, their numerators are unlike parts, and therefore 
 cannot be added to or subtracted from each other directly, 
 any more than yards and dollars. (Art. 101, Def. 14.) 
 
 158. To add Fractions which have a Common Denom- 
 inator. 
 
 1. What is the sum of f yard, f yard, and I yard ? 
 
 ANALYSIS. 3 eighths and 5 eighths OPERATION. 
 
 yard are 8 eighths, and 7 are 15 eighths, f + f + i= V , or l| y. 
 < qual to I J yard. (Art. 147.) For, since 
 
 the given fractions have a common denominator, their numerators 
 are like numbers. Hence, the 
 
 KFLE. Add the numerators, and place the sum over the 
 common denominator. 
 
 NOTE. The answers should be reduced to the lowest terms ; and If 
 'improper fractions, to whale or mixed numbers. 
 
 Add the following fractions . 
 
 2. f, f, and f 5. -rtfr, -flfr, -&, and ft%. 
 
 3- A, A> A, and ft. 6. &, rfft, |, and 
 
 4- A. A, A, and H- 7- m> -Hi, m and 
 
 158. How a<W fractions that have a common denominator T
 
 ADDITION OF FRACTIONS. 115 
 
 159. To add Fractions which have Different Denomina- 
 
 tors. 
 
 8. Find the sum of of a pound, f of a pound, and f of 
 a pound. 
 
 ANALYSIS. As these fractions OPBKATION. 
 
 have different denominators, 4 x 5 x 8=160, com. d. 
 
 their numerators cannot be added 1x5x8= 40, 1st nu. 
 
 in their present form. (Art. 157.) 2x4x8=: 64, 2d " 
 
 We therefore reduce them to a 5x4x5 100, 3d " 
 
 common denominator, which is Sum of nu., 204. Hence 
 
 160. Then i=tfft, i=-ftfc, and 4<D 64 100 204 
 
 ' = |. Adding the numerators, 7- + ~7~ + ~ ~ 7~; or l- 
 
 loo loo 160 160 
 and placing the sum over the 
 
 common denominator, we have f$, or i^,, the answer required. 
 For, reducing fractions to a common denominator does not alter 
 their value ; and when reduced to a common denominator, the nu- 
 merators are like numbers. (Arts. 153, 156.) 
 
 Or, we may reduce them to the least common denominator, which 
 is 40, and then add the numerators. (Art. 154.) Hence, the 
 
 EULE. Reduce the fractions to a common denominator, 
 and place the sum of the numerators over it. 
 
 Or, reduce the fractions to the least common denomina- 
 tor, and over this place the sum of the numerators. 
 
 REMARKS. i. The fractional and integral parts of mixed num- 
 bers should be added separately, and the results be united. 
 
 Or, whole and mixed numbers may be reduced to improper frac- 
 tion*, then be added by the rule. (Art. 148.) 
 
 2. Compound and complex fractions must be reduced to simple 
 ones ; then proceed according to the rule. (Ex. 20, 28.) 
 
 (For Addition of Denominate Fractions, see Art. 315.) 
 
 9. What is the sum of 5, 2 J, and io| ? 
 
 ANALYSIS. Reducing these fractions to a common 5 =e 
 
 denominator 12, we have, 5 = 5 ; 2^=2,\; and ioj= 2l = 2 4 
 ioyV, Adding the numerators. 4 twelfths and 9 3 o 
 
 twelfthsare if i-j\. i and 10 are n and 2 are 13 and ___ *~ " 
 
 5 are 18. Ans. i8rV- Or, s=f or ft; 2i=f$ ; and Ans. 18-^ 
 iol=\^. Now f $ + f I + - L ,= W, or i8-,V Ans. 
 
 159. How when they have not T Note. How add whole and mixed numbers ? 
 Compound and complex fractious* f
 
 116 ADDITION OF FRACTION'S. 
 
 Add the following: 
 
 (10.) (ii.) (12.) (13.) 
 
 '* 4* 27-A ft 
 
 4 Si A tt H 
 
 (i S .) (16.) (17.) (18.) (19.) 
 
 19^ 244 207! 175! 450 
 
 47i iH- 62| 207 67^ 
 
 68$ 68^ 49^ 368^ 
 
 20. What is the sum of of f , J- of 7, and of pf ? 
 
 21. What is the sum of f of -f-, |- of f, and f of 7 ? 
 
 22. What is the sum of f of 4^, f of 3, and - of 10^ ? 
 
 23. A beggar received $if from one person, $2^ from 
 another, and $3! from another : how much did he receive 
 from all ? 
 
 24. If a man lays up $43! a month, and his son $27^, 
 how much will both save ? 
 
 25. What is the sum of f , f , J, and f pound ? 
 
 26. A shopkeeper sold 17^ yards of muslin to one cus- 
 tomer, 8^ yards to another, 25^ to another : how many 
 yards did he sell to all ? 
 
 27. A farmer paid $i8f for hay, $45-fV for a cow, $150! 
 for a horse, and $275 for a buggy: how much did he give 
 
 for all? 
 
 .1 fi a 
 
 28. What is the sum of 2f, , and ^. , 
 
 53 4 
 
 ANALYSIS. Re. lucing the complex fractions to simple ones, we 
 
 have, **=l+s=-?e ; - 5 i=V+3=* 5 and -=3-*-4=A, or i Now, 
 53 4 
 
 .,. Add ^ *, and 2Jt 30. Add U, f, and 3. 
 
 23 7 573
 
 SUBTRACTION OF FKACTIONS. 117 
 
 SUBTRACTION OF FRACTIONS. 
 
 160. Subtraction of Fractions embraces tiao 
 classes of examples, viz.: those which have a common 
 denominator, and those which have different denominators. 
 
 161. To subtract Fractions which have a Common 
 
 Denominator. 
 
 1. What is the difference between || and ^f ? 
 ANALYSIS. 13 sixteenths from 15 sixteenths 
 
 ., x , j T-I OPERATION. 
 
 leave 2 sixteenths, the answer required. For, 
 
 since the given fractions have a common denomi- _ __ 3 __ _ 
 
 nator, their numerators, we have seen, are like JO 1 6 1 6 
 numbers. (Art. 156.) Hence, the 
 
 RULE. Take the less numerator from the greater, and 
 place the difference over the common denominator. 
 
 2. From f$ take ft. 5. From fi| take iff 
 
 3. From ff take $f. 6. From f$| take f-|. 
 
 4. From m take ^V 7- From T ^& take ^ 
 
 162. To subtract Fractions which have Different 
 Denominators. 
 
 8. From of a pound, subtract | of a pound. 6 24 
 
 ANALYSIS. Since these fractions have different j 28 
 
 denominators, their numerators cannot be subtracted 3 21 
 
 in their present form. We therefore reduce them to a "7 = To 
 common denominator, which is 28 ; then subtracting as 
 
 above, have g, the answer required. (Art. 153.) Ans 
 
 Hence, the 28 
 
 RULE. Reduce the fractions to a common denominator, 
 and over it place the difference of the numerators. 
 
 161. How subtract fractions that have a common denominator? 162. How 
 when they have different denominators ? Hem. How subtract mixed numbered 
 Compound and complex fractions ? A proper fraction from a whole number T
 
 118 8UBTBACT10K OF FRACTIONS. 
 
 REMARKS. i. The fractional and integral parts of mixed numbers 
 should be subtracted separately, and the results be united. 
 
 Or, they may be reduced to improper fractions, then apply the rule. 
 
 Compound and complex fractions should be reduced to simple ones, 
 and all fractions to their lowest terms. 
 
 2. A. proper fraction may be subtracted from a wJtole number by 
 taking it from a unit; then annex the remainder to the whole 
 number minus 1. 
 
 Or, the whole number may be reduced to a fraction of the same 
 denominator as that of the given fraction ; then subtract according 
 to the rule. (Art. 151.) 
 
 3. The operation may often be shortened by finding the least com 
 mon denominator of the given fractions. (Art. 154.) 
 
 (For subtraction of Denominate Fractions, see Art. 317.) 
 
 9. What is the difference between 1 2\ pounds and 5$ 
 pounds ? 
 
 ANALYSIS. The minuend 12^=12^. Now i2| sJ=6 a , Ans. 
 Or, i2i=^, or 4 4 a ; and 5$=^. Now^ a 4 a =Y. which reduced 
 to a mixed number, equals 6J, the same as before. 
 
 (10.) (II.) (12.) (I 3 .) (I 4 .) 
 
 From I 5% 7i 2 3f 
 
 Take f 3* 5f '5* H 
 
 (15.) (16.) (17.) (18.) (19-) 
 From sff yH 6 f! 9$f 
 
 Take 3 # 4H 3*1 5H 7 T Vs 
 
 20. From a box containing 56^ pounds of sugar, a grocer 
 took out 23! pounds : how many pounds were left in the box? 
 
 21. From a farm containing 165^ acres, the owner sold 
 78^ acres : how much land had he left ? 
 
 22. From 13 subtract f. 
 
 ANALYSIS. ist. Reducing 13 to sevenths, we have I3=V-, and 
 s 7 L 5 = * 7 &, or 12? Ans. 
 
 Or, 2d. Borrowing i from 13, and reducing it to ?ths, we have 
 13=12^, and 12} $=12$, Ans.
 
 8UBTBACTIOH OF FRACTIONS. 119 
 
 23. From 46 take 7^. 24. From 58 take 2of. 
 
 25. From 84} take 41. 2-6. From 150^ take 83. 
 
 27. From no take 7^^-. 28. From 1000 take 999!$ 
 
 29. Subtract of of 3 from | of f of i^. 
 
 ANALYSIS. Reducing i 4 __i 4. *3_3_ 12 
 
 the compound fractions ~ r 7 I *~~2'52~~5~~2O 
 
 to simple ones, then to j j. i I $ i S 
 
 a common denominator, -of -of 3 == of-of- = = 
 
 the subtrahend becomes 34 $414 20 
 
 fs, the minuend ft. ^ j^ 
 
 > = ?(>. Ana. '* 20* 
 
 (3-) (3I-) (32.) (33-) 
 
 From f of | fofft A of 4* -^ of 28 
 Take iof-J f of A f of 3 I of 4 | 
 
 34- From of 62-^ subtract -^ of i6. 
 
 35. A's farm contains 256! acres, B's 431-^ acres: what 
 is the difference in the size of their farms ? 
 
 36. If from a vessel containing 230-^ tons of coal, 119^ 
 tons are taken, how much will there be left ? 
 
 37. A man bought a cask of syrup containing 58$ 
 gallons ; on reaching home he found it contained only 1 7$ 
 gallons : how many gallons had leaked out ? 
 
 38. A lady having $100, paid $8J for a pocket hand- 
 kerchief, $15^ for a dress hat, $46! for a cloak: how 
 much had she left ? 
 
 39. From subtract . 
 
 6 5 
 
 ANALYSIS. Reducing the complex 5$_TT ,_ TT_SS 
 
 fractions to simple ones, then to a ~g~ ^T~ r() ~i2 off 
 
 common denominator, the minuend , 
 
 becomes f$, the subtrahend 5- (Arts, -l^*-:-^ -g= $ 
 
 152, I53-) Now, $ it = S3, or 5 "TTH is 
 
 B, Ant. Ans ' etr 4* 
 
 40. From - take -A 41. From take . 
 
 58 3 22
 
 120 MULTIPLICATION OF FRACTIONS. 
 
 MULTIPLICATION OF FRACTIONS. 
 CASE I. 
 
 163. To multiply a Fraction by a WTiole Number. 
 
 Ex. i. What will 4 pounds of tea cost, at | of a dollar 8 
 pound ? 
 
 ist METHOD. Since i pound costs $%, 4 pounds ist OPERATION. 
 will cost 4 times as much, and 4 times $ are &ff X 4 $^* 
 \ a =$2^, which is the answer required. For, and $-^=$2^ 
 multiplying the numerator, multiplies the frac- 
 tion. (Art. 144 Prin. 5.) 
 
 2d METHOD. If we divide the given de- *d OPERATION. 
 
 nominator by the given number of pounds, $^-7-4 = $^, or $2^ 
 we have f-5-4=$f, or $2^, which is the true 
 
 answer. For, dividing the denominator, multiplies the fraction. 
 (Art. 144, Prin. 8.) Hence, the 
 
 KULE. Multiply the numerator ~by the whole number. 
 Or, divide the denominator by it. 
 
 REMAKES. i. When the multiplicand is a mixed number, th 
 fractional and integral parts should be multiplied separately, and 
 the results be united. 
 
 Or, the mixed number may be reduced to an improper fraction, 
 and then be multiplied as above. 
 
 2. If a fraction is multiplied by its denominator, the product will 
 be equal to its numerator. For, the numerator is both multiplied 
 and divided by the same number. (Art. 86, a.) Hence, 
 
 3. A fraction is multiplied by a number equal to its denominator 
 by cancelling the denominator. Thus, f x 8=5. (Art. 121.) 
 
 4. In like manner, a fraction is multiplied by any factor of its de- 
 nominator by cancelling that factor. 
 
 2. Multiply 2 7 1- by 6. 
 
 ANALYSIS. Multiplying the fraction and integer OPERATION. 
 Separately, we have 6 times = V or 3$ ; and 6 
 times 27 = 162. Now 162 + 3^=165!, the answer. 
 Or, thus : 27|=H a ; and-4* * 6= 8 |, or 165!, Ans. An*. 1 65$ 
 
 163. How multiply a fraction by a whole number ? Upon what doea the first 
 method depend ? The second ? Bern. How multiply a mixed number by a 
 whole one ? How multiply a fraction by a number equal to its denominator 1 
 How by any factor in ite denominator ?
 
 MULTIPLICATION OF FRACTIONS. 121 
 
 (3-) (4-) (5-) (6.) (7-) 
 
 Mult. |f -& 2 3 f 35| 4 8f- 
 
 By 7 9 8 10 12 
 
 (8.) (9.) (10.) (ii.) (12.) 
 
 fi fi M 9*A 85* 
 
 By 48 100 78 26 48. 
 
 13. Multiply fjf- by 239. 
 
 14. What cost 8 barrels of cider, at $7$ a barrel ? 
 
 15. At $25! each, what will 9 chests of tea cost? 
 
 16. What will 25 cows come to, at $48f apiece? 
 
 17. What cost 27 tons of hay, at $29^ a ton ? 
 
 1 8. What cost 35 acres of land, at $45 per acre ? 
 
 19. At $34!: apiece, what cost 50 hogsheads of sugar. 
 
 20. At $45 1 apiece, what will 100 coats come to ? 
 
 21. What cost 6 dozen muffs, at $75^ apiece ? 
 
 CASE II. 
 165. To Multiply a Wliole Number by a Fraction. 
 
 DBF. Multiplying by a Fraction is taking a 
 certain part of the multiplicand as many times as there 
 are like parts of a unit in the multiplier. 
 
 But to find a given part of a number, we divide it into 
 as many equal parts as there are units in the denominator, 
 and then take as many of these parts as are indicated by 
 the numerator. (Art. 143.) That is, 
 
 To multiply a number by , divide it by 2. 
 
 To multiply a number by %, divide it by 3. 
 
 To multiply a number by f, divide it by 4 for , and 
 multiply this quotient by 3 for |, etc. Hence, 
 
 REMARKS. r. Multiplying a whole number by a fraction i? 
 the same as finding a fractional part of a number, which the pupil 
 dhould here review with care. (Art. 143.) 
 
 Dtf. What is it to multiply by a fraction ? How multiply by 1 By } ? By J f
 
 122 MULTIPLICATION OF FRACTIONS. 
 
 2. When the multiplier is i, the product is equal to the multipli- 
 cand. 
 
 When the multiplier is greater than i, the product is greater than 
 t.ie multiplicand. 
 
 When the multiplier is less than i, the product is less than the 
 multiplicand. 
 
 1. What will of a ton of iron cost, at 855 a ton ? 
 
 ist METHOD. Sinco i ton costs 55, | of a ton OPERATIC*. 
 
 will cost times $55, or of $55. But J of $55 $4)55 
 
 = 3 times of $55. Now \ of $55=855-^-4, or 
 $13^ ; and 3 times $I3$=$4ll, the answer required. 
 For, by definition, multiplying by a fraction is Ane 
 taking apart of the multiplicand as many times as 
 there are like parts of a unit in the multiplier. Now dividing the 
 Avhole number by 4, takes i fourth of the multiplicand once ; and 
 multiplying this result by 3, repeats this part 3 tunes, as indicated by 
 the multiplier. (Art. 143.) 
 
 2d METHOD. } of $55=i of 3 times $55. But $rt 
 
 $55 x 3 = $165, and $i6s-:-4 $41^, the same as 
 before. For, i of 3 times a number is the eame as \ / " 
 
 3 times i of it. Hence, the 
 
 Ans. $414 
 
 RULE. Divide the whole number by the denominator of 
 the fraction, and multiply by the numerator. 
 
 Or, Multiply the ivhole number by the numerator of the 
 fraction, and divide by the denominator. 
 
 REMARKS. i. The fraction may be taken for the multiplicand, 
 and the whole number for the multiplier, at pleasure, without affect- 
 ing the result. (Art. 45.) 
 
 2. When the multiplier is a mixed number, multiply by the frac- 
 tional and integral parts separately, and unite the results. 
 
 Or, reduce it to an improper fraction ; then proceed according to 
 the rule. (Ex. 2.) 
 
 2. What will 8| tons of copper ore cost, at $365 a ton ? 
 ANALYSTS. 8} =*$ ; now 4 6 a tons will come to *<* times $365 ; and 
 
 $365 *" = $3 1 39- An*. 
 
 165. How is a whole number multiplied by a fraction ? Hem. What is the 
 operation like ? When the multiplier is i, what is the prodnct ? W hen the mul- 
 tiplier is greater than i, what f When less, what? Hem How multiply a whola 
 by a mixed number ?
 
 MULTIPLICATION OF FRACTIONS. 123 
 
 (30 (4.) (5-) (6.) (7.) 
 
 Mult. 65 89 96 87 100 
 
 By j & _J| _4j _5* 
 
 8. What cost T 7 ff of an acre of land, at $38 an acre ? 
 
 9. At $29 a ton, what cost 8| tons of hay ? 
 
 10. What cost 28 tons of lead, at $223 per ton? 
 
 n. What is fj'of 845? 15. Multiply 79 by 7$. 
 
 12. What is T 7 ^ of 1876 ? 16. Multiply 103 by 9$. 
 
 13. What is |4 of 1000? 17. Multiply 1001 by 21-^. 
 
 14. What is of 2010 ? 18. Multiply 1864 by 37^. 
 
 CASE III. 
 166. To multiply a Fraction by a Fraction. 
 
 i. What will | of a pound of tea cost, at $^ a pound ? 
 
 ANALYSIS. of a pound will cost I sixth OPERATION. 
 
 as much as i pound ; and of $, 9 - is $ 6 V "& x f~ or *l 
 Again, of a pound will cost 5 times as BY CANCELLATION. 
 
 much as ; and 5 times $& are $$8=$J, 3> j^_ x 5, i_ . . 
 
 the answer required. 2, IE 6, 2 
 
 Or, having indicated the multiplication, caned the common fac- 
 tors 3 and 5, and then multiply the numerators together and the de- 
 nominators ; the result is $J, the same as before 
 
 The reason is this: Multiplying the denominator of the multipli- 
 cand , 9 by 6 the denominator of the multiplier, the result, $,&,, is 
 5 times too small ; for, we have multiplied by instead of of a 
 unit, the true multiplier. To correct this, we must multiply this re- 
 sult by 5, which is done by multiplying its numerator by 5. (Art. 144.) 
 
 The object of cancelling common factors is twofold: it shortens the 
 
 operation, and gives the answer in the lowest terms. Hence, the 
 
 / 
 KULE. Cancel the common factors ; then multiply the 
 
 numerators together for the new numerator, and the denom- 
 inators/or the new denominator. (Art. 144, Prin- 9.) 
 
 166. How multiply a fraction by a fraction ? Object of cancelling ? Ecm. How 
 multiply compound fractions ? Mixed numbers ! Complex fractions ?
 
 124 MULTIPLICATION OF FRACTIONS. 
 
 REMARKS. i. Mixed numbers should be reduced to improper 
 fractions, and complex fractions to simple ones ; then apply the rule. 
 
 "2. Multiplying by a fraction, reducing a compound fraction to a 
 timpte one, and finding a fractional part of a number, are identical 
 operations. (Arts. 143, 149, 166.) 
 
 2. Multiply f of | of by \ of f. 
 
 SOLUTION. It is immaterial as to the result, whether $ 2 
 
 the fractions are arranged in a horizontal or in a per- 24, 
 
 pendicular line, with the numerators on the right and 6 5 
 
 the denominators on the left. (Art. 122, Rent.) 3. i 
 
 J 
 
 Perform the following multiplications : 
 
 3- T X TJT 7- T X TJ II. if X -y X ^ 
 
 4 1 TTf X T5" " TT X yy 12. -g- X -j-jj- X ^- 
 
 15. Multiply f of ^ of 25 by f of ^ of f . 
 
 16. Multiply | of A of 33i by of ^ of i8|. 
 
 17. If i quart of cherries costs \ of f of 45 cents, what 
 will | of A of a quart cost ? 
 
 1 8. What will 5^ barrels of chestnuts cost, at 6f dollars 
 a barrel ? Ans. $36. 
 
 (19.) (20.) (21.) (22.) 
 
 Mult. 24^ 37^ 62^ 165^ 
 
 By 84 i8 31* 92! 
 
 23 
 
 el 2\ 
 
 What is product of ~ multiplied by . 
 
 21 . 21 2 | 8 8 
 - --=.- -7-3 = -. 
 
 . 
 ANALYSIS. ,-= i-6 = .and 
 
 64 24* 3 
 
 7,21 8. 7 7 
 
 Cancelling common factors, \re have X - = --- = -.Ana. 
 
 3>2* 9,3 3x3 9' 
 
 24. Multiply ?i by ^. 25. Multiply ^ by ^.
 
 MULTIPLICATION OF FRACTIONS. 125 
 
 167. The preceding principles may be reduced to the 
 following 
 
 GENERAL RULE. 
 
 I. Reduce ivhole and mixed numbers to improper frac- 
 tions, and complex fractions to simple ones. 
 
 II. Cancel the common factors, and place the product of 
 the numerators over the product of the denominators. 
 
 EXAMPLES. 
 
 1. What cost of a yard of calico, at $ a yard ? 
 
 2. "What cost f of a pound of nutmegs, at $| a pound ? 
 
 3. At $f a gallon, what will of a gallon of molasses 
 cost? 
 
 4. Multiply 23! by 6\. 7. Multiply pi-f by 43$. 
 
 5. Multiply 4 2 T 2 T by 8f 8. Multiply i6 4 | by 75^. 
 
 6. Multiply 65! by gf. 9. Multiply 2oof by 86 T 7 ^. 
 
 10. What is the product of f of of of f into y of -^ 
 
 1 1. What is the product of \ of 3^ into -| of 19^ ? 
 
 12. When freight is $i| per hundred, what will it cos 4 
 to transport 27 hundredweight of goods from New York 
 to Chicago ? 
 
 13. What will 1 6 quarts of strawberries come to, at 
 26^ cents per quart ? 
 
 14. At $3^ a barrel, what will be the cost of 47! barrels 
 of cider? 
 
 15. Bells are composed of ^ tin and f copper: how 
 many pounds o. each metal does a church-bell contain 
 which weighs 3750 pounds ? 
 
 1 6. What will 45 men earn in 15^ days, if each earns 
 $2f per day? 
 
 17. How many feet of boards will a fence 603-^ rods 
 long require, allowing 74^ feet of boards to a rod ? 
 
 167. What Is the general rule for multiplying fractions ?
 
 126 DIVISION OF FRACTIONS. 
 
 DIVISION OF FRACTIONS. 
 
 CASE I. 
 168. To divide a Fraction by a Wliole Number. 
 
 1. If 4 yards of flannel cost $f, what will i yard cost? 
 ist METHOD. i yard will cost i fourth as 
 
 much as 4 yards ; and dividing the numerator 
 
 by 4, we have $-*-4=$f, the answer required. ^---4. __ ^ 2 ^^ 
 
 For, dividing the numerator by any number, 9 
 
 divides the fraction by that number. 
 
 2d METHOD. Multiplying the denominator ad METHOD. 
 
 by the divisor 4 (the number of yards), the 
 result is - 3 ^ = f , the same as before. For, 0x4 ^6 
 
 multiplying the denominator by any number, g 
 
 divides the fraction by that number. (Art. 144, Ans. = $f 
 Prin. 7.) . Hence, the 
 
 EULE. Divide the numerator by the whole number. 
 Or, multiply the denominator by it. 
 
 REMARK. When the dividend is a mixed number, it should be 
 reduced to an improper fraction ; then apply the rule. 
 
 2. Divide $f by 9. Ans. $f -^9=$^. 
 
 Perform the following divisions : 
 3 . 
 4- U-9- 8. W-f-67. 12. 
 
 5 . 
 
 6. 
 
 9 . 
 
 14. 
 
 15. A man having | of a barrel of flour, divided it 
 equally among 5 persons : what part of a barrel did each 
 receive ? 
 
 16. If 12 oranges cost $ T Vb> what will i orange cost? 
 
 17. If 7 writing-books cost $|, what will that be apiece? 
 
 18. If 6 barrels of flour cost $45 f, what will i barrel cost ? 
 
 168. How divide a fraction by a whole number ? Upon what does the first 
 method depend? The second? Which is preferable? Bern. How is a mixed 
 number divided by a whole one ?
 
 DIVISION OF FRACTIONS. 
 
 127 
 
 Perform the following divisions : 
 
 19. 8o T 4 j-^i2. 23. 865tV-S2. 27. 
 
 20. 763^-^ J 4- 24. 490/^40. 28. 46841^68. 
 
 21. 28 4 --f-2O. 25. 758^f 48. 29. 7896-1-7-25. 
 
 22. 5lf 4-27. 26. 975iiy-7-63. 30. 9684^-^-84. 
 
 31. If, 5 yards of cloth cost $42 f, what will i yard cost. 
 
 32. If 6 horses cost $756^, what will i horse cost? 
 
 33. If 45 Ibs. of wool cost $54|, what will i Ib. cost ? 
 
 CASE II. 
 168. To divide a Whole If umber by a Fraction. 
 
 i. At $| a yard, how many yards of cashmere can you 
 buy for $20 ? 
 
 ANALYSIS. At $V a yard, $20 will OFEBATTOIT. 
 
 buy as many yards as there are 20x4=80 y. at $. 
 
 fourths in $20, and 4 times 20 are 80. 80-1-3 = 26^ y. at $J. 
 
 But the price $J, is 3 times aa much Or, 20 X $=26$ y. at $J. 
 as $\; therefore, at $J, you can buy 
 
 only i as many yards as at $| ; and ^ of 80 yards is 26 } yards, 
 the answer required. For, multiplying the whole number by 
 the denominator 4, reduces it to fourths, which is the same denomi- 
 nator as the given divisor. But when fractions have a common 
 denominator, their numerators are like numbers; therefore, onw 
 may be divided by the other, as whole numbers. (Art. 156.) But 
 multiplying the whole number 20, by the denominator 4, and dividing 
 the product by the numerator 3, is the same as inverting the 
 fractional divisor, and then multiplying the dividend by it. Hence, the 
 
 RULE. Multiply the whole number by the fraction in- 
 verted. (Art 165.) 
 
 REMARKS. i. When the divisor is a mixed number it should be 
 reduced to an improper fraction ; then divide by the rule. (Ex. 16.) 
 
 2 A fraction is inverted, when its terms are made to exchange 
 places. Thus, $ inverted becomes $. 
 
 3 After the denominator is inverted, the common factors shoulJ 
 be cancelled. (Art. 149, n.) 
 
 168, a. How divide a whole number by a fraction T How does it appear that 
 this process will give the true answer?
 
 128 DIVISION OF FBACTION8. 
 
 Perform the following divisions : 
 
 2 - 95-^1- 5- 175-^-nr- 8. 576-1-^. 
 
 3. 168-=-^. 6. 261-*-^. 9. 1236-^- 
 
 4. 245-^. 7. 348-7-3.. 10. 6240-^-^. 
 
 ii. How many yards of muslin, at $| a yard, can be 
 bought for $19 ? 
 
 ANALYSIS. At $^ a yard, $19 will buy as many yards as there 
 are thirds in 19, and 19 x $=57. Therefore $19 will buy 57 yards. 
 
 12. 27 =how many times ? 14. 38=how many times |? 
 
 13. 53 = how many times ? 15. 67 =how many times -j^? 
 
 1 6. How many cloaks will 45 yards of cloth make, each 
 containing 4^ yards ? 
 
 ANALYSIS. Reducing the divisor to an improper fraction, we have 
 $}%, and 45 yds. x f= 9 9 a , or 10 cloaks., An. 
 
 17. Divide 88 by lof. 19. Divide 785 by 62-}. 
 
 18. Divide 100 by 12^. 20. Divide 1000 by 87^. 
 
 CASE III. 
 
 169. To divide a Fraction by a Fraction, when they have a 
 Common, Denominator. 
 
 REMARK. This case embraces two classes of examples: 
 First, those in which the fractions have a common denominator. 
 Second, those in which they have different denominators. 
 
 2 1. At $f apiece, how many melons can be bought for & ? 
 
 ANALYSIS. Since $| will buy i melon, $ will buy OPERATION. 
 as many as $| are contained times in $J ; and 3 ^~J~Y :=:: 1 
 eighths are in 7 eighths, 2 times and i over, or 2^ times. Ans. 2$ m. 
 (Art. 156.) Hence, the 
 
 RULE. Divide the numerator of the dividend by that 
 of the divisor. 
 
 NOTE. When two fractions have a common denominator, their 
 numerators are like numbers, and the quotient is the same as if they 
 were whole numbers. (Arts. 64, 156.) 
 
 22. Divide ff by ~fa. 24. Divide f| by J. 
 
 23. Divide ff by jj. 25. Divide j| by |f 
 
 169. How divide a fraction by a fraction when they have a common denominator t
 
 DIVISION OF FRACTIONS. 12S 
 
 170. To divide a Fraction by a Fraction, when they have 
 Different Denominators. 
 
 1. At $| a pound, how much tea can be bought for $ ? 
 IST ANALYSIS. $f will buy as many pounds ^=.^=~^ 
 
 as $J are contained times iu $f. $=$*. Re- 
 
 ducing J and $ to a common denominator, they 
 become -& and -,%,and their numerators like num- 
 fors. Hence, -&-5--&=9-s- 8, or i\. Am. ii pound. 
 
 REMARKS. i. In reducing two fractions to a common denomina- 
 tor, we multiply the numerator of each into the denondnator of the. 
 other, and the two denominators together. But in dividing, no use 
 ia made of the common denominator ; hence, in practice, multiply- 
 ing the denominators together may be omitted. (Art. 153.) 
 
 20 ANALYSIS. $=$! At $^ a pound, $i will buy as many 
 pounds as $} are contained times in $i. Now i -*-*=$-*-*, and 
 ?,-*-$= |, or f pounds, the quotient being the divisor inverted. Again, 
 if $i will buy J pounds, $f will buy $ of f pounds ; and I of i = , 
 Or i J pounds, the same as before. 
 
 REMARKS. 2. In this analysis, it will be seen, by inspection, that 
 the numerator of each fraction is also multiplied into the denomi- 
 nator of the other. Both of these solutions, therefore, bring the 
 same combinations of terms, and the same result, as inverting the di- 
 visor, and multiplying the dividend by it. (Art. 166.) Hence, the 
 
 KULE. Reduce the fractions to a com. denominator, and 
 divide the numerator of the dividend by that of the divisor. 
 Or, multiply the dividend by the divisor inverted. 
 
 NOTES. i. The first method is based upon the principle that 
 numerators of fractions having a com. denom. are like numbers. 
 
 The second is evident from the fact that it brings the same com- 
 fcinations as reducing the fractions to a common denominator. 
 
 The divisor is inverted for convenience in multiplying, 
 
 2. After the divisor is inverted, the common factors should be 
 cancelled, before the multiplication is performed. 
 
 3. Mixed numbers should be reduced to improper fractions, com- 
 pound and complex fractions to simple ones. 
 
 4. Those expressions which have fractional denominators, are re- 
 reduced to simple fractions by the above rule. Ex. 14. (Art 141. Rem.) 
 
 170. How when they have not? Note. Upon what principle is the first method 
 based ? The second ? Show this coincidence. What ie done with mixed num- 
 bers, compound, and complex fractions.
 
 130 
 
 DIVISION OF FRACTIONS. 
 
 Perform the following divisions : 
 
 2. 
 
 jr 3 
 
 4. -fa-*?. 
 
 5- r~~A* 
 
 6. 
 
 7- 
 8- 
 9- 
 
 27! 
 
 14. Reduce ^ to a simple fraction. 
 i of 
 
 SOLUTION. The given expression is equivalent to 275-^-10^, and ia 
 reduced to a simple fraction by performing the division indicated. 
 Ans. f or 2\. (Art. 141, Rem.) 
 
 is4 
 
 1 7. Reduce -~^ 
 
 I I-J 1 2O-5- 
 
 16. Reduce T - 18. Reduce - 
 
 15. Reduce - 
 
 19. Divide 777^ dollars by 129! dollars. 
 
 20. How many rods in 2320^ feet, at 16^ feet to a rod ? 
 
 21. How many times 30^ sq. yards in 320^ sq. yards? 
 
 22. What is the quotient of f off of 4} divided by f of f ? 
 
 23 2 5 
 
 SOLUTION. - of - of 44 divided by - of = 
 34 7 s 
 
 2 2 5 5 
 
 When the perpendicular form is adopted, the 
 divisor must bj inverted, before its terms are arranged. 
 
 23. Divide f of f of by of f 
 
 24. Divide of f of | by of f . 
 
 25. Divide f of f of 4^ by of | of 4$. 
 
 i g 
 
 26. What is the quotient of f divided by -= 
 
 f 3i 
 
 J 
 
 ANALTSIS. The dividend | = 
 
 Q 
 
 x | ; the divisor -v 
 
 -v 
 
 Q 
 
 = - X 
 
 16 
 
 XT O T 
 
 27. What is quotient of ~ divided by -- ? 
 ^r rr
 
 DIVISION OF FRACTIONS. 131 
 
 171. The preceding principles may be reduced to the 
 following 
 
 GENERAL RULE. 
 
 Reduce ivhole and mixed numbers to improper fractions, 
 compound and complex fractions to simple ones, and mul- 
 tiply the dividend by the divisor inverted. 
 
 NOTE. After the divisor is inverted, the common factors should 
 be cancelled. 
 
 EXAMPLES. 
 
 1. If a young man spends $2^ a month for tobacco, in 
 what time will he spend $13^ ? (Art. 48, Note 3.) 
 
 2. If a family use 5^ pounds of butter a week, how long 
 will 45 1 pounds last them ? 
 
 3. If $|; will buy i yard of gingham, how much will 
 
 4. How many tons of coal, at $7^ a ton, can be bought 
 for $125!? 
 
 5. How many times will a keg containing 13! gallons 
 of molasses fill a measure that holds f of a gallon ? 
 
 6. What is the quotient of 24-^ divided by 8f ? 
 
 7. What is the quotient of ff divided by H ? 
 
 8. What is the quotient of f divided by $f ? 
 
 9. What is the quotient of 45! divided by 25^? 
 
 10. How much tea, at $i|- a pound, can be bought 
 for$ 75 f? 
 
 11. How many acres can be sowed with 57^ bushels of 
 oats, allowing i| bushel to an acre?. 
 
 12. A man having 57^ acres of land, wished to fence it 
 into lots of 5^ acres : how many lots could he make ? 
 
 13. How many yards of cloth, at $6|, can yon buy for 
 
 14. What is the quotient of f of f of If -=-| of | of z\ ? 
 
 15. What is the quotient of f of | of 8f -=-4f } ? 
 
 171. What is the general rule for dividing fractions t
 
 132 DIVISION OF FRACTIONS. 
 
 QUESTIONS FOR REVIEW. 
 
 1. A book-keeper adding a column of figures, made the 
 I3sulfc $563!; proving his work, he found the true amount 
 to be $607^ : how much was the error ? 
 
 2. A merchant paid two bills, one $278!, the other 
 $34o|, calling the amount $638^ : what should he have 
 paid ? What the error ? 
 
 3. What is the sum of 35^ and 23! minus 8$? 
 
 4. A speculator bought two lots of land, one containing 
 47-f 3 ^ acres, the other 63* acres: after selling 78$ acres, 
 how many had he left ? 
 
 1 2 5 
 
 5. What is the sum of -^ plus ? 
 
 3* 3t 
 
 6. What is the sum plus plus r ? 
 
 4 * if * 5i 
 
 7. A man owning f of a ship worth $48064, sold of 
 his share. What part of the ship did he sell ; what part 
 does he still own, and what is it worth ? 
 
 8. A farmer owning 75! acres, sold 31^ acres, and after- 
 ward bought 42! acres : how many acres did he then have ? 
 
 3 .5 
 
 o. What is the difference between ~ and ~ ? 
 
 I 6 
 6 s ?- 
 
 10. What is the difference between =- and *?. ? 
 
 3i o 
 
 11. What is the difference between -~ and - ;v ? 
 
 12. If it requires i-J bushels of wheat to sow an acre, 
 how many bushels will be required to sow 28$ acres ? 
 
 13. How many feet in 148$ rods, allowing 16$ feet to 
 a rod? 
 
 14. If a pedestrian can walk 45^ miles in i day, how 
 far can lie walk in i8f days ? 
 
 15. What will 37$ barrels of apples come to, at $2^ per 
 barrel ?
 
 QUESTIONS FOB EEVIEW. 133 
 
 1 6. The sum of two numbers is 68||, and the difference 
 between them is 13! : what are the numbers? 
 
 17. What is the product of -f into -f ? 
 
 3* 4i 
 
 1 8. What is the product of ^ into ^? 
 
 6| 12 
 
 19. What is the product of I into ? f ? 
 
 I2| 2\ 
 
 20. How many days' work will 100 men perform in \$ 
 of a day ? 
 
 21. A man owning -^ of a section of land, sold -J of his 
 share for $izf : what is the whole section worth, at that 
 rate? 
 
 22. How many times is -^ of f of 5 J contained in 23! ? 
 
 23. Divide of i8| by f of f of f of 31$. 
 
 24. If a gang of hands can do ^f of a job in 5! days, 
 what part of it can they do in i day ? 
 
 25. If | of a yard of satin will make i vest, how many 
 vests can be made from 31^ yards ? 
 
 26. How many oil-cans, each containing if gallon, can 
 be filled from a tank of 6 if gallons ? 
 
 27. If a man walks 3^ miles an hour, how long will it 
 take him to walk 45^ miles ? 
 
 28. By what must Jf be multiplied to produce 15!? 
 
 29. How many bushels of apples, at f of a dol., are 
 required to pay for 6 pair of boots, at $6 J ? 
 
 30. A farmer sold 330^ pounds of maple sugar, at i6| 
 cents a pound, and took his pay in muslin, at 22^ cents a 
 yard: how many yards did he receive? 
 
 31. Divide the quotient of 12^ divided by 3^ by the 
 quotient of 6-=- 
 
 32. What is the quotient of -f -r ? 
 
 TT 
 
 5 I. 
 
 33. What is the quotient of 12^ times =* 
 
 4
 
 134 FRACTIONAL RELATION 
 
 FRACTIONAL RELATION OF NUMBERS. 
 
 172. That Numbers may bo compared with each other frac- 
 tionally, they must be so far of the same nature that one may prop 
 erly be said to be a part of the other. Thus, an inch may be com- 
 pared with afoot ; for one is a twelfth part of the other. But it can- 
 not be said that &foot is any part of an hour ; therefore the former 
 cannot be compared with the latter. 
 
 173. To find what part one number is of another. 
 
 1. What part of 4 is i ? 
 
 ANALYSIS. If 4 is divided into 4 equal parts, one of those parts is 
 called i fourth. Therefore, i is \ part of 4. 
 
 2. What part of 6 is 4 ? 
 
 ANALYSIS. i is of 6, and 4 is 4 times , or $ of 6. But f =$ 
 (Art. 146) ; therefore, 4 is f of 6. Hence, the 
 
 RULE. Make the number denoting the part the numera- 
 tor, and that with which it is compared the denominator. 
 
 NOTE. i. This rule emb races four classes of questions : 
 i st. What part one whole number is of another. 
 2d. What part & fraction is of a whole number. 
 3d. What part a whole number is of a fraction. 
 4th. What part one fraction is of another. 
 
 2. When complex fractions occur, they should be reduced to 
 timple ones, and all answers to the lowest terms. (Art. 146.) 
 
 3. What part of 75 is 15 ? Of 84 is 30 ? 
 
 4. Of 91 is 63? 6. Of 8iisi8? 8. Of 256 is 72? 
 
 5. Of 48 is 72 ? 7. Of ioo is 75 ? 9. Of 375 is 425 ? 
 
 10. What part of i week is 5 days ? 
 
 11. A man gave a bushel of chestnuts to 17 boys: what 
 part did 5 boys receive ? 
 
 12. At $13 a ton, how much coal can be bought for $10 ? 
 
 13. A father is 51 years old, and his son's age is 17: 
 what part of the father's age is the son's ?
 
 OF NUMBERS. 135 
 
 14. If 8 pears cost 35 cents, what will 5 pears cost ? 
 
 ANALYSIS. 5 pears are $ of 8 pears ; hence, if 8 pears cost 35 
 ceats, 5 pears will cost 3 of 35 cents. Now i of 35 cents is 4! cents, 
 and 5 eighths are 5 times 4 2 cents, which are zil cents. 
 
 15. If 5 bar. of flour cost $45, what will 28 bar. cost? 
 
 16. If 50 yds. of cloth cost $175, what will 17 cost? 
 
 17. If 25 bu. of apples cost $30, what will no bu. cost? 
 r8. What part of 5 is f ? 
 
 ANALYSIS. Making the fraction which denotes the part the nu- 
 merator, and the whole number the denominator, we have a fraction 
 to be divided by a whole number. For, all denominators may be con- 
 sidered as divisors. Thus, J-5-5=i a u> Ana. (Art. 142.) 
 
 19. What part of 25 is |? 21. What part of 30 is -^ ? 
 
 20. What part of 35 is - 1 / ? 22. What part of 40 is -^ ? 
 
 23. If 5 acres of land cost $100, what will acre cost ? 
 
 ANALYSIS. i acre is of 5 acres, and $ of an acre is J of , or -fa 
 of 5 acres. Hence, 5 of an acre will cost .ft, of $100. Now ^ of $100 
 is $5 ; and 3 twentieths are 3 times 5 or $15. 
 
 24. When coal is $95 for 15 tons, what will f ton cost? 
 
 25. If 19 yards of silk cost $60, what will yard cost? 
 
 26. What part of f is 2 ? 
 
 ANALYSIS. Making the whole number which denotes the part, the 
 numerator, and the fraction the denominator, we have a whole num- 
 ber to be divided by a fraction. Thus, 2-7-^=2 x ^= 1 3 a , Ans. 
 
 27. What part of is 8 ? 29. What part of f is 1 1 ? 
 
 28. What part of | is 12 ? 30. What part of T 7 V is 20 ? 
 3 1 . What part of f is f ? 
 
 ANALYSIS. Making the fraction denoting the part the numerator, 
 and the other the denominator, we have a fraction to be divided by 
 a fraction. Thus, $-=-5 = ! x $-=$, Ans. (Art. 170.) 
 
 32. What part of f is $ ? 34. What part of |4 is 
 
 33. What part of ff is H ? 35. What part of H is |-f ? 
 36. 6 is what part of 25 ? 
 
 ANALYSIS. Reducing the mixed number to an improper fraction. 
 we have &i = V. and V^-25=:i, Ans.
 
 130 FRACTIONAL RELATION. 
 
 37. What part of 100 is 12^? 40. Of 100 is 62^? 
 
 38. What part of 100 is 33^? 41. Of 100 is i8J? 
 
 39. What part of 100 is i6f ? 42. Of 100 is 87^? 
 43. 12^ is what part of iSf ? 
 
 ANALYSIS. Reducing the mixed numbers to improper fractions 
 we have, i2^= 2 /, and iS}= 1 f-. Now 2 a i -j- 1 4 i =f, Ans. 
 
 44. What part of 62^ is i8| ? 45. Of 87^ i 
 
 46. At $g a pound, how much tea will $f buy ? 
 
 47. At $ per foot, how many feet of land can be 
 bought for *f| ? 
 
 48. A lad spent i8f cents for candy, which was 62^ 
 cents a pound : how much did he buy ? 
 
 49. A can do a certain job in 8 days, and B in 6 days ' 
 what part will both do in i day ? 
 
 50. What part of 4 times 20 is 9 times 16 ? 
 
 51. What part of 75 x 18 is 105-7-25 ? 
 
 52. What part of (68 24) x 14 is 168 H- 12 ? 
 
 174. To find a Number, a Fractional Part of it being given. 
 
 Ex. i. 9 is of what number ? 
 
 ANALYSIS. Since 9 is i third, 3 thirds or the whole number must 
 be 3 times 9 or 27. Therefore, 9 is a third of 27. 
 Or, thus : g is of 3 times 9, and 3 x 9=27. Therefore, etc. 
 
 2. 2 1 is f of what number ? 
 
 ANALYSIS. Since | of a certain number is OPERATION. 
 21 units, % or the whole number must be as many 21-7-^=21 X^ 
 units as J are contained times in 21 ; and 2i-i-J= 21x^=28. 
 21 x $=28, the answer required. For, a whole 
 number is divided by a fraction by multiplying the former by the 
 latter inverted. (Art. 168, a.) 
 
 Or, thus : Since 21 is 5 of a certain number, I fourth of it is I third 
 of 21, or 7. Now as 7 is i fourth of the number, 4 fourths must be 
 4 times 7 or 28, the same as before. Hence, the 
 
 RULE. Divide the number denoting the part by the 
 fraction. 
 
 Or, Find one part as indicated by the numerator of tM 
 fraction, and multiply this by the denominator.
 
 OF KUMBEKS. 137 
 
 NOTE. The learner should observe the difference between finding 
 i of a number, when J or the whole number is given, and when only 
 or a part of it is given. In the former, we divide by the denomi- 
 nator of the fraction ; in the latter, by the numerator, as in the 
 tecond analysis. If he is at a loss which to take for the divisor, let 
 him substitute the word parts for the denominator. 
 
 3. 56 is f of what? 7. 436 is f of what? 
 
 4. 68 is | of what? 8. 456 is of what? 
 
 5. 85 is f of what ? 9. 685 is -^ of what ? 
 
 6. 1 15 is f of what ? 10. 999 is ^ of what ? 
 
 1 1. A market man being asked how many eggs he had, 
 replied that 1 26 was equal to -^ of them : how many had he '* 
 
 12. If -j^- of a ship is worth $8280, what is the whole 
 worth ? 
 
 13. A commander lost f of his forces in a battle, and 
 had 9500 men left: how many had he at first? 
 
 14. %% is \ of what number? 
 
 ANALYSIS. fg is i of 4 times fg ; and 4 times $=2|. 
 
 J 5- fi i s f f what number ? 
 
 ANALYSIS. Since ii=$ of a certain number, \ of that number 
 must be'i of fi, and \ of H=A or \. Now if | of the number=i, 
 $ must equal 7 times \=\, or i 
 
 7, 21 * 7 
 
 Or, dividing H by f we have X =-, or i 
 4, ^o> A 4 
 
 1 6. || is | of what number ? 
 17. 1% is f of what number? 
 1 8. i8f is f of what number? 
 
 ANALYSIS. 18} =^5 . Since * 4 *=$, i = 1 4 , and t=*-l*, or 30, 
 Or, ^-j-frr 1 ^ x g= 4 o Q =30, the same as before. 
 
 J 9- 3?i is f of what number? 
 
 20. 66f is f of what number ? 
 
 21. 48 is | of f of what number ? 
 
 ANALYSIS. f of J=f. The question now is, 48 is f of what 
 number? Ana. 48 x \, or 108.
 
 138 FRACTIONAL RELATIONS. 
 
 22. 112 is f of f of what number? 
 
 23. In - of 120 how many times 15 ? 
 
 ANALYSIS. f I2 is r 3a J an( i 5 arc 7 times 13 i or 93 ^. Now 
 =rY a or 6*, Ana. 
 
 24. How many yards of brocatelle, at 89 a yard, can be 
 bought for | of $100 ? 
 
 25. A man paid -^ of $280 for 84 arm-chairs: what was 
 that apiece ? 
 
 26. 90 is of how many times 17 ? 
 
 ANALYSIS. As 90 is ? of a certain number, 1 is ^ of 90, which is 
 15 ; and ? are 7 times 15 or 105. Now 17 is in 105, 6 -ft- times. 
 Therefore, etc. 
 
 27. 125 is f of how many times 20 ? 
 
 28. A man paid 60 cents for his lunch, which was T 6 ff of 
 his money, and spent the remainder for cigars, which were 
 5 cents each: how much money had he; and how many 
 cigars did he buy ? 
 
 29. ^ of 1 10 is ^ of what number ? 
 
 ANALYSIS. ^ of no is n, and , 9 o, 9 times n or 99. Now, since 
 99 is ? of a number, | of it must be ^ of 99, which is n, and f must 
 be 7 times n or 77. Therefore, etc. 
 
 30. | of 126 is | of what number ? 
 
 31. | of 90 is | of how many times 1 1 ? 
 
 ANALYSIS. of 90 is 50. Now as 50 is f of a number, is | of 
 50 or 10 ; | is 8 times 10 or 80. Finally, n is contained in 80, 7ft- 
 times. Therefore of 90 is of 7 ft- times n. 
 
 32. | of 96 is -^ of how many times 20 ? 
 
 33. of 120 is f of how many sevenths of 56 ? 
 
 ANALYSIS. I of 120 is 100. If 100 is J of a number, is \ of the 
 number; now \ of 100 is 25, and 5s 9 times 25 or 225. Finally, | 
 of 56 is 8, and 8 is contained in 225, 28^ times. Therefore, of 120 
 is J of 28 i times | of 56. 
 
 34. f of 35 is I of how many tenths of 120 ?
 
 DECIMAL FRACTIONS. 
 
 175. Decimal Fractions are those in which the 
 unit is divided into tenths, hundredths, thousandths, etc. 
 . They arise from continued divisions by 10. 
 
 If a unit is divided into ten equal parts, the parts are 
 called tenths. Now, if one of these tenths is subdivided 
 into ten other equal parts, eacli of these parts will be one- 
 tenth of a tenth, or a hundredth. Thus, -fo-7- I0 or fV f 
 - I ^=- I J ir . Again, if one of these himdredths is subdivided 
 into ten equal parts, each of these parts will be one-tenth of 
 a hundredth, or a thousandth. Thus, -j^u-r io=rJ<rs> e tc 
 
 NOTATION OF DECIMALS. 
 
 176. If we multiply the unit i by 10 continually, it 
 produces a series of whole numbers which increase regu- 
 larly by the scale of 10 ; as, 
 
 I, 10, 100, 1000, i oooo, I ooooo, i oooooo, etc. 
 
 Now if we divide the highest term in this series by 10 
 continually, the several quotients will form an inverted 
 series, which decreases regularly by ten, and extends from 
 the highest term to i, and from i to J s , T ^, T^OTT* and so 
 on, indefinitely ; as, 
 
 i ooooo, i oooo, 1000, 100, 10, i, -j^-, -j^, T^OTT, etc. 
 
 177. By inspecting this series, the learner will perceive 
 that the fractions thus obtained, regularly decrease toward 
 the right by the scale of 10. 
 
 If we apply to this class of fractions the great law of 
 Arabic Notation, which assigns different values to figures, 
 
 175. What are decimal fractions ? How do they arise ? Explain this upon 
 the blackboard. 177. By what law do decimals decrease ?
 
 .140 DECIMAL FRACTIONS. 
 
 according to the place they occupy, it follows that a figure 
 standing in the first place on the right of units, denotes 
 tenths, or i tenth as much as when it stands in units' 
 place ; when standing in the second place, it denotes hun- 
 dredths, or i tenth as much as in the first place; when 
 standing in the third place, it denotes thousandths, etc., 
 each succeeding order below units being one tenth the 
 value of the preceding. Hence, 
 
 178. Fractions which decrease by the scale of ten, may 
 be expressed like whole numbers ; the value of each figure 
 in the decreasing scale being determined by the place it 
 occupies on the right of units. Thus, 3 and 5 tenths may 
 be expressed by 3.5 ; 3 and 5 hundredths by 3.05 ; 3 and 
 5 thousandths by 3.005, etc. 
 
 178, Decimals are distinguished from whole numbers 
 by a decimal point. 
 
 NOTES. i. The decimal point commonly used (.), is a period. 
 2. This class of fractions is called decimals, from the Latin decem, 
 ten, which indicates both their origin and the scale of decrease. 
 
 179. The Denominator of a decimal fraction is 
 always 10, 100, 1000, etc.; or i with as many ciphers 
 annexed to it as there are decimal places in the given 
 numerator. Conversely, 
 
 The Numerator of a decimal fraction, when written 
 alone, contains as many figures as there are ciphers in the 
 denominator. Thus ^, -^ T ^ nfJ expressed decimally, 
 are .5, .05, .005, etc. If the ciphers in .05 and .005 are 
 omitted, each becomes 5-tenths. 
 
 What place do tenths occupy ? Hundredths, thousandths, etc. f 178. How is 
 the value of decimal figures determined ? 178, a. How are decimals distinguished 
 from whole numbers ? Note. What is the decimal point ? From what does this 
 rlass of fractions receive its name ? 179. What is the denominator? How do 
 the number of ciphers in the denominator and the decimal places in the numera- 
 tor compare ?
 
 DECIMAL FRACTIONS. 141 
 
 180. The different orders of decimals, and their relative 
 position, may be seen from the following 
 
 TABLE. 
 
 2 =* 
 
 ; .a 
 
 rf a -g 
 
 fl 
 
 
 
 
 >s 
 
 
 
 
 tr. 
 
 o 
 
 ^ 
 
 2 
 
 
 00 
 
 .a 
 
 | 
 
 fl 
 
 g 
 
 o 
 
 rfl 
 
 =' 
 
 1 
 
 1 
 
 "S 
 
 "3 
 5 
 
 OD 
 
 ~ 
 
 5 
 
 s 
 
 T3 
 
 g 
 
 p 
 
 o 
 
 
 
 1 
 tJ 
 
 "s 
 2 
 
 E3 
 
 a 
 
 o 
 
 ^ 
 
 1 
 
 1 
 
 Q 
 
 a 
 
 9) 
 
 EH 
 
 B 
 
 1 
 
 a 
 6 
 EH 
 
 - 
 
 
 B 
 
 1 
 
 d 
 
 
 
 EH 
 
 
 
 B 
 
 1 
 
 
 
 3 
 
 2 
 
 6 
 
 7 
 
 2 
 
 5 
 
 4 
 
 5 
 
 8 
 
 W X ^ - 03 
 
 *~ 'S C< s IS 
 
 ill i 1 s 
 
 ^ ^ J ^ ri 
 
 ^ K EH ^i B EH 
 
 842567 
 
 V 
 
 Integers. Decimals. 
 
 NOTES. i. A Decimal and an Integer written together, are called 
 ft Mixed Number ; as 35.263, etc. (Art. 101, Def. 12.) 
 
 2. A Decimal and a Common Fraction written together, are called 
 a Mixed Fraction ; as .6^, .33 $. 
 
 181. Since the orders of decimals decrease from left to 
 right by the scale of i o, it follows : 
 
 First. Prefixing a cipher to a decimal diminishes its value 
 io times, or divides it by io. Thus .7=-^; 7=T L ff> 
 
 Annexing a cipher to a decimal does not alter its value. 
 Thus .7, .70, .700 are respectively equal to T 
 
 These effects are the reverse of those produced by annexing and 
 prefixing ciphers to whole numbers. (Arts. 57. 79.) 
 
 Second. Each removal of the decimal point one figure 
 io the right, increases the number 10 times, or multiplies 
 it by io. Each removal of the decimal point one figure 
 to the left, diminishes the number io times, or divides it 
 by io. Hence. 
 
 180. Name the orders of decimals toward the right. Name the orders of 
 Integers toward the left. 181. How is a decimal figure affected by moving it one 
 place to the right? How, if a cipher is prefixed to it? How if ciphers are 
 annexed ?
 
 142 DECIMAL FRACTIONS. 
 
 182. To write Decimals. 
 
 RULE. Write the figures of the numerator in their order, 
 assigning to each its proper place below units, and prefix to 
 them the decimal point. 
 
 If the numerator has not as many figures as required, 
 supply the deficiency by prefixing ciphers. 
 
 "Write the following fractions decimally : 
 
 ! rV 6 - 
 
 2- iVff- 7- 
 
 3. T^T. 8. 4 T V 13- i0io s oo 8 oo- 
 
 4- rVcr- 9- 2 i-dhy 14- 
 
 5. yfo. 10. 8 4T 4 S V 15. 
 
 17. Write 6 hundredths ; 63 thousandths; 109 ten 
 thousandths. 
 
 1 8. Write 305 thousandths; 21 hundred-thousandths; 
 95 millionths. 
 
 19. Write 4 thousandths ; 1 08 ten-thousandths; 46 hun- 
 dreths; 65 millionths; 1045 ten-millionths. 
 
 20. Write sixty-nine and four thousandths; ten and 
 seventy-five ten -thousands; 160 and 6 millionths. 
 
 21. Write 53 ten-thousandths; 63 and 28 hundred- 
 thousandths; 352 ten-millionths. 
 
 183. To read Decimals expressed by Figures. 
 
 RULE. Read the decimals as whole numbers, and apply 
 to them the name of the lowest order. 
 
 NOTE. In case of mixed numbers, read the integral part as if it 
 Btood alone, then the decimal. 
 
 Or, pronounce the word decimal, Jhen read the decimal figures aa 
 if they were whole numbers. 
 
 182. What is the rnle for writing decimals? Note. If the numerator has not 
 as many figures as there are ciphers in the denominator, what is to be done? 
 183. How are decimals read ? Note. In case of a mixed number, how ?
 
 
 DECIMAL FRACTIONS. 143 
 
 Or, having pronounced the word decimal, repeat the names of the 
 decimal figures in their order. Thus, 275.468 is read, " 275 and 468 
 thousandths;" or " 275, decimal four hundred and sixty-eight;" or 
 " 275, decimal four, six, eight" 
 
 i. Explain the decimal .05. 
 
 ANALYSIS. Since the 5 stands in the second place on the right of 
 the decimal point, it is equivalent to T fSr>, and denotes 5 hundredth^ 
 of one, or 5 such parts as would be obtained by dividing a unit into 
 100 equal parts. 
 
 Read the following examples : 
 
 (') ('.) (3-) 
 
 .36 2.751 32.862 
 
 479 4-8465 40.0752 
 
 .0652 7- 2 5 02 5 57.00624 
 
 .00316 8.400452 81.20701 
 
 183, <( Decimals differ from common fractions in three 
 respects, viz. : in their origin, their notation, and their limitation. 
 
 ist. Common fractions arise from dividing a unit into any number 
 of equal parts, and may have any number for a denominator. 
 
 Decimals arise from dividing a unit into ten, one hundred, one 
 thousand, etc., equal parts ; consequently, the denominator is always 
 10, or some power of 10. (Arts. 55, n. 179.) 
 
 2d. In the notation of common tractions, both the numerator and 
 denominator are written in full. 
 
 In decimals, the numerator only is written ; the denominator is 
 understood. 
 
 3d. Common fractions arc universal in their application, embracing 
 all classes of fractional quantities from a unit to an infinitesimal. 
 
 Decimals are limited to that particular class of fractional quanti- 
 ties whosa orders regularly decrease in value from left to right, by 
 the scale of 10. 
 
 NOTE. The question is often asked whether the expressions -| a -, 
 T^U TiiVo. etc., are common or decimal fractions. 
 
 All fractions whose denominator is written under the numerator, 
 fulfil the conditions of common fractions, and may be treated as such. 
 But fractions which arise from dividing a unit into 10, 100, 1000, eta, 
 equal parts, answer to the definition of decimals, whether the denom- 
 inator is expressed, or understood. 
 
 183, a. llow do decimals differ from common fraction.- ?
 
 144 DECIMAL FBACTIONS. 
 
 REDUCTION OF DECIMALS. 
 184. To Reduce Decimals to a Common Denominator, 
 
 1. Eeduce .06, 2.3, and .007 to a common denominator. 
 ANALYSIS Decimals containing the same number .06 0.060 
 
 of figures, have a common denominator. (Art. 179.) 2 .? 2 , oo 
 By annexing ciphers, the number ot decimal fig- _ 
 
 ares in each may be made the same, without altering 
 their value. (Art. 181.) Hence, the 
 
 RULE. Make the number of decimal figures the same in 
 each, by annexing ciphers. (Art. 181.) 
 
 2. Reduce .48 and .0003 to a common denominator. 
 
 3. Reduce 2 to tenths ; 3 to hundredths ; and .5 to 
 thousandths. Ans. 2.0 or f; 3.00 or f&; .500. 
 
 185. To Reduce Decimals to Common Fractions. 
 
 1. Reduce .42 to a common fraction. 
 
 ANALYSIS. The denominator of a decimal is i, with as many 
 ciphers annexed as there are figures in its numerator; therefore the 
 denominator of .42 is too. (Art. 179.) Ans, .42=-,^-. Hence, the 
 
 RULE. Erase the decimal point, and place the denomi- 
 nator lender the numerator. (Art. 179.) 
 
 2. Reduce .65 to a common fraction ; then to its lowest 
 terms. Ans. .65 =&, and -r^=i- 
 
 Reduce the following decimals to common fractions : 
 
 3. .128 7. .05 n. .0007 15. .200684 
 
 4. .256 8. .003 12. .04056 16. .0000008 
 
 5. .375 9. .0008 13. .00364 17. .12400625 
 
 6. .863 10. .0605 14. .00005 18. .24801264 
 
 186. To Reduce Common Fractions to Decimals. 
 i. Reduce f to a decimal fraction. 
 
 ANALYSIS \ equals of 3. Since 3 cannot be divided OPERATION 
 
 by 8; we annex a cipher to reduce it to tenths. Now} 8)3.000 
 
 0/30 tenths is 3 tenths, and 6 tenths over. 6 tenths Ans -Z7C 
 reauced to hundredths =60 hundredths, and J of 60 
 hundredths=7 hundredths and 4 hundredths over. 4 hundredth* 
 
 reduced to thousandths =40 thousandths, and i of 40 thousandths 
 =5 thousandths. Therefore, =.375. Hence, the
 
 DECIMAL FRACTIONS 145 
 
 RULE Annex ciphers to the numerator and divide by the 
 denominator. Finally, point off as many decimal figures in 
 the result as there are ciphers annexed to the numerator. 
 
 NOTE. If the number of figures in the quotient is less than tha 
 number of ciphers annexed to the numerator, supply the deficiency 
 by prefixing ciphers. 
 
 Demonstration A fraction indicates division, and its value is the 
 numerator divided by the denominator. (Arts. 134, 142.) Now, an- 
 nexing one cipher to the numerator multiplies the fraction by 10 ; 
 annexing two ciphers, by 100, etc. Hence, dividing the numerator 
 with one, two, or more ciphers annexed, gives a quotient, 10, roo. 
 etc., times too large. To correct this error the quotient is dividad 
 by 10, 100, looo, etc. But dividing by 10, 100, etc., is the same as 
 pointing off an equal number of decimal figures. (Art. 181.) 
 
 Reduce the following fractions to decimals : 
 
 2. i 6. | 10. $ 14. .g-^ 
 
 3- f 7- fV " s 3 S J 5- -jfzr 
 
 4- I 3. -I 12. ^ 1 6. jd-s 
 
 5- t 9- -h 3 J 3- TTO- 17- xMir 
 
 1 8. Reduce f to the form of a decimal. 
 
 ANALYSTS. Annexing ciphers to the numerator OPERATION. 
 and dividing by the denominator, as before, the quo- 3)2.000 
 tient consists of 6 repeated to infinity, and the re- .6666 etc, 
 maiuder is always 2. Therefore cannot be exactly 
 expressed by decimals. 
 
 19. Reduce 7 5 T to the form of a decimal. 
 
 ANAi/ys'B. Having obtained th~ee quotient OPERATION. 
 
 figures 135, the remainder is 5, the saoie as the 37(5.000000 
 original numerator; consequently, by annex- . I^I "55 etc. 
 
 ing ciphers to it, and continuing the division, 
 we obtain the same set of figures as before, repeated to infinity. 
 Therefore ^ cannot be exactly expressed by decimals. 
 
 187. When the numerator, with ciphers annexed, is 
 exactly divisible by the denominator, the decimal is called 
 a terminate decimal. 
 
 185. How reduce a decimal to a common fraction ? 186. How reduce a com- 
 mon fraction to a decimal ? Note. If th<? number of flares in the quotient i 
 t'lan that in the numerator, what is to be done ? Explain the reason for pointing 
 off the quotient.
 
 146 ADDITION OF DECIMALS. 
 
 When ifc is not exactly divisible, and the same figure a 
 set of figures continually recurs in the quotient, the deci 
 mal is called an interminatc or cii culatiny decimal. 
 
 The figure or set of figures repeated is called the repetend 
 Thus, the decimals obtained in the last two examples ar< 
 interminate, because the division, if continued forever, will 
 leave a remainder. The repetend of the i8th is 6; thai 
 of the i gth is 135. 
 
 NOTES. i. After the quotient has been carried as far as desirably 
 the sign ( + ) is annexed to it to indicate there is still a remainder. 
 
 2. If the remainder is such that the next quotient figure would be 
 5, or more, the last figure obtained is sometimes increased by i, and 
 the sign ( ) annexed to show that the decimal is too large. 
 
 3. Again, the remainder is sometimes placed over the divixor and 
 annexed to the quotient, forming a mixf.d fraction. (Art. i8c, n.) 
 Thus if jj- is reduced to the decimal form, the result may be expressed 
 by .6666+ ; by .6667 ; or by .6666 f. 
 
 (For the further consideration of Circulating Decimals, the student 
 is referred to Higher Arithmetic.) 
 
 Reduce the following to four decimal places : 
 20. $ 22. f 24. T \ 26. f 
 
 2L I 2 3 . A 25- & 2 7 . ft 
 
 Reduce the following to the decimal form : 
 
 28. 75$ 30. 261^ 32. 465^0 34- 74*VV 
 
 29. i 3 6| 31. 346^4. 33 . 523^ 35. 956^ 
 
 ADDITION OF DECIMALS. 
 
 188. Since decimals increase and decrease regularly by 
 the scale of 10, it is plain they may be added, subtracted, 
 multiplied, and divided like whole numbers. 
 
 Or, they may be reduced to a common denominator, then 
 be added, subtracted, and divided like Common Fractions. 
 (Arts. 156, 184.) 
 
 187. When the numerator with ciphers annexed is exactly divisible by the 
 denominator, what SB the decimal called ? If not exactiv divisible, what ? What 
 is the figure or set of figures repeated called?
 
 ADDITION OF DECIMALS. 147 
 
 189. To find the Amount of two OP more Decimals. 
 
 i. Add 360.1252, 1.91, 12.643, and 152.8413. 
 
 ANALYSIS. Since units of the same order or OPERATION. 
 
 like numbers ouiv can be added to each other, we 360.1252 
 
 reduce the decimals to a common denominator by 1 .9 1 oo 
 
 annexing ciphers; or, which is the same, by 12.6430 
 
 writing the decimals one under another, so that 152.8413 
 
 the decimal points shall be in a perpendicular Ans. $27.5, los 
 line. (Arts. 28, n. 156.) Beginning at the ri jhi, 
 we add each column, and set down the result as in whole numbers, 
 and for the same reasons. (Art. 29, n.) Finally, we place the deci- 
 mal point in the amount directly under those in the numbers added. 
 (Art. 178, a) Hence, the 
 
 RULE. I. Write the numbers so that the decimal points 
 shall stand one under another, with tenths under tenths, etc. 
 
 II. Beginning at the right, add as in whole numbers, and 
 place the decimal point in the amount under those in the 
 numbers added. 
 
 NOTE. Placing tenths under tenths, hundredths under hundredths, 
 etc., in effect, reduces the decimals to a com. denominator; hence 
 the ciphers on the right may be omitted. (Arts. 181, 184.) 
 
 (') (3) (4-) ( 5 .) 
 
 41.3602 416.378 36.81045 4-83907 
 
 4.213 85.1 .203 .293 
 
 61.46 .4681 5-3078 .40 
 
 375-265 4.38 87.69043 5.1067 
 
 482.2982 Ans. 375- 2 956 9.25 3.75039 
 
 6. AVI uit is the sum of 41.37 1 + 2.29 + 73.402 + 1.729 ? 
 
 7. What is the sum of 823.37 + 7.375 + 61.1 +.843 ? 
 
 8. What is the sum of .3925 + .64 + .462 + . 7 +.56781 ? 
 
 9. AVhat is the sum of 86 005+4.0003 + 2.00007 ? 
 to. AVhat is the sum of 1.713 + 2.30 + 6.400+27.004? 
 
 11. Add together 7 tenths; 312 thousandths; 46 hun 
 credths; 9 tenths; and 228 ten-thousandths. 
 
 12. Add together 23 ten-thousandths; 23 hundred- 
 thousandths; 23 thousandths; 23 hundredths; and 23. 
 
 1^9. How are decimals added f How point off the amount ?
 
 \ 
 
 148 SUBTRACTION OF DECIMALS. 
 
 13. Add together five hundred seventy-five and seven- 
 tenths; two hundred fifty-nine ten-thousandths; five- 
 millionths; three hundred twenty hundred-thousandths. 
 
 14. A farmer gathered iy| bushels of apples from one 
 tree ; 8^ bushels from another; 10^ bushels from another ; 
 and i6| bushels from another. Required the number of 
 bushels he had, expressed decimally. 
 
 15. A grocer sold 7-^ pounds of sugar to one customer; 
 11.37 pounds to another; icf pounds to another; 25^ 
 pounds to another; and 21.375 pounds to another: how 
 many pounds did he sell to all ? 
 
 SUBTRACTION OF DECIMALS. 
 
 190. To find the Difference between two Decimals. 
 
 j. What is the difference between 2.607 an( l -7 2 35 ? 
 
 ANALYSIS. We red uce the decimals to a common OPEKATIOH. 
 
 denominator, by annexing ciphers, or by writing the 2.607 
 
 game orders one under another. (Art. 189, n.) For, *7 2 35 
 
 units of the game order or like numbers only, can Ans. 1.8835 
 be subtracted one from the other. 
 
 Beginning at the right, we perceive that 5 ten-thousandths can 
 not be taken from o ; we therefore borrow ten, and then proceed in 
 all respects as in whole numbers. (Art. 38.) Hence, the 
 
 RULE. I. Write the less number tender the greater} so 
 that the decimal points shall stand one under the oilier, 
 with tenths under tenths, etc. 
 
 II. Beginning at the right hand, proceed as in subtract- 
 ing whole numbers, and place the decimal point in the 
 remainder tinder that in the subtrahend. 
 
 NOTE. Writing the same orders one under another, in effect 
 reduces the decimals to a common denominator. (Art. 189, n.) 
 
 (*) (3-) (4.) (5-) 
 
 From 13.051 7-0392 20.41 85.3004 
 
 Take 5.22 .4367 1 3-4 2 S 6 7-35 2 46 
 
 190. How are decimals subtracted ? How point off the remainder T
 
 MULTIPLICATION OFDECIMALS. 149 
 
 Perform the following subtractions : 
 
 6. 13.051 minus 5.22. 12. 10 minus 9.1030245. 
 
 7. 7.0392 minus 0.43671. 13. 100 minus 99.4503067. 
 
 8. 20.41 minus 3.0425. 14. i minus .123456789. 
 
 9. 85.3004 minus 7.35246. 15. i minus .98764321. 
 
 10. 93.38 minus 14.810034. 16. o.i minus .001. 
 
 11. 3 minus 0.103784. 17. o.oi minus .oooor. 
 
 18. From 100 take i thousandth. 
 
 19. From 45 take 45 ten-thousandths. 
 
 20. From i ten-thousandth take 2 millionths. 
 
 21. A man having $673.875, paid $230.05 for a horse: 
 how much had he left ? 
 
 22. A father having 504.03 acres of land, gave 100.45 
 acres to one son, 263.75 acres to another: how much had 
 he left ? 
 
 23. What is the difference between 203.007 and 302.07 ? 
 
 24. Two men starting from the same place, traveled in 
 opposite directions, one going 571.37 miles, the other 
 501.037 miles: how much further did one travel than the> 
 other ; and how far apart were they ? 
 
 MULTIPLICATION OF DECIMALS. 
 191. To find the Product of two or more Decimals. 
 
 1. What is the product of 45 multiplied by .7 ? 
 
 ANALYSIS. -T^^s- Now -^-times 45 = Vcf =315-5-10, OPERATION. 
 or 31 5, Ana. In the operation, we multiply by 7 in- 45 
 stead of -fo ; therefore the produc* is 10 times too large. .7 
 
 To correct this, we point off i figure on the right, -II.G Ant. 
 which divides it by 10. (Arts. 79, 143, 165.) 
 
 2. What is the product of 9.7 multiplied by .9 ? 
 ANALYSIS. 9.7=?$, and .9=-,V Now -f {J times ?= OPKRATIOK. 
 
 ^1=873-:- loo, or 8.73, Ans. In this operation we 9-7 
 
 also multiply as in whole numbers, and point off 2 .9 
 
 figures on the right of the product for decimals, which 8.73 
 divides it by 100. (Art. 79.)
 
 150 MULTIPLICATION OF DECIMALS. 
 
 BEM. By inspecting these operations, we see that each answer 
 contains as many decimal figures as there are- decimal places in both 
 factors. Hence, the 
 
 KULE. Multiply as in whole numbers, and from the 
 right of the product, point off as many figures for decimals, 
 as there are decimal places in both factors. 
 
 NOTES. i. Multiplication of Decimals is based upon the eame 
 principles as Multiplication of Common Fractions. (Art. 166.) 
 
 2. The reason for pointing off the product is this: The product of 
 any two decimal numerators is as many times too large as there arc 
 units in the product of their denominators, and pointing it off divides 
 it by that product. (Art. 79.) For, the product of the denominators 
 of two decimals is always i with as many ciphers annexed as there 
 are decimal places in both numerators. (Arts. 57, 179, 186, Dem.) 
 . 3. If the produ, has not as many figures as there are decimals in 
 both factors, supply the deficiency by prefixing ciphers. 
 
 2. Multiply .015 by .03. 
 
 SOLUTION. .015 x ,03=.ooo45. The product requires 5 decimal 
 places ; hence, 3 ciphers must be prefixed to 45. 
 
 (3-) (4-) (5-) (60 
 
 Multiply 29.06 .07213 .000456 4360.12 
 
 By .005 .OO2I -0037 5.000 
 
 7. 4.0005 x .00301. ii. 0.0048 x .0091. 
 
 8. 5.0206x4.0007. 12. 15.004 x .10009. 
 
 9. 3.0004x106. 13. 6.0103 x .00012. 
 10. 7.2136x100. 14. 20007 x .000001. 
 
 15. If i box contains 17.25 pounds of butter, how many 
 pounds will 25 boxes contain ? 
 
 1 6. What cost 20. <; barrels of flour, at $10.875 a barrel ? 
 
 17. If one acre produces 750.5 bushels of potatoes, how 
 much will .625 acres produce ? 
 
 18. What cost 53 horses, at $200.75 apiece? 
 
 19. Multiply 28 hundredths by 45 thousandths. 
 
 20. What cost 73.25 yards of cJoth, at $9.375 per yard ? 
 
 191. How are decimals multiplied ? How point off the product ? Note. Ex- 
 plain the reason for pointing off. If the product doea not contain as many 
 figure? as there are dccimnl^ in the factors, what is to be done ? 192. How mul- 
 tiply \ decimal by 10, 100, etc.
 
 MULTIPLICATION OF DECIMALS. 151 
 
 21. Multiply 5 tenths by 5 thousandths. 
 
 22. Multiply two hundredths by two ten-thousandths. 
 
 23. Multiply seven hundredths by seven millionths. 
 
 2 4.. Multiply two hundred and one thousandths by three 
 millionths. 
 
 25. Multiply five hundred- thousandths by six thous- 
 andths. 
 
 26. Multiply four millionths by sixty-three thousandths. 
 
 27. Multiply a hundred by a hundred-thousandth. 
 
 28. Multiply one million by one millionth. 
 
 29. Multiply one millionth by one billionth. 
 
 192. To multiply Decimals by 10, 100, 1000, etc. 
 
 30. Multiply .43215 by 1000. 
 
 . Removing a figure one place to the left, we have seen 
 multiplies it by 10. But moving the decimal point one place to the 
 right has the same effect on the position of the figures ; therefore, it 
 multiplies them by 10. For the same reason, moving the decimal 
 point two places to the right multiplies the figures by 100, and so on. 
 In the given example the multiplier is 1000 ; we therefore move the 
 decimal point three places to the right, and have 432.15, the answer 
 required. Hence, the 
 
 RULE. Move the decimal point as many places toward 
 the right as there are ciphers in the multiplier. (Art. 181.) 
 
 31. Multiply 32.0505 by ico. 
 
 32. Multiply 8.00356 by 1000. 
 
 33. Multiply 0.000243 by 10000. 
 
 34. Multiply 0.000058 by 100000. 
 
 35. Multiply 0.000005 by 1000000. 
 
 36. If a newsboy makes $0.005 on each paper, what is 
 Irs profit on 10000 papers? 
 
 37. What is the profit on 100000 eggs, at $0.006 apiece? 
 
 38. If a farmer gives 4.25 bushels of apples for one yard 
 of cloth, how many bushels should he give for 6.5 yards ? 
 
 39. If a man walks 3.75 miles an hour, how far will he 
 walk in 17.5 hours?
 
 152 DIVISION OF DECIMALS. 
 
 DIVISION OF DECIMALS. 
 193. To divide one Decimal by another. 
 
 1. How many times .2 in .3 ? 
 
 ANALYSIS. These decimals have a com. de- .2).8 
 nom. ; hence, we divide as in Common Fractions, Anv~Al\m &. 
 and the quotient is a whole number. (Art. 169, n.) 
 
 2. How many times .08 in .7 ? 
 
 ANALYSIS. Reduced to a common denominator, the .o8).7ooo 
 given decimals become .08 and .70. Now .08 is in .70, A g 
 8 times, and .06 rem. Put the 8 in units' place. Re- 
 duced to the next lower order, .06=. 060, and .08 is in .060, .7 of a 
 time, and. 004 rem. Put the 7 in tentlis' place. Finally, .08 is in .0040, 
 .05 of a time. Write the 5 in hundredth*' place. Ana. 8.75 times. 
 
 3. How many times is .5 contained in .025 ? 
 ANALYSIS. Reduced to a common denominatoi, ,5oo).O25oo 
 
 .5=. 500, and .O25=.o25. Since 500 is not contained 
 
 .. , ... 
 
 -i IT -/l/fco O.Os i. 
 
 in .025, put a cipher in units place, ana reduce to 
 the next lower order. But .500 is not contained in .0250. Put a 
 cipher in tenths' place, and reducing to the next order, .500 is in 
 .02500, .05 of a time. Write the 5 in hundredths' place. 
 
 REM. When two decimals have a com. denom. the quotient figures 
 thence arising are whole numbers, as in common fractions. (Art. 169, n.) 
 
 If ciphers are annexed to the remainder, the next quotient figure 
 will be tenths, the second hundredths, &c. Hence, the 
 
 EULW Reduce the decimals to a common denominator, 
 and divide the numerator of the dividend by that of the 
 divisor, placing a decimal point on the right of the quotient. 
 
 Annex ciphers to the remainder, and divide as before. 
 The figures on the left of the decimal point are whole num- 
 bers j those on the right, decimals. 
 
 Or, divide as in whole numbers, and from the right of the 
 quotient, point off as many figures for decimals as the deci- 
 mal places in the dividend exceed those in the divisor. 
 
 NOTES. i. If there are not figures enough in the quotient for the 
 decimals required by the second method, prefix ciphers. 
 
 193. How divide decimals? If there is a remainder? Rem. When two decimals 
 have a com. denom., what is the quotient ? When ciphers are annexed to tha 
 remainder, what ? When the ad method is used, how point off T
 
 DIVISION OF DECIMALS. 153 
 
 2. If there is a remainder after the required number of decimals is 
 found, annex the sign + to the quotient. 
 
 2. Divide .063 by 9. 4. Divide 642 by 1.07. 
 
 3. Divide .856 by .214. 5. Divide 4.57 by n. 
 
 Perform the following divisions : 
 
 6. 78.4-^-2.6. 14. .03753-^.00006, 
 
 7. 8.45 -r 3.5. 15. I2H-I.2. 
 
 8. 1.262 9/7. l6. I.2-T-.I2. 
 
 9. .4625 .65. 17. .12-:- 12. 
 
 10. 97.68100. 18. .00001-7-5. 
 
 11. 6.75-:-iooo. 19. .00005-:-.]'. 
 
 12. .576-^10000. 20. .0003-7- .00000 6. 
 
 13. 45.30-7-3020. 21. .27H-IOOOOOO. 
 
 22. If 2.25 yards of cloth make i coat, how many coata 
 can be made of 103.5 yards ? 
 
 23. How many rods in 732.75 feet,at 16.5 feet to a rod ? 
 
 24. At $18.75 apiece, how many stoves can be bought 
 for 8506.25 ? 
 
 194. To Divide Decimals by 10, 100, 1000, etc. 
 
 25. "What is the quotient of 846.25 divided by 100? 
 
 ANALYSIS. Moving the decimal point one place to the left divides 
 a number by 10. For the same reason, moving the decimal point 
 two places to the left, divides it by 100. In the given question, re- 
 j loving the decimal point two places to the left, we have 8.4625, the 
 rnswer required. Hence, the 
 
 RULE. Move the decimal point as many places toward 
 the left as there are ciphers In the divisor. (Art. 181.) 
 
 26. Divide 4375.3 by 1000. 28. Divide 2.53 by 100000. 
 
 27. Divide 638.45 by 10000. 29. Divide .5 by 1000000. 
 
 30. Bought 1000 pins for $.5 : what was the cost of each ? 
 
 31. If a man pays $475 for 10000 yards of muslin, what 
 is that a yard ? 
 
 194. How divide decimals by 10, 100, 1000, etc.?
 
 195. United States Money is the national cur- 
 rency of the United States, and is often called Federal 
 Money. It is founded upon the Decimal Notation, and is 
 thence called Decimal Currency. 
 
 Its denominations are eagles, dollars, dimes, cents, and 
 mills. 
 
 TABLE. 
 
 10 mills (m.) are i cent, - - - - - ft. 
 
 10 cents " I dime, d. 
 
 10 dimes " i dollar, - - dol. or $. 
 
 10 dollars " I eagle, - - - - E 
 
 
 
 NOTATION OF UNITED STATES MONEY. 
 
 196. The Dollar is the unit; hence, dollars are 
 whole numbers, and have the sign ($) prefixed to them. 
 
 In $i there are 100 cents; therefore cents are liun- 
 dredths of a dollar, and occupy hnndredths' place. 
 
 Again, in $i there are 1000 mills; hence, mills are 
 thousandths of a dollar, and occupy thousandths' place. 
 
 No FES. i. The origin of the sign ($) has been variously er- 
 plaine 1. Some suppose it an imitation of the two pillars of Her- 
 cules, connected by a scroll found on the old Spanish coins. Others 
 think it is a modified figure 8, stamped upon these coins, denoting 
 8 reals, or a dollar. 
 
 A more plausible explanation is that it is a monogram of United 
 States, the curve of the U being dropped, and the S written over it. 
 
 195. What !s United States money? Upon what founded? What sometimes 
 called? The denominations ? Repeat the Table. 196. What is the unit ? What are 
 dimes? Cents? Mills? Note. What is the origin of the sign $ ? The meaning 
 of dime? Cent? Mill? 197. How write United States money? Note. IIow arc 
 eagles and dimes expressed? If the number of cents is less than 10, what mnst 
 be done ? Why ? If the mill.' are 5 or more, what considered ? If less, what ?
 
 UNITED STATES MONEY. 155 
 
 2. The term Dime is the French dixieme, a tentfi; Cent from the 
 Latin centum, a hundred ; aud Jtfi'W from the Latin mille, a thousand. 
 
 3. United States money was established by act of Congress, in 
 1786. Previous to that, pounds, shillings, pence, etc., were in use. 
 
 197. To express United States money, decimally. 
 Ex. i. Let it be required to write 75 dollars, 37 cents, 
 and 5 mills, decimally. 
 
 ANALYSIS. Dollars are integers; we therefore write the 75 dol- 
 lars as a whole number, prefixing the ($), as $75. Again, cents are 
 hundredths of a dollar ; therefore we write the 37 cents in the first 
 two places on the right of tho dollars, with a decimal point on their 
 left as $75.37. Finally, mills are thousandths of a dollar ; and writing 
 the 5 mills in the first place on the right of cents, we have $75.375, 
 the decimal required. (Art. 179.) Hence, the 
 
 RULE. Write dollars as ivhole numbers, cents as hun- 
 dredths, and mills as thousandths, placing the sign ($) be- 
 fore dollars, and a decimal point between dollars and cents. 
 
 NOTES. I. Eagles and dimes are not used in business calcula- 
 tions ; the former are expressed by lens of dollars ; the latter by tens 
 of cents. Thus, 15 eagles are $150, and 6 dimes are 60 cents. 
 
 2. As cents occupy two places, if the number to be expressed is less 
 than 10, a cipher must be prefixed to the figure denoting them. 
 
 3. Cents are often expressed by a common fraction having 100 for 
 its denominator. Thus, $7.38 is written $7 iVfl. and i 8 rc ^d " 7 and 
 -, 3 ,, 8 (T dollars." 
 
 Mills also are sometimes expressed by a common fraction. Thus, 
 12 cts. and 5 mills are written $0.125, or $0.12^ ; 18 cts. and 7^ mills 
 are written $0.1875, or $o.i8J, etc. 
 
 4. In biisiness calculations, if the mills in the result are 5 or more^ 
 they are considered a cent ; if less than 5, they are omitted. 
 
 1. Write forty dollars and forty cents. 
 
 2. "Write five dollars, five cents and five mills. 
 
 3. Write fifty dollars, sixty cents, and three mills. 
 
 4. Write one hundred dollars, seven cents, five mills. 
 
 5. Write two thousand and one dollars, eight and a half 
 cents. 
 
 6. Write 7 hundred and 5 dollars and one cent. 
 
 7. Write 84 dollars and 12^ cents. 
 
 8. Write 5 and a half cents; 6 and a fourth cents; n 
 and three-fourths cents, decimally.
 
 15tf UNITED STATES MOXEY. 
 
 9. Write 7 dollars, 31 and a fourth cents, decimally. 
 
 10. Write 19 dollars, 31 J cents, decimally. 
 
 11. Write 14 eagles and 8 dimes, decimally. 
 
 12. Write 5 eagles, 5 dollars, 5 cents, and 5 mills. 
 
 198. To read United States money, expressed decimally. 
 RULE. Call the figures on the left of the decimal point,dol- 
 
 lars ; those in the first two places on the r ight, cents ; the next 
 figure, mills ; the others, decimals of a mill. 
 
 The expression $37.52748 is read 37 dollars, '52 cts., 7 
 mills, and 48 hundredths of a mill. 
 
 NOTE. Cents and mills are sometimes read as decimals of a do?lar. 
 Thus, $7.225 may be read 7 and 225 thousandths dollars. 
 
 Read the following: 
 
 1. $204.30 5. $78.104 9. $1100.001 
 
 2. $360.05 6. 890.007 10. $7.3615 
 
 3. 500.19 7. $1001.10 ii. $8.0043 
 
 4. $61.035 8. $1010.01 12. $10.00175 
 
 REDUCTION OF UNITED STATES MONEY. 
 CASE I. 
 
 199. To reduce Dollars to Cents and Mills. 
 
 1. In $67 how many cents? 
 
 ANALYSIS. As there arc zoo cents in every OPERATION. 
 
 dollar, there must be 100 times as many cents 67 X 100 6700 
 as dollars in the given sum. But to multiply A US. 6700 cts. 
 by loo we annex two ciphers. (Art. 57.) 
 
 2. In $84, how many mills? 
 
 ANALYSIS. For a like reason, there are 84 X 1000 = 84000 
 1000 times as many mills as dollars, or 10 Ans. 84000 mill*, 
 times as many mills as cents. Hence, the 
 
 RULE. To reduce dollars to cents, multiply them by 100. 
 To reduce dollars to mills, multiply them by 1000. 
 To reduce cents to mills, multiply them by 10. 
 
 198. How read United States money T
 
 UNITED STATES MONEY. 157 
 
 NOTE. Dollars and cents are reduced to cents ; also dollars, cents, 
 and mills, to mills, by erasing the sign of dollars ($,), and the deci- 
 mal point. 
 
 2. Reduce $135 to cents. 6. Reduce 97 cents to mills. 
 
 3. Reduce $368 to mills. 7. Reduce $356.25 to cents. 
 
 4. Reduce $100 to mills. 8. Reduce $780.375 to mills. 
 
 5. Reduce $1680 to cents. 9. Reduce $800.60 to mills. 
 
 CASE II. 
 
 200. To reduce Cents and Mills to Dollars, 
 i. In 6837 cents how many dollars? 
 
 ANALYSIS. Since 100 cents make I dollar, OPERATION. 
 6837 cents will make as many dollars as 100 is I ) 68 137 
 contained times in 6837, and 6837-:- 100=68, $68.37 -A.US. 
 
 and 37 cents over. (Art. 79.) 
 
 In like manner any number of mills will make as many dollars as 
 looo is contained times in that number. Hence, the 
 
 RULE. To reduce cents to dollars, divide them by 100. 
 
 To reduce mills to dollars, divide them by 1000. 
 
 To reduce mills to cents, divide them by 10. (Art. 194.) 
 
 NOTE. The first two figures cut off on the right are cents, the 
 next one mills. 
 
 2. Reduce 1625 cts. to dols. 6. Change 89567 cts. to dols. 
 
 3. Reduce 8126 m's to dols. 7. Change 94283 m's to dols. 
 
 4. Reduce 10000 m's to dols. 8. Change 85600 m's to cents. 
 
 5. Reduce 9265 m's to cents, o. Change 263475 m's to dols. 
 
 10. A farmer sold 763 apples, at a cent apiece : ho\V 
 many dollars did they come to ? 
 
 u. A market woman sold 5 hundred eggs, at 2 cents 
 each : how many dollars did she receive for them ? 
 
 12. A fruit dealer sold 675 watermelons at 1000 mills 
 apiece : how many dollars did he receive for them ? 
 
 199. How reduce dollars to cents? To mills? How cents to mills ? Note. How 
 dollars to cents and mills? aoo. How reduce cents and mills to dollars? JVofe 
 What are the fisares cut off?
 
 158 UtfllED STATES HONEY. 
 
 ADDITION OF UNITED STATES MONEY. 
 
 201. United Slates Money, we have seen, is founded 
 upon the decimal notation; hence, all its operations ara 
 precisely the same as the corresponding operations in 
 Decimal Fractions. 
 
 202. To find the Amount of two or more Sums of Money. 
 
 1. What is the sum of $45.625 ; $109.07 ; and $450.137 ? 
 
 ANALYSIS. Units of the same order only can be OPERATION. 
 
 added together. For convenience in adding, we there- $45.625 
 
 fore write dollars under dollars, cents under cents, etc., 109,07 
 
 with the decimal points in a perpendicular line. Begin- 450. 137 
 
 Hing at the ricrht, we add the columns separately, $604.8^2 
 placing the decimal point in the amount under the 
 
 points in the numbers added, to distinguish the dollars from cents 
 and mills. (Art. 197.) Hence, the 
 
 EULE. Write dollars under dollars, cents under cents, 
 etc., and proceed as in Addition of Decimals. 
 
 NOTE. If any of the given numbers have no cents, their place 
 should be supplied by ciphers. 
 
 2. A man paid $13.62^ for a barrel of flour, $25.25 for 
 butter, $9.75 for coal : what did he pay for all ? 
 
 3. A farmer sold a span of horses for $457.50, a yoke of 
 oxen for $235, and a cow for $87.75 : now much did he 
 receive for all ? 
 
 4. What is the sum of $97.87^; $82.09; $20. 12^? 
 
 5. What is the sum of $81.06; $69.18; $67.16; $7.13? 
 
 6. What is the sum of $101.101; $210.10^; $450.27^? 
 
 7. Add $7 and 3 cents; $10; 6 cents; i8f cents. 
 
 8. Add $68 and 5 mills; 87^ cents; 31! cents. 
 
 9. A man paid $8520.75 for his farm, $1860.45 for hia 
 stock, $1650.45 for his house, and $1100.07 f r his furni- 
 ture : what was the cost of the whole ? 
 
 ?nj. What is pnld of operations in United StiteR Money? 203. How add 
 Unite J States Money? Note. If any of the numbers have no cents, how proceed*
 
 UNITED STATES MONET. 159 
 
 10. A lady paid $31^ for a dress, $15 J- for trimmings, 
 and $7|- for making: what was the cost of her dress ? 
 
 11. A grocer sold goods to one customer amounting to 
 $17.50, to another $30.18^, to another $21.06-}, and to 
 another $51.73 : what amount did he sell to all ? 
 
 12. A young man paid $31.58 for a coat; $11.63 f r a 
 rest, $14.11 for pants, $10.50 for boots, $7^ for a hat, and 
 $if for gloves : what did his suit cost him ? 
 
 SUBTRACTION OF UNITED STATES MONEY. 
 
 203. To find the Difference between two Sums of Money. 
 
 1. A man having $1343.87^, gave $750.69 to tlie Patriot 
 Orphan Home : how much had he left ? 
 
 ANALYSIS. Since the same orders only can be sub- OPEBATIOX. 
 
 traded one from the other, for convenience we write $1343.875 
 
 dollars under dollars, cents under cents, etc. Begin- 750.69 
 
 ning at the right, we subtract each figure separately, &CQ., jgT 
 and place the decimal point in the remainder under 
 that in the subtrahend, for the same reason as in subtracting deci- 
 mals. (Art. 190.) Hence, the 
 
 KULE. Write the less number under the greater, dollars 
 under dollars, cents under cents, etc., and proceed as in 
 Subtraction of Decimals. (Art, 190.) 
 
 NOTE. If only one of the given numbers has cents, their place in 
 the other should be supplied by ciphers. 
 
 2. A man having $861.73, lost $328.625 in gambling: 
 how much had he left ? 
 
 3. If a man's income is$i75o,and his expenses $1145.37!, 
 how much does he lay up ? 
 
 4. A gentleman paid $1500 for his horses, and $975 \ for 
 his carriage : what was the difference in the cost ? 
 
 5. A merchant paid $2573^ for a quantity of tea, und 
 sold it for $3158! : how much did he make ? 
 
 203. How subtract United States money? Note. If either number has no 
 cents, what is to be done ?
 
 ICO UNITED STATES MONEY. 
 
 6. If I pay $5268 for a farm, and sell it for 84319.67, 
 how much shall I lose by the operation ? 
 
 7. From 673 dols. 6J cents, take 501 dols. and 10 cents. 
 
 8. From ion dols. \z\ cents, take 600 dols. and 5 cents. 
 
 9. From i dollar and i cent, subtract 5 cents 5 mills. 
 10. From i\ dols. subtract 7^ cents. 
 
 n. From 500 dols. subtract 5 dols. 5 cents and 5 mills. 
 
 12. A young lady bought a shawl for $35^, a dress for 
 $23^, a hat for $iof, a pair of gloves for $i, and gave 
 the clerk a hundred dollar bill: how much change ought 
 she to receive ? 
 
 MULTIPLICATION OF UNITED STATES MONEY. 
 
 204. To multiply United States Money. 
 
 1. What will 9~ yards of velvet cost, at $i8- per yard ? 
 
 ANALYSIS. 9^ yards will cost 9^ times as much as $18.21; 
 
 i yard. Now 9^ yds.=g.5 yds., and $i8j=$i8.25. _ - 
 We multiply in the usual way, and since there aro 
 three, decimal figures in both factors, we point off three 
 
 in the product for the same reason as in multiplying I 4 2 5 ^ 
 
 decimals. (Art. 191.) Hence, the $ J 73'375 
 
 RULE. Multiply, and point off the product, as in Multi- 
 plication of Decimals. (Art. 191.) 
 
 NOTES. i. In United States Money, as in simple numbers, the 
 multiplier must be considered an abstract number. 
 
 2. If either of the given factors contains a common fraction, it ia 
 generally more convenient to change it to a decimal. 
 
 2. "What will 65 barrels of flour cost, at $11.50 a barrel? 
 
 3. "What will 145.3 pounds of wool cost, at $i. 08 a pound? 
 
 4. What cost 75 pair of skates, at $3.87^ a pair ? 
 
 5. What cost 63 gallons of petroleum, at 95 centa 
 a gallon ? 
 
 6. At $4.17^ a barrel, what will no barrels of apples cost? 
 
 804. How multiply United States money ? Note, What must the multiplier be T
 
 UNITED STATES MONEY. J.C1 
 
 Perform the following multiplications : 
 
 7. $5i$xi.9$. ii. $765.40$ x 6.05. 
 
 8. $6.07^- x 2.3 J. 12. 6.07 x. 008. 
 
 9. $10.05 x 6-$> I 3- $-5 x 1000. 
 10. $100.031 x 3.105. 14. $1.01 ix. ooi. 
 
 15. What cost 35 pounds of raisins, at i8f cents a 
 pound ? 
 
 16. What cost 51 pounds of tea, at $1.15^ a pound? 
 
 17. What cost 150 gallons of milk, at 37 cts. a gallon ? 
 
 18. What cost 12 dozen penknives, at 31^ cents apiece? 
 
 19. What cost 13 boxes of hutter, each containing i6 
 pounds, at 37 cents a pound? 
 
 20. A merchant sold 12 pieces of cloth, each containing 
 35 yards, at $4-| a yard : what did it come to ? 
 
 21. What is the value of 21 bags of coffee, each weigh- 
 ing 55 pounds, at 47^ cents a pound ? 
 
 22. What cost 55 boxes of lemons, at $3^ a box ? 
 
 23. The proprietor of a livery stable took 37 horses to 
 board, at $32 a month: how much did he receive in 12 
 months ? 
 
 DIVISION OF UNITED STATES MONEY. 
 
 205. Division of United States money, like simple divi- 
 sion, embraces two classes of problems : 
 
 First. Those in which both the divisor and dividend are money. 
 
 Second. Those in which the dividend is money and the divisor is 
 an abstract number, or regarded as such. 
 
 In the former, a given sum is to be divided into parts, the value of 
 each part being equal to the divisor ; and the object is to ascertain 
 the number of parts. Hence, the quotient is times or an abstract 
 number. (Art. 63, .) 
 
 In the latter, a given sum is to be divided into a given number of 
 equal parts indicated by the divisor; and the object is to ascertain 
 the value or number of dollars in each part. Hence, the quotient is 
 money, the same as the dividend. (Art. 63, 6.) 
 
 905. What docs Division of U. 8. money embrace T The flrst class ? The second I
 
 162 UNITED STATES MONET. 
 
 206. To Divide Money by Money, OP by an Abstract Number 
 
 i. How many barrels of flour, at $9.56 a barrel, can be 
 bought for $262.90 ? 
 
 ANALYSIS. At $9.56 a barrel, $262.90 will OPERATION. 
 
 buy as many barrels as $9.56 are contained $ 
 times in $262.90, or 27.5 barrels. As the 1912 
 
 divisor and dividend both contain cents, 
 
 they are the same denomination ; therefore, 6602 
 
 the quotient 27, is a whole number. Annex- 
 
 ing a cipher to the remainder, the next quo- 
 tient figure is tenths. (Art. 193, Hem.) 
 Hence, the 
 
 EULE. Divide, and point off the quotient, as in Division 
 of Decimals. (Arts. 64, 193.) 
 
 NOTES. i. In business matters it is rarely necessary to carry the 
 quotient beyond mills. 
 
 2. If there is a remainder after all the figures of the dividend have 
 been divided, annex ciphers and continue the division as far as de- 
 sirable, considering the ciphers annexed as decimals of the dividend. 
 
 2. If 63 caps cost $123.75, what will i cap cost? 
 
 3. If 165 lemons cost $8.25, what will i lemon cost? 
 
 4. Paid $852.50 for 310 sheep : what was that apiece ? 
 
 5. If 356 bridles cost $1040, what will i bridle cost? 
 
 Find the results of the following divisions: 
 
 6. $i7.5o-r-$.i75 10. $.oo5-r-$.o5 
 
 7. $365.07-^- $1.01 n. $ioi-=-$i.oi 
 
 8. $1000-1-25 cts. 12. $500^ $.05 
 
 9. $.25-r$25 13. $I200-H$.002 
 
 14. If 410 chairs cost $1216, what will i chair cost? 
 
 15. At $9$ a ton, how much coal will $8560 buy? 
 
 16. When potash is $120.35 P er ton, how much can be 
 bought for $35267.28 ? 
 
 17. If 2516 oranges cost $157.25, what will one cost? 
 *8. Paid $273.58 for 5000 acres of land: what was that 
 
 per acre ? 
 
 ao6. flow divide money? Note. If there ie a remainder, bow proceed ?
 
 UNITED STATES MONET. 
 
 COUNTING-ROOM EXERCISES. 
 
 163 
 
 207. The Ledger is the principal look of accounts 
 kept by business men. It contains a brief record of their 
 monetary transactions. All the items of the Day Book are 
 transferred to it in a condensed form, for reference and 
 preservation, the debits (marked Dr.) being placed on the 
 left, and the credits (marked Or.) on the right side. 
 
 208. Balancing an Account is finding the differ- 
 ence between the debits and credits. 
 
 Balance the following Ledger Accounts : 
 
 (') 
 
 DR. 
 
 CR. 
 
 DR. 
 
 CR. 
 
 $8645.23 
 
 $2347.19 
 
 $76421.26 
 
 $43261.47 
 
 160.03 
 
 141.07 
 
 2406.71 
 
 728.23 
 
 2731.40 
 
 2137.21 
 
 724.05 
 
 6243.41 
 
 4242.25 
 
 3401.70 
 
 86025.21 
 
 75-69 
 
 324-31 
 
 2217.49 
 
 9307.60 
 
 53268.75 
 
 33i3-i7 
 
 168.03 
 
 685.17 
 
 3102.84 
 
 429.18 
 
 329.17 
 
 61.21 
 
 456.61 
 
 4536.20 
 
 2334-67 
 
 7824.28 
 
 32921.70 
 
 641.46 
 
 4506.41 
 
 60708.19 
 
 6242.09 
 
 182.78 
 
 239.06 
 
 1764.85 
 
 20374-34 
 
 2634.29 
 
 2067.12 
 
 32846.39 
 
 290.25 
 
 3727-34 
 
 675.89 
 
 385-72 
 
 4536.68 
 
 840.68 
 
 1431.07 
 
 23.64 
 
 42937-74 
 
 6219.77 
 
 4804.31 
 
 6072.77 
 
 8i9-35 
 
 4727.91 
 
 6536.48 
 
 50641.39 
 
 30769.27 
 
 23-45 
 
 720.34 
 
 2062.40 
 
 8506.35 
 
 650.80 
 
 36-45 
 
 301-53 
 
 97030.48 
 
 3267.03 
 
 7050.63 
 
 26.47 
 
 876.20 
 
 680.47 
 
 904.38 
 
 805.03 
 
 89030.50 
 
 64.38 
 
 78.05 
 
 2403.67 
 
 384.06 
 
 8350.60 
 
 307.63 
 
 10708.79 
 
 70207.48 
 
 207. What is the ledger ? What does it contain ? How balance an account ?
 
 104 BILLS. 
 
 MAKING OUT BILLS. 
 
 209. A Hill is a written statement of goods sold, ser- 
 vices rendered, etc., and should always include the price 
 of each item, the date, and the place of the transaction. 
 
 A Sill is Htceipted, when the person to whom it 
 is due, or his agent, writes on it the words " Received 
 payment," and his name. 
 
 209, . A Statement of Account is a copy of the 
 items of its debits and credits. 
 
 NOTES. i. The abbreviation Dr. denotes debit or debtor; Or., 
 credit or creditor ; per, by ; the character @, stands for at. Thus, 5 
 books, @ 3 shillings, signifies, 5 books the price of which is 3 shil- 
 lings apiece. 
 
 2. The learner should carefully observe the form of Bills, the 
 place where the date and the names of the buyer and seller are 
 placed, the arrangement of the items, etc. 
 
 Copy and find the amount due on the following Bills : 
 
 (i.) PHILADELPHIA, March ist, 1872. 
 
 Hon. HENRY BARNARD, 
 
 To J. B. LIPPINCOTT & Co., Dr. 
 
 For 6 Webster's Dictionaries, 4to., @ $12.50 
 " 8 reams paper, @ $3.75 
 " 36 slates, @ $0.27 | 
 
 CKEDIT. 
 
 $75 
 30 
 9 
 
 oo 
 oo 
 
 72 
 
 $114 
 
 72 
 
 
 By 10 School Architecture, 
 " 8 Journals of Education, 
 
 @ $545 
 @ $3-75 
 
 $54 
 30 
 
 50 
 
 oo 
 
 | 
 
 50 
 
 
 Balance, \ 
 
 
 
 i $30 22 
 
 209. What Is a bill ? 209, a. A statement of account ? Where is the date 
 placed? (See form.) The name of buyer? Seller? How is the payment of a 
 bill shown t Note. What does Dr. denote f Cr. t
 
 BILLS. 165 
 
 NEW YORK, April id, 1871. 
 
 Mrs. W. C. GAEFIELD, 
 
 Bought of A. T. STEWART. 
 
 28 yds. silk, 
 35 yda. table liiien, 
 6 pair gloves, 
 43-^ yds. muslin, 
 I dew. pr. cotton hose, 
 
 Amount, 
 
 @ $3-50 
 
 @$2.I2i 
 
 $i-75 
 @ $0.33 
 <& $0.80 
 
 ! 
 
 
 
 
 
 
 
 Received Payment, 
 
 A. T. STEWART. 
 
 (3-) KEW ORLEANS, J/ay 5^, 1872. 
 
 GEORGE PEABODY, ESQ., 
 
 Bought of JACOB BARKER. 
 
 Feb. i 
 
 I? 
 March 3 
 
 25 
 April 30 
 
 75 bis. pork, @, $25.00 
 160 bis. flour. @ $8.75 
 500 gals, molasses, @ $0.93 
 75 boxes raisins, @ $5.37 i 
 256 gals, kerosene, @ 0.87 j 
 
 Amount, 
 
 | 
 
 
 
 
 
 
 
 Received Payment, 
 
 J. BARKER, 
 By JOHN HOWARD. 
 
 (A\ BOSTON, May 23^, 1871. 
 
 Messrs. FAIRFIELD & WEBSTER, 
 
 To H. W. HALL, Dr. 
 
 For 319 yds. broadcloth, 
 " 416 yds. cassimere, 
 " I no yds. mnslin, 
 " 265 yds. ticking, 
 
 Amount, 
 
 @ $5-87i 
 
 @ $2.10 
 (^ $0.28 
 $047 
 
 
 
 
 
 
 
 
 Received Payment by Note, 
 
 GEORGE ANDEEBON, 
 for H. W. HALL-
 
 166 BILLS. 
 
 5. James Brewster bought of Horace Foote & Co., New 
 Haven, May i6th, 1867, the following items: 175 pounds 
 of sugar, at 17 cents ; 5 gallons of molasses, at 63 cents ; 
 3 boxes of raisins, at $6 ; 15 pounds of tea, at $i| : what 
 was the amount of his bill? 
 
 6. Georgo Bliss & Co. bought of James Henry, Cincin- 
 nati, June 3d, 1867, 1625 bushels of wheat, at $1.95 ; 130 
 barrels of flour, at $n; 265 pounds of tobacco, at 48 
 cents; and 1730 pounds of cotton, at 2 7^ cents: what was 
 the amount of their bill ? 
 
 7. Pinkney & Brother sold to Henry Rutledge, Rich- 
 mond, July isth, 1867, i shawl, $450; 19 yards of silk, at 
 $3.63 ; 1 6 yards point lace, at Si i ; 6 pair gloves, at $2.05 ; 
 and 12 pair hose, at 87-^ cents: required the amount. 
 
 8. Bought 37 Greek Readers, at $1.85; 60 Greek 
 Grammars, at $1.45; 75 Latin Grammars, at $1.38; 25 
 Virgils, at $3.62 ; 14 Iliad, at $3.28: required the amount 
 
 BUSINESS METHODS. 
 
 210. Calculations in United States money are the same 
 as those in Decimals ; consequently, in ordinary business 
 transactions no additional rules are required. By Reflec- 
 tion and Analysis, however, the operations may often be 
 greatly abbreviated. (Arts. 100, 211-214.) 
 
 1. What will 265 hats cost, at $7 apiece ? 
 
 ANALYSIS. 265 hats will cost 265 times as much as one hat ; arid 
 $7x265 = 11855. 
 
 Or thus : At $i each, 265 hats will cost $265 ; hence, at $7, they 
 will cost 7 times $265, and $265 x 7=$i8ss. 
 
 NOTES. i. Here the price of one and the number of articles are given, 
 to find their cost; hence, it is a question in Multiplication. (Art. 51.) 
 
 2. A mechanic sold 12 ploughs for $114: what was the 
 price of each ? 
 
 ANALYSIS. The price of i plough is A as much as that of 13 
 ploughs; and $114-^12 = 19.50.
 
 COUNTING-ROOM EXERCISES. 167 
 
 NOTE. 2. Here the number of articles and their cost are given, 
 to find the price of one, which is simply a question in Division. 
 
 3. A farmer paid $921.25 for a number of sheep valued 
 at $2.75 apiece: how many did he buy? 
 
 ANALYSIS. At $2.75 apiece, $921.25 will buy as many sheep as 
 $2.75 is contained times in $921.25 ; and $g2i.25-r-$2.75 = 335. 
 
 NOTE. 3. Here the whole cost and the price of one are given, to 
 find the number, which is also a question in Division. 
 
 4. What will 287 chairs cost, at $2f apiece? 
 
 5. What will 75 sofas cost, at $57.50 apiece? 
 
 6. What will 350 table books cost, at 12-^ cents apiece? 
 
 7. What cost 119 tons of coal, at 87^ a ton ? 
 
 8. Paid $84 for 252 pounds of butter: what was that a 
 pound ? 
 
 9. If 8 cords of wood cost $46.06, what will i cord cost? 
 
 10. If 46 acres of laud are worth $1449, what is i acre 
 worth ? 
 
 11. How many Bibles at $5^ can be bought for $561 ? 
 
 12. How many vests at $8^, can be bought for $1262.25 ? 
 
 13. How much will it cost a man a year for cigars, 
 allowing he smokes 5 a day, averaging 6^ cents each, and 
 365 days to a year? 
 
 14. A lady bought 18 yards of silk, at $2.12^ a yard; 14 
 yards of delaine at 63 cents; 12 skeins of silk, at 6\ cents 
 a skein : what was the amount of her bill ? 
 
 15. What will it cost to build a railroad 265 miles long, 
 at $11350 per mile? 
 
 1.6. A farmer sold his butter at 34 cents a pound, and 
 received for it $321.67^: how many pounds did he sell ? 
 
 17. The cheese made of the milk of 53 cows in a season 
 was sold for $1579.40, at 20 cents a pound: how many 
 pounds were sold ; and what the average per cow ? 
 
 1 8. A merchant sold 3 pieces of muslin, each containing 
 45 yards, at 40^ cents a yard, and took his pay in wheat 
 at $i a bushel: how much wheat did he receive?
 
 168 COUNTING-ROOM EXEKCISES. 
 
 19. If a man earns $9^ a week, and spends $4^, ho\ 
 much will he lay up in 52 weeks ? 
 
 20. If a man drinks 3 glasses of liquor a day, costing 10 
 cents a glass, and smokes 5 cigars at 6 cents each, how 
 many acres of land, at $i an acre, could he buy for the 
 sum he pays for liquor and cigars in 35 years, allowing 
 365 days to a year ? 
 
 21. A grocer bought 175 boxes of oranges, at $6.37^, 
 and sold the lot for $637.50: what did he make, or lose? 
 
 211. To find the Cost of a number of articles, the Price of 
 one being an Aliquot part of $f. 
 
 i. What will 168 melons cost, at 12^ cents apiece ? 
 
 ANALYSIS. By inspection tlio learner will perceive 
 that 12^ cents=| dollar. Now. at i dollar apiece, OPERATION. 
 168 melons will cost $168. Bat the price is i of a 8)6S 
 dollar apiece; therefore, they will cost \ of $168, $21 Ans. 
 which is $21 Hence, the 
 
 RULE. Take such a part of the given number as indi- 
 cated by the aliquot part of & i, expressing the price <f one; 
 the result will be the cost. (Arts. 105, 270.) 
 
 2. "What will 265 pounds of raisins cost, at 25 cents? 
 
 3. What cost 195 pounds of butter, at 33 cents? 
 
 4. What cost 352 skeins of silk, at 6 cents ? 
 
 5. What cost 819 shad, at 50 cents apiece? 
 
 6. At i2\ cents a dozen, what will 100 dozen eggs cost ? 
 
 7. At 20 cents a yard, what will 750 yards of calico cost ? 
 
 8. At i6| cents, what will 1250 pine apples come to? 
 
 9. What cost 1745 yards of delaine, at 33 \ cents ? 
 10. At 8 cents apiece, what will 375 pencils cost? 
 u. At 25 cents apiece, what will 1167 slates cost? 
 
 12. How much will 175 dozen eggs cost, at 20 cts. a doz. ? 
 13 What will 219 slates cost, at 12$ cents each? 
 
 14. At i6| cts. apiece, what will 645 melons cost? 
 
 15. At 33! cts. apiece, how much will 347 penknives cost ?
 
 COUNTING-BOOM EXERCISES. 169 
 
 212. To find the Cost of a number of articles, the Price of 
 
 one being $1 plus an Aliquot part of $1. 
 
 16. What cost 275 pounds of tea, at $1.25 a pound? 
 
 ANALYSIS. The price of i pound $1.25=$! + $i. At 4)275 
 $i a pound, the cost of 275 pounds would be $275. But AflTT 
 
 the price is fi + $.V; therefore 275 pounds will cost ; 
 
 once $275 + i of $275 = $343-75- Hence, the $343-75 
 
 RULE. To the number of articles, add \, -J, of itself, 
 as the case may be; the sum will be their cost. 
 
 17. At $1.25 per acre, what will 168 acres of land cost? 
 
 18. At $1.50 apiece, what will 365 chairs cost ? 
 
 19. What cost 512 caps, at $i apiece ? 
 
 20. At $i.i6| apiece, what cost 12 dozen fans? 
 
 21. A man sold 200 overcoats at a profit of $1.20 apices: 
 how much did he make ? 
 
 213. To find the Number of articles, the Cost being given, 
 
 and the Price of one an Aliquot part of $1. 
 
 22. How many spellers, at \z\ cents apiece, can be 
 bought for $25 ? 
 
 ANALYSIS. 12^ cents=$^; therefore $25 will $. I2 i :<pj 
 buy as many spellers as there are eighths in $25, $25 -:-$- = 200 
 and 25-^-^=200. Ans. zoo spellers. Hence, the ' 
 
 RULE. Divide the cost of the whole by the aliquot part 
 0/$i, expressing the price of one. 
 
 23. How many yards of flannel, at 50 cents, can be 
 bought for $6.83 ? 
 
 24. How many pounds of candy, at 33^ cents, can be 
 purchased with $375 ? 
 
 25. Paid $450 for cocoa-nuts which were 25 cents each, 
 how many were bought ? 
 
 26. At 20 ,?ents each, how many pine-apples can be pur- 
 chased for $538 ? 
 
 211. How fina the cos." of articles by aliquot parts of $i ? 212. How when the 
 price is $i plus an aliquot part of $i 213. How flnd the number of article^. 
 when cost i# given, nnd the nrice of one i? an r.liquot part of $i ?
 
 170 COUtfTING-ROOil EXEKC1SES. 
 
 214. To find the Cost of articles, sold by the 100, or 1000. 
 
 1. What will 1765 oranges cost, at $6.125 a hundred? 
 
 ANAIYBIS. At $6.125 f r each orange, 1765 OPERATION. 
 oranges would cost 1765 times $6 125, and $6 125 x $6.125 
 
 1765 = 110810.625. But the price is $6.125 tor a 1765 
 
 hundred ; therefore this product is 100 times too $108 1062? 
 large. To correct it, we divide by 100, or remove 
 the decimal point two places to the left. (Art. 181.) 
 
 In like manner if the price is given by the 1000, we multiply the 
 price and number of articles together, and remove the decimal point 
 in the product three places to the left, which divides it by 1000. 
 Heuce, the 
 
 RULE. Multiply the price and number of articles 
 together, and divide the product by 100 or 1000, as the 
 case may require. (Art. 181.) 
 
 NOTE In business transactions, the letter C is sometimes puv 
 for hundred; and M for thousand. 
 
 2. What will 4532 bricks cost, at $17.25 per M. ? 
 
 3. What cost 1925 pounds of maple sugar, at $12.50 per 
 hundred? 
 
 4. What cost 25268 feet of boards, at $31.25 per thous- 
 and ? 
 
 5. At $5f per hundred, how much will 20345 pounds 
 of flour come to ? 
 
 6. At $6.25 per hundred, what will 19263 pounds of 
 codnsh come to ? 
 
 7. What cost 10250 envelopes, at $3.95 per thousand? 
 
 8. What cost 1275 oysters, at $1.75 per hundred? 
 
 9. What cost 13456 shingles, at $7.45 per M. ? 
 io. What cost 82 rails, at $5! a hundred ? 
 
 n. What cost 93 pine apples, at $15.25 a hundred? 
 12. What cost 355 feet of lumber, at $45 per thousand 9 
 
 214. How find the cost of articles eold by the 100 or 1000 f Note. What does C 
 etaadfor? ThatM?
 
 COMPOUND NUMBERS. 
 
 215. Simple Numbers are those which contain 
 units of one denomination only ; as, three, five, 2 oranges, 
 4 feet, etc. (Art. 101.) 
 
 216. Compound Numbers are those which con- 
 tain units of two or more denominations of the same 
 nature; as, 5 pounds and 8 ounces; 3 yards, 2 feet and 4 
 inches, etc. 
 
 But the expression 2 feet and 4 pounds, is not a com- 
 pound number; for, its units are of unlike nature. 
 
 NOTES. i. Compound Numbers are restricted to the divisions of 
 Money, Weights, and Measures, and are often called Denominate 
 Numbers. 
 
 2. For convenience of reference, the Compound TaUes are placed 
 together. If the teacher wishes to give exercises upon them as they 
 are recited, he will find examples arranged in groups ooi responding 
 with the order of the Tables in Arts. 276, 279. 
 
 MOIS'EY. 
 
 217. Money is the measure or standard of value. It 
 is often called currency, or circulating medium, and is of 
 two kinds, metallic and paper. 
 
 Metallic Money consists of stamped pieces of metal, 
 called coins. It is also called specie, or specie currency. 
 
 Paper Money consists of notes or Mils issued by the 
 Government and Banks, redeemable in coin. It is often 
 called paper currency. 
 
 ais. Wliat are simple numbers ? 216. Compound? Give an example of each. 
 Note. To what p.-e compound numbers restricted? 217. What is moacy? Me- 
 tallic money ? Paper money ?
 
 172 COMPOUND NUMBERS. 
 
 UNITED STATES MONEY. 
 
 218. United States Money is the national cur- 
 rency of the United States, and is often called Federal 
 Money. Its denominations are eagles, dollars, dimes, cents, 
 and mills., 
 
 TABLE. 
 
 10 mills (in) are i cent, ct. 
 
 10 cents " i dime, d. 
 
 10 dimes " i dollar, ... - dol. or . 
 
 10 dollars " i eagle - - - - - Z7. 
 
 219. The Metallic Currency of the United States 
 consists of gold and silver coins, and the minor coins.* 
 
 1. The gold coins are the double eagle, eagle, half eagle, quarter 
 eagle, three dollar piece, and dollar. 
 
 The dollar, at the standard weight, is the unit of value. 
 
 2. The silver coins are the dollar, "trade" dollar, half dollar, 
 quarter dollar, and dime. 
 
 3. The minor coins are the $-cent and ycent pieces, and the cent. 
 
 220. The iveiyht and jmrity cf the coins of the 
 United States are regulated by the laws of Congress. 
 
 1. The standard weight of the gold dollar is 25.8 gr. ; of the quar- 
 ter eagle, 64 5 gr. ; of the 3-dollar piece, 77.4 gr. ; of the half eagle, 
 129 gr. ; of the eagle, 258 gr. ; of the double eagle, 516 pr. 
 
 2. The we'g7it of the dollar is 412 J grains, Troy; the "trade" 
 dollar, 420 grains ; the half dollar, i2i grams ; the quarter dollar 
 and dime, one-half and one-fifth the weight of the half dollar. 
 
 3. The weight of the 5-cent piece is 77. 16 grains, or 5 grams ; of 
 the 3-cent piece, 30 grains ; of the cent, 48 grains. 
 
 4. The standard purity of the gold and silver coins is nine-tenths 
 pure metal, and one-tenth alloy. The alloy of gold coins is silver &nd 
 copper; the silver, by law, is not to exceed one-tenth of the whole 
 alloy. The alloy of silver coins is pure copper. 
 
 219. Of what does the metallic currency of tha United States consist T What 
 are the gold coins ? The silver ? The minor coins 1 220. IIow is the weight and 
 purity of United States coins rejrulated ? 
 
 * Act ef Congrats, March $d, 1^73.
 
 COMPOUND NUMBERS. 173 
 
 5. The five-cent and three-cent pieces are composed of one-fourth 
 nickel anJ tJiree-fourths copper ; the cent, of 95 parts copper and 5 
 parts of tin and zinc. They are known as nickel, and bronze coins. 
 
 NOTES. i. The Trade dollar is so called, because of its intended 
 use for commercial purposes among the great Eastern nations. 
 
 2 . The gold coins are a legal tender in all payments; the silver coins, 
 for any amount not exceeding $5 in any one payment; the minor 
 coins, for any amount not exceeding 25 cents in any one payment. 
 
 3. The diameter of the nickel 5-cent piece is two centimeters, and 
 its weight 5 grams. These magnitudes present a simple relation of 
 the Metric weights and measures to our own. 
 
 4. The silver 5-cent and 3-cent pieces, the bronze 2-cent piece, 
 the old copper cent and half -cent are no longer issued. Mills were 
 never coined. 
 
 221. The Paper Currency of the United States 
 consists of Treasury-notes issued by the Government 
 known as Greenbacks, and Bank-notes issued by Banks. 
 
 NOTE. Treasury notes less than $i,'are called Fractional Cur- 
 rency. 
 
 ENGLISH MONEY. 
 
 222. English Money is the national currency of 
 Great Britain, and is often called Sterling Money. The 
 denominations are pounds, shillings, pence, and. fat things. 
 
 TABLE. 
 
 4 farthings (qr. or far.) are i penny, - - d. 
 
 12 pence " i shilling, - 8. 
 
 20 shillings " i pound or sovereign, - - 
 
 21 shillings " i guinea, ------ g. 
 
 NOTES. i. The gold coins are the sovereign and half sovereign. 
 The pound sterling was never coined. It is a bank note, and is 
 represented by the sovereign. Its legal value as fixed by Congress is 
 $4.8665. This is its intrinsic value, as estimated at the U. S. Mint. 
 
 What is the alloy of gold coins ? Of silver? 221. Of what does U. S. paper 
 currency consist? 222. English money? The denominations? The Table? 
 Note. Is the pound a coin ? IIow represented ? What is the value of a pound ?
 
 174 COMPOUND NUMBERS. 
 
 2. The silver coins aro the crown (53.); the hall-crown (2s. 6d.), 
 the florin (as ) ; the shilling (i2d.) ; the six-penny, four-penny, and 
 three-penny pieces. 
 
 3. The copper coins are the penny, half-penny, and farthing. 
 
 4. Farthings are commonly expressed as fractions of a penny, as 7$d. 
 
 5. The oblique mark (/) sometimes placed between shillings and 
 pence, is a modification of the long /. 
 
 CANADA MONEY. 
 
 223. Canada Money is the legal currency of the 
 Dominion of Canada. Its denominations are dollars, 
 cents, and mills, which have the same value as the cor- 
 responding denominations of U. S. money. Hence, all the 
 operations in it are the same as those in U. S. money. 
 
 NOTE. The present system was established in 1858. 
 
 FRENCH MONEY. 
 
 224. Frencli Money is the national currency of 
 France. The denominations are the franc, the decline, 
 and centime. 
 
 TABLE. 
 
 10 centimes - - - - are i decime. 
 10 declines - - - - " i franc. 
 
 NOTES. i. The system is founded upon the decimal notation ; 
 hence, all the operations in it are the same as those in U. S. money. 
 
 2. The franc is the unit; decimes are tenths of a franc, and 
 centimes hundredths. 
 
 3. Centimes by contraction are commonly called cents. 
 
 4. Decimes, like our dimes, are not used in business calculations; 
 they are expressed by tens of centimes. Thus, 5 decimes are ex- 
 jressed by 50 centimes; 63 francs, 5 deci roes, and 4 centimes are 
 written, 63.54 francs. 
 
 5. The legal value of the franc in estimating duties, is 19.3 cents, 
 its intrinsic value being the same. 
 
 Shilling ? Florin ? Crown ? How are farthings often written ? 223. What la 
 Canada money ? Iff denominations ? Their valne ? 224. French money ? Its 
 denominations The Table ? JVbfc. T\is unit ? The value of a franc ?
 
 WEIGHTS. 
 
 225. Weight is a measure of the force called gravity, 
 by which all bodies tend toward the center of the earth. 
 
 226. Net Weight is the weight of goods without the 
 bag, cask, or box which contains them. 
 
 Gross Weight is the weight of goods with the 
 bag, cask, or box in which they are contained. 
 
 The weights in use are of three kinds, viz : Troy, Avoir- 
 dupois, and Apothecaries' Weight 
 
 TROY WEIGHT. 
 
 227. Troy Weight is employed in weighing gold, 
 silver, and jewels. The denominations are pounds, ounces* 
 pennyweights, and grains. 
 
 TABLE. 
 
 24 grains (gr.) are i pennyweight, - - - pwt. 
 
 20 pennyweights " i ounce, oz. 
 
 12 ounces " i pound, lb. 
 
 NOTE. The unit commonly employed in weighing diamonds, 
 pearls, and other jewels, is the carat, which is equal to 4 grains. 
 
 228. The Standard Unit of weight in the United 
 States, is the Troy pound, which is equal to 22.794377 cubic 
 inches of distilled water, at its maximum density (39.83 
 Fahrenheit),* the barometer standing at 30 inches. It is 
 exactly equal to the Imperial Troy pound of England, the 
 former being copied from the latter by Captain Kater.f 
 
 NOTE. The original element of weight is a grain of wheat taken 
 from the middle of the ear or head. Hence the name grain as a 
 unit of weight. 
 
 225. What is weight ? 22i. Troy weight ? The denomination* ? The table? 
 
 227. The standard unit of weight ? Note. The original element of weight T 
 
 228. Avoirdupois weight ? The denominations ? The Tahle ? 
 
 * Hassler. t Professor A. D. Bache.
 
 176 COMPOUND NUMBERS. 
 
 AVOIRDUPOIS WEIGHT. 
 
 229. Avoirdupois Weight is used in weighing 
 all coarse articles ; as, hay, cotton, meat, groceries, etc., 
 und all metals, except gold and silver. The denominations 
 are tons, hundreds, pounds, and ounces. 
 
 TABLE. 
 
 16 ouncss (oz.} - are i pound, Ib. 
 
 100 pounds, - - - - " I hundredweight, - - - coot. 
 
 20 cwt., or 2000 Ibs., " i ton, T. 
 
 Tlie following denominations are sometimes used : 
 
 1000 ounces are i cubic foot of water. 
 100 pounds " i quintal of dry fish. 
 196 pounds " i barrel of flour. 
 200 pounds " i barrel of fish, beef, or pork, 
 280 pounds " i barrel of salt. 
 
 NOTES. i. The ounce is often divided into halves, quarters, etc. 
 2. In business transactions, the dram, the quarter of 25 Ibs., and 
 t}\BJi7-kin of 56 Ibs., are not used as units of Avoirdupois Weight. 
 
 HEM. In calculating duties, the law allows 112 pounds to a 
 hundredweight, and custom allows the same in weighing a few 
 coarse articles ; as, coal at the mines, chalk in ballast, etc. In all 
 departments of trade, however, both custom and the law of most of 
 the States, call 100 pounds a hundredweight. 
 
 230. The Standard Avoirdupois pound is equal 
 to 7000 grains Troy, or the weight of 27.7015 cubic inches 
 of distilled water, at its maximum density (39.83 Fah.), the 
 barometer being at 30 inches. It is equal to the Imperial 
 Avoirdupois pound of England. 
 
 231. Comparison of Avoirdupois and Troy Weight. 
 
 7000 grains equal I Ib. Avoirdupois. 
 5760 " " i Ib. Troy. 
 
 437 $ " " i oz. Avoirdupois. 
 480 " " i oz. Troy. 
 
 229. To what Is the Avoirdupois pound equal? 230. What is net weight? 
 Gross weight?
 
 COMPOUND NUMBERS. 177 
 
 APOTHECARIES' WEIGHT. 
 
 232. Apothecaries' Weight is used by physicians 
 in prescribing, and apothecaries in mixing, dry medicines. 
 
 20 grains (gr.) are i scruple, - - - - sc., or 3 . 
 
 3 scruples " i dram, dr., or 3 . 
 
 8 drams " I ounce, .... oz., or | . 
 
 12 ounces " i pound, - - - - lb., or ft>. 
 
 NOTE. The only difference between Troy and Apothecaries' 
 
 weight is in the subdicision of the ounce. The pound, ounce, and 
 grain are the same in each. 
 
 232, a. Apothecaries 9 Fluid Mcasu re is use'd 
 i;i mixing liquid medicines. 
 
 60 minims, or drops (ttj. or gtt.) are i fluid drachm, - - /3 . 
 
 8 fluid drachms " i fluid ounce, - - / . 
 
 16 fluid ounces " i pint, - . . . . 0. 
 
 8 pints " i gallon, ----- Cong. 
 
 NOTE. Gtt. for guttae, Latin, signifying drops ; 0, for oct'triits, 
 Lathi for one-eighth ; and Cong, conyiarium, Latin for gallon. 
 
 MEASURES OF EXTEISTSIOK 
 
 233. Extension is that which has one or more of 
 the dimensions, length, breadth, or thickness; as, line-*. 
 Kiirfaces, and solids. 
 
 A line is that which has length without breadth. 
 
 A surface is that which has length and breadth with- 
 out thicJcness. 
 
 A solid is that which has length, breadth, and thickness. 
 
 NOTE. A measure is a conventional standard or unit by which 
 values, weights, lines, surfaces, solids, etc., are computed. 
 
 234. The Standard Unit of length is the yard, which is de- 
 termined from the scale of Troughton, at the temperature of 62 
 Fahrenheit. It is equal to the British Imperial yard. 
 
 232. In what ia Apothecaries' Weiprht used ? Table ? 233. What is extension T 
 A line ? A surface ? A solid ? Not*. A measure ? 214. What is the standard 
 tmit of length ?
 
 178 COMPOUND NUMBERS. 
 
 LINEAR MEASURE. 
 
 235. Linear Measure is used in measuring that 
 which has length without breadth; as, lines, distances. 
 It is often called Long Measure. The denominations are 
 kagues, miles, furlongs, rods, yards, feet, and inches. 
 
 TABLE. 
 
 12 inches (in.) are i foot, ........ ft. 
 
 3 feet " i yard, ........ yd. 
 
 5 1 yds. or 16.} ft. " i rod, perch, or pole, - - - r. or p. 
 40 rods . " i furlong, ....... fur. 
 
 8 fur. or 320 rods " i mile, ...... - - m. 
 
 3 miles " i league, ....... I. 
 
 The following denominations are used in certain cases : 
 
 4 in. i hand, for measuring the height of horses. 
 9 in. = i span. 
 
 1 8 in. = i cubit. 
 
 6 ft. = i fathom, for measuring depths at sea. 
 3.3 ft. = i pace,* for measuring approximate distances. 
 
 5 pa. = i rod, " ^ " 
 ill-statute mi. = i geographic or nautical mile. 
 60 geographic, or 
 
 , ,, 
 
 6 9 i-itatute m. nearly, } = I de S ree on the e 1 uator - 
 360 degrees = i circumference of the earth. 
 
 A knot, used in measuring distances at sea, is equivalent to a nau- 
 tical mile. 
 
 (For the English and French methods of determining the standard 
 of length, see Higher Arith.) 
 
 NOTES. i. The original element of linear measure is a grain or 
 kernel of barley. Thus, 3 barley-corns were called an inch. But 
 the barley-corn, as a measure of length, has fallen into disuse. 
 
 2. The inch is commonly divided into halves, fourths, eighths, or 
 tenths ; sometimes into twelfths, called lines. 
 
 3. The mile of the Table is the common land mile, and contains 
 5280 ft. It i0 called the statute mile, because it is recognized by law, 
 both in the United States and England. 
 
 235. For what is linear measure used ? The denominations ? The Table T 
 Note. Tbe original element of linear measure? How is the inch commonly divided? 
 * A military pace or step is variously estimated 2\ and 3 foot.
 
 COMPOUND NUMBERS. 179 
 
 238. The Linear Unit employed by surveyors is 
 Gunter's Chain, which is 4 rods or 66 ft. long, and is sub- 
 
 iivided as follows : 
 
 
 
 7.92 inches (in.) are i link, I. 
 
 25 links ' i rod or polo - - - r, 
 
 4 rods " i chain, ch. 
 
 80 chains " i milo m. 
 
 NOTE. Gfunter's chain is so called from the name of its inventor. 
 Engineers of the present day commonly use a chain, or measuring 
 tupj 100 feet long, each foot being divided into tenths. 
 
 CLOTH MEASURE. 
 
 237. Cloth Measure is used in measuring those 
 articles of commerce whose length only is considered ; as, 
 cloths, laces, ribbons, etc. Its principal unit is the linear 
 yard. This is divided into halves, quarters, eighths, and 
 sixteenths. 
 
 TABLE. 
 
 , 3 ft. or 36 in. are i yard, - - - - yd. 
 
 18 in., " i half yard - - - i yd. 
 
 9 in., " i quarter yard - - i yd. 
 
 4' in., " i eighth " - - i yd. 
 
 2\ in., " i sixteenth" - - -^ yd. 
 
 NOTES. r. The old Ells Flemish, English, and French, are no 
 longer used in the United States ; and the nail (2\ inches), as a unit 
 ot measure, is practically obsolete. 
 
 2. In calculating duties at the Custom Houses, the yard is divided 
 into tenths and hundredths. 
 
 SQUARE MEASURE. 
 
 238. Square Measure is used in measuring sur- 
 faces, or that which has length and breadth without thick- 
 ness; as, land, flooring, etc. Hence, it is often called 
 land or surface measure. The denominations are acres 
 sjuare rods, square yards, square feet, and square inches. 
 
 2-jS. WTint is the linear unit commonly employed by surveyors ? z-^. Cloth 
 .Tisasnre? Its principal unit? How is the vard divided? The Table? 238. Square 
 measure? The denominations? ThcTablo?
 
 180 
 
 COMPOUND 
 
 TABLE. 
 
 144 sqnare inches (sq. in.) are i square foot, - - aj. ft. 
 9 square feet " i square yard, - - sq. yd. 
 
 3o^fcq. yards, or ) I i sq. rod, perch 
 
 272! sq. feet, j" /or pole, - - - sq. r. 
 
 160 square rods " i acre, A. 
 
 640 acres " i square mile, - - sq. m. 
 
 229. The Unit of Land Measure is the Acre. 
 and is subdivided as follows : 
 
 i acre, 
 
 - P- 
 
 - nq. c. 
 
 - A. 
 
 625 sq. links are i pole or sq. rod, - 
 
 16 poles " i square chain. - 
 
 10 sq. chains, or ) 
 160 sq. rods ) 
 
 NOTES. i. The Rood of 40 sq. rods is no longer used as a unit of 
 measure, 
 
 2. A Square, in Architecture, is 100 square feet. 
 
 239. a. The public lands of the United States are divided into 
 Toicmships, Sections, and Quarter-sections. 
 
 A Township is 6 miles square, and contains 36 sq. miles. 
 
 A Section is i mile square, and contains 640 acres. 
 
 A Quarter-section is 160 rods square, and contains 160 acres. 
 
 240. A Square is a rectilinear figure 
 which has four equal sides, and four riff fit 
 any lea. Thus, 
 
 A Square IncJt is a square, each side 
 of which is i inch in length. 
 
 A Square Yard is a square, each side 
 of which is i yard in length. 
 
 This measure is called Square Measure, 
 because its measuring unit is a square. 
 
 241. A Rectangle or Rectangular Figure is one which 
 !'.as/<??/r sides and four right angles. When all the sides are equal, 
 it is called a square; when the opposite sides only are equal.it is 
 tailed an oblong or parallelogram. 
 
 242. The A rea of a figure is the quart tity of surface it contains, 
 end is often called its superficial contents. 
 
 Note. How are the Government lands divided? How ranch land in n town- 
 chip ? In a section ? In a quarter-section ? 240. What is a square ? A square 
 inch ? A ponare yard ? Why is sqnare measure f>o called ? 241. What is a rect- 
 angular flgpre? 242. What is the area of a figure? 
 
 
 
 
 
 
 
 
 
 
 9 eq. ft. = i sq. yd.
 
 COMPOUND NUMBERS. 181 
 
 243. The n rca of all rectangular surfaces is found by multiplying 
 Hie length and breadth together. 
 
 244. The area and one side being given, the otlier side is found 
 y dividing the area by the given side. (Art. 93.) 
 
 CUBIC MEASURE. 
 
 245. Cubic Measure is used in measuring solids, 
 or that which has length, breadth, and thickness; us, 
 timber, boxes of goods, the capacity of rooms, ships, etc. 
 Hence it is often called Solid Measure. The denomina- 
 tions are cords, tons, cubic yards, cubic feet, and cubic inches. 
 
 TABLE. 
 
 1728 cubic inches (cu. in.) are i cubic foot, cu. ft. 
 27 cubic feet " I cubic yard, cu. yd. 
 
 128 cubic feet " i cord of wood, G'. 
 
 245, ft A Cord of wood is a pile 8 ft. long, 4 ft. wide, and 4 
 ft. high; for 8 X4 X4=i28. 
 
 A Cord Foot, is one foot in length of such a pile ; hence, 8 cord 
 ft. = i cord of wood. 
 
 245, b. A Register Ton is the standard for estimating the 
 capacity or tonnage of vessels, and is 100 cu. ft. 
 
 A Shippiny Ton, used in estimating cargoes, in the U. S., is 
 <o cu. ft. ; in England, 42 cu. ft. 
 
 NOTE. The ton of 40 ft. of round, or 50 ft. of hewn timber is sel. 
 ilora or never used. 
 
 246. A Cube is a regular solid bounded 3 x s*3=27 cu. ft 
 by six equal squares called its faces. Hence, 
 
 its length, breadth, and thickness are equal 
 to each other. Thus, 
 
 A Cubic Inch is a cube, each side of 
 v.-hich is a square inch ; a Cubic Yard is a 
 cube, each side of which is a square yard, etc. 
 
 This Measure is called Cubic Measure, 
 because its measuring unit is a cube. 
 
 
 
 243. How is the area of rectangular surfaces found ? 245. Cubic measure ? The 
 denominations ? Table ? Note. Describe a cord of wood ? A cord foot ? A 
 register ton ? Shipping ton ? t+6. What is a cube ? A cubic inch ? Yard i
 
 182 COMPOUND NUMBERS. 
 
 247. -4 rectangular body is one bounded by six rectangular sides, 
 eacli opposite pair beiug equal and parallel; as, boxes of goods, 
 blocks of hewn stone, etc. 
 
 When all the sides are equal, it is called a cube ; when the opposite 
 sides only are equal, it is called a parallclopiped. 
 
 248. The contents or solidity of a body is the quantity 
 of matter or space it contains. 
 
 249 The contents of a rectangular solid are found, by 
 multiplying the length, breadth, and thickness together. 
 
 MEASURES OF CAPACITY. 
 
 250. The capacity of a vessel is the quantity of space 
 included within its limits. 
 
 Measures of Capacity are divided into two classes, 
 dry and liquid measures. 
 
 DRY MEASURE. 
 
 251. Dry Measure is used in measuring grain, 
 fruit, salt, etc. The denominations are chaldrons, bushels, 
 pecks, quarts, said, frints. 
 
 TABLE. 
 
 2 pints (pt.) - - are i quart, qt. 
 
 8 quarts - . - " I peck, pk. 
 
 4 pecks, or 32 qts., " I bushel, bu. 
 
 36 bushels - - - "i chaldron, .... ch. 
 
 252. The Standard Unit of Dry Measure is the 
 bushel, which contains 2150.4 cubic inches, or 77.6274 Ibs. 
 avoirdupois of distilled water, at its maximum density. 
 
 247. What is a rectangular body? When all the sides are equal, what called? 
 When the opposite sides only are equal ? 248. What are the contents of a solid 
 body? 240. How find the contents of a rectangular solid? zw. The rapacity 
 of a vessel? 2*1. Dry measure? The denominations ? The Table? 252. The 
 standard unit of dry measure?
 
 COMPOUND NUMBERS. 
 
 183 
 
 It is a cylinder i8 in. in diameter, and 8 in. deep, the 
 same as the old Winchester bushel of England.* The 
 British Imperial Bushel contains 2218.192 cu. inches. 
 
 NOTES. i. The dry quart is equal to i\ liquid quart nearly. 
 2. The chaldron is used for measuring coke and bituminous coal. 
 
 253. The Standard Eushel of different kinds of grain, seeds, 
 etc., according to the laws of New York, is equal to the following 
 number of pounds : 
 
 32 Ibs. = i bu. of oats. 
 
 44 Ibs. = i 
 48 Ibs. = i 
 
 55 Ibs. r= i 
 
 56 Ibs. = i 
 
 Timothy seed. 
 ( buckwheat, or 
 l barley. 
 
 flax-seed. 
 
 rye. 
 
 58 Ibs. = i bu. of corn. 
 
 ( wheat, peas, 
 < potatoes, or 
 ( clover-seed. 
 j beans, or blue- 
 I grass seed, 
 loo Ibs. = i cental of grain. 
 
 60 Ibs. = i 
 
 62 Ibs. = i 
 
 NOTES. i. The cental is a standard recently recommended by 
 the Boards of Trade in New York, Cincinnati, Chicago, and other 
 large cities, for estimating grain, seeds, etc. Were this standard gen- 
 erally adopted, the discrepancies of the present system of grain deal- 
 ing would be avoided. 
 
 2. BusJiels are cJuinged to centals, by multiplying them by the 
 number of pounds in one bushel, and dividing the product by 100. 
 f he remainder will be hundredth of a cental. 
 
 LIQUID MEASURE. 
 
 254. Liquid Measure is used in measuring milk, 
 wine, vinegar, molasses, etc., and is often called Wine 
 Measure. The denominations are hogsheads, barrels, gal- 
 lons, quarts, pints, and gills. 
 
 TABLE. 
 
 4 gills (ffi ) are i pint, pt. 
 
 2 pints " i quart, qt. 
 
 4 quarts " i gallon, ... - . - gal. 
 
 31^ gallons " i barrel, bar. or bbl. 
 
 63 gallons " i hogshead, .... Jihd. 
 
 254. Liquid measure ? The denominations? Table? 
 * Professor A. D. Bache.
 
 1C4 COMPOUND XUMBE11S. 
 
 255. The Standard Unit of Liquid Measure is tha 
 gallon, which contains 231 cubic inches, or 8.338 Ibs. 
 avoirdupoia of distilled water, at its maximum density. 
 The British Imperial Gallon contains 277.274 cu. inches. 
 
 NOTES. i. The barrel and Tiogshead, as units of measure, are 
 chiefly used in estimating the contents of cisterns, reservoirs, etc. 
 
 2. Beer Measure is practically obsolete in this country. The old 
 beer gallon contained 282 cubic inches. 
 
 CIRCULAR MEASURE. 
 
 256. Circular Measure is used in measuring 
 angles, land, latitude and longitude, the motion of the 
 heavenly bodies, etc. It is often called Angular Measure. 
 The denominations are signs, degrees, minutes, and seconds. 
 
 TABLE. 
 
 60 seconds (") are i minute, - - - ' 
 
 60 minutes " i degree, - - - o , or deg. 
 
 30 degrees " i sign, - - - - g. 
 
 12 signs, or 360 " i circumference, - dr. 
 
 NOTE. Sij is are used in Astronomy as a measure of the Zodiac. 
 
 257. A Circle is a plane figure bounded by a curve line, every 
 part of which is equally distant from a point within called the 
 center. 
 
 The Circumference of a circle is the curve line by which it is 
 bounded. 
 
 The Diameter is a straight line dra%vn through the center, ter- 
 minating at each end in the circumference. 
 
 The IttKfius is a straight line drawn 
 from the center to the circumference, and 
 is equal to half thu diameter. 
 
 An Arc is any part of the circumfer- 
 ence. 
 
 In the adjacent figure, A D E B F is 
 the circumferenca ; C the center ; A B the 
 diameter ; C A, C D. C E, etc., are radii ; 
 A D, D E, etc., are arcs. 
 
 255. The standard of Liquid measure ? Note. What of Beer measure ? 256. In 
 \vhatisCircularmeasureused? The denominations? The Table? 257. What 
 It a circle? The circumference? Diameter? Radius? An arc?
 
 COMPOUND XUMBEES. 1S 
 
 253. A Plane Angle is the quantity of divergence of two 
 etra-igat lines starting from the same point. 
 
 Tne Lines which form the angle are 
 called the sides, and the point from which 
 they start, the vertex. Thus, A is the ver- 
 tex of the angle B A C, A B and A C the 
 sides. 
 
 259. A Perpendicular is a straight 
 line which meets another straight line so as 
 to make the two adjacent angles equal to 
 each other, as A B C, A B D. 
 
 Each of the two lines thus meeting is 
 perpendicular to the other. 
 
 C ii D 
 
 260. A Ttiylit Angle is one of the two equal angles formed 
 by the meeting of two straight lines which are perpendicular to 
 each other. All other angles are called oblique. 
 
 260. (i. The Pleasure of an angle is the arc of a circle included 
 between its two sides, as the arc D E, in Fig., Art. 258. 
 
 261. A Dct/rec. is one 36oth part of the circumference of a circle. 
 It is divided into 60 equa.1 parts, called minutes j the minute is 
 divided into 60 seconds, etc. Hence, the length of a degree, minute, 
 etc., varies according to the magnitude of different circles. 
 
 The length of a degree of longitude at the equator, also the aver- 
 age length of a degree of latitude, adopted by the U. S. Coast Sur- 
 vey, is 69.16 statute miles. At the latitude of 30 it is 59.81 miles, 
 at 6o 3 it is 34.53 miles, and at 90 it is nothing.* 
 
 262. A Semi-circumference is half a circumference, or 180. 
 A Oiifidratit is one-fourth of a circumference, or 90. 
 
 263. If two diameters are drawn perpendicular to each other, 
 they will form four right angles at the center, and divide the cir- 
 cumference into four equal parts. Hence, 
 
 A right angle contains 90 ; for the quadrant, which measures it, 
 is an arc of 90. 
 
 238. A plane angle ? The sides ? The vertex ? 259. A perpendicular ? 260. A 
 right angle ? 260, a. What is the measure of an angle ? 261. What is a degree? 
 Upon what does the length of a degree depend ? What its length at the equator ? 
 262. What is a semi-circumference ? How many degrees does it contain ? A 
 quadrant? 263. If two diameters are drawn perpendicular to each other, what 
 Is the result ? How many degrees in a right angle ? 
 
 * Encyclopedia Britannica,
 
 186 COMPOUND X UMBERS. 
 
 MEASUREMENT OF TIME. 
 
 264. Time is a portion of duration. It is divided 
 into centuries, years, months, weeks, days, hours, minutes, 
 and seconds. 
 
 TABLE. 
 
 60 seconds (sec.) are i minute 
 
 - m. 
 
 60 minutes 
 
 i hour - 
 
 h. 
 
 24 hours 
 
 i day 
 
 - d. 
 
 7 days 
 
 i week . 
 
 - w. 
 
 365 days, or ) 
 52 w. and id.} 
 
 i common year 
 
 - c.y. 
 
 366 days " 
 
 i leap year - 
 
 - i.y. 
 
 12 calendar months (mo.) " 
 
 i civil year - 
 
 - ?/ 
 
 ico years " 
 
 i century 
 
 - c. 
 
 . In most business transactions 30 days are considered a 
 Four weeks are sometimes called a lunar month. 
 
 265. A Civil Year is the year adopted by govern- 
 ment for the computation of time, and includes both com- 
 mon and leap years as they occur. It is divided into 12 
 calendar months, as follows : 
 
 January (Jan.) ist mo., 31 d. 
 
 February (Feb.) 2d " 28 d. 
 
 March (Mar.) 3d " 31 d. 
 
 April (Apr.) 4th " 30 d. 
 
 May (May) 5th " 31 d. 
 
 June (Jane) 6th " 30 d. 
 
 July (July) 7th mo., 31 d. 
 
 August (Aug.) 8th " 31 d. 
 September (Sep.) gth " 30 d. 
 October (Oct.) loth " 31 d. 
 November (Nov.) nth " 30 d. 
 December (Dec.) i2th " 31 d. 
 
 NOTES. i. The following couplet will aid the learner in remem- 
 bering the months that have 30 days each : 
 
 " Thirty days hath September, 
 April, June, and November." 
 
 All the other months have 31 days, except February, which in 
 common years has 28 days ; in leap years, 29. 
 
 266. Time is naturally divided into days and years. 
 The former are measured by the revolution of the earth on 
 its axis ; the latter by its revolution around the sun. 
 
 264. What is time T The denominations f The Table ?
 
 COMPOUND LUMBERS. 
 
 187 
 
 267. A Solar year is the time in which the earth, 
 starting from one of the tropics or equinoctial points, re- 
 volves around the sun, and returns to the same point. It 
 is thenc3 called the tropical or equinoctial year, and is 
 equal to 365d. $h. 48m. 49.7 sec.* 
 
 NOTE. i. The excess of the solar above the common year is 6 
 hours or \ of a day, nearly ; hence, in 4 years it amounts to about I 
 day. To provide for this excess, i day is added to every 4th year, 
 which is called Leap year or Bissextile. This additional day is given 
 to February, because it is the shortest month. 
 
 2. Leap year is caused by the excess of a solar above a common, 
 year, and is so called because it leaps over the limit, or runs on I day 
 more than a common year. 
 
 3. Every year that is exactly divisible by 4, except centennial 
 years, is a ieap year ; the others are common years. Thus, 1868, '72, 
 etc., are leap years; 1869, '70, '71, are common. Every centennial 
 year exactly divisible by 400 is a leap year ; the other centennial 
 j ears are common. Thus, 1600 and 2000 are leap years; 1700, 1800, 
 and 1900 are common. 
 
 268. An Apparent Solar Dai/ is the time between 
 the apparent departure of the sun from a given meridian 
 and his return to it, and is shown by sun dials. 
 
 A Trite or Wean solar day is the average length of 
 apparent solar days, and is divided into 24 equal parts, 
 called hours, as shown by a perfect clock. 
 
 269. A Civil Day is the day adopted by govern- 
 ments for business purposes, and corresponds with the 
 mean solar day. In most countries it begins and ends at 
 midnight, and is divided into two parts of 12 hours each; 
 the former being designated A. M.; the latter, p. M. 
 
 NOTE. A meridian is an imaginary circle on the surface of the 
 earth, passing through the poles, perpendicular to the equator. 
 A. M. is an abbreviation of ante meridies, before midday ; P. M., of 
 post meridies, after midday. 
 
 266. How is time naturally divided ? How is the former caused ? The latter ? 
 * Laplace^Somcrville, BaSly's Tables.
 
 188 
 
 COMPOUND NTJMBEIiS. 
 
 MISCELLANEOUS TABLES. 
 
 12 things are i dozen. 
 12 doz. " i gross. 
 
 24 slieets are i quire of paper. 
 20 quires " i ream. 
 
 2 leaves are I folio. 
 
 4 leaves " I quarto or 4to. 
 
 8 leaves " i octavo or 8vo. 
 
 12 gr.?ss are i great gross. 
 20 things " i score. 
 
 2 reams are i bundle. 
 5 bundles " i bale. 
 
 12 leaves are i duodecimo or i2ma 
 1 8 leaves " i eighteen rao. 
 24 leaves " i twenty-four mo. 
 
 NOTE. The terms folio, quarto, octavo, etc , denote the number 
 of leaves into which a sheet of paper is folded in making books. 
 
 270. Aliquot Parts of a Dollar, or 100 cents. 
 
 50 cents = $i 
 33!$ cents = $i 
 25 cents = $i 
 20 cents = $5 
 16] cents = $k 
 
 12^ cents = 
 
 10 cents 
 
 8\ cents = 
 
 64 cents = 
 
 5 cents = 
 
 271. Aliquot Parts of a Pound Sterling. 
 
 10 shil. = 
 
 6s. 8d. = : V 
 
 5 shil. -&\ 
 
 4 shil. =} 
 
 33. 4d. = 
 2s. 6d. = 
 2 shil. = 
 is. 8d. = 
 
 272. Aliquot Parts of a Pound Avoirdupois. 
 
 pound. 
 
 12 ounces = J pound. 
 
 8 - =i 
 
 si " =4 " 
 
 4 ounces 
 2 " 
 i " 
 
 273. Aliquot Parts of a Year. 
 
 9 months = 5 year. 
 8 " =5 " 
 6 " =i " 
 4 " =4 " 
 
 3 months = i year. 
 
 2 = 
 
 274. 
 
 o/" a Month. 
 
 20 days = ^ month. 
 
 is =i 
 
 10 " =4 " 
 6 " =4 " 
 
 5 days = ^ month, 
 3 " =-Ar " 
 
 2 " =^
 
 KEDUCTION. 
 
 275. Reduction is changing a number from one 
 denomination to another, without altering its value. It is 
 either descending or ascending. 
 
 Reduction Descending is changing higher de- 
 nominations to lower ; as, yards to feet, etc. 
 
 Reduction Ascendiny is changing lower denomi- 
 nations to higher ; as, feet to yards, etc. 
 
 276. To reduce Higher Denominations to Lower. 
 
 1. How many farthings are there in 23, 75. 5^d. ? 
 ANALYSIS. Since there are 203. in a OPERATION. 
 
 pound, there must be 20 times as many 23, 78. 5d. I far. 
 
 shillings as pounds, plus the given shil- 20 
 
 lings. Now 20 times 23 are 460, and "77 ~ 
 
 4603. + 7s.:=467S. Again, since there are 
 
 I2d. in a shilling, there must be 12 
 
 times as many pence as shillings, plus the 5"9 u. 
 
 given pence. But 12 times 467 are 5604, 4 
 
 and 5&04d. + 5d. = 56oo.d. Finally, since 22437 ^' J[X ' Ans. 
 there are 4 farthings in a penny, there 
 
 must be 4 times as many farthings as pence, plus the given farthings. 
 Now 4 times 5609 are 22436, and 22436 far. + i far.=22437 tar 
 Therefore, in 23, 73. s^d. there are 22437 far. Hence, the 
 
 RULE. Multiply the highest denomination by the number 
 required of the next lower to make a unit of the higher, and 
 to the product add the lower denomination. 
 
 Proceed in this manner with the successive denomina- 
 tions, till the one required is reached. 
 
 2. How many pence in 8, IDS. 7d. ? Ans. 2047d. 
 
 3. How many farthings in 123. pd. 2 far.? 
 
 4. How miny farthings in 41, 53. 4^d. ? 
 
 27-;. What is Reduction? TIow mrtnv kinds? Descending? Asrenrtin"? 
 276. Hov nre higher denomination* reduced to lower? Explain Es. i from th 
 blackboard 't
 
 190 SEDUCTION. 
 
 277. To reduce Lower Denominations to Higher. 
 
 5. In 22437 farthings, how many pounds, shillings, 
 pence, and farthings ? 
 
 ANALYSIS. Since iu 4 farthings there OPERATION. 
 
 is I penny, in 22437 farthings there are 4)22437 far. 
 
 as many pence as 4 farthings are con- i2^^6oQd I far 
 
 tained times in 22437 farthings, or 
 5609 pence and i farthing over. Again, 
 
 since in 12 pence there is I shilling, in < 2 3> 7& 
 
 5609 pence theru are as many shillings AllS. 23, 78. 5d. I far. 
 as 12 pence are contained times in 
 
 5609 pence, or 467 shillings and 5 pence over. Finally, since in 
 20 shillings there is i pound, in 467 shillings there are as many 
 pounds as 20 shillings are contained times in 467 shillings, or 
 23 pounds and 7 shillings over. Therefore, in 22437 farthings there 
 are 23, 73. sd. i far. Hence, the 
 
 RULE. Divide the given denomination by the number re- 
 quired of this denomination to make a unit of the next higher. 
 
 Proceed in this manner with the successive denomina- 
 tions, till the one required is reached. The last quotient) 
 with the several remainders annexed, will be the answer. 
 
 NOTE. The remainders, it should be observed, are the same 
 denomination as the respective dividends from which they arise. 
 
 278. PROOF. Reduction Ascending and Descending 
 prove each other ; for, one is the reverse of the other. 
 
 6. In 2047 pence how many pounds, shillings, and pence ? 
 
 7. In 614 farthings, how many shillings, pence, etc.? 
 
 8. How many pounds, shillings, etc., in 39610 farthings ? 
 
 9. An importer paid 27, 135. 8d. duty on a package of 
 English ginghams, which was 4d. a yard: how many 
 yards did the package contain ? 
 
 10. A railroad company employs 1000 men, paying each 
 43. 6d. per day : what is the daily pay-roll of the company ? 
 
 11. Reduce 18 Ibs. 6 ounces troy, to pennyweights. 
 
 277. How are lower denominations reduced to higher ? 27". Proof? Explain 
 Ex. 5 from the blackboard.
 
 REDUCTION. 191 
 
 12. Reduce 32 Ibs. 9 oz. 5 pwt. to pennyweights. 
 
 13. How many pounds and ounces in 967 ounces troy? 
 
 14. How many pounds, etc., in 41250 grains? 
 
 15. How many rings, each weighing 3 pwt., can bo 
 made of i Ib. 10 oz. 9 pwt. of gold ? 
 
 1 6. What is the worth of a silver cup weighing 10 oz. 
 16 pwt., at \z\ cents a pennyweight? 
 
 17. Reduce 165 Ibs. 13 oz. Avoirdupois to ounces. 
 
 18. Reduce 210 tons 121 pounds 8 ounces to ounces. 
 
 19. In 4725 Ibs. how many tons and pounds? 
 
 20. In 370268 ounces, how many tons, etc. ? 
 
 21. What will 875 Ibs. 1 1 oz. of snuff come to, at 5 cents 
 an ounce ? 
 
 22. A grocer bought i ton of maple sugar, at 10 cents 
 a pound, and sold it at 6 cents a cake, each cake weighing 
 4 oz. : what was his profit ? 
 
 23. In 250 Ibs. 6 oz. Apothecaries' weight, how many 
 drams ? 
 
 24. In 2165 scruples, how many pounds ? 
 
 25. How many feet in 1250 rods, 4 yards, and 2 feet? 
 For the method of multiplying by 5^ or 16^, see Art. 165. 
 
 Ans. 20639 feet. 
 
 26. How many feet in 365 miles 78 rods and 8 feet? 
 
 27. In 5242 feet, how many rods, yards and feet? 
 
 REMARK. In order to divide the second 5)1:242 ft. 
 
 dividend 1747 yards by 5.^, the number x , - ,., 
 
 of yards in a rod,' we reduce both the 5?J I 747 J l k 
 
 divisor and dividend to halves; then ' 
 
 divide one by the other. Thus, 5^=n I 03494 half yds, 
 halves ; and 1747=3494 halves ; and n is Ans. 317 r. 3^ yd. I ft. 
 contained in 3404, 317 times and 7 over. 
 
 But the remainder 7, is halves; for the dividend was halves; and 
 7 half yards=3^ yards. (Art. 168, a, 160.) 
 
 28. Reduce 38265 feet to miles, etc. 
 
 29. How many rods in 461 leagues, 2 miles, 6 furlongs?
 
 192 EEDUCTIOJT. 
 
 30. Reduce 7 m. 6 fur. 23 r. 5 yds. 8 in. to inches, and 
 prove the operation. 
 
 31. Eeduce i in. 7 fur. 39 r. 5 yds. i ft. 6 in. to inches, 
 ' and prove the operation. 
 
 32. If a farm is 2\ miles in circumference, what will it 
 cost to enclose it with a stone wall, at $1.85 a rod? 
 
 33. At 6 \ cents a mile, what will be the cost of a trip 
 round the world, allowing it to be 8299.2 leagues ? 
 
 34. How many eighths of a yard in a piece of cloth 
 5 7 yards long ? 
 
 35. How many sixteenths of a yard in 163 yards ? 
 
 36. In 578 fourths, how many yards? 
 
 37. In 1978 sixteenths, how many yards? 
 
 38. How many vests will i6| yards of satin make, allow- 
 ing \ yard to a vest ? 
 
 39. A shopkeeper paid $2.50 for i8| yards ribbon, and 
 making it into temperance badges of -J- yard each, sold 
 them at \2\ cents apiece: what was his profit? 
 
 40. How many square rods in 43816 square feet? 
 
 41. How many acres, etc., in 25430 square yards ? 
 
 42. Reduce 160 acres, 25 sq. rods, 8 sq. ft. to sq. feet. 
 
 43. Reduce too sq. miles to square rods? 
 
 44. A man having 64 acres 8 sq. rods of land, divided 
 it into 6 equal pastures: how much land was there in 
 each pasture ? 
 
 45. My neighbor bought a tract of land containing 15^ 
 acres, at $500 an acre, and dividing it into building lots 
 of 20 square rods each, sold them at $250 apiece : how 
 much did he make ? 
 
 46. Reduce 85 cubic yards, 10 cu. feet to en. inches. 
 
 47. Reduce 250 cords of wood to cu. feet. 
 
 48. Reduce 18265 cu - inches to cu. feet, etc. 
 
 49. Reduce 8278 cu. feet to cords. 
 
 50. Reduce 164 bu. i pk. 3 qts. to quarts. 
 
 51. Reduce 375 bu. 3 pks. i qt. i pt. to pints.
 
 REDUCTION. 193 
 
 52. Reduce 184 chaldrons 17 bu. to bushels. 
 
 53. In 8147 quarts how many bushels, etc. ? 
 
 54. A seedsman retailed 75 bu. 3 pk. of clover seed at 
 17 cents a quart : what did it come to ? 
 
 55. A lad sold 2 bu. i pk. 3 qts. of chestnuts, at 8 cents 
 a pint: how much did he get for them ? 
 
 56. In 98 quarts i pt. 2 gills, how many gills? 
 
 57. In 150 gals. 3 qts., how many quarts? 
 
 58. In 45 barrels, 10 gals. 3 qts., how many quarts? 
 
 59. How many gills in 17 hhds. 10 gals. 3 qts. ? 
 
 60. How many gallons, etc., in 86673 pints ? 
 
 6 1. How many bottles holding i pt. each are required 
 to hold a barrel of wine ? 
 
 62. What vvill a hogshead of alcohol come to, at 6 cents 
 a gill? 
 
 63. Reduce 30 days 5 hours 42 min. 10 sec. to seconds. 
 
 64. Reduce 17 weeks 3 days 5 hr. 30 min. to minutes. 
 
 65. Reduce a solar year to seconds. 
 
 66. Reduce 6034500 sec. to weeks, etc. 
 
 67. Reduce 5603045 hours to common years. 
 
 68. Reduce 10250300 minutes to leap years. 
 
 69. An artist charged me 75 cents an hour for copying 
 a picture; it took him 27 days, working io hours a day: 
 wliat was his bill ? 
 
 70. In 48561 seconds, how many degrees? 
 
 71. In 65237 minutes, how many signs? 
 
 72. In 237, 40', 31", how many seconds? 
 
 73. In 360, how many seconds? 
 
 74. How many dozen in 1965 things? 
 
 75. How many eggs in 125 dozen? 
 
 76. How many gross in 100000? 
 
 77. How many pens in 65 gross? 
 
 78. How many pounds in 3 score and 7 pounds? 
 
 79. How many sheets of paper in 75 quires? 
 So. How many quires of paper in i oooo sheets ?
 
 194 
 
 APPLICATIONS OF 
 
 APPLICATIONS OF WEIGHTS AND MEASURES. 
 
 4 yards. 
 
 279. The use of Weights and Measures is not 
 
 confined to ordinary trade. They have other and im- 
 portant applications to the farm, the garden, artificers' 
 work, the household, etc. 
 
 THE FARM AND GARDEN. 
 
 280. To find the Contents of Rectangular Surfaces. 
 
 1. How many square yards in a strawberry bed, 4 yards 
 long, and 3 yards wide ? 
 
 ILLUSTRATION. Let the bed be rep- 
 resented by the adjoining figure; its 
 length being divided into four equal . 
 parts, and its breadth into three, each 'g 
 denoting linear yards. The bed, ob- >> 
 viously, contains as many square yards 
 RB them are squares in the figure. 
 Now as there are 4 squares in i row, in 
 3 rows there must be 3 times 4 or 12 
 squares. Therefore, the bed contains 12 sq. yards. Hencj, the 
 
 RULE. Multiply the length by the breadth. (Art. 243.) 
 
 NOTES. i. Both dimensions should be reduced to the sam-e 
 denomination before they are multiplied. 
 
 2. The area and one side of a rectangular surface being given, the 
 other side is found by dividing the area by the given side. 
 
 3. One line is said to be multiplied by another, when the number 
 of units in the former are taken as many times as there are like u n its 
 in the latter. (Art. 47, n.) 
 
 2. How many acres in afield 40 rd. long and 31 rd. wide ? 
 
 wide? 
 
 3. If the width of an asparagus bed is 66 feet, what 
 must be the length to contain \ of an acre ? 
 
 279. What is sairt of the application* of Weight* and Measures? 280. How 
 fin 1 the area of a rectangular surface 1 Note. If t;.e dimensions are in different 
 denominations, what is 'to be done ? If the area and one side are given ?
 
 WEIGHTS AND MEASURES. 19fi 
 
 4. What is the length of a pasture, containing 15 acres, 
 whose width is 33 J rods ? 
 
 5. A speculator bought 30 acres of land at $50 per 
 acre, and sold it in villa lots of 5 rods by 4 rods, at $200 a 
 lot: what was his profit? 
 
 6. A garden 230 ft. long and 125 ft. wide, has a gravel 
 walk 6 ft. in breadth extending around it: how much 
 land does the walk contain ? 
 
 7. What is the difference between 2 feet square and 
 2 square feet ? 
 
 8. A farmer having 3 acres of potatoes, sold them at 
 25 cents a bushel in the ground ; allowing a yield of 2\ bu. 
 to a sq. rod, how many bushels did the field produce ; and 
 what did he receive for them ? 
 
 281. To find the number of Hills, Vines, etc., in a given 
 field, the Area occupied by each being given. 
 
 9. How many hills of corn can a farmer plant on 2 acres 
 of ground, the hills being 4 ft. apart ? 
 
 ANALYSIS. Since the hills are 4 ft. apart, each hill will occupy 
 4 x 4 or 16 sq. ft. Now 2 acres= 160 sq. rods x 272.25 x 2=87120 sq. ft. 
 Therefore, he can plant as many hills as 16 is contained times ip 
 87120; and 87120-^ 16=5445 hills. Hence, the 
 
 RULE. Divide tJie area planted by ilw area occupied ly 
 each hill or vine. 
 
 10. How many bulbs will a lady require for a crocus 
 bed, 1 2 ft. long and 4 ft. wide, if planted 6 inches apart ? 
 
 11. What number of grape Vines will be required for 
 $ of an acre, if they are set 3 ft. apart ? What will it cost 
 to set them at $5.25 per hundred ? 
 
 12. How large an orchard shall I require for 100 apple 
 trees, allowing them to stand 33 ft. apart ? 
 
 281. Hew find the number of hills, vines, etc., In a given field? 284. How n* 
 Uooring, plastering, etc., estimated?
 
 196 A !> P L I C A 1 1 IT 8 Jf 
 
 ARTIFICERS' WORK. 
 
 282. Flooring, plastering, painting, papering, roofing, 
 paving, etc., are estimated by the number of sq.ft. or sq. yds. 
 in the area, or by the "square" of 100 sq. feet. 
 
 13. What will be the cost of flooring a room 18 ft. long 
 and 1 6 ft. 6 in. wide, at i8| cts. a sq. foot? 
 
 SOLUTION. 18 ft. x 16^=297 sq. ft. ; and i8J cts. x 297= $55.68 J. 
 
 14. If a school house is 60 ft. long and 45 ft. wide, what 
 will be the expense of the flooring, at 61.08 per sq. yard ? 
 
 15. What will it cost to plaster a ceiling 18 ft. 6 in. long, 
 and 15 ft. wide, at 83.20 per square of 100 sq. ft, 
 
 1 6. What will be the cost of flagging a side-walk 206 ft 
 long and 9^ ft. wide, at $1.85 per sq. yard ? 
 
 17. What will be the expense of painting a roof 60 ft 
 long and 25 feet wide, at $1.50 per "square" ? 
 
 1 8. What cost the cementing of a cellar 65 ft by 25 ft, 
 at $.25 per square yard ? 
 
 MEASUREMENT OF LUMBER. 
 
 283. To find the Contents of Boards, the length and 
 
 breadth being given. 
 
 DBF. A Standard Board, in commerce, is i in. thick. Hence, 
 A Board Foot is i foot long, i foot wide, aud i inch thick. 
 A Cubic Foot is 12 board feet. Hence, the 
 
 EULE. Multiply the length in feet by the width in inches, 
 and divide the product by 12; the result will be board feet. 
 
 NOTES. i. If a board tapers regularly, multiply by the mean 
 width, which is ludf the sum of the two ends. 
 
 2. Shingles are estimated by the thousand, or bundle. They are 
 commonly 18 in. long, and average 4 in. wide. It is customary to 
 allow looo to a square of 100 feet. 
 
 3. The contents of boards, timber, etc., were formerly computed 
 by duodecimals, which divides t\\efoot into 12 in., the inch into 12 
 sec., etc. ; but this method is seldom used at the present day. 
 
 283. The thickness of a t-tumlard board ? What is a board foot T A cubic 
 foot ? lJ<nv find the contents of a board ? If the board is taperiug, how ?
 
 WEIGHTS A5JD MEASURES. 10? 
 
 19. If a board is 10 ft. long, and i ft. 3 in. wide, w,hat 
 are its contents, board measure? 
 
 SOLUTION. iox 15 = 150; and 150-4-12 = 12^ board ft., Ans. 
 
 20. What are the contents of a board 13 ft. long, and 
 i ft. 5 in. wide ? 
 
 SOLUTION. 13 x 17=221 ; and 221-5-12=18 fe board ft. 
 
 21. Required the contents of a board 14 ft. long, i ft 
 4 in. wide ; and its value at i\ cts. a foot ? 
 
 22. Required the contents of a tapering board 15 ft 
 long, 14 in. wide at one end, and 6 in. at the other. 
 
 23. Required the contents of a stock of 9 boards 14 ft 
 long, and i ft. 2 in. wide. 
 
 24. The sides of a roof are 45 ft. long and 20 ft. wide ; 
 what will it cost to shingle both sides, at $15.45 per M., 
 allowing 1000 shingles to a square ? 
 
 284. To find the Cubical Contents of Rectangular Bodies, 
 
 REM. Round Timber, as masts, etc., is estimated in cubic feet. 
 Hewn Timber, as beams, etc., either in board or cubic feet. 
 Sawed Timber, as planks, joists, etc., in board feet. 
 
 25. How many cubic feet are there in a rectangular 
 block of marble 4 ft. long, 3 ft. wide, and 2 ft. thick ? 
 
 4 feet. 
 
 ILLUSTRATION. Let the block be 
 represented by the adjoining diagram ; 
 its length being divided into four 
 equal parts, its bread tli three, and its 
 thickness into two, each denoting linear 
 feet. 
 
 In the upper face of the block, there are 3 times 4, or 12 sq. ft. Now 
 if the block were I foot thick, it would evidently contain i time as 
 many cubic feet as there are square feet in its upper face ; and I time 
 4 into 3=12 cubic feet. But the given block is 2 ft. thick ; therefore 
 it contains 2 times 4 into 3=24 cu. ft. Hence, the 
 
 RULE. Multiply tlio length, breadth, and thickness 
 together. (Art. 249.) 
 
 284. How are hewn timber, etc., es imaied? How joists, studs, etc. ? 285. How
 
 198 APPLICATIONS OF 
 
 285. To find the Contents of Joists, ic., 2, 3, 4, &c., in. thick. 
 
 RULE. Multiply the width l>y such a part of the length, 
 as the thickness is of 12 ; the result will be board feet. 
 
 NOTES. i. The approximate contents of round titriber or lorjs may 
 be found by multiplying \ of the mean circumference by itself, and 
 this product by the length. 
 
 2. Cubic feet are reduced to Board feet by multiplying them by 12. 
 For, I foot board measure is 12 inches square, and i in. thick ; there- 
 fore, 12 such feet make i cubic foot. Hence, 
 
 If one of the dimensions is inches, and the other two arc feet, the 
 product will be in Board feet. 
 
 3. To estimate hay. 
 
 A ton of hay upon a scaffold measures about 500 cu. ft. ; when in 
 a mow, 400 cu. ft. ; and in well settled stacks, 10 cu. yards. 
 
 4. To find the weight of coal in bins. 
 
 A ton (2000 Ibs.) of Lehigh white ash, egg size, measures 34 V cu. ft. 
 A ton of white ash Schuylkill, " " 35 cu. ft. 
 
 A ton of pink, gray and red ash, " " 36 cu. ft. 
 
 26. How many cu. feet in a stick of hewn timber 20 ft. 
 long, i ft. 3 in. wide, i ft. 4 in. thick ? Ans. 33^ en. ft 
 
 27. How many board feet in a joist 18 ft. long, 5 in. 
 wide, and 4 in. thick ? How many cubic feet ? 
 
 SOLUTION. 4 is of 12 ; and i of 18 ft. is 6 ft. Now 5 x6=3o 
 bd. ft. ; and 30-^-12=2^ cu. ft. 
 
 28. What cost 45 pieces of studding n ft. long, 4 in. 
 wide, and 3 in. thick, at 3 cts. a board foot ? 
 
 29. At 45 cts. a cu. foot, what will be the cost of a beam 
 32 ft. long, i ft. 6 in. wide, and i ft. i in. thick ? 
 
 30. What is the worth of a load of wood 8 ft long, 4 ft 
 wide, and 5 ft. high, at $3! a cord ? 
 
 31. How many tons of white ash, e-s Sch. coal, will a bin 
 10 ft long, 8 ft. wide, and 8 ft high, contain. Ans. i8f T. 
 
 find the contents of a rectangular body ? Note. How find the contents of round 
 timber? When the contents and two of the dimensions are given, how flTirt the 
 other dimension? TTnw reduce cubic feet of timber to beard feet? Why 
 1. <>:M! iHiinoihion is inches, and the other two ft., what is the product T
 
 WEIGHTS AXD MEASURES. 199 
 
 32. How many cords of wood in a tree 60 feet high, 
 whose mean circumference is 10 feet? 
 
 33. Hov many tons of hay in a mow 22 ft. by 20 ft, 
 and 15 ft. high ? 
 
 34. At 818.50 a ton, what is the worth of a scaffold of 
 hay 30 ft. long, 12 ft. wide, and 10 ft. high? 
 
 286. Stone masonry is commonly estimated by the 
 perch of 25 cubic feet 
 
 Excavations and embankments are estimated by the 
 cubic yard. In removing earth, a cu. yard is called a load. 
 
 Brickwork is generally estimated by the 1000, but 
 sometimes by cubic feet 
 
 NOTES. i. A perch strictly 8peaking=24| cu. feet being 16^ ft. 
 long, ii ft. wide, and i ft. high. 
 
 2. The average size of bricks is 8 in. long, 4 in. wide, and 2 in. thick. 
 
 In estimating the labor of brickwork by cu. feet, it is customary to 
 measure the length of each wall on the outside ; BO allowance is 
 made for windows and doors or for corners. 
 
 In finding the exact number of bricks in a building, we should 
 make a deduction for the windows, doors, and corners ; also an allow- 
 ance of -,' u of the solid contents for the space occupied by the mortar. 
 
 35. What will be the cost of digging a cellar 45 feet 
 long, 24 feet wide, and 8 feet deep, at 35 cents a cu. yard ? 
 
 36. At $5.25 a perch (25 cu. ft), what cost the labor 
 of building a cellar wall, i ft. 6 in. thick, and 8 ft high, 
 the cellar being 22 by 45 feet? 
 
 37. How many bricks of average size will it take to 
 build the walls of a house 50 ft long, 35 ft wide, 21 ft. 
 high, and i ft. thick, deducting -^ of the contents for the 
 space occupied by the mortar, but making no allowance 
 for windows, doors, or corners ? 
 
 38. At 45 cents a cubic yard, what will it cost to fill in 
 a street 800 ft long, 60 ft. wide, and 4^ ft. below grade ? 
 
 286. How are stone masonry, excavations, etc., vthjamci T How brir.U work? 
 tfot*. The overage eixfl of bricks ?
 
 200 APPLICATIONS OK 
 
 THE HOUSEHOLD. 
 
 287. To find the Quantity of material of given width required 
 to line or cover a given Surface. 
 
 39. How many yards of serge yd. wide are required to 
 line a cloak containing 7$ yds. of camlet i-J yd. wide ? 
 
 ANALYSIS. The surface to be lined=7J yds. x i\=^ x L f sq. 
 yards : i yd. of the liniug=i yd. x f = sq. yd. Now as f sq. yard 
 of camlet requires i yard of lining, the whole cloak will require aa 
 many yards of lining as f is contained times in J , 5 6 i ; and iftf-t-^ 
 L r^ x 2=^- or 15^ yards, the serge required. Hence, the 
 
 RULE. Divide the number of square yards or feet in the 
 given surface by the number of square yards or feet in a 
 yard of the material used. 
 
 40. How many yards of Brussels carpeting yd. wide, 
 are required for a parlor floor 21 ft. long and 18 ft. wide? 
 
 41. How many yards of matting 1 J yd. wide will it take 
 to cover a hall 16 yds. long and 3-^ yds. wide ? 
 
 42. How much cambric -J yd. wide is required to line a 
 dress containing 15 yds. of silk yd. wide ? 
 
 43. How many sods, each being 15 in. square, will be 
 required to turf a court yard 25 by 20 feet? 
 
 44. How many yards of silk yd. wide are required to 
 line 2 sets of curtains, each set containing 10 yds. of 
 brocatelle i-J yd. wide? 
 
 45. How many marble tiles 9 in. square, will it take to 
 cover a hall floor 48 ft. long and 7^ ft. wide ? 
 
 46. How many rolls of wall paper 9 yds. long and i ft. 
 wide, are required to cover the 4 sides of a room 18 by 16 ft. 
 and 9 ft. high, deducting 81 sq. ft. for windows and doors? 
 
 47. How many gallons of water in a rectangular cistern, 
 whose length is 8 ft, breadth 6 ft., and depth 5 ft. ? 
 
 NOTE. Cubic measure is changed to gallons or bushels, by reducing 
 the former to cu. inches, and diciding the result by 231, or 2150.4, aa 
 the case may be. (Arts. 252, 255.) 
 
 *8/. How find the quantitj of material required to line or cover a given purfaca !
 
 WEIGHTS AND MEASURES. 
 
 48. A man constructed a cistern containing 20 hogs- 
 heads, the base being 6 ft- square : what was its depth ? 
 
 49. How many bushels of wheat will a bin 6 ft. long, 
 \ ft. wide, and 3 ft. deep, contain ? 
 
 50. How many cu. feet in a vat containing 50 hogsheads ? 
 
 NOTE. Gallons and bushels are reduced to cubic inches by multi- 
 plying them by 231 or 2150.4, as the case may be. 
 
 51. A man having 10 bu. of grain, wishes to store it in 
 a box 5 ft. long and 3 ft. wide : what must be its height ? 
 
 52. A man built a cistern in his attic, 5 ft. long, 4 ft. 
 wide, and 3 ft. deep : what weight of water will it contain, 
 allowing 1000 oz. to a cu. ft. ? 
 
 288. To change Avoirdupois Weight to Trey, and Troy 
 Weight to Avoirdupois. 
 
 53. If the internal revenue tax is 5 cents per ounce 
 Troy, on silver plate exceeding 40 ounces, what will be the 
 tax on plate weighing 7 Ib. 8 oz. Avoirdupois ? 
 
 ANALYSIS. 7 Ib. S oz. Avoirdupois =120 oz. : and 120 oz. X437^ = 
 52500 grains. Now 52500 gr.-f- 480=: 109! oz. Troy. Again, 109} oz. 
 40 oz. (exempt)=69| oz. ; and 5 cts. x 6gf:=$3.461. Ans. 
 
 54. How many spoons, each weighing i oz. Avoirdupois, 
 can be made from i Ib. 9 oz. 17 pwt. 12 gr. Troy? 
 
 ANALYSIS. i Ib. 9 oz. 17 pwt. 12 gr.= 10500 grains; and 10500 
 rs.-t-437.5 grs. (i oz. Avoir.)=24 spoons. Ans. Hence, the 
 
 RULE. Reduce the given quantity to grains; then reduce 
 t)w grains to the denominations required. (Art. 231.) 
 
 55. If a family have 1 2 Ib. 6 oz. Avoir, of silver plate, what 
 will be the tax, exempting 40 oz., at 5 cts. per oz. Troy ? 
 
 56. A lady bought a silver set weighing n Ibs. 4 oz. 
 Troy : how many pounds Avoir, should it weigh ? 
 
 57. How many rings, each weighing 3^ pwt., can be 
 made from a .bar of gold weighing i pound Avoirdupois? 
 
 58. What is the value of a silver pitcher weighing 2 Ib. 
 8 oz. Avoir., at $2 per ounce Troy? 
 
 59. A miner sold a nugget of gold weighing 3$ Iba 
 Avoir., at $17 per oz. Troy : what did he get for it ?
 
 DENOMINATE FRACTIONS. 
 
 289. A Denominate Fraction, is one or more of 
 the equal parts into which a compound or denominate 
 number is divided. 
 
 Denominate Fractions are expressed either v.r 
 common fractions, or as decimals. The former are usually 
 termed denominate fractions; the latter, denominate 
 decimals. 
 
 REDUCTION OF DENOMINATE FRACTIONS. 
 CASE I. 
 
 290. To reduce a Denominate Fraction from Higher 
 
 denominations to Lower. 
 
 1. What part of an inch is -fe of a yard ? 
 
 ANALYSIS. ^ of i yard equals ^ of 2 yards. 2 yd. x 3 = 6 ft. 
 Reducing the numerator to inches, we have 6 ft. X 1 2 7 2 in. 
 2 yards- 2 x 3 x 12, or 72 inches ; and 9 ^ of 72 ^ns. * or 4 in. 
 in. is H or inch. 
 
 Or, denoting the multiplications, and cancelling the factors 
 common to the numerators and denominators, we have 
 
 2 2 X ? . 2X3X12. 2 X $ X 12 ?. 
 
 yd. = - 2. ft = - !L m. = 2. ~_ = 3 in. Hence, the 
 96 96 96 96, 4 4 
 
 RULE. Reduce the numerator to the denomination re- 
 quired, and place it over the given denominator; cancdliity 
 the factors common to both. (Art. 276.) 
 
 NOTE. The steps in this operation are the same as those in 
 Reduction Descending. 
 
 2. Reduce ^ of a bushel to the fraction of a quart. 
 
 3. Reduce ^5- to the fraction of a penny. 
 
 4. Reduce -^^ of a day to the fraction of an hour. 
 
 5. What part of an ounce is -^ of a pound ? 
 
 6. What part of a square inch is -yfj of a sq. foot ? 
 
 289. What are denominate fractions? How are they expressed? 290. Row 
 ttr denominate fractions reduced from hijrher denominations to lower? Note 
 What are the etepn in this operation like? Explain Ex, t from the blackboard.
 
 DENOMINATE FB ACTIONS. 
 
 CASE II. 
 
 293. To reduce a Denominate Fraction from Lower 
 denominations to Higher. 
 
 7. What part of a yard is | of an inch ? 
 
 ANALYSIS. Reducing I yard to fourths of an inch, we have I yd. 
 = 36 in. ; and 36 in. x 4=144 fourths inch. Now 3 fourths are T f 3 ol 
 144 fourths ; therefore f of an inch is T^T or 4 1 s of a yard. Hence, the 
 
 RULE. Reduce a unit of the denomination in which the 
 required fraction is to be expressed, to the same denomination 
 as the given fraction, and place the numerator over it. 
 
 8. What part of a dollar is of a mill ? 
 
 9. What part of a pound is of a farthing ? 
 10. What part of a Troy ounce is -^ of a grain ? 
 n. Reduce f of a gill to the fraction of a gallon. 
 
 12. Reduce of a rod to the fraction of a mile. 
 
 1 3. Reduce of a pound to the fraction of a ton. 
 
 CASE III. 
 
 292. To reduce a Denominate Fraction from higher 
 to a IVliole Number of lower denominations. 
 
 14. Reduce f of a yard to feet and inches. 
 
 ANALYSIS. Reducing the given fraction to the next lower denomi- 
 nation, we have yard x 3= 1 6 i ft. or 2 ft. and I ft. remainder. 
 
 Again, reducing the remainder to the next lower denomination, 
 we have ft. x 12= V- or 4a i Q - Therefore, yards equals 2 ft. 4^ in. 
 (Arts. 147, 276.) Hence, the 
 
 RULE. Multiply the given numerator by the number 
 required to reduce the fraction to the ne.it lower denomina- 
 tion, and divide the product by the denominator. (Art. 276.) 
 
 Multiply and divide the successive remainders in the 
 same manner till the lowest denomination is reached. The 
 several quotients will be the answer required. 
 
 291. now are denominate fractions reduced from lower denominations to 
 higher? 2^2. How arc denominate fractions reducwl from higher to wLoie 
 numbers of lower denominations T
 
 204 DENOMINATE FRACTIONS. 
 
 15. Eeduce of an eagle to dollars and cents. Ans. $7.50 
 
 16. Reduce f of a pound sterling to shillings and pence 
 
 17. How many pecks and quarts in -ff of a bushel ? 
 
 1 8. How many pounds in f of a ton ? 
 
 19. How many ounces, etc., in | of a Troy pound ? 
 
 20. What is the value of -^ of a mile ? 
 
 21. What is the value of f| of an acre ? 
 
 22. Keduce | of a cord to cu. feet 
 
 CASE IV. 
 
 293. To reduce a Compound Number from lower to a 
 Denominate Fraction of higher denominations. 
 
 23. Reduce 2 ft. 3 in. to the fraction of a yard. 
 ANALYSIS. 2 ft. 3 in.=27 inches; and i yard=36 inches. The 
 
 question now is, what part of 36 in. is 27 in. ? But 27 is f J or J of 36. 
 (Art. 173.) Therefore 2 ft. 3 in. is ^ of a yard. Hence, tho 
 
 RULE. I. Reduce the given number to its lowest denom- 
 ination for the numerator. 
 
 II. Reduce to the same denomination, a unit of the 
 required fraction, for the denominator. 
 
 NOTES. i. If the lowest denomination of the given number con- 
 tains a fraction, the number must be reduced to the parts indicated 
 by the denominator of the fraction. 
 
 2. If it is required to find ichnt part one compound number is of 
 another which contains different denominations, reduce both to tht 
 lowest denomination mentioned in either, and proceed as above. 
 
 24. Reduce 2 qt. i pt. 3! gills to the fraction of a gal. ? 
 
 25. What part of a pound Troy is 5 oz. 2 pwt. ic gr. ? 
 
 26. Reduce 3 pk. 2 qt. i pt. to the fraction of a bushel 
 
 27. Reduce 6 cwt. 48 Ib. to the fraction of a ton. 
 
 28. What part of a square yard is 5 sq. ft. 2of sq. inches ? 
 
 29. What part of 3, IDS. 2d. is 75. 9d. 2 far. ? 
 
 30. What part of a week is 3 days, 5 hrs. 40 min ? 
 
 3 1. What part of a cord is 25^ cu. feet of wood ? 
 
 32. What part of 5 miles 2 fur. 14 rods is 2 m. i fur. 2 rods? 
 
 293. How arc denominate numbers reduced from lower denominations to 
 fractions of a higher denomination ? Explain Ex. 23 from tho blackboard.
 
 DENOMINATE DECIMALS. 205 
 
 CASE V. 
 
 294. To reduce a Denominate Decimal from a higher to a 
 Compound Number of lower denominations. 
 
 33. Reduce .89375 gallon to quarts, pints, and gills? 
 
 ANALYSIS. This example is a case of 
 
 Reduction Descending. (Art. 276.) We '89375 gal. 
 
 therefore multiply the given decimal of a 4 
 
 gallon by 4 to reduce it to the next lower 3.5 7500 qt. 
 
 denomination, and pointing off the pro- 2 
 
 duct as in multiplication of decimals, the , 
 
 result is 3 qts. and .57500 qt. In like 
 manner, we multiply the decimal .57500 - . 
 
 qt. by 2 to reduce it to pints, and the .60006 gi. 
 
 result is i pt. and .15000 pt. Finally, we Ans. 3 qt. I pt. 0.6 gi. 
 multiply the decimal .15000 pt. by 4 to 
 
 reduce it to gills, and the result is .6 gill. Therefore, .89375 gal. 
 equals 3 qt. i pt. 0.6 gi. Hence, the 
 
 RULE. Multiply the denominate decimal by the number 
 required of the next lower denomination to make one of the 
 given denomination, and point off the product as in multi' 
 plication of decimals. (Arts. 191, 276.) 
 
 Proceed in this manner with the decimal part of the suc- 
 cessive products, as far as required. The integral part of 
 the several products will be the answer. 
 
 34. What is the value of .125445 in shillings, etc. ? 
 
 35. What is the value of .91225 of a Troy pound? 
 
 36. Reduce .35 mile to rods, etc. 
 
 37. How many quarts and pints in .625 of a gallon? 
 
 38. How many minutes and seconds in .65 1 degree ? 
 
 39. How many days, hours, etc., in .241 week ? 
 
 40. What is the value of .25256 ton ? 
 
 41. What is the value of .003 of a Troy pound ? 
 
 42. What is the value of 5.62542 ? 
 
 294. How are denominate decimals reduced from higher denominations to 
 whole numbers of a lower denomination ? Explain Ez. 33 upon the blackboard
 
 206 DENOMINATE DECIMALS. 
 
 CASE VI. 
 
 295. To reduce a Compound Number from lower to a 
 Denominate Decimal of a higher denomination. 
 
 43. Reduce 3 pk. 2 qt. i pt. to the decimal of a bushel. 
 
 ANALYSIS. 2 pints make i quart ; hence OPERATION. 
 
 there is \ as many quarts as pints, and of 21 pt. 
 
 i qt. is .5 qt., which we write as a decimal on o c o f 
 
 the right of the given quarts. In like manner _ 1 
 
 there is as many pecks as quarts, and \ of 
 
 2.5 is .3125 pk., which we place on the right Ans. .828125 bu. 
 of the given pecks. Finally, there is | as 
 
 many bushels as pecks, and ^ of 3.3125 is .828125 bu. Therefore, 
 3 pk. 2 qt. i pt. equals .828125 bushel. Hence, the 
 
 RULE. Write the numbers in a column, placing the 
 lowest denomination at the top. 
 
 Beginning with the lowest, divide it by the number re- 
 quired of this denomination to make a unit of the next 
 higher, and annex the quotient to the next denomination. 
 
 Proceed in this manner with the successive denomina- 
 tions, till the one required is reached. 
 
 44. Reduce 6 oz. 10 pwt. 4 gr. to the decimal of a pound. 
 
 45. Reduce 10 Ib. to the decimal of a ton. 
 
 46. Reduce 3 fur. 25 r. 4 yd. to the decimal of a mile. 
 
 47. Reduce 9.6 pwt. to the decimal of a Troy pound. 
 
 48. What decimal part of a barrel are 15 gal. 3 qt. 
 
 49. What decimal part of 2 rods are 2\ fathoms ? 
 
 50. What decimal part of 3, 33. 6d. are 158. io.$d. ? 
 
 51. What part of a barrel are 94.08 Ibs. of flour? 
 
 52. Reduce 7.92 yds. to the decimal of a rod. 
 
 53. Reduce i day 4 hr. 10 sec. to the decimal of a week. 
 
 54. Reduce 45 sq. rods 25 sq. ft. to the decimal of an acre. 
 
 55. Reduce 53^ cu. feet to the decimal of a cord. 
 
 295. flow arc compound numbers reduced from lower denominations tf 
 denominate decimals of a higher? Explain Ex. 43 upon the blackboard?
 
 METEIO WEIGHTS 
 MEASUEES. 
 
 296. Metric Weight sand Measures are founded 
 upon the decimal notation, and are so called because their 
 primary unit or lase is the Meter.* 
 
 297. The Meter is the unit of length, and is equal to 
 one ten-millionth part of the distance on the earth's surface 
 from the equator to the pole, or 39.37 inches nearly. 
 
 NOTES. i. The term meter is from the Greek metron, a measure. 
 2. The standard meter is a bar of platinum deposited in the 
 archives of Paris. 
 
 298. The Metric System employs five different units or 
 denominations and seven prefixes. 
 
 The units are the me'ter, li'ter, gram, ar, and ster.\ 
 
 299. The names of the higher denominations are formed 
 by prefixing to the unit the Greek numerals, dele' a 10, 
 hek'to too, kil'o 1000, and myr'ia 10000 ; as, dek'a-me'ter, 
 10 meters; hek'to-me'ter, 100 meters, etc. 
 
 The names of the lower denominations, or divisions 
 of the unit, are formed by prefixing to the unit the Latin 
 numerals, dec'i (des'-ee) . i, cen'ti (cent'-ee) .01, and mil'li 
 (mil'-lee) .001; as, dec'i-me'ter, T ^ meter; cen'ti-me'ter, 
 meter ; mil'li-me'ter, T ^ meter. 
 
 NOTE. The numeral prefixes are the key to the whole system ; 
 their meaning therefore should be thoroughly understood. 
 
 296. Upon what are the Metric Weights and Measures founded? Why eo 
 called ? 297. What is the meter 299. How are the names of the lower denom- 
 inations formed ? The higher ? 
 
 * This system had its origin in France, near the close of the last century. Ita 
 simplicity and comprehensiveness have secured its adoption in nearly all the 
 countries of Europe and South America. Its use was legalized in Great Britain 
 in 1864 ; and in the United States, by Act of Congress, 1866. 
 
 t The spelling, pronunciation, and abbreviation of metric terms in this work, 
 arf> the same as adopted by the American Metric Bureau, Boston, aud the, 
 Mevrolo;;ical Society, New York,
 
 208 METRIC OB DECIMAL 
 
 LINEAR MEASURE. 
 
 300. The Unit of Length is the METER, which is 
 equal to 39.37 inches.* 
 
 The denominations are the mil'li-me'ter, cen'ti-me'ter, 
 dec'i-me'ter, meter, dek'a-me'ter, hek'to-me'ter, kil'o-me'ter, 
 and myr'ia-me'ter. 
 
 TABLE. 
 
 10 mtt'ttme'ters (mm.) make i cen'ti-me'ter - - . cm. 
 
 10 cen ti me'ters " i dec'i-me'ter - - dm. 
 
 10 dec'i-me ters " i ME'TER - - m. 
 
 10 me'ters " i dek'a-me ter - - Dm. 
 
 10 dek'a-me'ters " I hek'to-me'ter - - Hm. 
 
 10 hek'to-me ters " i kil'o-me'ter - Km. 
 
 10 kil'o-me'ters " i myr'ia-me'ter - Mm. 
 
 NOTES. i. The Accent of each unit and prefix is on the first 
 syllable, and remains so in the compound words. 
 
 To abbreviate the compounds, pronounce only the prefix and the 
 first letterof the unit ; as, centim, million, centil, decig, hektog, etc. 
 
 2. The meter, like our yard, is used in measuring cloths, laces, 
 moderate distances, etc. 
 
 For long distances the kilometer (3280 ft. 10 in.) is used ; and 
 for minute measurements, the centimeter or millimeter. 
 
 3. Decimeters, dekameters, hektometers, like our dimes and eagles, 
 are seldom used. 
 
 SQUARE MEASURE. 
 
 301. The Unit for measuring surfaces is the Square 
 Meter, which is equal to 1550 sq. in. 
 
 The denominations are the sq. cen'ti-me'ter, sq. dec'i- 
 me' tjbr, and sq. me' ter. 
 
 100 sq. cen'tim. make I sq. dec'i-me'ter - - sq. dm. 
 100 sq. dec'im. " i Sq. Me* ter - - sg. m. 
 
 NOTE. The square meter is used in measuring floorings, ceilings, 
 etc.; square deci-mtters and eenti-meters, for minute surfaces. 
 
 300. What is the unit of Linear Meaeare ? Its denominations ? Recite the 
 Table. 301. What is the unit for measuring surfaces ? Recite the table. 
 
 * Authorized by Act of Congress, 1866.
 
 WEIGHTS AND MEASURES. 209 
 
 302. The Utt it for measuring land is the Ar, which 
 is equal to a square dekameter, or 119.6 sq. yards. 
 
 The only subdivision of the Ar is the cen'tar ; and 
 the only multiple is the liek'tar. Thus, 
 
 100 cent'ars (ca.) make i Ar - - - a. 
 100 are " i hek'tar - - Ha. 
 
 NOTES. i. The term ar is from the Latin area, a surface. 
 
 2. In Square Measure, it takes roo units of a lower denomination 
 to make one in the next higher ; hence each denomination must have 
 two places of figures. In this respect centars correspond to cents. 
 
 CUBIC MEASURE. 
 
 303. The Unit for measuring solids is the Cubic 
 Meter, which is equal to 35.316 cu. ft. 
 
 TABLE. 
 
 1000 cu. mil'lim. make i cu. cen'ti me'ter - cu. cm. 
 1000 cu. cen'tim. " i cu. dec'i-me'ter - cu. dm. 
 1000 cu. dec'im. " i Cu. Me'ter - cu. m. 
 
 NOTES. i. The cubic meter is used in measuring embankments, 
 excavations, etc. ; cubic centimeters and millimeter tf.for minute bodies. 
 
 2. Since it takes 1000 units of a lower denomination in cubic 
 measure to make one of the next higher, it is plain that, like mills, 
 each denomination requires three places of figures. 
 
 304. In measuring wood, the Ster, which is equal to 
 a cubic meter, is sometimes used. 
 
 The only subdivision of the ster is the dec'i-ster ; and 
 the only multiple, the dck'a-sfer. Thus, 
 
 10 dec'i-sters make i Ster - - - s. 
 10 sters " i dek'a-ster - Ds. 
 
 NOTES. i. The term ster is from the Greek stereos, a solid. 
 2. In France, firewood is commonly measured in a cubical box, 
 whose length, breadth, and height are each i meter. 
 
 3. As the ster is applied only to wood, and probably will never 
 come into general use, its divisions and multiples may be omitted. 
 
 Note. How many places of figures does each denomination occupy? Why? 
 
 302. What is> the unit for measuring l.inrt ? Its divisions? Its multiples? 
 
 303. What are the units for menf-urnis* solids? Recite the Table. How many 
 places does each denomination in cubic measure occupy ? Why ?
 
 210 METRIC OB DECIMAL 
 
 DRY AND LIQUID MEASURE. 
 
 305. The Unit of Dry and Liquid Measure 
 
 is the Liter (lee'ter], which is equal to a cubic decim, or 
 1.0567 liquid quart or 0.908 dry quart. 
 
 The denominations are the mil'li-li'ter, cen'ti-li'ter, 
 dec'i-li'ter, liter, dek'a-li'ter, hek'to-li'ter, kil'o-li'tei: 
 
 10 mil'li-li'ters (ml.) make i cen'ti-li'ter - - - d. 
 
 10 cen'ti-li'ters " i dec'i-li'ter - - - dl. 
 
 10 dec'i-li'ters " i Li'ter - - - - I. 
 
 10 li'ters " i dek'a-liter - - - Dl. 
 
 10 dek'a-li'ters " i hek'to-li'ter - - HI. 
 
 10 hek to-li'ter " i kil'o-li'ter - - - Kl. 
 
 10 kil'o liters " i myr'ia-li'ter - - Ml. 
 
 NOTES. i. The liter is used in measuring milk, wine, etc. For 
 small quantities, the centiliter and millUiter are employed ; and for 
 large quantities, the dekaliter. 
 
 2. For measuring grain, etc., the hektoliter, which is equal to 
 2.8375 bushels, is commonly used. 
 
 3. The term liter is from the Greek litra, a pound. 
 
 WEIGHT. 
 
 306. The Unit of Weight is the Gram, which is 
 equal to 15.432 grains. 
 
 The denominations are the mil'li-gram, cen'ti-gram, 
 dec'i-gram, gram, dek'a-gram, hek'to-gram, kil'o-gram, 
 myr'ia-gram, and ton'neau or ton. 
 
 10 milli -grams (ing. 
 
 ) make i centi-gram - - 
 
 eg. 
 
 10 centi grams 
 
 " i dec'i-gram - - 
 
 dg. 
 
 10 dec i-grams 
 
 " i Grain 
 
 ff- 
 
 10 grams 
 
 " i dek'a-gram - - 
 
 Dg. 
 
 10 dek'a-grams 
 
 " i hekto-gram - - 
 
 Hg. 
 
 10 hek to-grams 
 
 " i Kil'o-grani 
 
 Kg. 
 
 10 kil o-grams 
 
 " i myr'ia-gram - - 
 
 Mg. 
 
 100 myr'ia-grams 
 
 " i TONNEAU or Ton 
 
 T. 
 
 305. What is the unit of Dry and Liquid Men^ure? Its denomlnatitmst Ra- 
 pnat the Table. 306. What is the unit of Weight? Its deuomiitutioimf
 
 WEIGHTS AXD MEASURES. 211 
 
 REMARK. As the quintal (10 Mg.) is seldom used, and millier 
 means the same as tonneau or ton, both these terms may be dropped 
 from the table. The metric ton is equal to 2204.6 Ibs. 
 
 NOTES. i. This Table is used in computing the weight of all 
 objects, from the minutest atom to the largest heavenly body. 
 
 2. The grum is derived from the Greek gramma, a rule or 
 standard, and is equal to a cubic centimeter of distilled water at its 
 greatest density, viz., at the temperature of 4 centigrade thermome- 
 ter, or 39.83 Fahrenheit, weighed in a vacuum. 
 
 3. The common unit for weighing groceries and coarse articles 
 is the kilogram, which is equal to 2.2046 pounds Avoirdupois. 
 
 4. Kilogram is often contracted into kilo, and tonneau into tan. 
 
 307. To express Metric Denominations decimally in terms 
 of a given Unit. 
 
 1. Write 7 Hm. o Dm. 9 m. 3 dm. 5 cm. decimally in 
 terms of a meter. 
 
 ANALYSIS. Metric denominations in- OPERATION. 
 
 crease and decrease by the scale of 10, and 79'35 m -> A&8* 
 correspond to the orders of the Arabic 
 
 Notation. We therefore write the given meters as units, the deka- 
 meters as tens, and the hektometers as hundreds, with the deci- 
 meters and centimeters on the right as decimals. Hence, the 
 
 RULE. Write the denominations above the given unit 
 in their order, on the left of a decimal point, and those 
 below the unit, on the right as decimals. 
 
 NOTES i. If any intervening denominations are omitted in the 
 given number, their places must be supplied by ciphers. 
 
 2. As each denomination in square measure occupies two places of 
 figures, in writing square decimeters, etc., as decimals, if the number 
 is less than 10, a cipher must be prefixed to the figure denoting them. 
 
 Thus, 17 sq. in., and 2 sq. dm. = 17.02 sq. meters. (Art. 302, ??.) 
 
 3. In like manner, in writing cubic decimeters, etc., as decimals, 
 if the number is less than 10, two ciphers must be prefixed to it. 
 
 Thus, 63 cu. m. and 5 cu dm. = 63.005 cu. meters. 
 
 Repeat the Table. Note. The common unit for weighing groceries, etc.? 
 307. How write metric qrnnHti'js In terms of a given unit 5- AV>V. 1C any clo- 
 uomiitur.ion is omitted, what is to be don ?
 
 212 METBIC OR DECIMAL 
 
 308. Metric denominations expressed decimally are read 
 like numbers in the Arabic notation. 
 
 Thus, the Answer to Ex. i, (709.35), is read, "seven hundred 
 and nine, and thirty-five hundredths meters, or seven hundred 
 nine meters, thirty-five," as we read $709.35, "seven hundred 
 nine dollars, thirty-five," omitting the name cents. 
 
 NOTE. The number of figures following the name of the unit, 
 shows the denomination of the last decimal figure. 
 
 If the name is required, it is better to use abbreviations of the 
 metric terms, as centims, millims, etc., than the English words 
 hundredth*, or thousandths of a meter, etc. The former are not only 
 significant of the kind of unit and the value of the decimal, but 
 are understood by all nations. 
 
 Write decimally, and read the following : 
 
 1. 5 Mm., 3 Km., 7 Hm., o Dm., 9 m., 8 dm., 4 cm., 
 5 mm. in terms of a meter, hektometer, kilometer, and 
 decimeter. 
 
 2. 4 Kg., 5 Hg., o Dg., 5 g, i dc., o eg., 8 mg. in terms 
 of a dekagram, centigram, hektogram, and kilogram. 
 
 3. Write and read 5 KL, o HI., 6 Dl., 3 1., o dl. 8 cl. 
 in terms of a hektoliter. Ans. 50.6308 hektoliters. 
 
 309. To reduce higher Metric Denominations to lower. 
 Ex. i. Eeduce 46.3275 kilometers to meters. 
 
 ANALYSIS From kilometers to meters OPERATION. 
 
 there are three denominations, and it 4"-3 2 75 
 
 takes 10 of a lower denomination to make Ioo 
 
 a unit of the next higher. We therefore Ans. 46327.5000 m. 
 
 multiply by 1000, or remove the decimal 
 
 point three places to the right. (Art. 181.) Hence, the 
 
 RULE. Multiply the given denomination by 10, 100, 
 1000, etc., as the case way require ; and point off the prod- 
 uct as in multiplication of decimals. (Art. 191.) 
 
 NOTE. It should be remembered that in the Metric System each 
 denomination of square measure requires two figures ; and each 
 denomination of cubic measure, three figures.
 
 WEIGHTS AND MEASUKES. 813 
 
 e. In 43.75 hectares, how many square meters? 
 
 3. Reduce 867 kilograms to grams. 
 
 4. Reduce 264.42 hectoliters to liters. 
 
 5. In 2561 ares, how many square meters? 
 
 6. In 865 2 cubic meters, how many cubic decimeters ? 
 
 7. Reduce 4256.25 kilograms to grams. 
 
 310. To reduce lower Metric Denominations to higher. 
 
 8. Reduce 84526.3 meters to kilometers. 
 
 ANALYSIS. Since it takes 10 linear units 
 to make one of the next higher denomination, 'ERATION. 
 
 it follows that to reduce a number from a IOOO ) 8 45 26 -3 m. 
 lower to the next higher denomination, it 84.5263 km. 
 
 must be divided by 10; to reduce it to the 
 
 next higher still, it must be again divided by 10, and so on. From 
 meters to kilometers there are three denominations ; we therefore) 
 divide by 1000, or remove the decimal point three places to the left. 
 The answer is 84.5263 km. Hence, the 
 
 RULE. Divide the given denomination by 10, 100, 1000, 
 etc., as the case may require, and point off the quotient as 
 
 in division of decimals. (Art. 194.) 
 
 t 
 9. In 652254 square meters, how many hectares? 
 
 10. Reduce 87 meters to kilometers. 
 
 n. In 1482.35 grams, how many kilograms? 
 
 12. In 39267.5 liters, how many kiloliters? 
 
 APPROXIMATE VALUES. 
 
 310 ft. In comparing Metric Weights and Measures with those 
 now in use, the approximate values are often convenient. Thus, 
 when no great accuracy is required, we may consider 
 
 i decimeter as 
 
 4 in. 
 
 i cu. meter or etere as j cord. 
 
 i meter " 
 
 40 in. 
 
 i liter 
 
 " i quart. 
 
 5 meters " 
 
 i rod. 
 
 i hectoliter 
 
 " 2^ bushels. 
 
 i kilometer " 
 
 $ mile. 
 
 i gram 
 
 " 15 i grains. 
 
 i sq. meter " 
 
 io sq. feet. 
 
 i kilogram 
 
 " 2 pounds. 
 
 i hectare " 
 
 2^ acres. 
 
 i metric ton 
 
 " 2200 pounds. 
 
 309. How reduce higher metric dtmoinmatioiia to lower ? 310. Lower to higher 1
 
 214 MKTEIC OR DECIMAL. 
 
 APPLICATIONS OF METRIC - WEIGHTS AND 
 MEASURES. 
 
 311. To add, subtract, multiply, and divide Metric Weights 
 
 and Measures. 
 
 EULE. Express the numbers decimally, and proceed 
 as in the corresponding operations of whole numbers and 
 decimals. 
 
 1. What is the sum of 7358.356 meters, 8.614 hecto- 
 meters, and 95 millimeters ? 
 
 SOLUTION. 7358.356 m. + 861.4 m., +.095 m.^Saig.Ssim. Ans. 
 
 2. What is the difference between 8.5 kilograms and 
 976 grams? ' 
 
 SOLUTION. 8.5 kilos .976 kilos=7.524 kilos. Ana. 
 
 3. How much silk is there in 1 2\ pieces, each contain- 
 ing 48.75 meters? 
 
 SOLUTION. 48.75 m. x 12.5=609.375 m. Ans. 
 
 4. How many cloaks, each containing 5.68 meters, can 
 be made from 426 meters of cloth ? 
 
 SOLUTION. 426 rn.-r-5.68 m.=75 cloaks. AJIS. 
 
 312. To reduce Metric to Common Weights and Measures. 
 
 i. Eeduce 5.6 meters to inches. 
 
 39.37 in. 
 ANALYSIS. Since in i meter there are 39.37 inches, c 6 m 
 
 in 5.6 meters there are 5.6 times 39.37 in. ; and 
 39-37 x 5 .6=220.472 in. Therefore, in 5.6 m. there 
 are 220.472 in. Hence, the 
 
 Ans. 220.472 in. 
 
 EULE. Multiply tJie value of th-e principal unit of the 
 Table by the given metric number. 
 
 NOTE. Before multiplying, the metric number should be reduced 
 to the same denomination as the principal vnit. whose value is 
 taken for the multiplicand. 
 
 311. How are Metric Weights and Measures added, cnbtrsetod, multiplied, 
 and divide-:? ? 312. How reduce metric to common weights and measures r
 
 WEIGHTS AND MEASURES. 2lo 
 
 2. In 45 kilos, how many pounds? Ans. 99.207 Ibs. 
 
 3. In 63 kilometers, how many miles? 
 
 4. Eeduce 75 liters to gallons. 
 
 5. Reduce 56 dekaliters to bushels. 
 
 6. Reduce 120 grams to ounces. 
 
 7. Reduce 137.75 kilos to pounds. 
 
 8. In 36 ares, how many square rods ? 
 
 ANALYSIS. In I are there are 1 19.6 sq. yds. ; hence in 36 ares there 
 are 36 times as many. Now 119.6x36=4305.6 sq. yds., and 4305.0 
 sq. yds.-r- 30^=142.33 sq. rods. Ans. 
 
 9. In 60.25 hektars, how many acres? 
 10. In 120 cu. meters, how many cu. feet ? 
 
 313. To reduce Common to Metric Weights and Measures. 
 
 n. Reduce 213 feet 4 inches to meters. 
 
 ANALYSIS 213 ft. 4 in.=256o in. Now, OPERAT10N 
 
 in 39.37 in. there is i meter; therefore, in ., 7 ) 2 =;6o!oo in. 
 2560 in. there are as many meters as 39.37 is ^"'^' / 3 \ 
 contained times in 2560; and 2560-7-39.37= Ans. 65.02 + 00. 
 
 65.02 + meters. Hence, the 
 
 RULE. Divide the given number by tlie value of the 
 principal metric unit of the Table. 
 
 NOTE. Before dividing, the given number should bo reduced t 
 the lowest denomination it contains ; then to the denomination IE 
 which the value of the principal unit is expressed. 
 
 1 2. In 63 yds. 3 qrs., how many meters ? 
 
 13. Reduce 13750 pounds to kilograms. 
 
 14. Reduce 250 quarts to liters. 
 
 15. Reduce 2056 bu. 3 pecks to kilohters. 
 
 16. In 3 cwt. 15 Ibs. 12 oz., how many kilos? 
 
 17. In 7176 sq. yards, how many sq. meters? 
 
 18. In 40.471 acres, how many hektars? 
 
 19. In 14506 cu. feet, how many cu. meters? 
 
 20. In 36570 cu. yards, how many cu. meters? 
 
 313. How reduce common to metric weights and measures t
 
 COMPOUND ADDITION. 
 
 314. To add two OP more Compound Numbers. 
 
 i. What is the sum of 13 Ib. 7 oz. 3 pwt 18 gr. ; 9 oz. 5 
 pwt. 6 gr. ; and 2 Ib. 8 oz. 8 pwt. 3 gr. ? 
 
 ANALYSIS. Writing the same denomi- OPERATION. 
 
 nations one under another, and beginning lb - oz - P^ S 1 "- 
 
 at the right, 3 gr. + 6 gr. + i8 gr. are 27 
 gr. The next higher denomination is 
 pwt., and since 24 gr.=i pwt., 27 gr. 
 
 must equal I pwt. and 3 grains. We set Ans. 1J o 17 3 
 the 3 gr. under the column added, be- 
 cause they are grains, and carry the i pwt. to the column of penny 
 weights, because it is a unit of that column. (Art. 30, n.) Next, i 
 pwt. + 8 pwt. + 5 pwt. + 3 pwt. are 17 pwt. As 17 pwt. is less than 
 an ounce or a unit of the next higher denomination, we set it under 
 the column added. Again, 8 oz. + 9 oz. + 7 oz. are 24 oz. Now as 
 12 oz.=i Ib., 24 oz. must equal 2 Ib. and o oz. We set the o under 
 the ounces, and carry the 2 Ib. to the next column. Finally, 2 Ib. 
 4-2 Ib. + 13 Ib. are 17 Ib., which we set down in full. Hence, the 
 
 RULE. I. Write the same denominations one under 
 another ; and, beginning at the right, add each column 
 separately. 
 
 II. If the sum of a column is less than a unit of the 
 next higher denomination, write it under the column added. 
 
 If equal to one or more units of the next higher denomi- 
 nation, carry these units to that denomination, and write 
 the excess under the column added, as in Simple Addition. 
 
 NOTES. i. Addition, Subtraction, etc., of Compound Numbers are 
 the same in principle as Simple Numbers. The apparent difference 
 arises from their scales of increase. The orders of the latter increase 
 by the constant scale of 10 ; the denominations of the former by a 
 variable scale. In both we carry for the number which it takes of a 
 lower order or denomination to make a unit in the next higher. 
 
 2. If the answer contains a fraction in any of its denominations, 
 except the lowest, it should be reduced to wfwle numbers of lower 
 denominations, and be added to those of the same name. 
 
 314. How are compound numbers added ? \ote. The difference between Com 
 pound and Simple Addition ? If the answer contains a fraction ?
 
 COMPOtTND ADDITION. 217 
 
 3 if they occur in the given numbers, it is generally best to 
 reduce them to lowor denominations before the operation is com- 
 menced, 
 
 2. What is the sum of 6 gal. 3 qt. i pt. 2 gi. ; 4 gal. 2 qt. 
 o pt i gi., and 7 gal. i qt. i pt. 3 gi. ? 
 
 Ans. 1 8 gal. 3 qt. i pt. 2 gi 
 
 (3-) (40 (5-) 
 
 
 
 8. 
 
 d. 
 
 far. 
 
 T. 
 
 cwt. 
 
 Ib. 
 
 oz. 
 
 bu. 
 
 pk. 
 
 qt. 
 
 7 
 
 16 
 
 6 
 
 I 
 
 3 
 
 6 
 
 42 
 
 4 
 
 14 
 
 2 
 
 5 
 
 i 
 
 5 
 
 8 
 
 2 
 
 7 
 
 
 
 26 
 
 7 
 
 21 
 
 3 
 
 6 
 
 2 
 
 7 
 
 9 
 
 3 
 
 15 
 
 17 
 
 14 
 
 8 
 
 8 
 
 i 
 
 7 
 
 6. A merchant sold 19! yd. of silk to one lady; i6| yd. 
 to a second, 20^ yd. to a third, and 17! to a fourth : how 
 much silk did he sell to all ? 
 
 7. If a dairy woman makes 315 Ib. 5 oz. of butter in 
 June, 275 Ib. 10 oz. in July, 238 Ib. 8 oz. in August, and 
 263 Ib. 14 oz. in September, how much will she make in 
 4 months ? 
 
 8. A ship's company drew from a cask of water 10 gals. 
 
 2 qts. i pt. one day, 12 gal. 3 qt. the second, 15 gal. i qt. 
 i pt. the third, and 16 gal. 3 qt. the fourth: how much 
 water was drawn from the cask in 4 days ? 
 
 9. A farmer sent 4 loads of wheat to market ; the ist 
 contained 45 bu. 3 pk. 2 qt. ; the 2d, 50 bu. i pk. 3 qt ; 
 the 3d, 48 bu. 2 pk. 5 qt. ; and the 4th, 51 bu. 3 pk. 5 qt. : 
 what was the amount sent ? 
 
 10. How much wood in 5 loads which contain re- 
 spectively i c. 28^ cu. ft.; i c. 47! cu. ft; i c. n cu. ft. 
 
 i c. no cu. ft; and i c. n-J cu. ft. ? 
 
 1 1. A miller bought 3 loads of wheat, containing 28 bu. 
 
 3 pks. ; 36 bu. i pk. 7 qts. ; 33 bu. 2 pks. 3 qts. respectively: 
 how many bushels did he buy ? 
 
 If any of the given numbers contain a fraction, how proceed? 315. How add 
 denominate fractions * 
 
 10
 
 218 COMPOUND ADDITION. 
 
 12. What is the sum of 13 m. 65 r. 3 yd. i ft. ; 12 m. 
 40 r. 2 yd. i ft. ; and 21 m. 19 r. i yd. 2 ft. ? 
 
 SOLUTION. In dividing the yards m. r. yd. ft. 
 
 by s (the number required to *3 5 3 I 
 
 ruake a rod), the remainder is i 12 40 2 I 
 
 yard. But i yd. equals i ft. 6 in. ; 21 19 i _ 2 
 
 we therefore add i ft. 6 in. to the 46 125 li i 
 
 feet and inches in the result. The I yd. = o I 6 in. 
 
 Ana. 5s 46 m. I25 r. i yd. 2 ft. 6 in. ^ s . 46 125 i 2 6 
 
 13. What is the sum of 8 w. 2^d. 7 h. 40 m. ; 4 w. 5^ d. 
 o h. 15 m. ; and 10 w. o d. 16 h. 3 m. ? 
 
 14. What is the sum of 15 A. no sq. r. 30 sq. yds.; 
 6 A. 45 sq. r. 16 sq. yds ; 42 A. 10 sq. r. 25 sq. yds. ? 
 
 315. To add two OP more Denominate Fractions. 
 
 15. What is the sum of f bu. pk. and f quart ? 
 ANALYSIS -Reducing the de- 4 bu> pk . x qi ,, t> 
 
 nominate fractions to whole num- f ^ _ ": 
 
 bers, then adding them, the 3 nf _ o iJL 
 
 amount is 3 pk. 6 qt. Oj^ pt., 
 
 CUU.I/ U.J-1. V J.O 3 I^JX- V V^W* V YO ,f u '> -rfC-l. **O. J j_ 
 
 (Art. 292.) Ttr 
 
 1 6. What is the sum of .125, .655. and .75d. ? 
 
 ANALYSIS. Reducing the denominate .125 2S. 6d. o far 
 decimals to whole numbers of lower .658. =o 7 3.2 
 
 denominations, then adding, the result .75d. o o 3 
 
 is 33. 2d. 2.2 far. Hence, the 
 
 jfift/G. 3 ~ &. 
 
 KULE. Reduce the denominate fractions, whether com- 
 mon or decimal, to whole numbers of lower denominations : 
 then add them like other compound numbers. 
 
 17. Add .75 T. .5 cwt. .25 Ib. 19. Add f Ib. ^ X oz. -^ pwt. 
 
 18. Add .25 bu. \ of 3 pk. 20. Add .15, .55. .8d. 
 
 21. Bought 4 town lots, containing \ acre; f acre; 
 i2i sq. rods; and 150.5 sq. rods respectively: how much 
 land was there in the 4 lots ? 
 
 22. Bought 3 loads of wood ; one containing cord, 
 another 75.3 cu. feet; the other ij cord: how much did 
 they all contain ?
 
 COMPOUND SUBTKACTIOK'. 
 
 316. To find the Difference between two Compound Numbers. 
 
 1. From 5 mi. 3 fur. n r. 4 ft. take 2 mi. i fur. 4 r. 9 ft. 
 ANALYSIS. We write the less number m. fur. r. ft. 
 
 under the greater, feet under feet, etc. S3 11 4 
 
 Beginning at the right, 9 ft. cannot be 2 I 4 9 
 
 taken from 4 feet ; we therefore borrow i Ans. 3 2 6 I li 
 rod, a unit of the next higher denomina- 
 tion. Now i rod, or i6V ft., added to 4 ft. are 20^ ft., and 9 ft. from 
 zo\ ft. leave n ft. Set the remainder under the term subtracted, 
 and carry i to the next term in the lower number, i rod and 4 r. are 
 5 r., and 5 r. from n r. leave 6 rods, i furlong from 3 fur. leaves 
 2 fur. ; and 2 m. from 5 m. leave 3 m. Therefore, etc. Hence, the 
 
 RULE. I. Write the less number under the greater, 
 placing the same denominations one under another. 
 
 II. Beginning at the right, subtract each term in the 
 lotver number from that above it, and set the remainder 
 under the term subtracted. 
 
 III. If any term in the lower number is larger than that 
 above it, borrow a unit of the next higher denomination and 
 add it to the upper term ; then subtract, and carry i to the 
 next term in the lower number, as in Simple Subtraction. 
 
 NOTES. i. Sorrowing in Compound Subtraction is based on the 
 same principle as in Simple numbers. That is, we add as 
 many units to the term in the upper number as are required of 
 that denomination to make a unit of the next higher. 
 
 2. If the answer contains a fraction in any of its denominations 
 except the lowest, it should be reduced to lower denominations, and 
 be united to those of the same name, as in Compound Addition. 
 
 2. From i mile, take 240 rods 3 yd. 2 ft. 
 
 3. From 14 Ib. 5 oz. 3 pwt. 16 gr., take 7 Ib. 6 pwt. 7 gr. 
 
 4. From 12 T. 9 cwt. 41 Ib., take 5 T. i cwt. 15 Ib. 
 
 5. Take 20 gal. 3 qt. i pt. 2 gi. from 29 gal. i qt. i pt. 
 
 316. How subtract compound numbers? Note. What is said of borrowing 
 If the answer contains a fraction, how proceed ?
 
 220 COMPOUND STTBTBACIION. 
 
 6. From 250 A. 45 sq. r. 150 sq. ft., take 91 A. 32 sq. t 
 200 sq. ft. 
 
 7. A miller has two rectangular bins, one containing 
 324 cu. ft. no cu. in.; the other, 277 cu. ft. 149 cu. in.: 
 #hat is the difference in their capacity ? 
 
 8. A man having 320 A. 50 sq. r. of land, gave his 
 eldest son 175 A. 29 sq. r., and the balance to his youngest 
 son : what was the portion of the younger ? 
 
 9. A merchant bought two pieces of silk ; the longer 
 contained 57 yd. 3 qr. ; the difference between them was 
 i if yards : what was the length of the shorter? 
 
 10. What is the difference between 10 in. 2 fur. 27 r. 
 i ft 7 in. and 6 m. 4 fur. 28 r. i yd. ? 
 
 317. To find the Difference between two Denominate Frac- 
 tions. 
 
 1. From of a yard, take -fj of a foot. 
 
 ANALYSIS. Reducing the denominate yd. = 2 f t. 6 in. 
 
 fractions to whole numbers, we have li ft. = o 1 1 in. 
 
 | yard = 2 ft. 6 in. and -^ foot = n in. ~~j~ i ft. TllT. 
 Ana. i ft. 7 in. Hence, the 
 
 RULE. Reduce the denominate fractions, whether com- 
 mon or decimal, to whole numbers of lower denominations ; 
 then proceed according to the rule above. 
 
 2. From .525 take .75 of a shilling. 
 
 3. From I bu. take -f of a peck. 
 
 4. From | of a gal., take i f of a pint. 
 
 5. A goldsmith having a bur of gold weighing 1.25 Ibs., 
 cnt off sufficient to make 6 rings, each weighing .15 
 ounce ; how much was left 9 
 
 6. If a man has .875 acre of land, and sells off two 
 building lots, each containing 12.8 sq. rods, how much 
 land will be left ? 
 
 7. From f of f cwt. of sugar, take 31.25 Ibs. 
 
 317. How subtract denominate fractions ?
 
 COMPOUND SUBTRACTION. 221 
 
 318. To find the Difference of Time between two Dates, 
 
 in years, months, and days. 
 
 i. What is the difference of time between July 4th, 
 1776, and April 3d, 1870? 
 
 ANALYSIS. We place the earlier date tin- y. m. d. 
 
 der the later, with the year on the left, the 1870 4 3 
 
 number of the month next, and the day of I77& 7 4 
 
 the month on the right. Since we cannot Ans. 93 8 29 
 take 4 from 3, we borrow 30 days, and 4 from 
 
 33 leaves 29. Carrying i to the next denomination, we proceed as 
 above. (Art. 264, n.) Hence, the 
 
 KULE. Set the earlier date under the later, the years on 
 the left, the month next, and the day on the right, and pro- 
 ceed as in subtracting other compound numbers. 
 
 NOTE. Centuries are numbered in dates, from the beginning of 
 the Christian era ; months, from the beginning of the year ; and 
 days, from the beginning of the month. 
 
 3. Washington was born Feb. 22d, 1732, and died Dec. 
 1 4th, 1799: at what age did he die? 
 
 319. To find the difference between two Dates in Days, 
 
 the time being less than a year. 
 
 i. A note dated Oct. 2oth, 1869, was paid Feb. icth, 
 1870 : how many days did it run ? 
 
 ANALYSIS. In Oct. it ran 3120=11 Oct. 31 20 n d. 
 
 days, omitting the day of the date ; in Nov., Nov. = 30 d. 
 
 30 days; in Dec., 31 ; in Jan., 31; and in Dec. 3 1 d. 
 
 Feb., 10, counting the day it was paid. Jan. 3 1 d. 
 
 Now ii + 30 +31 + 31 + 10=113 days, the p e b. =10 d. 
 
 number required. Hence, the * i it d 
 
 RULE. Set doion the number of days in each month and 
 part of a month between the two dates, and the sum will be 
 the number of days required. 
 
 NOTE. The day on which a note or draft is dated, and that on 
 which it becomes due, must not both be reckoned. It is customary 
 to omit the former, and count the latter. 
 
 318. How find the difference between two dates in years? 319. In days!
 
 222 COMPOUND SUBTRACT 10 If. 
 
 2. What is the number of days between Nov. loth, 1869, 
 and March 3d, 1870? 
 
 3. A person started on a journey Aug. ipth, 1869, and 
 returned Nov. ist, 1869: how long was he absent? 
 
 4. A note dated Jan. 3ist, 1870, was paid June 3oth, 
 1870: how many days did it run ? 
 
 5. How many days from May 2ist, 1868, to Dec. 3ist, 
 following ? 
 
 6. The building of a school-house was commenced April 
 i st, and completed on the loth of July following : how 
 long was it in building ? 
 
 320. To find the difference of Latitude OP Longitude. 
 
 DBF. i. Latitude is distance from the Equator. It is reckoned 
 in degrees, minutes, etc., and is called North or South latitude, ac- 
 cording as it is north or south of the equator. 
 
 2. LoHf/ifnde is the distance on the equator between a con <//- 
 tionnl or fixed meridian and the meridian of a given place. It is 
 reckoned in degrees, minutes, etc., and is called East and West longi- 
 tude, according as the place is east or west of the fixed meridian, xmtil 
 i8o D or half the circumference of the earth is reached. 
 
 7. The latitude of New York is 40 42' 43" N. ; that of 
 New Orleans, 29 57' 30" N.: how much further north is 
 New York than New Orleans ? * 
 
 ANALYSIS. Placing the lower latitude N. Y. 40 42' 43" 
 under the higher, and subtracting as in the Jf. Q. 29 57' 30" 
 preceding rule, the Ans. is 10 45' 13". Ans. 10 Zq' 17" 
 
 8. The latitude of Cape Horn is 55 59' S., and that of 
 Cape Cod 42 i' 57.1" N. : what is the difference of lati- 
 tude between them ? 
 
 ANALYSIS. As one of these capes is north of the equator and the 
 other smith, it is plain that the difference of their latitude is the sum 
 of the two distances from the equator. We therefore add the two 
 latitudes together, and the result is 98 o' 57 i'. 
 
 320. Def. What is latitude ? Longitude ? How find the difference of latitude 
 or longitude befween two places ? 
 
 * The latitude and longitude of the places in the United States here given are 
 taken from the American Almanac, 1854 ; those of places in foreign lands, montly 
 from Bo'-vditch'rf Navigator.
 
 COMPOUND SUBTRACTION. 223 
 
 9. The longitude of Paris is 2 20' East from Green- 
 wich; that of Duolin is 6 20' 30'' West: what is the dif- 
 ference in their longitude ? 
 
 ANALYSIS. As the longitude of one of these places 2 20' oo" 
 
 is East from Greenwich the standard meridian, and 6 20' 30" 
 
 that of the other West, the difference of their longi- 7T5 ~, r, 
 
 tude must be the sum of the two distances from the 
 standard meridian. Ana. 8" 40' 30". Hence, the 
 
 RULE. I. If both places are the same side of the equator, 
 cr the standard meridian, subtract the less latitude or 
 longitude from the greater. 
 
 II. If the places are on different sides of the equator, or 
 the standard meridian, add the two latitiides or longitudes 
 together, and the sum will be the answer. 
 
 10. The longitude of Berlin is 13 24' E. ;* that of 
 New Haven, Ct., 72 55' 24" "W. : what is the difference? 
 
 11. The longitude of Cambridge, Mass., is 71 7' 22"; 
 that of Charlottesville, Va., 78 31' 29": what is the 
 difference ? 
 
 12. The latitude of St. Petersburg is 59 56' north, that 
 of Rome 41 54' north : what is their difference ? 
 
 13. The latitude of Albany is 4? 39' 3", that of Rich- 
 mond 37 32' 17": what is the difference? 
 
 14. The latitude of St. Augustine, Flor., is 29 48' 30" ; 
 that of St. Paul, Min., is 44 52' 46": what is the dif- 
 ference ? 
 
 15. The longitude of Edinburgh is 3 12' W.; that of 
 Vienna 16 23' E.f : required their difference? 
 
 1 6. The latitude of Valparaiso is 33 2' S. ; that of Ha- 
 vana is 23 9' N.: what is the difference? 
 
 17. The latitude of Cape of Good Hope is 34 22' S. ; 
 that of Gibraltar, 36 7' N. : what is their difference ? 
 
 18. The longitude of St. Louis is 90 15' 16"; that of 
 Charleston, S. C., 79 55' 38": what is the difference? 
 
 * Encyc. Brit. * Encyc. Amer.
 
 COMPOUND MULTIPLICATION. 
 
 321. To multiply Compound Numbers, 
 i. A farmer raised 6 acres of wheat, which yielded 15 bn. 
 3 pk. i qt. per acre: how much wheat had he ? 
 
 ANALYSIS. 6 acres will produce 6 times as OPERATION. 
 
 much as i acre. Beginning at the right, 6 times bu- pk- 1*- 
 
 1 qt. are 6 qts. As 6 quarts are less than a 
 peck, the next higher denomination, we set the 
 
 6 under the term multiplied. 6 times 3 pk. are Ans. 94 2 6 
 18 pk. Since 4 pk.=i bu., 18 pecks=4 bu. and 
 
 2 pk. over. Setting the remainder 2 under the term multiplied, and 
 carrying the 4 bu. to the next product, we have 6 times 15 bu. go 
 bu., and 4 bu. make 94 bu. Therefore, etc. Hence, the 
 
 RULE. I. Write the multiplier under the lowest denomi- 
 nation of the multiplicand, and, beginning at the right, 
 multi2ily each term in succession. 
 
 II. If the product of any term is less than a unit of the 
 next higher denomination, set it under the term multiplied. 
 
 III. If equal to one or more units of the next higher de- 
 nomination, carry these units to that denomination, and 
 write the excess under the term multiplied. 
 
 NOTES. i. If the multiplier is a composite number, multiply by 
 one of the factors, then this partial product by another, and so on. 
 
 2. If a fraction occurs in the product of any denomination except 
 tlnj lowest, it should be reduced to lower denominations, and be 
 united to those of the same name as in Compound Addition. (Art. 314.) 
 
 (2.) (3-) 
 
 Mult. 12 T. 7 cwt. 1 6 Ib. 21, 138. 8|d. 
 
 By 8 7_ 
 
 <j.. What is the weight of 10 silver spoons, each weigh- 
 ing 3 oz. 7 pwt. 13 gr. ? 
 
 321. How are compound numbers' multiplied? Nrce. If the multiplier is a 
 composite number, how proceed 1 When a fraction occurs in aiy7 denomination 
 except tire last, how ?
 
 COMPOUND MULTIPLICATION. 225 
 
 (50 (6.) 
 Mult. 9 oz. 13 pwt. 7 gr. by 18. Mult. 10 r. i yd. i ft. by 7. 
 
 9 oz. 13 pwt. 7 gr. 10 r. i yd. i ft. o in. 
 
 3 7 
 
 2lb. 4 19 21 71 3^ i 
 
 6 i) T l.= i 6 
 
 14 Ib. 5 oz. 19 pwt. 6 gr. Ans. Ans. 71 r. 3 yd. 2 ft. 6 in. 
 
 7. If a family use 27 gal. 2 qt. i pt. of milk in a month, 
 how much will they use in a year ? 
 
 8. If a man chops 2 cords 67 cu. feet of wood per day, 
 how much will he chop in 9 days ? 
 
 9. What cost 27 yards of silk, at 173. 7^d. sterling per 
 yard? 
 
 10. If a railroad train goes at the rate of 23 m. 3 fur. 
 21 r. an hour, how far will it go in 24 hours? 
 
 n. How much corn will 63 acres of land produce, at 
 30 bu. 3 pk. per acre ? 
 
 1 2. How many cords of wood in 1 7 loads, each contain- 
 ing i cord 41 cu. ft.? 
 
 13. How much hay in 12 stacks of 5 tons, 237 Ibs. each? 
 
 14. How much paper is required to print 20 editions of 
 a book, requiring 65 reams, 7 quires, and 10 sheets each ? 
 
 15. If a meteor moves through 5 23' 15 ''in a second, 
 how far will it move in 30 seconds ? 
 
 1 6. If the daily session of a school is 5 h. 45 min., how 
 many school hours in a term of 15 weeks of 5 days each ? 
 
 17. A man has u village lots, each containing 12 sq. r. 
 4 sq. yd. 6 sq. ft. : how much do all contain ? 
 
 1 8. If i load of coal weighs i T. 48^ Ib., what will 72 
 loads weigh ? 
 
 19. How many bushels of corn in 12 bins, each con- 
 taining 130 bu. 3 pk. and 7 qfc. ? 
 
 20. A grocer bought 35 casks of molasses, each contain' 
 ing 55 gal. 2 qt. i pt. : how much did they all contain?
 
 COMPOUKD DIVISION. 
 
 322. Division of Compound Numbers, like Simple 
 Division, embraces two classes of problems : 
 
 First. Those in which the dividend is a compound number, and 
 the divisor is an abstract number. 
 
 Second. Those in which both the divisor and dividend are com- 
 pound numbers. In the former the quotient is a compound number. 
 In the latter, it is times, or an abstract number. (Art. 64.) 
 
 323. To divide one Compound Number by another, or by an 
 Abstract number. 
 
 Ex. i. A dairy-woman packed 94 Ib. 2 oz. of butter in 
 6 equal jars : how much did each jar contain ? 
 
 ANALYSIS. The number of parts is given, OPERATION. 
 
 to find the value of each part. Since 6 jars 6)94 Ib. 2 OZ. 
 
 contain 94 Ibs. 2 oz., i jar must contain ^ Ans. je lb.il OZ. 
 of 94 Ibs. 2 oz. Now % of 94 Ibs. is 15 Ibs. 
 
 and 4 Ibs. remainder. Reducing the remainder to oz., and adding 
 the 2 oz., we have 66 oz. Now ^ of 66 oz. is n oz. (Art. 63, 6.) 
 
 2. A dairy-woman packed 94 Ibs. 2 oz. of butter in jars 
 of 15 Ibs. ii oz. each : how many jars did she have? 
 
 ANALYSIS. Here the size of each 94 Ib. 2 oz. = 1506 oz. 
 
 part is given, to find the number of i e Ib. 1 1 oz. = 251 oz. 
 
 parts in 94 Ib. 2 oz. Reduce both O Z.)l5o6 OZ. 
 
 numbers to oz., and divide as in sim- 
 
 ., . 
 
 pie numbers. (Art. 63, a.) Hence, the -*** 
 
 fi,ULE. I. When the divisor is an abstract number, 
 Beginning at the left, divide each denomination in suc- 
 
 cession, and set the quotient tinder the term divided. 
 
 If there is a remainder, reduce it to the next lower de- 
 
 nomination, and, adding it to the given units of this de- 
 
 nomination, divide as before. 
 II. When the divisor is a compound number, 
 Reduce the divisor and dividend to the lowest denomina- 
 
 tion contained in either, and divide as in simple numbers. 
 
 322. How many classes of examples docs division of compound numbers em 
 brace? The first? The second? 323. What is the rule ?
 
 TIME AND LONGITUDE. 227 
 
 NOTE. If the divisor is a composite number, we may divide by it* 
 factors, as in simple numbers. (Art. 77.) 
 
 3. Divide 29 fur. 19 r. 2 yd. i ft. by 7. 
 
 4. Divide 54 gal. 3 qt. i pt. 3 gi. by 8. 
 
 5. A miller stored 450 bu. 3 pks. of grain in 18 equal 
 bins : how much did he put in a bin ? 
 
 6. A farm of 360 A. 42 sq. r. is divided into 23 equal 
 pastures: how much land does each contain? 
 
 7. How many spoons, each weighing 2 oz. 10 pwt., can 
 be made out of 5 Ib. 6 oz. of silver ? 
 
 8. How many iron rails, 18 ft. long, are required for a 
 railroad track 15 miles in length? 
 
 9. How many times does a car-wheel 15 ft. 6 in. in cir- 
 cumference turn round in 3 m. 25 r. 10 ft. ? 
 
 10. How many books, at 48. 6d. apiece, can you buy for 
 2, 143. 3d.? 
 
 ii If 6 men mow 86 A. 64 sq. rods in 6 days, how 
 much will i man mow in i day? 
 
 12. A farmer gathered 150 bu. 3 pk. of apples from 24 
 trees : what was the average per tree ? 
 
 Y 
 
 COMPARISON OF TIME AND LONGITUDE. 
 
 324. The Earth makes a revolution on its axis once in 24 
 hours ; hence -* 1 ,- part of its circumference must pass under the sun 
 in i hour. But the circumference of every circle is divided into 
 360 , and v l 4 - of 360 is 15. It follows, therefore, that 15 of longi- 
 tude make a difference of i hour in time. 
 
 Again, since 15" of longitude make a difference of i hour in time, 
 15 of longitude (,/u of 15") will make a difference of i minute (fa of 
 an hour) in time. 
 
 In like manner, 15" of longitude (-^ of 15'), will make a difference 
 of i second iu time. Hence, the following 
 
 TABLE. 
 
 15 of longitude are equivalent to i hour of time. 
 
 J5' " i minute " 
 
 15" * * " i second "
 
 228 TIME AND LONGITUDE. 
 
 CASE I. 
 
 325. To find the Difference of Time between two places, 
 the difference of Longitude being given. 
 
 Ex. i. The difference of longitude between New York 
 and London is 73 54' 3": what is the difference of time? 
 
 ANALYSIS. Since 15 of Ion. are equiv- OPERATION. 
 
 alent to i hour of time, the difference of 15 j 73 54' 3" 
 
 time must be -,V part as many hours, min- . 
 
 A , ' Ans. 4h. 55 m. 36.23. 
 
 utes, and seconds as there are degrees, 
 
 etc., in the dif. of Ion. ; and 73 54' 3"-j-i5=4 h. 55' 36.2 " Hence, the 
 
 RULE. Divide the difference of longitude by 15, and the 
 degrees, minutes, and seconds of the quotient will be the 
 difference of time in hours, minutes, and seconds. (Art. 3 23.) 
 
 2. The difference of longitude between Savannah, Ga., 
 and Portland, Me., is 10 53' 2": what is the difference in 
 time? 
 
 i 3' 30" W., that of 
 , noon in Boston what 
 is the time at Detroit ? 
 
 4. The longitude of Philadelphia is 75 9' 54", that of 
 Cincinnati 84 27': when it is noon at Cincinnati what is 
 the time at Philadelphia ? 
 
 5. The Ion. of Louisville, Ky., is 85 30', that of Bur- 
 lington, Vt., 73 10': what is the difference in time? 
 
 6. When it is noon at Washington, what is the time of 
 (by at all places 22 30' east of it? What, at all places 
 22 30' west of it? 
 
 7. How much earlier does the sun rise in New York, 
 Ion. 74 3", than at Chicago, Ion. 87 35'? 
 
 8. How much later does the sun set at St. Louis, whoso 
 longitude is 90 15' 16" W., than at Nashville, Tenn., 
 whose longitude is 86 49' 3" ? 
 
 325. How find the difference of time between two places, the difference ol 
 longitude being given? 
 
 3. The longitude of Boston is 7 
 Detroit is 83 2' 30" W. : when it is
 
 AND LONGITUDB. 
 
 CASE II. 
 
 326. To find the Difference of Longitude between two 
 places, the difference of Time being given. 
 
 9. A whaleman wrecked on an Island in the Pacific, 
 found that the difference of time between the Island and 
 San Francisco was 2 hr. 27 min. 54! sec. : how many de- 
 grees of longitude was he from San Francisco ? 
 
 ANALYSIS. Since 15 of Ion. are equira OPERATION. 
 
 lent to i hour of time, 15' of Ion. to i min. 2\\. 27 m. 54.6 860. 
 
 of time, and 15" to i sec. of time, there must t r 
 
 be 15 times as many degrees, minutes, and - 
 
 seconds in the difference of longitude as *** 3" 5 39 
 there are hours, minutes, and seconds in the 
 difference of time ; and 2 h. 27 m. 54.6 s. x 15=36 58' 39". Hence, the 
 
 RULE. Multiply the difference of time fyi i$>*nd the 
 hours, minutes, and seconds of the product will be the 
 difference of longitude in degrees, minutes, and seconds. 
 
 10. The difference of time between Richmond, Va., ana 
 Newport, R, I., is 24 min. 36 sec. : what is the difference 
 of longitude ? 
 
 1 1. The difference of time between Mobile and Galveston 
 is 27 min. sec.: what is the difference of longitude? 
 
 12. The difference of time between Washington and 
 San Francisco is 3 hr. i min. 39 sec. : what is the dif- 
 ference in longitude ? 
 
 13. The distance from Albany, N. Y., to Milwaukee, is 
 nearly 625 miles, and a degree of longitude at these places 
 is about 44 miles : how much faster is the time at Albany 
 than at Milwaukee ? 
 
 14. The distance from Trenton, N. J., to Columbus, 0.. 
 is nearly 400 miles, and a degree of longitude at these 
 places is about 46 miles: when it is noon at Columbus 
 what is the time at Trenton ? 
 
 326. How find the different of longitude, when the difference of time U given*
 
 PERCENTAGE. 
 
 327. Per Cent and Rat e Per Cent denote hun- 
 dredths. Thus, i per cent of a number is y^ part of that 
 number ; 3 per cent, yf^, &c. 
 
 328. Percentage is the result obtained by finding a 
 certain ger cent of a number. 
 
 NOTE. The term per cent, is from the Latin per, by and centum, 
 hundred. 
 
 NOTATION OF PER CENT. 
 
 329. The Sign of Per Cent is an oblique line 
 between two ciphers (%) ; .as 3$, 15$. 
 
 NOTE. The sign (%}, is a modification of the sign division (-*-), 
 the denominator 100 being understood. Thus 5$ =-,-u= 5-7-100. 
 
 330. Since per cent denotes a certain part of a hundred, 
 it may obviously be expressed either by a common fraction, 
 whose denominator is 100, or by decimals, as seen in the 
 following 
 
 TABLE. 
 
 3 per cent is written .03 
 
 7 per cent " .07 
 
 10 per cent .10 
 
 25 per cent " .25 
 
 50 per cent " .50 
 
 loo per cent " i.oo 
 
 125 per cent " 1.25 
 
 300 per cent " 3.00 
 
 ^ per cent is written .003 
 
 | per cent -0025 
 
 f per cent " -0075 
 
 f per cent .006 
 
 2^ per cent " .025 
 
 7 per cent " .074 
 
 31 i per cent -3125 
 
 per cent " 1.125 
 
 NOTES. i. Since hundredths occupy two decimal places.it follows 
 hat every per cent requires, at least, two decimal figures. Hence, 
 
 327. What do the termc per cent and rate percent denote ? 328. What is fi<- 
 ceutage? Note. Prom what are the terms per cent and percentage derived? 
 32q. What 'is the sign of per cent? 330. How may per cent be expressed? 
 /Vote. How many decimal places does it require? Why? If the tnven per cent 
 Is lc?? than IP, what is to be done? What is 100 i>cr cent of a nnmher?
 
 PERCENTAGE. 231 
 
 If the given per cent is less than 10, a cipher must be prefixed to 
 the figure denoting it. Thus, 2 % is written .02 ; 6 %, .06, etc. 
 
 2. A hundred per cent of a number is equal to the number itself; 
 for {ftll is equal to i. Hence, 100 per cent is commonly written i.oo. 
 
 If the given per cent is 100 or over, it may be expressed by an 
 integer, a mixed number, or an improper fraction. Thus, 125 pei 
 cent is written 125 %, 1.25, or |gg. Hence, 
 
 331. To express Per Cent, Decimally, 
 
 Write the figures denoting the per cent in the first two 
 places on the right of the decimal point ; and those denoting 
 parts of i per cent, in the succeeding places toward the right. 
 
 NOTES. i. When a given part of i per cent cannot be exactly 
 expressed by one or two decimal figures, it is generally written as a 
 common fraction, and annexed to the figures expressing the integral 
 per cent. Thus, 4}, c / is written .04^, instead of .043333 + 
 
 2. In expressing per cent, when the decimal point is used, the 
 words per cent and the sign (%) must be omitted, and vice versa. 
 Thus. .05 denotes 5 per cent, and is equal to TOO or -fa ; but .05 per 
 cent or .05$ denotes T Sn of T u, and is equal to TO&OTT or 
 
 Express the folloAving per cents, decimally : 
 
 1. 2%, 6%, 8%, 14%, 20%, 35^, 60%, 12%. 
 
 2. 80^, 101%, 104%, 150%, 210$, 300$. 
 3- i- 
 
 332. To read any given Per Cent, expressed Decimally. 
 
 Read the first two decimal figures as per cent; and those 
 on the right as decimal parts of i per cent. 
 
 NOTE. Parts of i per cent, when easily reduced to a common 
 fraction, are often read as such. Thus .105 is read 10 and a half per 
 cent ; .0125 is read one and a quarter per cent. 
 
 Read the following as rates per cent : 
 
 4. .05; .07; .09; .045; .0625; .1875; - I2 5; - I6 5; - 2 7- 
 
 5. .10; .17; .0825; .05125; .33^; .i6f; .75375. 
 
 6. i.oo; 1.06; 2.50; 3.00; 1.125; I -7 2 5; ^Sl- 
 
 331. How express per cent, decimally? Note. When a part of i per cent, can- 
 not be exactly expressed by one or two decimal figures, how is it commonly 
 written f 332. How read a given per cent, expressed decimally ? 

 
 232 PERCENTAGE. 
 
 333. To change a given Per Cent from a Decimal to a 
 
 Common Fraction. 
 
 7. Change 5^ to a common fraction. Ans. TW A- 
 
 8. Change .045 to a common fraction. Ans. - 2 . 
 
 RULE. Erase the decimal point or sign of per cent (), 
 and supply the required denominator. (Art. 179.) 
 
 NOTE. When a decimal per cent is reduced to a common fraction, 
 then to its lowest terms, this fraction, it should be observed, will 
 express an equivalent rate, but not the rate per cent. 
 
 Change the following per cents to common fractions : 
 
 9. 5 percent; io#; 4^; 20^; 25^; 50$; 75$. 
 10. 6^ per cent; 12^: 8$; 33^; 62-^. 
 n. \ per cent; ; $#; #; f#; $#; -&<; 25^. 
 
 334. To change a common Fraction to an equivalent Per Cent. 
 
 12. To what per cent is of a number equal ? 
 
 ANALYSIS. Per cent denotes hundredths. The I i .00 -f- 3 
 question then is, how is i reduced to hundredths? r oo _i. * ,^i 
 Annexing cipher's to the numerator, and dividing 
 
 by the denominator, we have ^=1.00-5-3 or .33 ^. Hence, the 
 
 RULE. Annex two ciphers to the numerator, and divide 
 ly the denominator. (Art. 186.) 
 
 13. To what % is \ equal ? i? |? -? f? f? |? 
 
 14. To what # is ^ equal P-^? ^?^? &? T 3 T? A? 
 
 15. To what ^ is | equal? J? |? f? 1? A? ? 
 
 335. In calculations of Percentage, four elements 01 
 parts are to be considered, viz. : the base, the rate per cent, 
 b.Q percentage, and the amount. 
 
 1. The base is the number on which the percentage is calculated. 
 
 2. The rntfe /jr Cfnt is the number which shows liow many 
 hundredths of the base are to be taken. 
 
 333. How change a given per cent from a decimal to a common- fraction ? 
 334. How change a common fraction to an equal per cent ? 335. How xnany parta 
 re to be considered in ralcnMions by percentage?
 
 PE 11C EXT AGE. 233 
 
 3. The percentage is the number obtained by taking that portion 
 of the base indicated by the rate per cent. 
 
 4. The amount is the base plus, or minus the percentage. 
 
 The relation between these parts is such, that if any two of them 
 are given, the other two may be found. 
 
 NOTES. i. The term amount, it will be observed, is here employed 
 in a modified or enlarged sense, as in algebra and other departments 
 of mathematics. Tliis avoids the necessity of an extra rule to meet 
 the cases in which the final result is less than the base. 
 
 2. The conditions of the question show whether the percentage is 
 to be added to, or subtracted from the base to form the amount. 
 
 3. The learner should be careful to observe the distinction between 
 percentage and per cent, or rate per cent. 
 
 Percentage is properly a product, of which the given per cent or 
 rate per cent, is one of the factors, and the base the other. This care 
 is the more necessary as these terms are often used indiscriminately. 
 
 4. The terms per cent, rate per cent, and rate, are commonly used 
 as synonymous, unless otherwise mentioned. 
 
 PROBLEM I. 
 
 336. To find the Percentage, the Base and Rate being 
 
 given. 
 Ex. i. What is 5 per cent of $600 ? 
 
 ANALYSIS. 5 per cent is .05 ; therefore 5 per $600 B. 
 
 cent of a number is the same as .05 times that .05 R. 
 
 number. Multiplying the base, $600, by the rate ^ns. $^o oo P 
 .05, and pointing off the product as in multipli- 
 cation of decimals, the result is $30. (Art. 191.) Hence, the 
 
 RULE. Multiply the base by the rate, expressed deci- 
 mally. 
 
 FORMULA. Percentage Base x Rate. 
 
 NOTES. i. When the rate is an aliquot part of 100, the percentage 
 may be found by taking a like part of the base. Thus, for 20 %, 
 take !; ; for 25%, take -\, etc. (Arts. 105, 270.) 
 
 2. When the bass is a compound number, the lower denominations 
 should be reduced to a decimal of the highest ; or the higher to the 
 lowest denomination mentioned ; then apply the rule. 
 
 3. Finding a per cent of a number is the same as finding a frac- 
 tional part of it, etc. The pupil is recommended to review with care, 
 Arts. 143, 165, 191 
 
 Explain them . Wha t is the relation of these parts The difference between per- 
 cen^ge and per cent ?
 
 234 PERCENTAGE. 
 
 3. 3$ of $807 ? ii. $$% of 1000 men ? 
 
 4. 5$ of 216 bushels? 12. io^$ of 1428 meters? 
 
 5. 8% of 282.5 yds. ? 13. 50% of $1715-57 ? 
 
 6. 4% of 2 1 6 oxen ? 1 4. |$ of 21.2? 
 
 7- 5?% of 150 yards ? 15. \% of 500 liters? 
 
 8. 16$ of $72.40? 1 6. f$ of 230 kilograms? 
 
 9. 12$ of 840 Ibs. ? 17. 100$ of 840 pounds ? 
 10. 14$ of 451 tons? 18. 200$ of $500? 
 
 19. A farmer raised 875 bu. of corn, and sold 9$ of it : 
 how many bushels did he sell ? 
 
 20. The gold used for coinage contains 10$ of alloy: 
 how much alloy is there in 3^ pounds of standard gold ? 
 
 21. A man having a hogshead of cider, lost 15$$ of it 
 by leakage : how many gallons did he lose ? 
 
 22. A garrison containing 4000 soldiers lost 21$ of them 
 by sickness and desertion : what was the number lost ? 
 
 23. A grocer having 1925 pounds of sugar, sold \2\ per 
 cent of it : how many pounds did he sell ? 
 
 ANALYSIS. 12.3 % is \- of 1005, and i<x>% of OPERATIOH. 
 
 a number is equal to the number itself ; there- 8)1925 Ibs. 
 
 fore TI\ per cent of a number is equal to \ of Ans. 240.62 < 
 that number, and ^ of 1925 Ibf ... is 240% Ibs. In 
 the operation we take ^ of the base. 
 
 Solve the next 9 examples by aliquot parts : 
 
 24. Find 25$ of $860. 26. 1 2^$ of 258 meters. 
 
 25. 10$ of 1572 pounds. 27. 20$ of 580 liters. 
 
 28. A drover' taking 2320 sheep to market, lost 25$ of 
 them by a railroad accident : how many did he lose ? 
 
 29. A farmer raised 468 bu. of corn, and 33^$ as many 
 oats as corn : how many bushels of oats did he raise ? 
 
 30. A young man having a salary of $1850 a year, spent 
 50 per cent of it : what were his annual expenses ? 
 
 31. What is 334$ of 1728 cu. feet of wood ? 
 
 32. What is 12^ per cent of 16, 8s. ? 
 
 336. How find the percpnta<ro vrhon the base and' rate arc piveti ? When thf 
 rate is an aliquot part of 100, how proceed ? When a compound number f
 
 PERCENTAGE. 235 
 
 PROBLEM II. 
 337. To find the Amount, the Base and Rate being given, 
 
 1. A commenced business Atith $1500 capital, and laiJ 
 up S% the first year: what amount was he then worth ? 
 
 ANALYSIS. Since he laid up 8f c , he was OPERATION. 
 
 worth his capital, $1500, plus 8f of itself. $1500 B. 
 
 But his capital is 100% or i time iieolf ; and 1.08, i H Ii. 
 
 100% +8%=io8% or 1.08; therefore he was 120.00 
 
 worth i. 08 times $1500. Now $1500 x 1.08 =; 1500 
 
 $1620. We multiply the base by i plus the $1620 oo Am't 
 given rate, expressed decimally. (Art. 191.) 
 
 2. B commenced business with $1800 capital, and 
 squandered 6% the first year: what was he then worth"? 
 
 ANALYSIS. As B squandered 6%, he was $1800 B. 
 
 worth his capital $1800, minus 6% of itself. .04 R. 
 
 But his capital is 100% or i time itself; and 7700 
 
 100% 6% =94% or .94. Therefore he had .94 T 62oo 
 
 times $1800 ; and $1800 x .94=$i692. Here we v . , 
 
 multiply the base by i minus the given rate, $ l6 9 2 - Am t 
 expressed decimally. (Art. 191.) Hence, the 
 
 EULE. Multiply the base by i plus or minus the rate, as 
 the case may require. The result will be the amount. 
 
 FORMULA. Amount =. Base x (i Rate). 
 
 NOTE. i. The character ( ) is called the double or amJnguous 
 sign. Thus, the expression $5 $3 signifies that $3 is to be added 
 to or subtracted from $5, as the case may require, and is read, 
 " $5 pl us or minus $3." 
 
 2. The rule is based upon the axiom that the whole is equal to 
 the sum of all its parts. 
 
 3. When, by the conditions of the question, the amount is to be 
 greater than the base, the multiplier is i plus the rate ; when the 
 amount is to be less than the base, the multiplier is I mi;ius the 
 rate. 
 
 337. How find the amount when the base and rate are given ? 338. How el*e 
 \ the amount found, when the base aud ratq arc given ?
 
 236 PERCENTAGE. 
 
 338. When the base and rate are given, the amount rujv 
 also be obtained by first finding the percentage, then adding 
 it to or subtracting it from the base. (Art. 336.) 
 
 3. and D have 1000 sheep apiece; if C adds 15% to 
 his flock, and D sells 1 2% of his, how many sheep will 
 each have ? 
 
 4. A merchant having $2150.38 in bank, deposited 
 1% more : what amount had he then in bank ? 
 
 5. If you have $3000 in railroad stock, and sell $% of it, 
 what amount of stock will you then have ? 
 
 6. The cotton crop of a planter last year was 450 bales; 
 this year it is 12 per cent more : what is his present crop ? 
 
 7. An oil well producing 2375 gallons a day, loses 15$ 
 of it by leakage : what amount per day is saved ? 
 
 8. A gardener having 1640 melons in his field, lost 20^' 
 of them in a single night: what number did he have 
 left? 
 
 9. A man paid $420 for his horses, and 12% more for 
 his carriage : what was the amount paid for the carriage ? 
 
 10. A man being asked how many geese and turkeys he 
 had, replied that he had 150 geese; and the number of 
 turkeys was 14$ less: how many turkeys had he? 
 
 it. A fruit grower having sent 2500 baskets of peaches 
 to New York, found 9% of them had decayed, and sold the 
 balance for 62 cts. a basket: what did he receive for his 
 peaches ? 
 
 12. A Floridian having 4560 oranges, bought 25% more, 
 and sold the whole at 4 cts. each : what did he receive 
 for them ? 
 
 13. If a man's income is $7235 a year, and he spends 
 33 J$ of it, what amount will he lay up ? 
 
 14. A man bought a house for $8500, and sold it for 
 20$ more than he gave : what did he receive for it ? 
 
 15. A merchant bought a bill of goods for $10000, and 
 s >ld them at a loss of 2\%\ what did he receive ?
 
 PEBCENTAGE. 237 
 
 PROBLEM III. 
 
 339. To find the Rate, the Base and Percentage being given ; 
 Or, to find what Per Cent one number is of another. 
 
 i. A clerk's salary, being $1500 a year, was raised $250*, 
 what rate was the increase ? 
 
 ANALYSIS. Iii this example $150x5 is the OPERATION. 
 
 base, and $250 the percentage. The question I5oo)$25o.oo P. 
 then is this: $250 is what per cant of $1500? Ans 
 
 Now $250 is i^ 5 ,, -,, of $1500; and $25o-j-$i5oo 
 =.16666, etc., or 16$$. The first two decimal figures denote the 
 per cent ; the others, parts of i#. (Arts. 331, 2.) Hence, the 
 
 EULE. Divide the percentage ly the base. 
 
 FORMULA. Rate = Percentage -j- Base. 
 
 NOTES. i. This prob. is the same as finding what part one number 
 is of another, then changing the common fraction to hundredths. 
 (Arts. 173, 186, 334.) 
 
 It is based upon the principle that percentage is a product ot 
 which the base is a factor, and that dividing a product by one of its 
 factors will give the other factor. (Art. 93.) 
 
 2. The number denoting the base is always preceded by the word 
 of, which distinguishes it from the percentage. 
 
 3. The given numbers must be reduced to the same denomination-, 
 and if there is a remainder after two decimal figures arc obtained, 
 place it over the divisor and annex it to the quotient. 
 
 2. What % of 15 is 2 ? 6. What % of 8 are 1 53. ? 
 
 3. What %<$ $20 are $5 ? 7. What % of 56 gals, are 7 qts.? 
 
 4. What % of 48 is 1 6 ? 8. What % are 5 dimes of $5 ? 
 
 5. What % of $5 are 75 cts.? 9. What % of f ton is \ ton ? 
 
 10. The standard for gold and silver coin in the U. S. 
 is 9 parts pure metal and i part alloy : what % is the alloy :' 
 
 11. From a hogshead of molasses 15 gals, leaked out 
 what per cent was the leakage ? 
 
 12. A grocer having 560 bbls. of flour, sold \ of it. 
 what per cent of his flour did he sell ? 
 
 13. A horse and buggy are worth $475; the buggy is 
 worth $i 10 ; what % is that of the value of the horse ? 
 
 330. How find the rate, when the base and percentage are given ? To what la 
 this problem equivalent ? JVbre. Upon what is it based ?
 
 238 PEBCENTAGE. 
 
 PROBLEM IV. 
 340. To find the Base, the Percentage and Rate being given. 
 
 1. A father gave his son $30 as a birthday present, 
 which was 6% of the sum he gave his daughter: how 
 much did he give his daughter ? 
 
 ANALYSIS. The percentage $30 is the product of OPERATION. 
 the base into .06 the rate; therefore $30-^.06 is the .o6)$3o.oo 
 other factor or bass ; and $30-7- .06 =$500, the sum he ^jsTlfecoo 
 gave his daughter. (Art. 193, n.) 
 
 Or, since $30 is 6 % of a number, i % of that number must be \ of 
 $30, which is $5 ; and 100% is 100 times $5 or $500. 
 
 It is more concise, and therefore preferable, to divide the per- 
 centage by the rate expressed decimally ; then point off the quotient 
 as in division of decimals. (Art. 193.) Hence, the 
 
 RULE. Divide the percentage ty the rate, expressed 
 decimally. 
 
 FORMULA. Base = Percentage -f- Rate. 
 
 NOTES. i. This problem is the same as finding a number when 
 a given per cent or a fractional part of it is given. (Arts. 174, 334.) 
 
 2. The rule, like the preceding, is based upon the principle that 
 percentage is a product, and the rate one of its factors. (Art. 335, n.) 
 
 3. Since the percentage is the same part of the base as the rate is 
 of 100, when the rate is an aliquot part of 100, the operation will to 
 shortened by using this aliquot part as the divisor. 
 
 2. 40 is i2^ N of what number? 
 SOLUTION. \2\%=\ and 40-74=40x8=320. An*. 
 
 3. 20=5$ of what number? Ans. 400. 
 
 4. 15 bushels=:6% of what number? 
 
 5. $29 = 8% of what? 
 
 6. 45 tons=25% of what? 
 
 7. 150=133^ of what? 
 
 8. 37-5=6^ of what? 
 
 9. 45 francs = 1 2\% of what? 
 
 10. 40 \% of what ? 
 
 11. 50 c,\&. \% of what ? 
 
 12. $ioo=|^ of what? 
 
 13. $35.203$ of what? 
 
 14. 68 yds.= 125$ of what ? 
 
 140. How flud the base, when the percentage and rate are given f Note. Upon 
 what does the rule depend f When the rate is an aliquot part of 100, how proceed f
 
 PERCENTAGE. 2 
 
 15. z% of $150 is (>% of what sum? 
 
 1 6. 12% of 500 is 60% of what number? 
 
 17. A paid a school tax of $50, which was i% on the 
 valuation of his property : what was the valuation ? 
 
 1 8. B saves 31^$ of his income, uud lays up $600: what 
 is his income ? 
 
 19. A general lost i6% of his army, 315 killed, no 
 prisoners, and 70 deserted: how many men had he? 
 
 20. According to the bills of mortality, a city loses 450 
 persons a month, and the number of deaths a year is \\% 
 of ics population : what is its population ? 
 
 PROBLEM V. 
 
 341. To find the Base, the Amount and Rate being given. 
 i. A manufacturer sold a carriage for $633, which was 
 more than it cost him : what was the cost ? 
 
 ANALYSIS. The amount received $633, is OPERATION. 
 
 equal to the cost or base plus $\% of itself. 1.055)^633.000 
 Now the cost is 100% or i time itself, and _Ans. $600 
 
 100% +5^ = 1. 05^; hence $633 equals 1.05^ 
 
 times the cost of the carriage. The question now is: 633 is 105-^$ 
 or 1.05 i times what number? If 633 is 1.05^ times a certain num- 
 ber, once that number is equal to as many units as 1.05! is contained 
 times in 633 ; and 633-7-1.055=600. Therefore the cost was $600. 
 
 2. A lady sold her piano for $628. 25, which was 12-^ 
 less than it cost her: what was the cost? 
 
 ANALYSIS. There being a loss in this case, OPERATION. 
 the amount received, $628.25, equals the cost 
 
 or base minus \z\% of itself. But the cost is Ans. $7 18 
 100^7 or i time itself, and i<X)% i.2\ % = .%}} ; 
 hence $628.25 equals .875 times the cost. Now if $628.25 equals 
 .87^ times the cost, once the cost must be as many dollars as .87-}- is 
 contained times in $628.25, or $718. Hence, the 
 
 RULE. Divide the amount by i plus or minus the rate,. 
 as the case may require. 
 
 FORMULA. Base = Amount -f- (i Rate}. 
 
 341. How find the base, the amount and rate being given? Note. Upon what
 
 240 PEBCENTAGE. 
 
 NOTES. i. This problem is the same as finding a number which 
 is a given per cent greater or less than a given number. 
 
 2. The rule depends upon the principle that the amount is a pro- 
 duct of which the base is one of the factors, and i plus or minu* the 
 rate, the other. 
 
 3. Thp nature of the question shows whether i is to be increased 
 or diminisfied by the rate, to form the divisor. 
 
 4. When the rate is an aliquot part of 100, the operation is often 
 shortened by expressing it as a common fraction. Thus25$>=:i; 
 and i or } + ^=f, etc. 
 
 3. What number is 8% of itself less than 351 ? A. 325. 
 
 4. "What number is $\% of itself more than 378 ? A. 400. 
 
 5. What number diminished 33^ of itself will equal 
 539i ? 
 
 6. 2275 is 2 5% more than what number? 
 
 7. is i2\% more than what number? 
 ANALYSIS. i2i$=i ; and \ -5-i=$ -4-t=f or f- An** 
 
 8. -f is 10% less than what number ? 
 
 9. A owns \ of a ship, which is 1 6f % less than B's part : 
 what part does B own ? 
 
 10. A garrison which had lost 28^ of its men, had 3726 
 left : how many had it at first ? 
 
 1 1. A merchant drew a check for $45 60, which was 25$ 
 more than he had in bank : how much had he 011 deposit ? 
 
 12. The population of a certain place is 8250, which is 
 20% more than it was 5 years ago : how much was it then ? 
 
 13. A man lays up $2010, which is 40% less than his in- 
 come : what is his income ? 
 
 14. A drover lost 10% of his sheep by disease, 15$ were 
 stolen, and he had 171 left : how many had he at first ? 
 
 15. The attendance of a certain school is 370, and 7 1 
 of the pupils are absent : what is the number on register? 
 
 1 6. An army having lost iof c in battle, now contains 
 5220 men : what was its original force ? 
 
 docs this rule depend ? How determine whether i is to be increased or dimin- 
 ished by the rate ? When the rate is au aliquot part, how proceed? To what U 
 this problem equivalent 1
 
 APPLICATIONS OF PEKCENTAGE. 
 
 342. The Principles of Percentage are applied 
 to two important classes of problems: 
 
 First. Those in which time is one of the elements o; 
 calculation ; as, Interest, Discount, etc. 
 
 Second. Those which are independent of time ; as, Com- 
 mission, Brokerage, and Profit or Loss. 
 
 COMMISSION AND BROKERAGE. 
 
 343. Commission is an allowance- made to agents, 
 collectors, brokers, etc., for the transaction of business. 
 
 JZroJierage is Commission paid a broker. 
 
 NOTES. i. An Agent is one who transacts business for another, 
 and is often called a Commission Merchant, Factor, or Correspondent. 
 
 2. A Collector is one who collects debts, taxes, duties, etc. 
 
 3. A Broker is one who buys and sells gold, stocks, bills of ex- 
 change, etc. Brokers are commonly designated by the department 
 of business in which they are engaged ; as, Stock-brokers, Exchange- 
 brokers, Note-brokers, Merchandise-brokers, Real-estate-brokers, etc. 
 
 4. Goods sent to an agent to sell, are called a consignment ; th 
 person to whom they are sent, the consignee ; and the person send- 
 ing them the consignor. 
 
 344. Commission and Brokerage are computed at a 
 certain per cent of the amount of business transacted. 
 Hence, the operations are precisely the same as those in 
 Percentage. That is, 
 
 The sales of an agent, the sum collected or invested by 
 lim, are the base. 
 
 The per cent for services, the rate. 
 The commission, the percentage. 
 The sales, etc., plus or minus the commission, theltmount. 
 
 342. To what two elapses of problems are the principles of percentage applied T 
 543. What is commission? Brokerage? JVofe. An agent? What called? A 
 collector? Broker? 344. How are commission and brokerage computed ? What 
 IB the base ? The rate ? The percentage ? The amount ? Note. The net proceeds f
 
 242 COMMISSION AND BROKERAGE. 
 
 NOTES. i. The rate of commission and brokerage varies. Com- 
 mission merchants usually charge about 2\ per cent for selling 
 goods, and 2^ per cent additional for guaranteeing the payment 
 Stock-brokers usually charge per cent on the par value of stocks, 
 without regard to their market value. 
 
 2. The net proceeds of a business transaction, are the gross amount 
 of sales, etc., minus the commission and other charges. 
 
 345. To flnd the Commission, the Sales and the Rate 
 being given. 
 
 Multiply the sales by the rate. (Problem I, Percentage.) 
 
 NOTES. i. When the amount of sales, etc., and the commission 
 are known, the net proceeds are found by subtracting the commission 
 from the amount of sales. Conversely, 
 
 2. When the net proceeds and commission are known, the amount 
 of sales, etc., is found by adding the commission to the net proceeds. 
 
 3. When both the amount of sales, etc., and the net proceeds are 
 known, the commission is found l\y subtracting the net proceeds 
 from the amount of sales. 
 
 4. In the examples relating to stocks, a share is considered $100, 
 unless otherwise mentioned. (Ex. 2.) 
 
 (For methods of analysis and of deducing the rules in Commission, 
 Profit or Loss, etc., the learner is referred to the corresponding 
 Problems in Percentage.) 
 
 1. A merchant sold a consignment of cloths for $358: 
 what was his commission at 2\ per cent ? 
 
 SOLUTION. Commission =$358 (sales) x .025 (rate) $8.95. Ans. 
 
 2. A broker sold 39 shares of bank stock: what was his 
 brokerage, at \ per cent ? 
 
 SOLUTION. 39 shares =$3900; and $3900 x .005 = 119.500. Am. 
 
 3. Sold a consignment of tobacco for $958.25 : what was 
 my commission at 3^ ? 
 
 4. A man collected bills amounting to $11268.45, and 
 charged zY/'c'- w ^ at was his commission; and how much 
 
 did he pay his employer ? 
 
 
 
 345. How find the ooimnisfion or brokerage, when the sales and the rate arc 
 2ft'*, How find the r.et proceeds ?
 
 COMMISSION AND BROKERAGE. 
 
 5. A commission merchant sold a consignment of goods 
 for $4561, and charged 2\% commission, and $% for 
 guaranteeing the payment : what were the net proceeds ? 
 6. An agent sold 1530 Ibs. of maple sugar at i6f cts., 
 for which he received z\% commission: what were the 
 net proceeds, allowing $7.50 for freight, and $3.10 for 
 storage ? 
 
 346. To find the Rate, the Saies and the Commission 
 
 being given. 
 Divide the commission by the sales. (Prob. III. Percentage.) 
 
 7. An auctioneer sold goods amounting to $2240, for 
 which he charged $53.20 commission : what per cent 
 was that ? 
 
 SOLUTION. Per cent = $53.20 (com.) -f- $2240 (sales) = .02375, or 
 2j per cent. (Art. 339, 331, n.) 
 
 8. A broker charged $19 for selling $3800 railroad 
 stock : what per cent was the brokerage ? 
 
 9. Received $350 for selling a consignment of hops 
 amounting to $7000: what per cent was my commis- 
 sion ? 
 
 10. An administrator received $118.05 f r settling an 
 estate of $19675 : what per cent was his commission ? 
 
 347. To find the Sales, the Commission and the Rate 
 
 being given. 
 
 Divide the commission by the rate. (Prob. IV, Per ct.) 
 
 n. An agent charged 2% for selling a quantity of 
 muslins, and received $93.50 commission : what was the 
 amount of his sales ? 
 
 SOLUTION. Sales= $93. 50 (com.)-?- .02 (rate)=$4675. Am. 
 
 346. How find the per cent commission, when the sales and the commission 
 are given ? 347. How find the amount of sales, when the commission and the 
 rate arc ffivon ?
 
 244 COMMISSION AND BKOKEBAGE 
 
 12. Received $45 brokerage for selling stocks, which 
 was \% of what was sold : what was the amount of stocks 
 sold? 
 
 13. A commission merchant charging 2\% commission, 
 and 2 \% for guaranteeing the payment, received $210.60 for 
 selling a cargo of grain : what were the amount of sales, 
 and the net proceeds ? 
 
 14. A district collector received $67.50 for collecting a 
 school tax, which was $\% commission : how much did he 
 collect, and how much pay the treasurer ? 
 
 15. An auctioneer received $135 for selling a house, 
 which was i$%: for what did the house sell; and how 
 much did the owner receive ? 
 
 348. To find the Sales, the net proceeds and per cent 
 commission being given. 
 
 Divide the net proceeds by i minus the rate. (Prob. V.) 
 
 1 6. An agent sold a consignment of goods at z\% com- 
 mission, and the net proceeds remitted the owner were 
 $3381.30 : what was the amount of. sales? 
 
 SOLUTION. Sales=$338i. 30 (net p.)-7-.g75 (i rate) =$3468. Ana. 
 
 17. A tax receiver charged 5% commission, and paid 
 $4845 net proceeds into the town treasury : what was the 
 amount collected? 
 
 NOTE. In this and similar examples, the pupil should observe 
 that the base or sum on which commission is to be computed is the 
 sum collected, and not the sum paid over. If it were the latter, the 
 agent would have to collect his own commission, at his own expense, 
 and his rate of commission would not be rSir. but T S. In the col- 
 lection of $100,000, this would cause an error of more than $350. 
 
 1 8. After retaining z\% for selling a consignment of 
 flour, my agent paid me $6664 : required the amount of 
 sales, and his commission. 
 
 348. How find the sales, etc., the net proceeds and the per cent commission 
 being given f
 
 COMMISSION AND BROKERAGE. 245 
 
 rg. After deducting \\% for brokerage, and $45.28 for 
 advertising a house, a broker sent the owner $15250: for 
 what did the house sell ? 
 
 20. An administrator of an estate paid the heirs $25686, 
 ""charging z\% commission, and $350 for other expenses: 
 
 what was the gross amount collected ? 
 
 349. To find the Sum Invested, the sum remitted and the 
 per cent commission being given. 
 
 21. A manufacturer sent his agent $3502 to invest i:i 
 wool, after deducting his commission of 3% : what sum di< i 
 he invest ? 
 
 ANALYSIS. The sum remitted $3502, includes both the sum in- 
 vested and the commission. But the sum invested is ioof c of itself, 
 andioo#+3# (the commission) - 103$. The question now is : 
 $3502 is 103% of what number? $35O2-^i.O3=$34oo, the sum in- 
 vested. (Art. 340, .) Hence, the 
 
 RULE. Divide the sum remitted by i plus the per cent 
 commission. (Prob. V, Per ct.) 
 
 NOTE. The learner will observe that the base in this and sim 
 ilar examples is the sum invested, and not the sum remitted. If it 
 were the latter, the agent would receive commission on his commis- 
 sion, which is manifestly unjust. 
 
 22. A teacher remitted to an agent $3131.18 to be laid 
 out in philosophical apparatus, after deducting 4^' com- 
 mission : how much did the agent lay out in appa- 
 ratus ? 
 
 23. If I remit my agent $2516 to purchase books, after 
 deducting 4$ commission, how much does he lay out in 
 books ? 
 
 24. Remitted $50000 to a broker to be invested in city 
 property, after deducting \\ c / r for his services: how much 
 did he invest, and what was his commission ? 
 
 340. How find the sum invested, the sum remitted and the per cent commis- 
 sion being given ?
 
 246 
 
 ACCOUNT OF SALES. 
 
 ACCOUNT OF SALES. 
 
 350. An Account of Sales is a written statement, 
 made by a commission merchant to a consignor, contain- 
 ing the prices of the goods sold, the expenses, and the net 
 proceeds. The usual form is the following : 
 
 Sales of Grain on acc't of E. D. BARKER, ESQ., Chicago. 
 
 DATE. 
 
 BUTEB. 
 
 DESCRIPTION. 
 
 BUSHELS. 
 
 PRICK. 
 
 EXTENSION. 
 
 IS?!. 
 
 April 3 
 
 J. Hoyt, 
 A. Woodruff, 
 Hecker & Co., 
 
 Winter wheat, 
 Spring " 
 Corn, 
 
 565 
 870 
 1610 
 
 @, $2.10 
 @ 1-95 
 @ 1.05 
 
 $1186.50 
 1696.50 
 1690.50 
 
 Gross amount, 
 
 Charges. 
 
 Freight on 3045 bu., at 20 cts., 
 Cartage " '' $15.30, 
 
 Storage " 38.75, 
 
 Commission, 2\ % , 
 
 $609.00 
 
 1530 
 
 38.75 
 
 "4-34 
 
 Net proceeds, 
 
 $4573.50 
 
 $777-39 
 $3796.11 
 
 YORK, July sth, 1871. 
 
 J. HENDERSON & Co. 
 
 Ex. 25. Make out an Account of Sales of the following: 
 
 James Penfield, of Philadelphia, sold on account of 
 J. Hamilton, of Cincinnati, 300 bbls. of pork to W. Gerard 
 & Co., at $27; 1150 hams, at $1.75, to J. Eamsey; 87*5 
 kegs of lard, each containing 50 lb., at 8 cts., to Henry 
 Parker, and 750 lb. of cheese, at 10 cts., to Thomas Young. 
 
 Paid freight, $65.30; cartage, $15.25 ; insurance, $6.45 ; 
 commission, at 2%. What were the net proceeds ? 
 
 26. Samuel Barret, of New Orleans, sold on account of 
 James Field, of St. Louis, 85 bales cotton, at $96.50; 
 63 barrels of sugar, at $48.25 ; 37 bis. molasses, at $35. 
 
 Paid freight, $45.50; insurance, $15; storage, $35.50; 
 >-\%- What were the net proceeds ? 
 
 350. What is an account of sales f
 
 PROFIT AND LOSS. 247 
 
 PROFIT AND LOSS. 
 
 351. Profit and Loss are the sums gained or lost in 
 business transactions. They are computed at a certain 
 per cent of the cost or sum invested, and the operations 
 are the same as those in Percentage and Commission. 
 
 The Cost or sum invested is the Base ; 
 
 The Per cent profit or loss, the Rate ; 
 
 The Profit or Loss, the Percentage ; 
 
 The Selling Price, that is, the cost plus or minus tho 
 profit or loss, the Amount. 
 
 352. To find the Profit or Loss, the Cost and the Per Cent 
 Profit or Loss being given. 
 
 Multiply the cost by the rate. (Problem 1, Per ct.) 
 
 NOTE. When the per cent is an aliquot part of 100, it is gener- 
 ally shorter, and therefore preferable to use the fraction. (Art. 336, n.) 
 
 1. A man paid $250 for a horse, and sold it at 15$ 
 profit: how much did he gain? Ans. $37.50. 
 
 2. A man paid $450 for a building lot, and sold at a 
 loss of n$: how much did he lose? Ans. $49.50. 
 
 3. Paid $185 for a buggy, and sold it 12% less than cost: 
 what was the loss ? 
 
 4. Paid $110 for a pair of oxen, and sold them at 20% 
 advance : what was the profit ? 
 
 5. A lad gave 87^ cts. for a knife, and sold it at 10% 
 below cost : how much did he lose ? 
 
 6. Bought a watch for $83^, and sold it at a loss of 20^: 
 what was the loss ? 
 
 7. Bought a pair of skates for $4.20, and sold them at 
 advance : required the gain ? 
 
 351. What are profit and loss? How reckoned? To what does the cost or 
 sum invested answer ? The per cent profit or loss ? The profit or loss* ? The 
 selling price? 352. How find the profit or loss, when the cost and per cent are 
 Note. When the per cent ie an aliquot part of 100, how proceed?
 
 248 PKOFIT AND LOSS. 
 
 353. To find the Selling Price, the Cost and Per Cent 
 Profit or Loss being given. 
 
 Multiply the cost by i plus or minus the per cent. (Prob. 
 II. Percentage.) 
 
 NOTE. When the cost and per cent profit or loss are given, the 
 telling price may also be found by first finding the profit or loss ; 
 then add it to or subtract it from the cost. (Art. 338.) 
 
 8. A man paid $300 for a house lot : for what must he 
 sell it to gain 20$! ? 
 
 ANALYSIS. To gain 20%, he must sell it for the cost plus 20 ( > r . 
 That is, selling pr.=f 300 (cost) x 1.20 (i + 20$)= $360. Ana. 
 
 9. A farmer paid $250 for a pair of oxen: for how 
 much must he sell them to lose 15$? 
 
 Selling pr.=$25o (cost) x .85 (1 15$) =$2 12.50. Am. 
 
 10. A and B commenced business with $2500 apiece. 
 A adds 11% to his capital during the first six months, and 
 B loses \i% of his: what amount is each then worth ? 
 
 n. A merchant paid $378 for a lot of silks, and sold 
 them at 20% profit : what did he get for the goods ? 
 
 12. If a man pays $2750 for a house, for how much 
 must he sell it to gain 1% ? 
 
 13. If a man starts in business with a capital of $8000, 
 and makes 19$ clear, how much will he have at the close 
 of the year ? 
 
 14. If a merchant pays 15 cts. a yard for muslin, how 
 must he sell it to lose 25%? 
 
 15. Bought gloves at $15 a dozen: how must I sell 
 them a pair, to lose 20%? 
 
 1 6. Paid $25 per dozen for pocket handkerchiefs: for 
 what must I sell them apiece to make 33^ per cent? 
 
 353. How find the selling price, when the cost and per cent profit or loss aw 
 (flven. Note. How else may the selling price be found ?
 
 PEOFIT AND LOSS. 249 
 
 17. Paid $196 for a piece of silk containing 50 yds.: 
 how must I sell it per yard to gain 25$ ? 
 
 18. Bought a house for $3850: how must I sell it to 
 make 12%%? 
 
 19. A speculator invested $14000 in flour, and sold at a 
 loss of 8-fyc : what did he receive for his flour ? 
 
 354. To find the Per Cent, the Cost and the Profit of 
 Loss being given. 
 
 Divide the profit or loss ly the cost. (Prob. Ill, Per ct.) 
 
 NOTE. When the cost and selling price are given, first find the 
 profit or loss, then the^er cent. (Art. 339.) 
 
 20. If I buy an acre of land for $320, and sell it for $80 
 more than it cost me, what is the per cent profit ? 
 
 SOLUTION. Per cent=$8o (gain)H-$320 (cost)=.25 or 25$. Ans. 
 
 21. A jockey paid $875 for a fast horse, and sold it so 
 as to lose $250 : what per cent was his loss ? 
 
 22. If I pay 22\ cts. a pound for lard, and sell it at 2\ 
 cts. advance, what per cent is the profit ? 
 
 23. If a newsboy pays 2\ cts. for papers, and sells them 
 at \\ cent advance, what per npnt is his profit? 
 
 24. If a speculator buys apples at $2.12^ a barrel, an$ 
 Bells them at $2.87^, what is his per cent profit? 
 
 ANALYSIS. $2.87^ 82.12^=8.75 profit per bl. Therefore, $.75 
 (gain)-5-$2.i2^ (co8t)=.35, 5 r or 35/V/6- (Art. 339, n.) 
 
 25. If I sell an article at double the cost, what per cent 
 is my gain ? 
 
 26. If I sell an article at half the cost, what per cent is 
 my loss ? 
 
 27. If I buy hats at $3, and sell at $5, what is the per 
 cent profit ? 
 
 28. If I buy hats at $5, and sell at $3, what is the per 
 cent loss ? 
 
 354. How find the per cent profit or loss, when the cost and profit or loes are
 
 250 PEOFIT AND LOSS. 
 
 29. If a man's debts are $3560, and he pays only $1780, 
 what per cent is the loss of his creditors ? 
 
 30. If | of an article be sold for its cost, what is the 
 per cent loss ? 
 
 ANALYSIS. If J are sold for its cost, $ must be sold for ^ of ^, 
 or the cost, and -J for or f the cost. Hence, the loss is ^ the 
 cost , and ^-*-$=.33^ or 
 
 31. If you sell \ of an article for | the cost of the whole, 
 what is the gain per cent ? 
 
 32. If I sell | of a barrel of flour for the cost of a barrel, 
 what is the per cent profit ? 
 
 33. If a milkman sells 3 quarts of milk for the price he 
 pays for a gallon, what per cent does he make ? 
 
 34. Bought 3 hhd. of molasses at 85 cts. per gallon, and 
 sold one hhd. at 75 cts., the other two at $i a gallon- 
 required the whole profit and the per cent profit? 
 
 ) 
 355. To find the Cost, the Profit or Loss and the Per Cent 
 
 Profit or Loss being given. 
 
 Divide the profit or loss ~by the given rate. (Problem IV, 
 Percentage.) 
 
 NOTE. When the per cent profit or loss is an aliquot part of 
 100, the operation may often be abbreviated by using this aliquot 
 part as the divisor. (Art. 341, n.) 
 
 35. A grocer sold a chest of tea at 25% profit, by which 
 he made $22^: what was the cost ? 
 
 SOLUTION. Cost=$22. 50 (profit)-*-. 25 (rate)=$9o. Ans. 
 Or, cost= $22.50-:-^= $90. Ans. (Art. 340, n.) 
 
 36. A speculator lost $1950 on a lot of flour, which was 
 20% of the cost: required the cost? 
 
 37. Lost 65 cents a yard on cloths, which was IT>% f 
 the cost: required the cost and selling price ? 
 
 ANALYSIS. The cost =65 cts. -H.I 3= $5 ; and $5 $.6s=$4.35 the 
 selling price. (Art. 340, n.) 
 
 355. How find the cost, when the profit or loss and the per cent profit or loss 
 are given,
 
 PEOFIT AND LOSS. 251 
 
 38. Gained $2^ per barrel on a cargo of flour, which 
 was 2^% : required the cost and selling price per barrel ? 
 
 AKALYSIS. 20^=4, and $2.50-7-^=112.50 cost, and $12.50+ 
 j>2.5o=$i5, selling price. 
 
 39. If I sell coffee at \o% profit I make 10 cts. a pound: 
 what was the cost ? 
 
 40. A man sold a house at a profit of 33$$, and thereby 
 gained $7500: required the cost, and selling price? 
 
 41. If I make 20% profit on goods, what sum must I lay 
 out to clear $3500, and what will my sales amount to ? 
 
 42. A and B each gained $1500, which was 12^% of A's 
 and 1 6% of B's stock : what was the investment of each ? 
 
 43. If a merchant sells goods at \Q% profit, what must 
 be the amount of his sales to clear $25000 ? 
 
 44. A market man makes \ a cent on every egg he 
 sells, which is 25% profit: what do they cost him, and 
 how sell them ? 
 
 356. To find the Cost, the Selling Price and the Per Cent 
 Profit or Loss being given. 
 
 Divide the selling price by i plus or minus the rate of 
 profit or loss, as the case may require. (Prob. V, Per ct.) 
 
 45. A jockey sold two horses for $168.75 eac ^ '> on one ne 
 made 1 2-^, on the other lost 1 2\% : what did each horse 
 cost him ? 
 
 SOLUTION. Cost of one=$i68.75 (sel. pr.)-=-i.i25 (i + rate)=$i5o. 
 Cost of others $168.75 (sel- pr.)-*-.875 (i rate)=$iq2.86. Ans. 
 
 46. By selling 568 bis. of beef at $15^ a barrel, a grocer 
 lost i2-|%: what was the cost? Ans. $10061. 71^. 
 
 47. Sold 5000 acres of land at $3^ an acre, and thereby 
 gained 22$: what was the cost? 
 
 48. Sold a case of linens for 27, ios., making a profit 
 of 25$: what was the cost? 
 
 3t6. How find the cost, when the sailing price and the per cent profit re given t
 
 252 PROFIT AND LOSS. 
 
 49. If a newsboy sells papers at 4 cts. apiece, he makes 
 ZZ\% ' what do they cost him ? 
 
 357. To Mark Goods so that a given Per Cent may be 
 deducted, and yet make a given Per Cent profit or loss. 
 
 50. Bought shoes, at $2.55 a pair: at what price must 
 they be marked that 1 5% may be deducted, and yet be 
 gold at 20% profit ? 
 
 ANALYSIS. The selling price is 120$ of $2.55 (the cost) ; $2.55 x 
 i.20=$3.o6, the price at which they are to be sold. But the marked 
 price is loofo of itself; and 100^ 15^=85^. The question novr 
 is, $3.06 are 85$ of what sum I If $3.06=-,^, Tiu=$3-o6H-85, or 
 $.036; and -H}X=$3.6o. Am. (Art. 340.) Hence, the 
 
 KULE. Find the selling price, and divide it by i minus 
 the given per cent to be deducted ; the quotient will be the 
 marked price. 
 
 51. Paid 56 cts. apiece for arithmetics: what must they 
 be marked in order to abate 5$, and yet make 25% ? % 
 
 52. If mantillas cost $24 apiece, at what price must they 
 be marked that, deducting 8$, the merchant may realize 
 ZZ\% profit? 
 
 53. When apples cost $3.60 a barrel, what must be the 
 asking price that, if an abatement of 12-^ is made, there 
 mil still be a profit of i6|^? 
 
 54. A merchant paid 87.^ cts. a yard for a case of linen, 
 which proved to be slightly damaged : how must he mark 
 it that lie may fall 25^, and yet sell at cost? 
 
 55. A goldsmith bought a case of watches at $60: how 
 must he mark them that, abating 4^', he may make 20', ? 
 
 56. If cloths cost a tailor $4.50 a yard, at what price 
 must he mark them, that deducting \Q% he will make 
 15 per cent profit? 
 
 357. When the selling price and the per cent profit or loss are given, how find 
 the per cent profit or loss at any proposed price ? How find what to mark goods. 
 that a given per cent may be deducted, and yet a given per cent profit or loss be
 
 PBOFIT AND LOSS. ^ 253 
 
 QUESTIONS FOR REVIEW. <\'j 
 
 1. A grocer paid 23 cts. a Ib. for a tub of butter weighing ^ 
 65 Ibs., and sold it at 18% profit: how much did he make?^ 
 
 2. Bought a house for $3865, and paid $1583.62 for 
 pairs: how much must I sell it for to mate i6f%?C 
 
 3. Paid $2847 for a case of shawls, and $956 for a case 
 of ginghams; sold the former at 22%% advance, and the 
 latter at i \% loss : what was received for both tCil/l - ^ 
 
 4. A jockey bought a horse for $125, Avhich he traded 
 for another, receiving $37 to. boot; he sold the latter at 
 25% less than it cost: what sum did he lose? G^ 
 
 5. What will 750 shares of bank stock cost, if the bro- 
 kerage is \% and the stock 3% above par ? o i "V l/l 
 
 6. Sold 3 tons of iron, at 15^ cts. a pound, and charged .* 
 $\% commission : what were the net proceeds ' . . 4U 
 
 7. Bought wood 'at $4-J a cord, and sold it for 1 6 1 : re- 
 quired the per cent profit ? 1 
 
 ~"S7 A shopkeeper buys thread at 4 cts. a spool, aivd sells 
 at 6% cts.: Avhat per cent is his profit and how much 
 would he make on 1000 gross? C> "* 
 
 9. Bought 1650 tons of ice, at $12 ; one half melted, and 
 sold the rest at $i a hund. : what per cent was the loss?Of- .:, 
 
 10. The gross proceeds of a consignment of apples was 
 $1863.75, and the agent deducted $96.9 1 for selling: re- 
 quired the per cent commission and the net proceeds? 
 
 11. A lady sold her piano for f of its cost: what per 
 cent was the loss ? 
 
 12. If I pay $2 for 3 Ibs. of tea, and sell 2 Ibs. for $3, 
 what is the per cent profit? 
 
 13. A man sold his house at 20$ above cost, and therely 
 made $1860: required the cost and the selling price.? 
 
 14. A miller sells flour at 15$ more than cost, and 
 makes $1.05 a bar.: what is the cost and selling price ? 
 
 15. Lost 25 cts. a pound on indigo, which was \2\% of 
 the cost : required the cost and the selling price.
 
 254 PKOFIT AND LOSS. 
 
 1 6. A commission merchant received $260 for selling 
 a quantity of provisions, which was 5%: required the 
 amount of sales and the net proceeds. 
 /- 1 7. A broker who charges \%, received $60 for selling a 
 quantity of uncurrent money : how much did he sell ? 
 
 18. A grocer makes $1.25 a pound on nutmegs, which 
 is 100% profit : what does he pay for them ? 
 
 19. A merchant sold a bill of white goods for $7000, 
 and made zzk% ' required the cost and the sum gained. 
 
 20. Sold a hogshead of oil at 93^ cts. a gallon, and 
 made i8f$: required the cost and the profit. 
 
 21. A man, by selling flour at $12^ a barrel, makes 25%: 
 what does the flour cost him ? 
 
 22. Made 12^ on dry goods, and the amount of sales 
 was $57725 : required the cost and the sum made. 
 
 23. Sold a quantity of metals, and retaining $\% com- 
 mission, sent the consignor $15246: required the amount 
 of the sale and the commission. 
 
 24. Sold 250 tons of coal at $6|, and made 12^' : what 
 per cent would have been my profit had I sold it for 
 $8 a ton ? 
 
 25. A merchant sold out for $18560, and made 15% on 
 his goods : what per cent would he have gained or lost by 
 selling for $15225 ? 
 
 26. Paid $40 apiece for stoves : what must I ask that I 
 may take oif 20%, and yet make 20$ on the cost ? 
 
 27. If a bookseller marks his goods at 25$ above cost, 
 and then abates 25%, what per cent does he make or lose-3 
 
 28. A grocer sells sugar at 2\ cts. a pound more than 
 oost, and makes 20% pro6t : required the selling price. 
 
 29. A man bought 2500 bu. wheat, at 3>if ; 3200 bu. corn, 
 at 87^ cts. ; 4000 bu. oats, at 25 cts.; and paid $450 freight: 
 he sold the wheat at 5$ profit, the corn at 11% loss, and 
 the oats at cost ; the commission on sales was 5% : what 
 was his per cent profit or loss ?
 
 .. 29- 
 
 INTEREST. 
 
 358. Interest is a compensation for the use of money. 
 It is computed at a certain per cent, per annum, and em- 
 braces five elements or parts: the Principal, the Rate, the 
 Interest, the Time, and the Amount. 
 
 359. The Principal is the money lent. 
 The JZate is the per cent per annum. 
 The Interest is the percentage. 
 
 The Time is the period for which the principal draws 
 interest 
 The Amount is the sum of the principal and interest. 
 
 NOTES. i. The term per annum, from the Latin per and annus, 
 signifies by the year. 
 
 2. Interest differs from the preceding applications of Percentage 
 only by introducing time as an element iu connection with per cent. 
 The terms rate and rate per cent always mean a certain number of 
 hundredths yearly, and pro rata for longer or shorter periods. 
 
 360. Interest is distinguished as Simple and Compound. 
 Simple Interest is that which arises from the 
 
 principal only. 
 
 Compound Interest is that which arises both from 
 the principal and the interest itself, after it becomes due. 
 
 361. The Legal Kate of interest is the rate allowed 
 by law. Rates higher than the legal rate, are called usury. 
 
 NOTES. i. In Louisiana the legal rate is - - - $%. 
 In the N. E. States, N. C., Penn., Del., Md., Va., W. Va., 
 Tenn., Ky., 0., Mo., Miss., Ark., Flor, la., 111., Ind., the 
 
 Dist. of Columbia, and debts due the United States, - b%. 
 
 In N. Y ., N. J., S. C., Ga., Mich., Min., and Wis., - - 1%, 
 
 In Alabama and Texas, 8$, 
 
 In Col , Kan., Neb., Nev., Or., Cal. and Washington Ter., - io$>. 
 
 l. In some States the law allows higher rates by special agreement. 
 
 3. When no rate is specified, it is understood to be the It gal rate 
 
 358. What is interest ? 359. The principal ? The rate ? The int. T The time I 
 Fbe aint. ? 360. What is simple interest ? Compound ?
 
 256 INTEREST. 
 
 THE SIX PER CENT METHOD. 
 
 362. Since the interest of $i at 6 per cent for 1 2 months 
 or i year, is 6 cents, for i month it is i-tivelfth of 6 cents 
 or \ cent ; for z months it is 2 halves or i cent; for 3 mos. 
 i^ cent, for 4 mos. 2 cents, etc. That is, 
 
 The interest of ^i, at 6 per cent, for any number of 
 months, is half as many cents as months. 
 
 363. Since the interest of $i at 6 per cent for 30 days 
 or i mo., is 5 mills or \ cent, for i day it is -^ of 5 mills 
 or mill ; for 6 days it is 6 times or i mill ; for 1 2 days, 
 2 mills ; for 15 days, z\ mills, etc. That is, 
 
 The interest of 81, at 6 per cent, for any number of days, 
 is i-sixth as many mills as days. Hence, 
 
 364. To find the Interest of $1, at 6 pep cent, for any 
 given time. 
 
 Take half the number of months for cents, and one sixth of 
 the days for mills. Their sum will be the interest required. 
 NOTES. i. When the rate is greater or less than 6 per cent, the 
 interest of $i for the time is equal to the interest at 6 per cent 
 increased or diminisJicd by such a part of itself as the given rate 
 exceeds or falls short of 6 per cent. Thus, if the rate is 7%, add ^ to 
 the int. at 6;? ; if the rate is 5$, subtract , etc. 
 
 2. In finding i-sixth of the days, it is sufficient for ordinary pur. 
 poses to carry the quotient to tenths or hundredths of a mill. 
 
 3. When entire accuracy is required, the remainder should be 
 placed over the 6, and annexed to the multiplier. 
 
 1. Int. of $i, at 1% for 9 m. 18 d. ? Ans. $.056. 
 
 2. Int. of $i, at 5$ for n m. 21 d. ? AH*. 8.04875. 
 
 3. What is the int. of $i, at 6% for 7 m. 3 d. ? 
 
 4. What is the int. of $i, at 6% for n m. 13 d. ? 
 
 Note. Wherein does interest differ from the preceding applications of per- 
 centage? 361. The legal rate? What is usury ? 362. What is the interest of $i 
 sit 6 per cent for months? 363. For daye? 364. How find the int. of $i for any 
 [riven time ? Note. When the rate is greater or less thau 6 per cent, to what is 
 the interest of $i eqnal ?
 
 INTEREST. 257 
 
 REM. The relation between the principal, the interest, the rate, 
 the time, and the amount, is such, that when any three of them are 
 given, the others can be found. The most important of these 
 problems are the following : 
 
 PROBLEM I. 
 
 365. To find the Interest, the Principal, the Rate, and 
 Time being given. 
 
 1. What is the interest of $150.25 for i year 3 months 
 and 1 8 days, at 6%? 
 
 ANALYSIS. The interest of $i for 15 m.=.o75 OPERATION. 
 
 iSd. =.003 Prin. $150.25 
 
 " i y. 3 m. 18 d. =To7lT .078 
 
 Now as the int. of $i for the given time and 120200 
 
 rate is $.078 or .078 times the principal, the int. 105175 
 
 of $150.25 must be .078 times that sum; and $ii.7iqco 
 $150.25 x .078 = $!!. 71950. Hence, the 
 
 RULE. Multiply the principal by the interest of $i for 
 the time, expressed decimally. (Art. 336.) 
 
 For the amount, add the interest to the principal, 
 
 REM. i. The amount may also be found by multiplying the 
 principal by i plus the interest of $i for the time. (Art. 337.) 
 
 2. When the rate is greater or less than 6%, it is generally best to 
 find the interest of the principal at 6% for the given time ; then add 
 to or subtract from it such a part of itself, as the given rate exceeds 
 or falls short of 6 per cent. (Art. 364.) 
 
 3. In finding the time, first determine the number of entire calendar 
 months ; then the number of days left. 
 
 4. In computing interest, if the mills are 5 or more, it is customary 
 to add i to the cents ; if less than 5, they are disregarded. 
 
 Only three decimals are retained in the following Answers, and 
 each is found by the rule under which the Ex. is placed. 
 
 2. What is the amt. of $150.60 for i y. 5 ra. 15 d.at 6% .' 
 ANALYSIS. Int. of Si for i y. 5 mos. or 17 mos., equals .085 ; 15 d. 
 
 it equals .0025 ; and .085 + .0025 = . 0875 the multiplier. Now $150.60 
 x.o875=$i3. 1775, int. Finally, $i5O.6o + $i3.i775=$i63. 7775, Ans. 
 
 ^5. How find the interest, when the principal, rate, and time are given? 
 How find the amount, when the principal and interest are given f Hem, How 
 else is the amount found ':
 
 258 INTEBEST. 
 
 3. Find the interest of $31.75 for r yr. 4 mos. at 6f c . 
 
 4. What is the ink of $49.30 for 6 mos. 24 d. at 6% ? 
 
 5. What is the int. of $51.19 for 4 mos. 3 d. at yfa ? 
 
 6. What is the int. of $142.83 for 7 mos. 18 d. at 5%? . 
 
 7. What is the int. of $741.13 for n mos. 21 d. at 6$? 
 
 8. What is the int. of $968.84 for i yr. lomos. 26d.-at6$? 
 
 9. What is the int. of $639 for 8 mos. 29 d. at 7%? 
 co. What is the int. of $741:13 for 7 mos. 17 d. at 
 
 11. What is the int. of $1237.63 for 3 mos. 3<i. at 8 
 
 12. What is the int. of $2046^ for 13 mos. 25 d. at 4 
 
 13. What is the int. of $3256.07 for i rn. and 3 d. at 
 
 14. Find the amount of $630.37^ for 9 mos. 15 d. at i 
 
 15. Find the amount of $75.45 for 13 mos. 19 d. at 
 T 6. Find the amount of $2831.20 for 2 mos. 3 d. at 
 
 17. Find the amount of $356.81 for 3 m. n d. at 
 
 18. Find the amount of $2700 for 4 mos. 3 d. at 
 
 19. Eequired the amount of $5000 for 33 days at 1 
 
 20. Eequired the int. of $12720 for 2 mos. 17 d. at 
 
 21. What is the amt. of $221.42 for 4 mos. 23 d. at 
 
 22. What is the int. of $563. 16 for 4 mos. at 2% a month ? 
 SUGGESTION. At 2 % a month, the int. of $i for 4 mos. is $.08. 
 
 23. What is the int. of $7216.31 for 3 mos. at \% a month? 
 
 24. Find the int. of $9864 for 2 mos. at 2\% a month ? 
 
 25. Find the amt. of $3540 for 17 mos. 10 d. at l\%t 
 
 26. What is the interest on $650 from April i7th, 1870, 
 to Feb. 8th, 1871, at 6^? 
 
 ANALYSIS. 1871 y. 2 m. 8 d. \ Int. $i for 9 m.=.o45 $650 
 
 1870 y. 4 m. 17 d. v " " 21 d. =.0035 .0485 
 
 Time, o y. g m. 21 d. \ Multiplier, .0485 $31-525 
 
 27. What is the interest on $1145 from July 4th, 1867, 
 to Oct. 3d, 1868, at -]%. 
 
 28. What is the interest on a note of $568.45 from May 
 2ist, 1861, to March 25th, 1862, at 5$? 
 
 29. Eequired the amount of $2576.81 from Jan. 
 1871, to Dec. 1 8th, 1871, at ^%.
 
 INTEREST. 259 
 
 METHOD BY ALIQUOT PARTS. 
 
 366. To find the Interest by Aliquot Parts, the Principal, 
 Rate, and Time being given. 
 
 i. What is the interest of $137 for 3 y. i m. 6 d. at 5^? 
 
 ANALYSIS. As fhe rate is 5 % , the interest for $ 1 3 7 prill . 
 
 i year is .05 (-[fh;) of the principal, ... .05 rate. 
 
 Now $137 x. 05 =$6.85, 1 2)16.85 int. i y. 
 
 Again, the int. for 3 y. is 3 times as much as for i y., 3 .7- 
 
 And $6.85 (int. i y.) x 3=120.55, - - - 20.55 int. 3 y. 
 The int. for i m. is -fa of int. for i yr. ; and 
 
 $6.85-t-i2=. 57 , 5) -57 " im. 
 The int. for 6 d. is i of int. for i m.; and 
 
 $-57--5=-4. -"4 " 6 d. 
 
 Adding these partial interests together, we have $21.234, Ans. 
 
 the answer required. Hence, the 
 
 KULE. For one year. Multiply the principal by the 
 rate, expressed decimally. 
 
 For two or more years. Multiply the interest for i year 
 by the number of years. 
 
 For months. Take the aliquot part of i year's interest. 
 
 For days. Take the aliquot part of i month's interest. 
 
 That is, for i m., take T T j of the int. for i y. ; for 
 i m., ; for 3 m., J, etc. 
 
 For i d., take ^ of the int. for i m. ; for 2 d., T x ^ ; for 
 6 d., ; for 10 d., , etc. 
 
 NOTE. This method has the advantage of directness for different 
 rates ; but in. practice, the preceding method is generally shorter 
 and more expeditious. 
 
 2. What is the int. of $143.21 for 2 y. 5 m. 8 d. at 
 
 3. What is the int. of $76.10 for i y. 3 m. 5 d. at 
 
 366. How find the interest for i year, at any given rate, by aliquot parts? 
 How for 2 or more years ? For months ? For days ? What part for i month * 
 Cor 3 mos.? For 6 mos.? For i day ? For 5 d.? Foriod,? For 15 d.? Forzod.T
 
 260 INTEREST. 
 
 4. What is the int. of $95.31 for 8 m. 20 d. at 
 
 5. What is the int. of $110.43 frr l J r - 6 m - I0 
 
 6. What is the int. of $258 for 3 yrs. 7 m at S 
 
 7. What is the interest of $205.38 for 5 yrs. at 
 
 8. Find the interest of $361.17 for n months at &%? 
 9. Find the interest of $416.84 for 19 days at 7$$? 
 
 10. At i\%, what is the int. of $385.20 for i yr. 13 d. 
 
 1 1. At 5^%, what is the int. of $1000 for" i y. i m. 3 d. ? 
 
 12. At Sf c , what is the int. of $1525.75 for 3 months? 
 
 13. Kequired the interest of $12254 for 2\ years at 8%? 
 
 14. What is the amount of $20165 f r 5 m - J 7 d. at 7$? 
 
 METHOD BY DAYS. 
 
 367. To find the Interest by Days, the Principal, Rate, 
 and Time being given. 
 
 i. What is the interest of $350 for 78 days at 6%? 
 
 ANALYSIS. The interest of $i at 6$ for i day $350 
 
 is of a mill or 5( f,,r, dollar. (Art. 363.) There- 78 
 
 fore for 78 days it must be 78 times gB L no = BoTf ^2800 
 
 dol., or sJfiu times the principal. Again, since 
 the int. of $i for 78 days is ^80 times the princi- 
 
 pal, the interest of $350 for the same time and 6{ooo)$27j3OO 
 rate must be sftijij times that sum. In the opera- Ans. $4.55 
 tion, we multiply the principal by 78, the numera- 
 tor, and divide by the denominator 6000. Hence, the 
 
 EULE. Multiply the principal by the number of days, and 
 divide by 6000. TJie quotient will be the interest at 6%. 
 
 For other rates, add to or subtract from the interest at 
 6 per cent, such a part of itself as the required rate is 
 greater or less than 6f c . (Art. 364.) 
 
 NOTE. This rule, though not strictly accurate, is generally used 
 by private bankers and money-dealers. It is based upon the suppo- 
 sition that 360 days are a year, which is an error of five days or Va 
 of a year. Hence, the result is -fa too large. When entire accuracy 
 is required, the result must be diminished by -fa part of itself. 
 
 367. How compute interest by days ? Note. Upon what is this method founded T
 
 INTEBEST. 261 
 
 2. A note for $720 was dated April 1 7th, 1871 : what was 
 the interest on it the i6th of the following July at 1% ? 
 Omitting the day of the first date. (Art. 339, n.) . $720 prin. 
 April has 30 days 17 d.= 13 d. 90 days. 
 
 May " 31 d. 6|ooo)64|8oo 
 
 June " 30 d. 6) 1 0.80 int. 6%. 
 
 July " i6d. i. 80 1%. 
 
 Time " =90 d. Ans. $12.60 int. 1%. 
 
 3. What is the interest of $517 for 33 days at 6%? 
 
 4. What is the interest of $208.75 f r 63 days at 6%? 
 
 5. What is the interest of $631.15 for 93 days at 7$? 
 
 6. Find the interest of $1000 for 100 days at 5%? 
 
 7". Find the amount of $1260.13 for 120 days at 6%? 
 
 8. Required the interest of $3568.17 for 20 days at 
 
 9. Required the amount of $4360.50 for 3 days at , ,_ 
 
 10. Find the interest of $5000 from May 2ist to the / $ /d* 
 5th of Oct. following at 6% ? 
 
 n. Find the interest of $6523 from Aug. i2th to the 
 5th of Jan. following at 7%? 
 
 12. Find the interest of $7510 from Jan. 5th to the 
 loth of the following March, being leap year, at 6% ? 
 
 PROBLEM II. 
 
 368. To find the Rate, the Principal, the Interest, and the 
 Time being given. 
 
 i. A man lent his neighbor $360 for 2 yrs. 3 mos., and 
 received $48.60 interest : what was the rate of interest ? 
 
 ANALYSIS. The int. of $360 for $360 x .01 =$3.60 int. i y. 
 i year at i % is $3.60; and for 2 } y. 3.60 x 2\ = $8.10 " 2\ y. 
 $8.10. Now, if $8.10 is \% of the $8.io)$48.6o 
 principal, $48 60 must be as many luTTT'-npr of 
 
 , -ii /fco. \J pel \>u. 
 
 per cent as $8.10 are contained 
 
 times in $48.60, which is 6. Therefore the rate was b%. Hence, the 
 
 RULE. Divide the given interest by the interest of the 
 principal for the tim,e, at i per cent. 
 
 368. How find the rate, when the principal, rate, and time are given f
 
 262 INTEREST. 
 
 NOTE. Sometimes the amount is mentioned instead of the prin. 
 cipal, or the intercut. In either case, the principal and interest may 
 be said to be given. For, the amt.=the prin. + int. ; hence, amt. 
 int.=the prin. ; and amt. prin.=the int. (Arts. 365, 101, Def. 17.) 
 
 2. If $600 yield $10.50 interest in 3 months, what is the 
 rate per cent ? 
 
 3. At what rate will $1500 pay me $52.50 interest 
 semi-annually ? 
 
 4. At what rate will $1000 amount to $1200 in 3 y. 4 m.? 
 
 5. A lad at the age of 14 received a legacy of $5000, 
 which at 21 amounted to $7800 : what was the rate of int. ? 
 
 6. A man paid $9600 for a house, and rented it for $870 
 a year: what rate of interest did he receive for his money ? 
 
 7. At what rate of int. will $500 double itself in 12 years ? 
 
 8. At what rate will $1000 double itself in 20 y. ? In 10 y. ? 
 
 9. At what rate will $1250 double itself in 14! years? 
 10. At what rate must $3000 be put to double itself in 
 
 1 6f years ? 
 
 PROBLEM III. 
 
 369. To find the Time, the Principal, the Interest, and the 
 Rate being given. 
 
 i. Loaned a friend $250 at 6%, and received $45 interest: 
 how long did he have the money ? 
 
 ANALYSIS. The int. of $250 at $250 x .o6 = $i5 int. for i y. 
 6% for i yr. is $15. Now if $15 $I5)$45 int. 
 
 int require the given principal ^wsTT^years. 
 
 i yr. at 6#, to earn $45 int., the 
 
 same principal will be required as many years as $15 are contained 
 times in $45; and $45-i-$i5=3. He therefore had the money 3 
 years. Hence, the 
 
 RULE. Divide tlie given interest by the interest of the 
 principal for i year, at the given rate. 
 
 Note. How find the principal, when the amount and interest are given ? How 
 find the interest, when the amount and principal are given ? 369. How fad thw 
 time, when the principal, interest, and rate are given ?
 
 INTEREST. 263 
 
 NOTES. i. If the quotient contains decimals, reduce them to 
 months and days. (Art. 294.) 
 
 2. If the amount is given instead of the principal or the interest, 
 find the part omitted, and proceed as above. (Art. 368. n.) 
 
 2. In what time will $860 amount to $989 at 6% ? 
 ANALYSIS. The amount $989 $86o=$i29.oo the int., and the 
 
 nt. of $860 for i year at 6f c is $51.60. Now $i29H-$5i.6o=:2.5, or 
 iV years. 
 
 3. In what time will $1250 yield $500, at 7% ? 
 
 4. In what time will $2200 yield $100, at 6% ? 
 
 5. In what time will $10000 yield $200, at 8$? 
 
 6. In what time will $700 double itself, at 6% ? 
 SOLUTION. The int. of $700 for i year at 6$, is $42 ; and $700-*- 
 
 $42 =i6| years. Ans. 
 
 7. How long must $i 200 be loaned at 7% to double itself? 
 
 8. In what time will $7500 amount to $15000, at 6$? 
 9. In what time will $10000 amount to $25000, at 8$? 
 
 PROBLEM IV. 
 
 370. To find the Principal, the Interest, the Rate, and the 
 Time being given. 
 
 i. What principal, at 6%, will produce $60 int. in 2\ yrs. ? 
 ANALYSIS. 2-J- y. 30 mos. ; therefore the 
 int. of $i for the given time at 6</c is 15 cents. 2\ v. = 3o m. 
 Now as $.15 is the int. of $i for the given \~ ~p 4j & JK 
 time and rate, $60 must be the int. of as many 
 
 dollars as $.15 are contained times in $60 ; 
 
 and $60 -H$. 1 5 = $400, the principal required. Am. $400 prin. 
 
 Hence, the 
 
 . Divide the ffiren inferfist by the interest of $* for 
 fie ffiren time and rate, expressed decimally. 
 
 2. At 6#, what principal will yield $100 in i year? 
 
 3. At 7$, what principal will yield $105 in 6 months? 
 
 370. How find the principal, when the interest, rate, and time are given ?
 
 64 INTEEEST. 
 
 4. What sum, at 5%, will gain $175 in i year 6 months? 
 
 5. What sum must be invested at 6% to pay the ground 
 rent of a house which is $150 per annum? 
 
 6. A gentleman wished to found a professorship with 
 annual income of $2800: what sum at 1% will produce it? 
 
 7. A man invested his money in 6% Government stocks, 
 and received $300 semi-annually : what was the sum 
 invested ? 
 
 8. A man bequeathed his wife $1500 a year: what sum 
 must he invest in 6% stocks, to produce this annuity ? 
 
 PROBLEM V. 
 
 371. To find the Principal, the Amount, the Rate, and the 
 Time being given. 
 
 1. What principal will amount to $508.20 in 3 y. at 6%: 
 
 ANALYSIS. 3 y.=3& m. ; therefore the 
 
 int. of $i for the given time at 6% is 18 cts., 3 years=36 m. 
 
 and the amt.=$i.i8. (Art. 365.) Now as j n t. $ I= <$. X 8 
 
 1.18 is the amt. of $i principal for the given amt $ I = l> I 1 8 
 
 time and rate, $508.20 must be the amount 
 
 of as many dollars principal as $1.18 are con- * * J j^5 - 2O 
 
 tained times in $508.20 ; and $5o8.2O-r-$i.i8 Ans. $430.68 
 
 =$430.68, the principal required. Hence, the 
 
 EULE. Divide the given amount by the amount of $i for 
 the given time and rate, expressed decimally. 
 
 2. What principal, at 6%, will amount to $250 in i year? 
 
 3. What principal, at 7$, will amount to $356 in 
 i year 3 months ? 
 
 4. What sum must a father invest at 6, for a son 19 
 years old, that he may have $10000 when he is 21 ? 
 
 5. What sum loaned at i% a month will amount to 
 $1000 in i year? 
 
 6. What sum loaned at 2% a month will amount to 
 $6252 in 6 mos. ? 
 
 371. How find the principal, when the amount, rate, and time are pivu f
 
 PROMISSORY iNOTES. 
 
 372. A Promissory Note is a written promise to 
 pay a certain sum at a specified time, or on demand. 
 
 NOTE. A Note should always contain the words " value re- 
 ceived ; ' otherwise, the holder may be obliged to prove it was 
 given foi a consideration, in order to collect it. 
 
 373. The person who signs a note is called the maker or drawer* 
 the person to whom it is made payable, the payee; and the person 
 who has possession of it, the holder. 
 
 374. A. Joint Note is one signed by'two-or more persons. 
 
 375. The Face of a Note is the sum whose payment is 
 promised. This sum should be written in wards in the body of thfa 
 note, and in figure* at the top or bottom. 
 
 376. When a note is to draw interest from its date, it should 
 contain the words " with interest ;" otherwise no interest can be 
 collected. For the same reason, when it is to draw interest from a 
 particular time after dtite, that fact should be specified in the note. 
 
 All notes are entitled to legal interest after they become due, 
 whether they draw it before, or not. 
 
 377. Promissory Notes are of two kinds ; negotiable, and non- 
 negotiable. 
 
 378. A Negotiable Note is a note drawn for the payment of 
 money to " order or bearer," without any conditions. 
 
 A Non-Negotiable Note is one which is not made payable to 
 " order or bearer," or is not payable in money. 
 
 NOTES. i. A note payable to A. B., or " order," is transferable by 
 indorsement; if to A. B, or "bearer," it is transferable by delivery. 
 Treasury notes and bank bills belong to this class. 
 
 2. If the words " order" and " bearer" are both omitted, the note 
 can be collected only by the party named in it. 
 
 379. An Indorser is a person who writes his name upon the 
 back of a note, as security for its payment. 
 
 380. The jHaturit}/ of a Note is the day it becomes legally 
 due. In most of the United States a note does not become legally due 
 
 372. What ia a promissory note ? Whnt T>nrtionlnr wnrrls phonlrt n note rnn- 
 Jaln ? 373. What is the one who signs it called ? The one to whom it is payable ? 
 The oce who has it? 374. A joint note? 375. The face of a note? 377. Of 
 how many kinds are notes ? 378. A negotiable note ? A non-negotiable note ? 
 Kate. How is the former transferable? Why is the latter not negotiable r 
 3; j. What is an indorscr ? 380. The maturity of a note ? 
 
 12
 
 266 INTEREST. 
 
 until three days after the time specified. These three days are called 
 days of grace. Hence, a note matures on the last day ot grace. 
 
 38X. When a note is given for any number of months, calendar 
 'months are always to be understood. 
 
 382. If a note is payable on demand, it is legally due as soon as 
 presented. If no time i& specified for the payment, it is understood 
 to be on demand. 
 
 383. A Protest is a written declaration made by a notary 
 public, that a note has been duly presented to the maker, and has 
 not been paid. 
 
 NOTE. A protest must be made out the day the note or draft 
 matures, and sent to the indorser immediately, to hold him responsible. 
 
 ANNUAL INTEREST. 
 
 384. When notes are made " with interest payable annually," 
 some States allow simple legal interest on each year's interest from 
 the time it becomes due to the time of final settlement. 
 
 385. To compute Annual Interest, the Principal, Rate, 
 and Time being given. 
 
 1. "What is the amount due on a note of 8500, at 6%,- 
 in 3 years with interest payable annually ? 
 
 SOLUTION. Principal, 500.00 
 
 Int. for i y. is $30 : for 3 y. it is $30 x 3, or 90.00 
 
 Int. on ist annual int. for 2 y. is 3-6o 
 
 2d " " " xy. is i. 80 
 
 The Amt. is $595 .40. Hence, the Ans. $595.40 
 
 RULE. Find the interest on the principal for the given 
 
 time and rate; also find the simple legal iniersst on each 
 
 year's interest for the time it has remained unpaid. 
 
 Tlie sum of the principal and its interest, with the 
 interest on the unpaid interests, will be the amount. 
 
 2. What is the amount of a note of $1000 payable in 
 4 years, with interest annually, at 7$? Ans. $1309.40 
 
 382. Wlien is a note on demand due? 383. What ia a protest? 385. How 
 compute annual interest?
 
 PARTIAL PAYMENTS. 
 
 386. Partial Payments are parts of a debt paid 
 at different times. The sums paid, with the date, are 
 usually written on the back of the note or other obliga- 
 tion, and are thence called indorsements. 
 
 UNITED STATES RULE. 
 
 387. To compute Interest on Notes and Bonds, when Partial 
 Payments have been made. 
 
 1. Find the amount of the principal to the time of the 
 first payment. If the payment equals or exceeds the interest, 
 subtract it from this amount, and, considering the remainder 
 a new principal, proceed as before. 
 
 II. If the payment is less than the interest, find the 
 amount of the same principal to the next payment, or to 
 the period when the sum of the payments equals or exceeds 
 the interest then due, and subtract the sum of the payments 
 from this amount. 
 
 Proceed in this manner with the balance to the time 
 of settlement. 
 
 NOTES. i. Tliis method was early inaugurated by the Supreme 
 Court of the United States, and is adopted by New York, Massa. 
 chusetts, and most of the States of the Union. CJiancellor Kent. 
 
 2. The following examples show the common forms of promissory 
 nr tes. The first is negotiable by indorsement; the second by transfer; 
 the third is a. joint note, but not negotiable. 
 
 $750. 
 
 WASHINGTON, D. C., Jan. "jth, 1870. 
 
 i. Four months after date, I promise to pay to the ordei 
 of George Green, seven hundred and fifty dollars, with 
 interest at 6^, for value received. 
 
 HENEY 
 
 386. What arc partial payments ? 387. What is the United States rule ?
 
 268 INTEREST. 
 
 On this note the following payments were indorsed. 
 
 June loth, 1870, $43. Feb. 1 7th, 1871, $15.45. Nov. 23d, 
 1871, $78.60. "What was due Aug. 25th, 1872 ? 
 
 OPERATION. 
 
 Principal, dated Jan. 7th, 1870, $750.00 
 
 Int. to ist payt. June loth, 1870 (5 m. 3 d.) (Art. 365), I 9- I 3 
 
 Amount, = 769.13 
 
 ist payment June loth, 1870, 43-OO 
 
 Remainder or new principal, =726.13 
 
 Int. from ist payt. to Feb. I7th, 1871 (8 m. 7 d.), 29.89 
 2! payt. less than int. due, $15.45 
 
 Int. on same prin. to Nov. 23d, 1871 (9 m. 6 d.), 33-4 
 
 Amount, = 789.42 
 
 3d payt. to be added to 2d, 678.60 = 94-5 
 
 Remainder or new principal, = 695.37 
 
 Int. to Aug. 25th, 1872 (9 m. 2 d.), 3 I -5 2 
 
 Balance due Aug. 25th, 1872, = $726.89 
 
 $1500. 
 
 NEW ORLEANS, July ist, 1869. 
 
 2. Two years after date, we promise to pay to James 
 Underbill or bearer, fifteen hundred dollars, with interest 
 at 7$, value received. 
 
 G. H. DENNIS & Co. 
 
 Indorsements: Received, Jan. 5th, 1870, $68.50. Aug. 
 8th, 1870, $20.10. Feb. nth, 1871, $100. How much 
 was due at its maturity ? 
 
 $930. 
 
 ST. Louis, March $t7i, 1860. 
 
 3. On demand, we jointly and severally promise to pay 
 J. C. Williams, nine hundred and thirty dollars, with 
 interest at 8$, value received. THOMAS BENTON. 
 
 HENKY VALENTINE. 
 
 Indorsements: Received Oct. loth, 1860, $20. Nov. 
 t6th, 1861, $250.13. June 2oth, 1862, $310: what was 
 iuc Jan. 3oth, 1863?
 
 PARTIAL PAYMEXT3. 269 
 
 MERCANTILE METHOD. 
 
 Sf f . When notes and interest accounts payable within a 
 year, receive partial payments, business men commonly 
 employ the following method : 
 
 Find the amount of the whole debt to the time of settle- 
 ment; also find the amount of each payment from the time 
 it was made to the time of settlement. 
 
 Subtract the amount of the payments from the amount of 
 .the debt; the remainder ivill be the balance due. 
 
 4. A debt of $720.75 was duo March isth, 1870, on 
 which the following payments were made : April 3d, $170 ; 
 Mtiy 2oth, $245.30; June 17^,887.50. How much was 
 due at 6%, Sept. 5th, 1870 ? 
 
 Principal dated March isth, 1870, $720.75 
 
 Int. to settlement (174 d.)=$72o.75 x .029, (Art. 367.) * 20.90 
 
 Amount, Sept. 5, '70, = 74 J '65 
 
 ist payt., $170. Time, 155 d. Amt , =$174.39 
 2d payt., $245.30. Time, 108 d. Amt., = 249.72 
 3d payt., $87.50. Time, 80 d. Amt., = 88.67 
 
 Amt. of the Payts., = 512.78 
 
 Balance due Sept. 5th, 1870, $228.87 
 
 5. Sold goods amounting to $650, to be paid Jan. ist, 
 1868. June loth, received $125; Sept. 1 3th, $75.50; Oct. 
 3d, $210: what was due Dec. 3ist, 1868, at 6% interest? 
 
 6. A note for $820, dated July 5th, 1865, payable in 
 i year, at 1% interest, bore the following indorsements : 
 Jan. loth, 1866, received $150; March 2oth, received 
 $73.10; May 5th, received $116; June 15^1, received 
 $141.50: what was due at its maturity? 
 
 7. An account of fnoo due March 3d, received the 
 following payments: June ist, $310; Aug. 7th, $119; 
 Oct. 1 7th, $200: what was due on the 27th of the next 
 Dec., allowing 7^ interest ?
 
 270 INTEREST. 
 
 CONNECTICUT METHOD. 
 
 389. I. "When the first payment is a year or more from 
 the time the interest commenced. 
 
 Find the amount of the principal to that time. If the 
 payment equals or exceeds the interest due, subtract it from 
 the amount thus found, and considering the remainder a neiv 
 principal, proceed thus till all the payments are absorbed. 
 
 II. When a payment is made before a year's interest has 
 accrued. 
 
 Find the amount of the principal for i year; also if the 
 payment equals or exceeds the interest due, find its amount 
 from the time it was made to the end of the year, and sub- 
 tract this amount from the amount of the principal; and 
 treat the remainder as a new principal. 
 
 But if the payment be less than the interest, subtract the 
 payment only, from the amount of the principal thus found, 
 and proceed as before. 
 
 NOTE. If the settlement is made in less than a year, find the 
 amount of the principal to the time of settlement ; also find the 
 amount of the payments made during this period to the same date, 
 and subtracting this amount from that of the principal, the remainder 
 will be the balance due. Kirby's Reports. 
 
 $650. 
 
 NEW HAVEN, April izth, 1860. 
 
 8. On demand, I promise to pay to the order of George 
 Selden, six hundred and fifty dollars, with interest, value 
 received. THOMAS SAWYER. 
 
 Indorsements: May ist, 1861, received $116.20. Feb. 
 loth, 1862, received $61.50, Dec. i2th, 1862, received 
 $12.10. June zoth, 1863, received $110: what was due 
 Oct. 2ist, 1863? 
 
 388. What is the mercantile method ? 389. What is the Connecticut method ?
 
 PARTIAL PAYMENTS. 271 
 
 Principal, dated April i2th, 1860, $650.00 
 
 Int. to ist payt. May ist, 1861 (i y. 19 d.), 41.06 
 
 Amount, May i, '61, = 691.06 
 
 ist payt. May ist, 1861, 116.20 
 
 Remainder or New Prin., May i, '61, 574-86 
 
 Int. to May i, '62, or i y. (2d payt. being short of i y.), 34-49 
 
 Amount, May i, '62, = 609.35 
 
 Amt. of 2d payt. to May i, '62 (2 m. 19 d.), 62.31 
 
 Rem. or New Prin., May i, '62, = 547-4 
 
 Ami., May i, '63 (i y.), = 5 79-86 
 
 3d payt. (being less than int. due), draws no int., 12.10 
 
 Bern, or New Prin., May 1/63, = 567-76 
 
 Amt., Oct. 21, '63 (5 m. 20 d.), 583.85 
 
 Amt. of last payt. to settlement (4 m. i d.), 112.22 
 
 Balance due Oct. 21, 63, = $47 1.63 
 
 NOTE. For additional exercises in the Connecticut rule, the 
 ptudent is referred to Art. 387. 
 
 VERMONT RULE. 
 
 390. I- " When payments are made on notes, bills, or similar 
 obligations, whether payable on demand or at MI specified time, ' with 
 interest,' such payments shall be applied; First, to liquidate the 
 interest that has accrued at the time of such payments ; and, 
 secondly, to the extinguishment of the principal." 
 
 II. " The annual interests that shall remain unpaid on notes, bills, 
 cr similar obligations, whether payable on demand at a specified 
 time, " with interest annually," shall be subject to simple interest 
 from the time they become due to the time of final settlement." 
 
 III. " If payments have been made in any year, reckoning from the 
 time such annual interest began to accrue, the amount of such pay- 
 ments at the end of such year, with interest thereon from the time 
 of payment, shall be applied ; First, to liquidate the simple interest 
 that has accrued from the unpaid annual interests. 
 
 " Secondly, To liquidate the annual interests that have become due; 
 " Thirdly, To the extinguwJiment of the principal." 
 
 391. The Rule of New Hampshire, when partial payments ar 
 made on notes " with interest annually," is essentially the same as 
 the preceding. But " where payments are made expressly on account 
 of interest accruing, but not then due, they are applied when the 
 interest falls due, without interest on such payments."
 
 272 
 
 PABTIAL PAYMENTS. 
 
 $1500. 
 
 BURLINGTON, Feb. x 
 9. On demand, I promise to pay to the order of Jam! 
 Sparks, fifteen hundred dollars, with interest annually, 
 received. AUGUSTUS WARREN. 
 
 Indorsements: Aug. ist, i865,received$i6o; July i2th. 
 1866, $125 ; June i8th, 1867, $50. Required the amouri 
 due Feb. ist, 1868. 
 
 Principal, $1500.00 
 
 Int. to Feb. i, '66 (i yr. at 6%), 90.00 
 
 Amount, = i59O.o<? 
 
 ist payment Aug. i, '65, $160.00 
 
 Int. on same to Feb. i, '66 (6 mos.), 4.80 164.80 
 
 Remainder or new principal, 1425.20 
 
 Int. on same to Feb. i, '67 (i yr.), 85.5 1 
 
 Amount, 
 
 2d payment July 12, '66, 
 Int. on same to Feb. i, '67 (6 m. 20 d.), 
 
 Remainder or new principal, 
 Int. on same to Feb. i, '68 (i yr.), 
 
 Amount, 
 
 3d payment June 18, '67, 
 Int. on same to Feb. i, '68 (7 m. 14 d.), 
 
 Bal. Feb. ist, 1868, = $1412.57 
 
 5125.00 
 4.16 
 
 129.16 
 
 $50.00 
 
 1.87 
 
 - 1381.55 
 82.89 
 
 = 146444 
 
 51-87 
 
 $2000. 
 
 CONCORD, Jan. i$th, 1869. 
 
 10. Two years after date, I promise to pay to the order 
 of Lewis Hunt, two thousand dollars, "with interest:" 
 the payee, Jan. isth, 1870, received, by agreement, $200 
 on account of interest then accruing. What was due on 
 this note Jan. 15th, 1871, by the Vt. and 1ST. H. rules? 
 
 By the Vt. rule, the bal.=$224O $212=^2028 ) . 
 " N. H. rule, " =$2240 $200=^2040 \ 
 
 In the former, interest is allowed on the payment from its date to 
 the settlement ; in the latter, it is not
 
 COMPOUND INTEREST. 
 
 392. Interest may be compounded annually, semi" 
 annually, quarterly, or for any other period at which the 
 interest is made payable. 
 
 393. To compute Compound Interest, the Principal, 
 the Rate, and Period of compounding being given. 
 
 1. What is the compound interest of $600 for 3 years, 
 at 6%'i 
 
 Principal, $600.00 
 
 Int. for ist year, $600 x .06, 36.00 
 
 Amt. for i y., or 2d prin., = 636.00 
 
 Int. for 2d year, $636 x .06, 38.1 6 
 
 Amt. for 2 yrs., or 3<i prin., = 674.16 
 
 Int. for 3d year, $674.16 x .06, 4-45 
 
 Amt, for 3 years, = 714.61 
 
 Original principal to be subtracted, 600.00 
 
 Compound int. for 3 years, = &i 14.61 
 Hence, the 
 
 RULE. I. Find the amount of the principal for the first 
 period. Treat this amount as a new principal, and find 
 the amount due on it for another period, and so on through 
 every period of the given time. 
 
 II. Subtract the given principal from the last amount, 
 and the remainder will be the compound interest. 
 
 NOTES. i. If there are months or days after the last regular 
 period at which the interest is compounded, find the interest on the 
 amount last obtained for them, and add it to the same, before sub- 
 tracting the principal. 
 
 2. Compound interest cannot be collected bylaw; but a creditor 
 may receive it, without incurring the penalty of usury. Saving* 
 Banks pay it to all depositors who do not draw their interest 
 when due. 
 
 2. What is the compound ink of $500 for 3 yrs. at 7% ? 
 
 3. What is the compound int. of 8750 for 4 yrs. at 
 
 393. How find compound interest, when the principal, rate, and time are given?
 
 274 
 
 COMPOUND INTEREST. 
 
 4. What is the com. int. of $1000 for 2 y. 7 m. 9 d. at 6? 
 
 5. What is the interest of $1360 for 2 years at 7^, 
 compounded semi-annually ? 
 
 6. What is the amount of $2000 for 2 years at ^'/com- 
 pounded quarterly ? 
 
 COMPOUND INTEREST TABLE. 
 
 Showing the amount of $i, at 3, 4, 5, 6, and 1% com- 
 pound interest, for any number of years from i to 25. 
 
 Yrs. 
 
 9*. 
 
 4%. 
 
 &. 
 
 6#. 
 
 H6. 
 
 I. 
 
 1.030 ooo 
 
 1.040 ooo 
 
 1.050000 
 
 1.060000 
 
 1.07 ooo 
 
 2. 
 
 1. 060 900 
 
 1.081 600 
 
 1. 102 500 
 
 1.123 600 
 
 1.14490 
 
 3- 
 
 1.092 727 
 
 1.124 864 
 
 I.I57625 
 
 1.191 016 
 
 1.22 504 
 
 4 
 
 1.125 59 
 
 1.169859 
 
 I.2I5506 
 
 1.262477 
 
 1.31079 
 
 5- 
 
 1.159274 
 
 1.216653 
 
 1.276 282 
 
 1.338226 
 
 1.40 255 
 
 6. 
 
 1.194052 
 
 1.265319 
 
 1.340 096 
 
 1.418519 
 
 1.50073 
 
 7- 
 
 1.229 874 
 
 I -3 I 593 2 
 
 1.407 100 
 
 1.503 630 
 
 1.60578 
 
 8. 
 
 1.266 770 
 
 1.368569 
 
 1-477455 
 
 1-593 848 
 
 I.7l8l8 
 
 9- 
 
 I-304773 
 
 1.423312 
 
 i-55 I 328 
 
 1.689479 
 
 1.83 845 
 
 10. 
 
 1-343 9 l6 
 
 1.480 244 
 
 1.628 895 
 
 1.790848 
 
 1.96715 
 
 ii. 
 
 1.384234 
 
 I-53945I 
 
 1.710339 
 
 1.898 299 
 
 2.10485 
 
 12. 
 
 1.425 761 
 
 i .60 1 032 
 
 1-795 856 
 
 2.012 196 
 
 2.25 219 
 
 13- 
 
 1.468534 
 
 1.665 74 
 
 1.885 649 
 
 2.132 928 
 
 2.4d 984 
 
 14- 
 
 1.512590 
 
 1.731 676 
 
 1.979932 
 
 2.26o 904 
 
 2-57 853 
 
 15- 
 
 1-557 9 6 7 
 
 1.800 944 
 
 2.078928 
 
 2.396558 
 
 2-75 903 
 
 1 6. 
 
 1.604 76 
 
 1.872981 
 
 2.182875 
 
 2.540352 
 
 2.95 216 
 
 17- 
 
 1.652848 
 
 1.947 900 
 
 2.292 018 
 
 2.692 773 
 
 3.15881 
 
 18. 
 
 1-702433 
 
 2.025 817 
 
 2.406 619 
 
 2 -854339 
 
 3-37 293 
 
 19. 
 
 1-753 5 6 
 
 2.106 849 
 
 2.526950 
 
 3.025 600 
 
 3.61 652 
 
 20. 
 
 1.806 in 
 
 2.191 123 
 
 2.653 2 98 
 
 3-207 135 
 
 3.86 968 
 
 21. 
 
 1.860295 
 
 2.278768 
 
 2-785 9 6 3 
 
 3-3995 6 4 
 
 4.14056 
 
 22. 
 
 1.916 103 
 
 2.369919 
 
 2.925 261 
 
 3-603537 
 
 4.43 040- 
 
 23- 
 
 i-9735 8 7 
 
 2.464 716 
 
 3-071524 
 
 3-8i975o 
 
 4-74052 
 
 2 4 . 
 
 2.032 794 
 
 2-563 304 
 
 3.225 100 
 
 4.048 935 
 
 5.07 236 
 
 25- 
 
 2.093 778 
 
 2.665 8 3 6 
 
 3-386355 
 
 4.291 871 
 
 5-42 743
 
 COMPOUND INTEHEST. 275 
 
 394. To find Compound Interest by the Table. 
 
 j. Yv r hat is the amount of $900 for 6 yrs. at 7$, the int. 
 Compounded annually ? What is the compound interest ? 
 
 SOLUTION Tabular amt. of $i for 6 yrs. at 1%, $1.50073 
 
 The principal, 90; 
 
 Ami. for 6 yrs., = 81350.65700 
 The principal to be subtracted from amt., 900 
 
 Compound Int. for 6 yrs., = $450.657 
 Hence, the 
 
 RULE. I. Multiply the tabular amount of $i for the given 
 time and rate by the principal ; the product will be the 
 amount. 
 
 II. From the amount subtract the principal, and the 
 remainder will be the compound interest. 
 
 NOTE. If the given number of years exceed that in the Table, 
 find the amount for any convenient period, as half the given years ; 
 then on this amount for the remaining period. 
 
 8. What is the interest of $800 for 9 years at 6, com- 
 pounded annually ? 
 
 9. What is the int. of $1100 for 12 years at 7^, com- 
 pounded annually ? 
 
 10. What is the int. of $1305 for 16 years at $%, com- 
 pounded annually? 
 
 11. What is the amount of $4500 for 15 years at 4^', 
 compounded annually ? 
 
 12. What is the amount of $6000 for 25 years at 7$, 
 compounded annually? 
 
 13. What is the amount of $3800 for 30 years at 6$, 
 compound interest? 
 
 14. What is the compound interest of $4240 at $% fcr 
 40 years ? 
 
 15. What is the amount of $1280 for 50 years at -]% 
 compound interest ? 
 
 1 6. What will $100 amount to in 60 years at 6% com- 
 pound interest?
 
 . 
 
 276 DISCOUNT. 
 
 DISCOUNT. 
 
 395. Commercial Discount is a deduction of a 
 certain per cent from the price-list of goods, the face of 
 bills, &c., without regard to time. 
 
 396. True Discount is the difference between a 
 debt bearing no interest and its present ivorth. 
 
 397. The Present Worth of a debt payable at a 
 future time without interest, is the sum, which, put at 
 legal interest for the given time, Avill amount to the debt. 
 
 397, a. To find tha Net Proceeds of Commercial Discount. 
 
 1. Sold goods marked 81560, art 20% discount, on 4m., 
 then deducted 5% for cash. Required the net proceeds? 
 
 ANALYSIS. $1560 x .20 = $312.00, and $1560 $312 = $1248. 
 1248 x .05 =$62.40, and 1248 $62.4o=$n85.6o, net. Hence, the 
 
 RULE. Deduct the commercial discount from the list 
 price, and from the remainder take the cash discount. 
 
 2. What is the net value of a bill of $3500, at 15$ dis- 
 count, and $% additional for cash? Ans. $2826.25. 
 
 3. What is the net value of a bill amounting to 85280, 
 at i z\% discount ? Ans. $4620. 
 
 398. To find the Present Worth of a debt, the Rate and 
 Time being given. 
 
 1. What is the present worth of $250.51, payable iu 
 8 months without interest, money being worth 6% ? 
 
 ANALYSIS. The amount of $i for 8 $i.o4 = amt. 2 
 
 mos., at 6fc, is $1.04 ; therefore $i is the $1.04)8250.51 debt 
 present worth of $1.04, due in 8 mos., &2AO 871; Ans 
 
 and$250.5i-H-$i.o4=$24O.875. Hence, the 
 
 RULE. Divide the debt by the amount of Si for the gicen 
 time and rate ; the quotient will be the present worth. 
 
 2. Find the present worth of $300, due in 10 m., when 
 interest is 1%. 
 
 395. What is discount? 396. Commercial? True? 397. Present worth? 
 397, a. How find net proceeds of commercial dUcotmt ? 398. Present worth ?
 
 DISCOUNT. 277 
 
 3. Find the present worth of $500, due in i year, when 
 interest is 8%. 
 
 4. What is the present worth of a note for $1250, pay- 
 able in 6 mos., interest being 6% ? 
 
 5. A man sold his farm for $2500 on i year without 
 interest: what is the present worth of the debt, money 
 being 7%? 
 
 6. What is the present worth of a legacy of 85000, 
 payable in 2 years, when interest is 6% ? 
 
 399. To find the True Discount, the Rate and Time 
 being given. 
 
 7. What is the true discount at 6^ on a note of 8474.03 
 due in 6 months and 3 days ? 
 
 AKALYSis._The amount of $i for $ I . o3o5 ) 474 .o 3 
 
 6 mos. and 3 days is $1.0305. (Art. 365.) - <?/ 
 
 Therefore, the present worth of the note 8400 
 
 is $474.03-^1.0305, or $460. (Art. 398.) 8474-03 466 = $I4.O3 
 But, by definition, the debt, minus the 
 
 present worth, is the true discount ; and $474.03 $460 (present 
 worth)=$i4.o3, the Ans. Hence, the 
 
 RULE. First find the present worth; then subtract it 
 from the debt. 
 
 8. ( Find the discount on $2560, due in 7 months, at 6f c . 
 
 9. Find the discount on $2819, due in 9 months, at $%. 
 
 10. Bought $2375 worth of goods on 6 months: what is 
 the present worth of the bill, at 8% ? The discount ? 
 
 11. At 6% what is the present worth of a debt of $3860, 
 half of which is due in 3 months and half in 6 months? 
 
 12. What is the difference between the int. of $6000 for 
 i y. at 6%, and the discount for i y. at 6^? 
 
 13. Bought a house for 85560 on i| year vithout 
 interest: what would be the discount at 7%, if paid down ? 
 
 14. If money is worth 7^, which is preferable, $15000 
 cash, or $16000 payable in a year without interest? 
 
 399. How find the true discount, when the rate and time are given ?
 
 278 BANKS AND BANK DISCOUNT. 
 
 BANKS AND BANK DISCOUNT. 
 
 400. Banks are incorporated institutions which deal 
 in money. They are of four kinds: banks of Deposit, 
 Discount, Circulation, and Savings. 
 
 401. A BanJs of Deposit is one that receives 
 money for safe keeping, subject to the order of the 
 depositor. 
 
 A Bank of Discount is one that loans money, 
 discounts notes, drafts, etc. 
 
 A Bank of Circulation is one that makes and 
 issues bills, which it promises to pay, on demand. 
 
 A Savings Bank is one that receives small sums 
 on deposit, and puts them at interest, for the benefit 
 of depositors. 
 
 402. Bank Discount is simple interest paid in 
 advance. 
 
 NOTES. i. A note or draft is said to bo discounted when the 
 interest for the given time and rate is deducted from the face of it, 
 and the balance paid to the holder. 
 
 2. The part paid to the holder is called the proceeds or avails of 
 the note; the part deducted, the dincount. 
 
 3. The time from the date when a note is discounted to its ma- 
 turity, is often called the Term of Discount. 
 
 403. To find the Bank Discount, the Face of a Note, the 
 Time, and the Rate being given. 
 
 i. What is the bank discount on $450 for 4 m. at 6%? 
 
 ANALYSIS. The int. of $i for 4 in. 3 d. is $.0205 ; and $450 x .0205 
 =$9.225, the bank discount required. (Art. 380.) Hence, the 
 
 EULE. Compute the interest on the face of the note at 
 the given rate, for 3 days more than the given time. 
 
 To find the proceeds : Subtract the discount from the 
 face of the note. 
 
 400. What is a bank? 401. A bank of deposit? Discount? Circulation? 
 Savings bank? 402. What is bank discount? 403. How find the bank discount, 
 when the face of a note, the rate and time are given ? How find the proceeds ?
 
 BANKS AND BANK DISCOUNT. 579 
 
 NOTE. If a note is on interest, find its amount at maturity, and 
 taking this as the face of the note, cast the interest on it as above. 
 
 )( 2. Find the term of discount and proceeds of a note for 
 $500, on 90 d., dated June eth, 1873, and discounted July 
 3d, 1873, at 1%. Ans. 65 d.; Pro., 1493.68. 
 
 3. Find the proceeds of a note of $730 due in 3m. at 6%. 
 
 4. Find the proceeds of a draft for $1000 on 60 d. at 6%. 
 
 5. Find the maturity, term of discount, and proceeds of 
 a note of $1740, on 6m., dated May i, '73, and dis. Aug. 2ist, 
 '? 3> at 5 %- Ans. Nov. 4th, '7 3 ; Time, 75 d. ; Proceeds, . 
 
 6. "What is the difference between the true and bank 
 discount on $5000 due in i year at 6%? / V *J u 
 
 7. A jobber buying $7500 worth of goods for cash, sold 
 them on 4 mos. at 12^ advance, and got the note dis- 
 counted at 7% to pay the bill: how much did he make ? 
 
 404. To find the Face of a Note, that the proceeds at Bank 
 Discount shall be a specified sum, the R. and T. being given. 
 
 8. For what sum must a note be drawn on 6 mos. that 
 at (>% bank discount, the proceeds shall be $500 ? 
 
 ANALYSIS. The bank discount of $i for 6 mos. 3 d. at 6^ is 
 $.0305 ; consequently the proceeds are $.9695. Now as $.9695 are 
 the proceeds of 551, $500 must be the proceeds of as many dollars as 
 $.9695 is contained times in $500; and $5oo-5-$.9695=$5i5.729, the 
 face of the note. Hence, tho 
 
 RULE. Divide the given proceeds ly the proceeds of $i 
 for the given time and r^te. 
 
 9. What must be the face of a note on 4 mos. that when 
 discounted Bt 1% the procoeds may be $750 ? 
 
 10. What was the face of a note on 60 days, the pro- 
 ceeds of which being discounted at 5$, were 81565 ? 
 
 11. If the avails are $2165.45, the time 4 mos., and the 
 rate of bank discount 8$, what must be the face of the note? 
 
 12. Bought a house for $7350 cash : how large a note on 
 4 mos. must I have discounted at bank 6% to pay this sum? 
 
 404. How find how large to make a note to raise a specified sam, when the rato 
 and time are given
 
 280 STOCK INVESTMENTS. 
 
 STOCK INVESTMENTS. 
 
 405. Stocks are the funds or capital of incorporated 
 companies. 
 
 An Incorporated Company is an association 
 authorized by law, to transact business. 
 
 NOTE. Stocks are divided into equal parts called shares, and tlio 
 owners of the shares are called stockholders. These shares vary from 
 $25 to $500 or $1000. They are commonly $100 each, and will be 
 so considered in the following exercises, unless otherwise stated. 
 
 406. Certificates of Stock are written statements, 
 specifying the number of shares to which holders are 
 entitled. They are often called scrip. 
 
 407. The JPar value of stock is the sum named on 
 the face of the scrip, and is thence called its nominal value. 
 
 The Market value is the sum for which it sells. 
 
 NOTES. i. When shares sell for their nominal value, they are at 
 par; when they sell for more, they are above par, or at & premium; 
 when they sell for less, they are below par, or at a discount. 
 
 2. When stocks sell at par, they are often quoted at 100 ; when 8 % 
 above par, at 108 ; when %% beloio par, at 92. The term pur, Latin, 
 signifies equal ; hence, to be at par, is to be on an equality. 
 
 408. The Gross Earnings of a Company are its entire 
 
 receipts. 
 
 The Net Earnings are the sums left after deducting all 
 expenses. 
 
 409. Instalments are portions of the capital paid by the 
 stockholders at different times. 
 
 410. Dividends are portions of the earnings distributed among 
 the stockholders. They are usually made at stated periods ; as, 
 annually ; semi-annually, etc. 
 
 411. A Bond is a writing under seal, by which a 
 party binds himself to pay the holder a certain sum, at or 
 before a specified time. 
 
 405. What are stocks ? An incorporated company ? Note. Into what are 
 Blocks divided ? What are stockholders ? 406. What are certificates of stock ? 
 
 407. What is the par value of stock? The market value? Note. When are 
 stocks at par? Above par? Below par? The meaning of the term par? 
 
 408. What are the gross earnings of a company ? The net earnings ? 405. Whul 
 ore instalments ? 410. Dividends? 411. What is a bond?
 
 BTOCK INVESTMENTS. 283 
 
 UNITED STATES BONDS. 
 
 412. United States Bonds are those issued by 
 Government, and are divided into two classes : those pay- 
 able at a given date, and those payable within the limits 
 of two given dates, at the option of the Government. 
 
 NOTE. The former are designated by a combination of the 
 numerals, which express the rate of interest they bear, and the year 
 they become due ; as " 6s of '81." 
 
 The latter are designated by combining the numerals expressing 
 the two dates between which they are to be paid ; as " 5-203." 
 
 413. A Coupon is a certificate of interest attached to 
 a bond, which, on the payment of the interest, is cut off 
 and delivered to the payor. 
 
 414. The principal U. S. bonds are the following : 
 
 1. "6s of '8 1," bearing 6$ interest, and payable in 1881. 
 
 2. " 5-203," bearing 6% interest, and payable in not less than 5 or 
 more than 20 years from their date, at the pleasure of the Govern- 
 ment. Interest paid semi-annually in gold. 
 
 3. " 10-403,' bearing 5 % interest, redeemable after 10 years from 
 their date, interest semi-annually in gold. 
 
 4. " 53 of '81," bearing 5 % interest, redeemable after 1881, interest 
 paid quarterly in gold. 
 
 5. "4>s of "86," bearing $\% interest, redeemable after 1886, the 
 interest paid quarterly in gold. 
 
 6. "43 of IQOI," bearing 4% interest, the principal payable after 
 1901, the interest paid quarterly in gold. 
 
 415. State, City, Railroad Bonds, etc., are payable at a specified 
 time, and are designated by annexing the numeral denoting the rate 
 of interest they bear, to the name of the State, etc., by which they 
 are issued ; as, New York 6s ; Georgia 73. 
 
 416. Computations in stocks and bonds are founded 
 upon the principles of percentage; the par value being 
 the base, the per cent premium the rate, the premium, etc., 
 the percentage, and the market value the amount. 
 
 412. What are U. S. bonds? How many classes? Note. How are the former 
 d?sia^ate:l ? The latter ? 413. What is a coupon ? 414. What is meant by U. S, 
 teof'si? By U. S. s-208 ? By U. S. 10-408? By new U. S. 5* of '81 ?
 
 282 STOCK INVESTMENTS 
 
 NOTE. The comparative profit of investments in U. S. Bonds de- 
 pends upon their market value, the rate of interest they bear, and 
 the premium on gold. That of Railroad and other Stocks-pon their 
 market value, and the per cent of their dividends 
 
 v 4- 
 
 417. To find the Premium, Discount, Instalment, or 
 Dividend, the Par Value and the Rate being given. 
 
 1. "What is the premium on 20 shares of the New Orleans 
 National Bank, at 8$ ? 
 
 ANALYSIS. 208. =$2000; and $2000 x .08 =$160, Ana. Hence, the 
 
 RULE. Multiply the par value by the rate, expressed 
 decimally. (Art. 336.) 
 
 2. What is the discount on 27 shares of Michigan 
 Central Eailroad at g% ? 
 
 3. What is the premium on three $1000 IT. S. 6s of '81, 
 when they stand at \i\% above par ? 
 
 4. The Maryland Coal Company called for an instalment 
 of 15%: what did a man pay who owned 35 shares? 
 
 5. What is the discount on $4000 Tennessee 6s, at IO;T/? 
 
 6. The Virginia Manufacturing Company declared a 
 dividend of 17^: to what sum were 28 shares entitled? 
 
 418. To find the Market Value of Stocks and Bonds, the 
 
 Rate and Nominal Value being given. 
 
 7. What is the market value of 45 shares of New York 
 Central, at A,% premium ? 
 
 ANALYSIS. 459. =4500; and $4500x1.04 (i + the rate)=$468o, 
 the value required. Hence, the 
 
 RULE. Multiply the nominal value by i plus or minus 
 the rate. 
 
 Or, multiply the market value of i share by the number 
 of shares. 
 
 NOTE. In finding the net value of stocks, the brokerage, postage 
 stamps, and other expenses must be deducted from the market value. 
 
 8. What is the worth of 814000 of Kentucky 6s, at 92 ? 
 
 417. How find the premium, etc., when the par value and rate are given?
 
 STOCK INVESTMENTS. 283 
 
 9. What is the worth of an investment of $9500 in 
 tJ. S. 5-203, at \z>\% premium ? 
 
 10. What is the net value of 58 shares of New Jersey 
 Central, at no; deducting the express and other charges 
 $1.89, and brokerage at \% ? 
 
 =ni. What will be realized from $7500 Texas 75, at 15$ 
 discount, deducting the brokerage at %f , and $11.39 f r 
 postage and other charges. 
 
 12. What will $10000 U. S. 5-203 cost at ^\% premium, 
 adding brokerage at \%, and other expenses $^.37^-? 
 
 13. What is the value in currency of $15750 gold coin, 
 the premium being 22-^? 
 
 14. A person has $10000 U. S. 5-203: what will he re- 
 
 ceive annually in currency, when gold is 1 2% premium ? 
 
 419. To find the Rate, the Par Value, and the Dividend, 
 Premium, or Discount being given. 
 
 15. The capital stock of a company is fiooooo; its 
 gross earnings for the year are $34500, and its expenses 
 $13500 : deducting from its net earnings $1000 as- surplus, 
 what per cent dividend can the company make ? 
 
 ANALYSIS. $13500 + $1000 = $14500, and $34500 $14500 = 
 $20000, the net earnings. Now 20000 -H$IOOOOO=. 20 or 20$, the 
 rate required. Hence, the 
 
 RULE. Divide the premium or discount, as the case may 
 be, by the par value. (Art. 339.) 
 
 16. Paid $750 premium for 50 shares of bank stock: 
 what was the per cent? 
 
 17. The discount on 100 shares of the Pacific Eailroad 
 is $625 : what is the per cent below par? 
 
 1 8. The net earnings of a company with a capital of 
 
 $480000 are $35000 ; reserving $3000 as surplus, what 
 per cent dividend can they declare ? 
 
 4:8. How find the market value when the rate and nominal valne are given? 
 419. How fiud the rate, when the premium, discount, or dividend are given?
 
 284 STOCK INVESTMENTS. 
 
 420. To find how much Stock can be bought for a specified 
 sum, the Rate of Premium or Discount being given. 
 
 19. How much of Kansas 6s, at 20$ discount, can be 
 bought for $5200 ? 
 
 ANALYSIS. $5200-=-. 80 (val. of $i 8tock)=$6soo, Ans. Hence, the 
 
 RULE. Divide the sum to be invested by i plus or minus 
 the rate, as the case may be. (Art. 349.) 
 
 Or, divide the given sum by the market value q/*$i of stock. 
 
 20. How many $100 IT. S. 10-403, at $% premium, can 
 be bought for $4200 ? 
 
 2 1. What amount of Virginia 6s, at 90, can be purchased 
 for $10800? 
 
 421. To find what Sum must be invested in Bonds to realize 
 a given income, the Cost and Rate of Interest being given. 
 
 22. What sum must be invested in Missouri 6s at 90, to 
 realize an income of $1800 annually? 
 
 ANALYSIS. At 6% the income of $i is 6 cts. ; and $1800-5- .06= 
 $30000, the nominal value of the bonds. Again, at .90, $30000 of bonds 
 will cost .90 times f 30000=$ 27000, the sum required. Hence, the 
 
 RULE. I. Divide the given income by the annual interest 
 9/ $i of bonds ; the quotient will be the nominal value of 
 the bonds. 
 
 II. Multiply the nominal value by the market value of$i 
 of bonds; the product will be the sum required. 
 
 23. What sum must be invested in IT. S. 5-205, at 106, 
 to yield an annual income of $2500 in gold? 
 
 24. How much must one invest in Wisconsin 8s, at 95, 
 to receive an annual income of $3000 ? Ans. $35625. 
 
 25. How much must be invested in Mississippi 6s, at 80, 
 to yield an annual interest of $4200 ? 
 
 420. How find the quantity of stock that can be bought for a specified sum, 
 when the rate is given ?
 
 EXCHANGE. 
 
 422. Exchange is a method of making payments 
 between distant places by Bills of Exchange. 
 
 423. A Bill of Exchange is an order or draft 
 directing one person to pay another a certain sum at a 
 specified time. 
 
 2vOTE. The person who signs the bill is called the drawer or 
 maker ; the one to whom it is addressed, the drawee; the one to 
 whom it is to be paid, the payee ; t'lie one who sends it, the remitter. 
 
 424. Bills of Exchange are Domestic or Foreign* 
 Domestic Hills are those payable in the country 
 
 where they are drawn, and are commonly called Drafts. 
 
 Foreign Bills are those drawn in one country and 
 payable in another. 
 
 425. A Sight Bill is one payable on its presentation. 
 
 A Time Bill is one payable at a specified time after 
 its date, or presentation. 
 
 426. The Par of Exchange is the standard by 
 which the value of the currency of different countries is 
 compared, and is either intrinsic or nominal. 
 
 An Intrinsic Par is a standard having a real and 
 fixed value represented by gold or silver coin. 
 
 A Nominal far is a conventional standard, having 
 any assumed value which convenience may suggest. 
 
 427. When the market price of bills is the same as the 
 face, they are at par; when it exceeds the face, they are 
 above par, or at a premium ; when it is less than the face, 
 they are below par, or at a discount (Art. 407, n.} 
 
 NOTES. i. The fluctuation in the price of bills from their par 
 value, is called the Course of Exchange. 
 
 422. What is exchange? 423. A bill of exchange? 424. What arc domestic 
 bills? Foreign hills T 425. What are eight hills? Time bills? 426. What i 
 the par of exchange ? An intrinsic par ? A nominal par ?
 
 286 DOMESTIC EXCHANGE. 
 
 2. The rate of Exchange between two places or countries depends 
 upon the circumstances of trade. If the trade between New York 
 and New Orleans is equal, exchange is at par. If the former owes 
 the latter, the demand for drafts on New Orleans is greater than the 
 supply; hence they are above par in New York. If the latter owes 
 the former, the demand for drafts is less than the supply, con- 
 sequently drafts on New Orleans are below par. 
 
 428. An acceptance of a bill or draft is an engage- 
 ment to pay it according to its conditions. To show this, 
 it is customary for the drawee to write the word accepted 
 across the face of the bill, with the date and his name. 
 
 NOTE. Bills of Exchange are negotiable like promissory notes, 
 and the laws respecting their indorsement, collection, protest, etc., 
 are essentially the same. 
 
 DOMESTIC OR INLAND EXCHANGE. 
 
 429. To find the Cost of a Draft, its Face and the Rate 
 
 of Exchange being given. 
 
 i. "What is the cost of a sight draft on New York, for 
 
 $4500, at z\% premium ? 
 
 ANALYSIS. At z\% premium, a draft of $i will $4500 
 
 cost $1 + 2^ cts.=$i.o25, or 1.025 times the draft. 1.025 
 
 Hence, the cost of $4500 draft will be 1.025 times 
 
 T , 
 
 *. , .. I 2.>O 
 
 its face ; and $4500 x 1.025 = $4612.50, 
 
 2. What is the cost of a sight draft on St. Louis of 
 $5740, at 4% discount? 
 
 ANALYSIS. At 4^ discount, a draft of $i will $5740 
 
 cost $14 cents=$O96, or .96 times the draft .06 
 
 Therefore, the cost of $5740 draft will be .96 times &_ 4 
 
 Its face ; and $5740 x .96 =$5 5 10.40, Hence, the 
 
 RULE. Multiply the face of the draft ly the cost of$i 
 of draft, expressed decimally. 
 
 427. Whan are bills at par ? When above par ? When below ? Note. What 
 is the fluctuation in the price of exchange called ? Upon what does the rate of 
 exchange depend ? Explain ? 428. The acceptance of a bill ? 429. How find tho 
 <:ost of a draft, when the face and rate are given ?
 
 DOMESTIC EXCHANGE. 287 
 
 NOTES. i. When payable at sight, the worth of $i of draft is $i 
 plus or minus the rate of exchange. 
 
 2. On time drafts, both the exchange and the bank discount are 
 computed on tbeir face. Dealers in exchange, however, make but, 
 one computation ; the rate for time drafts being enough less than 
 night drafts to allow for the bank discount. 
 
 .V merchant in Galveston bought a draft of 62000 on 
 Philadelphia at 60 days sight: what was the draft worth, 
 the premium being z\%-> an( l the bank discount (F%. * 
 
 3. What cost a sight draft of $3560, at 2% discount? 
 
 4. Eequired the worth of a draft for $4250 on Chicago, 
 at 90 days, sight drafts being \% discount and interest 1%. 
 
 5. The Bank of New York having declared a dividend 
 of 4%, a stockholder living in Savannah drew on the bank 
 for his dividend on 50 shares, and sold the draft at \\% 
 premium : how much did he realize from the dividend ? 
 
 430. To find the Face of a Draft, the Cost and Rate of 
 Exchange being given. 
 
 7. Bought a draft in Omaha on New York, payable in 
 90 days, for $3043.50, exchange being $% premium, and 
 interest 6 C / : what was the face of the draft ? 
 
 ANALYSIS. The cost of $r of | x _l_ ,^jjj.j o - 
 
 draft is $i plus the rate, minus j)i g on $j for 
 the bank discount for the time. ^ ^ 
 
 Now Si + 3 = *i.Q3 ; the bank 0st f $' 
 
 discount on $i for 93 d. is $0.0155, 
 
 and $1.03 - $0.0155 = $1.0145. Face of dft.= $3000. 
 
 Now if $1.0145 w 'll b u y $ x f 
 
 draft, $3043.50 will buy a draft of as many dollars as 1.0145 is con- 
 tained times in $3043.50. $3O43.5o-=-i.oi45 = $3Ooo, the draft re- 
 quired. Hence, the 
 
 R ULE . Divide the cost of the draft by the cost of $i 
 of draft, expressed decimally. 
 
 8. What is the face of a sight draft purchased for $1250, 
 the premium being 2 \% ? 
 
 430. How find the face of a draft, the cost and rate twin? given?
 
 288 FOBEIGN EXCHANGE. 
 
 _--9. What is the face of a draft at 60 days sight, purchased 
 for $1500, when interest is &%, and the premium 2 c / c t 
 
 10. What is the face of a sight draft, purchased for 
 82500, the discount being $\% ? 
 
 __.! i. What is the face of a draft on 4 in., bought for $3600, 
 the int. being d%, and exchange 2% discount? 
 
 12. A merchant of Natchez sold a draft on Boston at 
 r^ prem., for $3806.25 : what was the face of the draft ? 
 
 FOREIGN EXCHANGE. 
 
 431. A Foreign Bill of Exchange is a Bill 
 drawn in one country and payable in another. 
 
 A Set of Exchange consists of three bills of the 
 same date and tenor, distinguished as the First, Second, 
 and Third of exchange. They are sent by different mails, 
 in order to save time in case of miscarriage. When one is 
 paid, the others are void. 
 
 432. The Legal far of Exchange between 
 Great Britain and the United States, is $4.8665 gold to the 
 pound sterling.* 
 
 433. To find the Cost of a Bill on England, the Face and 
 the Rate of Exchange being given. 
 
 i. What is the cost of the following bill on London, at 
 $4.8665 to the sterling? 
 
 i2s. 
 
 NEW YORK, July 4^, 1874. 
 
 At sight of this first of exchange (the second and third 
 of the same date and tenor unpaid), pay to the order of 
 Henry Crosby, three hundred and fifty-four pounds, twelve 
 shillings sterling, value received, and charge the same to 
 the account of 0. J. KING & Co. 
 
 To GEORGE PEABODY, Esq., London. 
 
 431. What is a foreign bill of exchange ? What is the legal par of exchange 
 with Great Britain T 
 
 * Act of Congress, March sd, 1873 To take effect Jan. ist, 1874.
 
 FOREIGN EXCHANGE. 289 
 
 ANALYSIS. Reducing the given shillings to OPERATION. 
 
 the decimal of a pound, 354, 128. = 354.6. 4.8665 
 
 (Art. 295.) Now if i is worth $4.8665, 354.6 _ 354-6 
 
 are worth 354 6 times as much; and $4.8665 x Ans. $1725.661 
 354.6=$i725.66i, the cost of the bill. 
 
 2. What is the value of a bill on England for 436, 53. 6d., 
 at $4.85^ to the sterling? 
 
 ANALYSIS. 436, 59. 6d.= 436.275, and the OPERATION. 
 
 market value of exchange is $4.8525 to the . $4.8525 
 
 Now $4. 8525 X436.275=$2ii7.o2, the cost of the 436.275 
 
 bill. Hence, the Ans. ' 
 
 RULE. Multiply the market value of i sterling by the 
 face of the bill ; the product will be its value in dollars 
 and cents. 
 
 NOTES. i. If there are shillings and pence in the given bill, 
 they should be reduced to the decimal of a pound. (Art. 295.) 
 
 2. Bills on Great Britain are drawn in Sterling money. 
 
 3. The New Par of Sterling Exchange $4.8665, is the intrinsic 
 value of the Sovereign or pound sterling, as estimated at the United 
 States Mint, and is 9^ % greater than the old par. 
 
 4. The Old Par, which assumed the value of the sterling to be 
 $4.44^, is abolished by law, and all contracts based upon it after 
 January ist, 1874, are null and void. 
 
 3. What is the cost of a bill on Dublin for 38:1, at 
 $4.87! to the sterling ? 
 
 4. What is the cost in currency of a bill on England for 
 750, exchange being at par, and gold 33$$ premium? 
 
 5. B owes a merchant in Liverpool 1500; exchange is 
 64.93!-; to transmit coin will cost 2$! insurance and freight; 
 and when delivered its commercial value is $4.80 to a 
 pound: which is the cheaper, to buy a bill, or send the 
 gold ? How much ? 
 
 6. A New Orleans merchant consigned 568 bales of cot- 
 ton, weighing 450 Ibs. apiece, to his agent in Liverpool; 
 the agent paid id. a pound freight, and sold it at i2d. a 
 
 433. How find the cost of a bill on England, when the face and rate are given?
 
 290 FOREIGN EXCHANGE. 
 
 
 
 pound ; he charged z\% commission, and 8, 6s. for stor- 
 age: what did the merchant realize for his cotton, ex- 
 change on the net proceeds being $4.91^ to the ? 
 
 434. To find Ihe Face of a Bill on England, the Cost and 
 the Rate of Exchange being given. 
 
 7. What is the face of a bill on England which cost 
 $1725.72, exchange being at par, or $4.8665 to the ? 
 
 ANALYSIS. Since $4.8665 buys i of * g/c/c - ^172 c 72 
 bill, $1725.72 will buy as many pounds as 
 $4.8665 are times in $1725.72, or 354.612 
 =354, I2s. 2fd. Hence, the -Ans. 354, J 2S. 2|d. 
 
 RULE. Divide the cost of the bill ~by the market value 
 ofi sterling ; the quotient will be the face of the bill 
 
 8. What is the face of a bill on London which cost 
 $2500, exchange being $4.88^ to the ? 
 
 3. What amount of exchange on Dublin can be obtained 
 for $3750, at $4.84^ to the sterling? 
 
 10. A merchant in Charleston paid $5000 for a bill on 
 London, at $4.87^ to the : what was the face of the bill ? 
 
 11. Paid $7500 for a bill on Manchester: what was the 
 face of it, exchange being $4.86^ to the ? 
 
 434, a. In quoting Exchange on Foreign Countries, the gen- 
 eral rule is to quote the money of one country against the money 
 of other countries. Thus, exchange is quoted 
 
 On Austria, in cents to the Florin (silver)r=$o.476. 
 
 On Frankfort, in cents to the Florin (gold) =$0.4 165. 
 
 On the German Empire, in cents to the Mark (gold)=$o 2382. 
 
 On North Germany, in cents to the new Thaler (silver)=; $0.714. 
 
 On Russia, in cents to the Rouble (silver) =$0.77 17. 
 
 NOTE. Bills of Exchange between the United States and foreign 
 countries are generally drawn on some of the great commercial 
 centers ; as London, Paris, Frankfort, Amsterdam, etc. 
 
 434. How find the face of a bill on England, when the cost and rate are given ?
 
 FOREIGN" EXCHANGE. 291 
 
 < 
 
 435. Sills on France are drawn in French cur- 
 rency, and are calculated at so many francs and centimes 
 to a dollar. 
 
 NOTE. Centimes are commonly written as decimals of a franc 
 Thus 5 francs and 23 centimes are written 5.23 francs. (Art. 224, n.\ 
 
 436. To find the Cost of a Bill on France, the Face ana 
 
 Rate of Exchange being given. 
 
 12. What is the cost of a bill on Paris of 1500 francs, 
 exchange being 5.25^ to a dollar ? 
 
 ANALYSIS. Since 5.25 francs will buy a 5.25 fr.) 1500.00 fr. 
 bill of $i, 1500 francs will buy a bill of as < ~&2&z 71 4- 
 
 many dollars as 5.25 is contained times in 
 1500; and 1500-7-5.25=1285.71, the cost required. Hence, the 
 
 RULE. Divide the face of the till by the number of franca 
 to $i exchange. 
 
 13. What is the cost of a bill of 3500 francs on Havre, 
 exchange being 5.18 francs to a dollar ? 
 
 437. To find the Face of a Bill on France, the Cost and 
 
 the Rate of Exchange being given. 
 
 14. A traveler paid $300 for a bill on Paris; exchange 
 being 5.16 francs to $i : what was the face of the b'ill ? 
 
 ANALYSIS. If $i will buy 5.16 francs, $300 will buy 300 times as 
 many ; and 5.16 fr. x 300=1548 francs, the face of the bill required. 
 Hence, the 
 
 RTJLE. Multiply the number of francs to $i exchange b$ 
 the cost of the bill. 
 
 15. Paid $2500 for a bill on Lyons, exchange being 
 5.22 francs to i : what was the face of the bill ? 
 
 1 6. A merchant paid $3150 for a bill on Paris, exchange 
 being 5.23 francs to $i : what was the face of the bill ? 
 
 435. How is exchange on France calculated f 436. How find the cost of a bill 
 on France, when the face and rate are given f 437. How find the face, when the 
 cost and rate are given ?
 
 ISTSURA^CE. 
 
 438. Insurance is indemnity for loss. It is dis- 
 tinguished by different names, according to the cause of 
 the loss, or the object insured. 
 
 439. Fire Insurance, is indemnity for loss by fire. 
 Marine Insurance, for loss by sea. 
 
 Life Insurance, for the loss of life. 
 Accident Insurance, for personal casualties. 
 Health Insurance, for personal sickness. 
 Stock Insurance, for the loss of cattle, horses, etc. 
 
 NOTES. i. The party who undertakes the risk is called the 
 Insurer or Underwriter. 
 2. The party protected by the insurance is called the Insured. 
 
 440. A Policy is a writing containing the evidence 
 and terms of insurance. 
 
 The Premium, is the sum paid for insurance. 
 
 NOTE. The business of insurance is carried on chiefly by In- 
 corporated Companies. Sometimes, however, it is undertaken by 
 individuals, and is then called out-door-insurance. 
 
 441. Insurance Companies are of two kinds : Stock Companies, 
 and Mutual Companies. 
 
 A Stock Insurance Company is one which has a paid up 
 capital, and divides the profit and loss among its stockholders. 
 
 A Mutual Insurance Company is one in which the losses 
 are shared by the parties insured. 
 
 442. Premiums are computed at a certain per cent of 
 the sum insured, and the operations are similar to those 
 in Percentage. 
 
 NOTE. Policies are renewed annually, or at stated periods, and 
 the premium is paid in advance. In this respect insurance differs 
 from commission, etc., which have no reference to time. 
 
 438. What is insurance? 439. Fire insurance? Marine? Life? Accident? 
 Health? Stock? 440. What is a policy ? The premium ? 441. How many kinds 
 of insurance companies'? A stuck insurance company? A mutual? 442. How 
 ere premiums computed ?
 
 INSUEANCE. 293 
 
 FIRE AND MARINE INSURANCE. 
 
 443. To find the Premium, the Sum insured and the Rate 
 for the period being given. 
 
 1. What premium must I pay per annum for insuring 
 $1750 on my house and furniture, at \%'t 
 
 ANALYSIS. $1750 x .005 (rate)=$8.75, Ans. Hence, the 
 RULE. Multiply the sum insured by the rate. (Art. 336.) 
 NOTE. The rate of insurance is sometimes stated at a certain 
 number of cents on $100. In such cases the rate should be reduced 
 lo the decimal of $i before multiplying. Thus, if the rate is 25 cents 
 on $100, the multiplier is written .0025. (Art. 295.) 
 
 2. At 35 cents on $100, what is the insurance of $1900? 
 
 3. At i-^fc, what is the premium on $2560? 
 
 4. At z\%, what is the premium on $3750? 
 
 5. At 25 cents on $100, what is the premium on $4280? 
 
 6. At 50 cents on $100, what is the premium on $5000? 
 
 7. At 3^', what is the cost of insuring $6175 ? 
 
 8. What is the premium for insuring a ship and cargo 
 \ ulued at $35000, at 2\ r / c ? 
 
 9. What is the cost of insuring a factory and its con- 
 d'lits, valued at $48250, at $\%, including $i-J for policy? 
 
 444. To find what Sum must be insured to cover both the 
 Property and Premium, the Rate being given. 
 
 10. For what must a factory worth $20709 be insured, 
 to cover the property and the premium of 2\% ? 
 
 ANALYSIS. The sum to be insured includes the property plus the 
 premium. But the property is 160% of itself, and the premium is 
 z}% of that sum ; therefore $20709 = 100 % 21%, or g?.V times the 
 r-i'.im to be insured. Now, if $20709 is .97^ times the required sum, 
 once that sum is $20709-^ .97!, or 21240, Ans. (Art. 337.) Hence, the 
 
 EULE. Divide the value of the property by i minus the rate. 
 
 NOTE. This and the preceding problem cover the ordinary cases 
 of Fire and Marine Insurance. Should other problems occur, they 
 may be solved like the corresponding problems in Percentage 
 
 How find the premium, when the sum insured and the rate are given f
 
 294 INSURANCE. 
 
 11. A merchant sent a cargo of goods worth $15275 to 
 Canton: what sum must he get insured at 3%, that he 
 may suffer no loss, if the ship is wrecked ? ^h.9) 
 
 12. A house and furniture are worth $27250 : what sum 
 must be insured at 2% to cover the property and premium? 
 
 13. What sum must be insured at *>% to cover the 
 premium, with a vessel and cargo worth $35250? 
 
 LIFE INSURANCE. 
 
 445. Life Insurance Policies are of different kinds, and 
 the premium varies according to the expectation of life. 
 
 ist. Life Policies, which are payable at the death of the party 
 named in the policy, the annual premium continuing through life. 
 
 2d. Life Policies, payable at the death of the insured, the 
 annual premium ceasing at a given age. 
 
 3d. Term Policies, payable at the death of the insured, if he 
 dies during a given term of years, the annual premium continuing 
 till the policy expires. 
 
 4th. Endowment Policies, payable to the insured at a given 
 age, or to his heirs if he dies before that age, the annual premium 
 continuing till the policy expires. 
 
 NOTE. The expectation of life is the average duration of the life 
 of individuals after any specified age. 
 
 1. What premium must a man, at the age of 25, pay 
 annually for a life policy of $5000, at ^\% ? 
 
 ANALYSIS. <$% =.045, and $5000 x . 045=1225, Ana. (Art. 443.) 
 
 2. What is the annual premium for a life policy of 
 2500, at 5$ ? 
 
 JL- 3. A man at the age of 35 years effected a life insurance 
 of $7500 for 10 yrs., at z\%\ what was the amt. of premium? 
 
 4. A man 65 years old negotiated a life insurance of 
 $8000 for 5 years, at 1 2%% : what was the amt. of premium ? 
 
 5. At the age of 30 years, a man got his life insured for 
 ' $75000, at 4^ per annum, and lived to the age of 70 : which 
 
 Was the greater, the sum he paid, or the sum received? 
 
 444. How find what sum must be insured to cover the property and preminm ?
 
 TAXES. 
 
 446. A Tax is a sum assessed upon the person or 
 property of a citizen, for public purposes. 
 
 A Property Tax is one assessed upon property. 
 
 A Personal Tax is one assessed upon the person, 
 and is often called a poll or capitation tax. 
 
 NOTE. The term poll is from the German polle, the head ; capita- 
 tion, from the Latin caput, the head. 
 
 447. Property is of two kinds, real and personal. 
 Heal Property is that which is fixed; as, lands, 
 
 houses, etc. It is often called real estate. 
 
 Personal Property is that which is movable; as, 
 money, stocks, etc. 
 
 448. An A.ssessor is a person appointed to appraise 
 property, for the purpose of taxation. 
 
 449. An Inventory is a list of taxable property, 
 with its estimated value. 
 
 450. Property taxes are computed at a certain per cent on the 
 valuation of the property to be assessed. That is, 
 
 The valuation of the property is the base ; the sum to be raised, 
 the percentage ; the per cent or tax on $i, the rate ; the sum collected 
 minus the commission, the net proceeds. 
 
 Poll tuxes are specific sums upon those not exempt by law, without 
 r<;gard to property. 
 
 451. To assess a Property Tax, the Valuation and the 
 Sum to be raised being given. 
 
 i. A tax of $6250 was levied upon a corporation of 6 per- 
 sons; A's property was appraised at $30000; B's, 837850; 
 C's, $40150 ; D's, $50000 ; E's, $55000 ; F's, $37000. What 
 was the rate of the tax, and what each man's share ? 
 
 446. What is n tax ? A property tax ? A personal tax ? Note. What is a per- 
 sonal tax called ? 447. Of how many kinds is property '! What is real property ? 
 l'i!!vonal? 44X. Whut is an assessor? 449. An inventory? 450. How are prop- 
 erty tuxes computed? How are poll taxes levied? 451. How is a property tax 
 assessed, when the valuation and the sum to he raised are given?
 
 396 
 
 TAXES. 
 
 ANALYSIS. The valuation=$30ooo + $37850 + $40150 + $50000 + 
 $55000 + $37000= $250000. The valuation $250000, is the base; 
 and the tax $6250, the percentage. Therefore $625o-i-$25oooo=.O25, 
 or z\fo, the rate required. 
 
 Again, since A's valuation was $30000, and the rate z\%, his tax 
 must have been "2.\ c /o of $30000; and $30000 x .025 = $750.00. Tho 
 tax of the others may be found in like manner. Hence, the 
 
 EULE. I. Make an inventory of all the taxable property. 
 
 II. Divide the sum to be raised by the amount of the 
 inventory, and the quotient will be the rate. 
 
 III. Multiply the valuation of each man's property by 
 the rate, and the product will be his tax. 
 
 NOTES. i. If a poll tax is included, the sum arising from the 
 polls must be subtracted from the sum to be raised, before it is 
 divided by the inventory. 
 
 2. If the tax is assessed on a large number of individuals, the 
 operation will be shortened by first finding the tax on $i, $2, $3, etc., 
 to $9 ; then on $10, $20, etc., to $90 ; then on $100, $200, etc., to $900, 
 etc., arranging the results as in the following 
 
 ASSESSORS' TABLE. 
 
 r * 
 
 $1 
 
 pays 
 
 $.025 
 
 $10 
 
 pay 
 
 $.25 
 
 $100 
 
 pay 
 
 $2.50 
 
 2 
 
 M 
 
 .050 
 
 20 
 
 a 
 
 .50 
 
 2OO 
 
 ( 
 
 5.00 
 
 3 
 
 tt 
 
 075 
 
 30 
 
 te 
 
 75 
 
 300 
 
 I 
 
 7-5 
 
 4 
 
 11 
 
 .IOO 
 
 40 
 
 a 
 
 I.OO 
 
 40O 
 
 t 
 
 IO.OO 
 
 5 
 
 It 
 
 .125 
 
 50 
 
 tt 
 
 1-25 
 
 500 
 
 ( 
 
 12.50 
 
 6 
 
 It 
 
 .150 
 
 60 
 
 it 
 
 1.50 
 
 1000 
 
 t 
 
 25.00 
 
 7 
 
 ft 
 
 175 
 
 70 
 
 a 
 
 i-75 
 
 2OOO 
 
 (( 
 
 50.00 
 
 8 
 
 it 
 
 .200' 
 
 80 
 
 tt 
 
 2.OO 
 
 3000 
 
 tt 
 
 75.00 
 
 9 
 
 tt 
 
 .225 
 
 9 
 
 <t 
 
 2.25 
 
 4OOO 
 
 ft 
 
 100.00 
 
 2. B's valuation $37850 $30000 + $7000 + $800 + $50, 
 
 By the table the tax on $30000=8750.00 
 
 " " 7000= 175.00 
 
 " " 800= 20.00 
 
 5= 5 
 
 Therefore, we have B's tax, =$946.25 
 
 Note. If a poll tax is included, how proceed ?
 
 TAXES. 297 
 
 3. Required C, D, E, and F's taxes, both by the rule 
 and the table. 
 
 4. A tax levied on a certain township was $16020; the 
 valuation of its taxable property was $784750, and the 
 number of polls assessed at $1.25 was 260. What was the 
 ' ue of tax ; and what was A's tax, who paid for 3 polls, 
 the valuation of his property being $7800 ? 
 
 5. The State levied a tax of $165945 upon a certain city 
 which contained 1260 polls assessed 75 cents each, an in- 
 ventory of $542.7600 real, and 72400 personal property. 
 What was G's tax, whose property was assessed at $15000 ? 
 
 6. What was H's tax, whose inventory was $10250, and 
 3 polls ? 
 
 7. A district school-house cost $2500, and the valuation 
 of the property of the district is $50000 : what is the rate, 
 and what A's tax, whose property is valued at $3400 ? 
 
 452. To find the Amount to be assessed, to raise a net 
 sum, and pay the Commission for collecting it. 
 
 8. A certain city required $47500 to pay expenses: what 
 amount must be assessed in order to cover the expenses, 
 and the commission of $% for collecting the tax ? 
 
 ANALYSIS. At 5$ for collection, $i assessment yields $.95; 
 therefore, to obtain $47500 net, requires as many dollars assessment 
 as $.95 are contained times in $47500; and $475OO-e-.95 (1 5^) = 
 $50000, the sum required to be assessed. (Art. 341.) Heiice, the 
 
 "RULE. Divide the net sum by i minus the rate; the 
 quotient will be the amount to be assessed. (Art. 348.) 
 
 9. What sum must be assessed to raise a net amount of 
 ^3500, and pay the commission for collecting, at 4%? 
 
 10. What sum must be assessed to raise a net sum of 
 $5260, and pay for the collection, at $\% commission ? 
 
 11. What sum must be assessed to raise a net amount 
 ^0500, and pay the commission for collecting, at 
 
 452. How find the amount to be assessed, to cover the sum to be raised and 
 the commission for collection ?
 
 DUTIES. 
 
 453. Duties are sums paid on imported goods, and 
 are often called customs. 
 
 454. A Custom House is a building where duties are re- 
 ceived, ships entered, cleared, etc. 
 
 455. A Fort of Entry is one where there is a Custom House. 
 The Collector of a Port is an officer who receives the duties, 
 
 has the charge of the Custom House, etc. 
 
 456. A Tariff is a list of articles subject to duty, stating the 
 rate, or the sum to be collected on each. 
 
 457. An Invoice is a list of merchandise, with the cost of the 
 several articles in the country from which they are imported. 
 
 458. Duties are of two kinds, specific and ad valorem. 
 
 459. A Specific Duty is a fixed sum imposed on 
 each article, ton, yard, etc., without regard to its cost. 
 
 An Ad Valorem Duty is a certain per cent on the 
 value of goods in the country from which they are 
 imported. 
 
 NOTE. The term ad valorem is from the Latin ad and valorem, 
 according to value. 
 
 460. Before calculating duties, certain allowances are made, 
 called tare, tret or draft, leakage, and breakage. 
 
 Tare is an allowance for the weight of the box, bag, cask, etc., 
 containing the goods. 
 
 Tret is an allowance in the weight or measure of goods for waste 
 or refuse matter, and is often called draft. 
 
 Leakage is an allowance on liquors in casks. 
 
 Breakage is an allowance on liquors in bottles. 
 
 NOTES. i. Tare is calculated either at the rate specified in the 
 invoice, or at rates established by Act of Congress. 
 
 2. Leakage is commonly determined by gauging the casks, and 
 Breakage by counting. 
 
 3. In making these allowances, if the fraction is lees than \ it is 
 rejected, if \ or more, i is added. 
 
 453. What are duties ? 454. A custom house? 455. What is a port of entry T 
 A collector ? 456. A tariff? 457. What is an invoice ? 458. Of how many 
 kinds are duties? 459. A specific duty? An ad valorem? 460. What ia tareT 
 Tret ? Leakage ? Breakage ?
 
 DUTIES. 299 
 
 PROBLEM I. 
 
 461. To calculate Specific Duties, the quantity of goods, 
 and the Sum levied on each article being given. 
 
 1. What is the specific duty on 12 casks of brandy, each 
 containing 40 gal., at $i- per gal., allowing 2% for leakage ? 
 
 ANALYSIS. 12 casks of 40 gallons each, =480.0 gal. 
 
 The leakage at z% on 480 gallons, = 9.6 gal. 
 
 The remainder, or quantity taxed, = 47-4 gal. 
 Now $1.50 x 470.4=$ 705.60, the duty required. Hence, the 
 
 RULE. Deduct the legal allowance for tare, tret, etc., 
 from the goods, and multiply the remainder by the sum 
 levied on each article. 
 
 2. What is the duty, at $1.25 a yard, on 65 pieces of 
 brocade silk, each containing 50 yards ? 
 
 3. What is the duty on 87 hhds. of molasses, at 20 cts. 
 per gallon, the leakage being 3% ? 
 
 4. What is the duty, at 6 cts. a pound, on 500 bags of 
 coffee, each weighing 68 Ibs., the tare being 2% ? 
 
 PROBLEM II. 
 
 462. To calculate Ail Valorem, Duties, the Cost of the 
 
 goods and the Rate being given. 
 
 5. What is the ad valorem duty, at 25$, on a quantity 
 of silks invoiced at $3500 ! J 
 
 ANALYSIS. Since the duty is 25$ ad valorem, $35 
 
 it is .25 times the invoice; and $3500 x .25 = 1875. .25 
 
 Hence, the Ans , $875.00 
 
 RULE. Multiply the cost of the goods by the given rate, 
 expressed decimally. 
 
 6. What is the duty, at ZZ\% a ^ valorem, on 1575 yds. 
 of carpeting invoiced at $1.80 per yard? 
 
 7. What is fhe ad valorem duty, at 40%, on no chests 
 of tea, each containing 67 Ibs., and invoiced at 90 cts. a 
 pound, the tare being 9 Ibs. a chest ? 
 
 461. How calculate specific duties when the quantity of goods and the sum 
 evied on eacli article are given ? 462. How ad valorem ?
 
 INTERNAL REVENUE. 
 
 463. Internal Revenue is the income of the 
 Government from Excise Duties, Stamp Duties, Licenses, 
 Special Taxes, Income Taxes, etc. 
 
 464. Excise Duties are taxes upon certain home productions,, 
 and are computed at a given per cent on their value. 
 
 465. Stamp Duties are taxes upon written instruments ; as, 
 notes, drafts, contracts, legal documents, patent medicines, etc. 
 
 466. A License Tax is the sum paid for permission to pursue 
 certain avocations. 
 
 467. Special Taxes are fixed sums assessed upon certain 
 articles of luxury; as, carriages, billiard tables, gold watches, etc. 
 
 Income Taxes are those levied upon annual incomes. 
 
 NOTE. In determining income taxes, certain deductions are made 
 for house rent, National and State taxes, losses, etc. 
 
 468. To compute Income Taxes, the Rate being given. 
 
 1. What is a man's tax whose income is $5675; the 
 rate being 5^', the deductions $1000 for house rent, $350 
 national tax, and -Si 100 for losses ? 
 
 ANALYSIS. Total deductions are $1000 + $350 + $iioo=$245o; 
 and $5675 $245O=$3225 taxable income. Now $% of $3225 = 
 3225 x .05 =$161.25, * ue t a x required^ Hence, the 
 
 KULE. From the income subtract the total deductions, 
 and multiply the remainder by the rate. 
 
 NOTE. If there are special taxes on articles of luxury, as carriages, 
 etc., they must be added to the tax on the income. 
 
 2. What was A's revenue tax for 1869, at 5% ; his income 
 being $4750; his losses $1185, and exemptions 1200 ? 
 
 3. In 1870, A had $10500 income; 35 oz. taxable plate 
 at 5 cts., i watch $2, and i carriage $2 : what was his tax 
 at 5%, allowing $2100 exemption? 
 
 463. What is internal revenue? 464. Excise duties? 463. Stamp duties? 
 466. A license tax? 467. Special taxes? Income taxes? 468. How compute 
 Income taxes, when the rate is given ?
 
 EQUATION OF PAYMENTS. 
 
 469. Equation of Payments is finding the average 
 time for payment of two or more sums due at different times. 
 
 The average time sought is often called the mean, or 
 ciuated time. 
 
 470. Equation of Payments embraces two classes of examples. 
 ist. Those in which the items or bills have the same date, but 
 
 different lengths of credit. 2d. Those in which they have different 
 dates, and the same or different lengths of credit. 
 
 PROBLEM I. 
 
 471. To find the Average Time, when the items have the 
 same date, but different lengths of credit. 
 
 i. Bought Oct. 3d, 1870, goods amounting to the fol- 
 lowing sums: $50 payable in 4 m., $70 in 6 m., and $80 in 
 8 m. : what is the average time at which the whole may be 
 paid, without loss to either party? 
 
 ANALYSIS. The int. on $50 for 4 m.=the int. $^0 X4=2OO 
 on $i for 50 times 4 m. or 200 m. Again the int. - o x 5 420 
 on $70 for 6 m.=.the int. on $i for 70 times 6 m., g Q x 5 = 640 
 or 420 in. Finally, the int. on $80 for 8 m.=the . 
 
 int. of $i for 80 times 8 m., or 640 m. Now 200 
 m. + 42o m. 4-640 m. = i26o m. ; therefore I am Ans. 6-j^ m. 
 entitled to the use of $i for 1260 m. But the sum 
 of the debts is $50 + $70 + $80= $200. Now as I am entitled to the 
 use of $i for 1260 m., 1 must be entitled to the use of $200 for ^ part 
 of 1260 in., and i26o-^2oo=6, 3 o m., or 6 m. 9 d., the average time re- 
 quired. Hence, the 
 
 RULE. Multiply each item by its length of credit, and 
 ill ride the sum of the products by the sum of the items. 
 The quotient will be the average time. 
 
 NOTES. i. When the date of the payment is required, add the 
 average time to the date of the transaction. Thus, in the preceding 
 Ex., the date of payt. is Oct. 3d + 6 m. 9 d., or April i2th, 1871. 
 
 469. What is equation of payments ? Note. What is the average time called T 
 470. How many classes of examples in Equation of Payments? 471. How find 
 the average time, when the Items have the same date, but different credit*
 
 302 EQUATION OF PAYMENTS. 
 
 2. This rule is applicable to notes as well as accounts. It ia 
 founded upon the supposition that bank discount is the same aa 
 simple interest. Though not strictly accurate, it is in general use. 
 
 3. If one item is cash, it has no time, and no product; but in find- 
 ing the sum of items, this must be added with the others. 
 
 4. In the answer, a fraction less than day, is rejected ; if | day 
 or more, i day is added. 
 
 2. Bought a house June 2oth, 1 870, for $3000, and agreed 
 to pay \ dpwn; | in 6 m., the balance in 12 m. : at what 
 date may the whole be equitably paid ? 
 
 3. A owes B $7ocy payable in 4 mos. ; $500, in 6 mos. ; 
 $800, in 10 mos.; and $1000 in 12 mos.: in what time 
 may the whole be justly paid? 
 
 4. Bought a bill of goods March roth, 1868, amounting 
 to $2500; and agreed to pay $500 cash, $750 in 10 days, 
 $600 in 20 days, $400 in 30 days, and $250 in 40 days. 
 At what date may I equitably pay the whole ? 
 
 5. A jobber sold me on the ist of March $12000 worth 
 of goods, to be paid for as follows : J cash, in 2 mos., in 
 4 mos., and the remaining - in 6 mos. When may 1 pay 
 the whole in equity ? 
 
 PROBLEM II. 
 
 472. To find the Average Titnc, when the items have 
 different dates, and the same or different lengths of credit. 
 
 6. Bought the following bills of goods: March loth, 
 1870, $500 on 2 m. ; April 4th, $800 on 4 m. ; June i5th, 
 1 1 ooo on 3 m. What is the average time? 
 
 ANALYSIS. We first find Mayio, oo x $500= oooo d. 
 when the items are due by A ^ 86 x $ 8oo - 68800 (1. 
 adding the time of credit to g t J2 g x $ IOOO = 128000 (1. 
 the date of each, and for con- -r - . 7-5- 
 
 venience place these dates 
 
 in a column. Taking the Average time, &5-11 d. 
 
 earliest date on which cither 
 
 item matures as a standard, we find the number of days from this 
 standard date to the maturity of the other items, and place them on 
 the right, with the sign ( x ) and the items opposite.
 
 EQUATION OF PAYMENTS. 303 
 
 Multiplying each item and its number of days together, the sum 
 of the products shows that the interest on the several items is equal 
 to the interest of $i for 196800 days. 
 
 Now if it takes 196800 days for $i to gain a certain sum, it will 
 take $2300, jAiu of 196800 d. to gain this sum ; and 196800-^-2300= 
 85^, or 86 days from the assumed date. Now May 10 + 86 d.=Aug. 
 4th, the date when the amt. is equitably due. Hence, the 
 
 RULE. I. Find the date when the several items become 
 due, and set them in a column. 
 
 Take the earliest of these dates as a standard, and set the 
 number of days from this date to the maturity of the other 
 items in another column on the right, with the items opposite. 
 
 II. Multiply each item by its number of days, and divide 
 the sum of the products by the sum of the items. The 
 quotient will be the average time of credit. 
 
 NOTES. i. Add the average time to the standard date, and the 
 result will be the equitable date of payment. 
 
 2. The latest date on which either item falls due may also be 
 taken as the standard ; and having found the average time, subtract 
 it from this date ; the result will be the date of payment. 
 
 3. The date at which each item becomes due, is readily found by 
 adding its time of credit to the date of the transaction. 
 
 7. A bought goods as follows: Apr. loth, $310 on 6 m. ; 
 May zist, $468 on 2 m. ; June ist, $520 on 4 m. ; July 8th, 
 6750 on 3 m. : what is the average time, and at what date 
 may the whole debt be equitably discharged ? 
 
 SUGGESTION. The second item matures earliest; this date is 
 therefore the standard. Ans. Av. time 59 d. ; date Sept. i8th. 
 
 8. Sold the following bills on 6 months credit: Jan. 
 i5th, $210; Feb. nth, $167; March 7^1,1320.25; April 
 2d, 8500.10: when may the whole be paid at one time? 
 
 9. Bought June 5th, on 4 m., groceries for $125 ; June 
 nst, $230.45; July 1 2th, $267; Aug. 2d, $860.80: what 
 is the amt. and time of a note to cover the whole ? 
 
 472. How find the average tiaie, when the items have different dates, and the 
 same or different credits ? Note. How find the date of the payment ? How find 
 the date at which each item becomes due ?
 
 304 
 
 AVERAGING ACCOUNTS. 
 
 AVERAGING ACCOUNTS. 
 
 473. An Account is a record of the items of debit 
 and. credit in business transactions. 
 
 NOTE. The term debit is from the Latin detritus, owed. 
 
 474. A Merchandise Balance is the difference 
 between the debits and credits of an account. 
 
 A Cash Balance is the difference between the debits 
 and credits, with the interest due on each. 
 
 NOTES. i. Bills of goods sold on time are entitled to interest 
 after they become due ; and payments made before they are due PTC 
 also entitled to interest. 
 
 475. To find the Average Time for paying the balance of 
 an Account which has both debits and credits. 
 
 i. Find the cash Bal. of the following Acct., and when due. 
 Dr. WM. GORDON in Acct. with JOHN KANDOLPH. Cr. 
 
 1870. 
 
 Feb. 10. 
 May ii. 
 
 July 26. 
 
 ForMdse., 4 m. 
 
 :: :: \: 
 
 $450.00 
 500.00 
 360.00 
 
 1870. 
 
 Mar. 20. 
 July 9. 
 Sept. 15. 
 
 By Sundries, 3 m. 
 " Draft, 60 d. 
 " Cash, 
 
 $325-00 
 150.00 
 400.00 
 
 ANALYSIS. Setting 
 down the date when each 
 item is due.take June loth, 
 the earliest of these dates, 
 as the standard, find the 
 number of days from this 
 standard to the maturity 
 of each item on both sides, 
 and place it on the right 
 of its date with the sign 
 ( x ), then the items, add- 
 ing 3 days grace to the 
 time of the draft. 
 
 Debits. 
 
 June 10, oo d. x $450 = ooooo d. 
 Aug. n, 62 d. x 500 = 31000 d. 
 Sept. 26, 108 d. x 360 = 38880 d. 
 
 Amt. debits, 81310, Int. 69880 d. 
 
 Credit*. 
 
 June 20, 10 d. x$325 = 3250 d. 
 
 Sept. 10, 92. d. x 150 = 13800 d. 
 
 " J 5> 97 d- x 400 = 38800 d. 
 
 Amt. credits, $875,1^.55850 d. 
 
 Cash bal. $435, " 14030 d. 
 
 $435)14030 d. = 32 d.+. 
 Ans. Bal. $435, due July 12, '70. 
 
 473. What is an account? Note. What is the meaning of the term debit J 
 . What is a merchandise balance f A cash balance ?
 
 AVERAGING ACCOUNTS. 305 
 
 Multiplying each item on both sides by its number of days, 
 the int. on the debits is equal to the int. of $i for 69880 days, 
 and the int. on the credits is equal to the int. of $i for 55850 days. 
 (Art. 474.) The balance of int. on the Dr. side is 69880 d. 55850 d. 
 = 14030 d. ; that is to the int. of $i for 14030 d. The balance of 
 items on the Dr. side is $1310 $875=$435. 
 
 Now if it takes 14030 days for $i to gain a certain sum, it will 
 take $435, 4 h- of 14030 d. and 14030-^-435=3215, or 32 d. But the 
 assumed standard June ioth + 32 d.=July i2th. Therefore the 
 balance due Randolph the creditor is $435, payable July I2th, 1870. 
 
 If the greater sum of items and the greater sum of products were 
 n opposite sides, it would be necessary to subtract the average time 
 from the assumed date. Hence, the 
 
 RULE. I. Set down the date when each item of debit and 
 credit is due; and assuming the earliest of these dates as a 
 standard, write the number of days from this standard 
 date to the maturity of the respective items, on the right, 
 with the sign ( x ) and the items themselves opposite. 
 
 II. Multiply each item by its number of days, and divide 
 the difference between the sums of products by that between 
 the sums of items ; the quotient will be the average time. 
 
 III. If the greater sum of items and the greater sum of 
 products are both on the same side, add the average time to 
 the assumed date; if on opposite sides, subtract it; and 
 the result will be the date when the balance of the account is 
 equitably due. 
 
 NOTES. i. The average time may be such as to extend to a date 
 either earlier or later than that of any of the items. (Ex. 2.) 
 
 2. In finding the maturity of notes and drafts, 3 days grace should 
 be added to the specified time of payment. 
 
 3. In finding the extension to which the balance of a debt is 
 entitled, when partial payments are made before it is due, 
 
 Multiply each payment by the time from its date to the maturity 
 of the debt, and divide the sum of the products by the balance 
 remaining unpaid. 
 
 4. When no time of credit is mentioned, the transaction is under- 
 stood to be for cash, and its payment due at once. 
 
 475. How find the average time for paying the balance, when there are both 
 debits and credits T
 
 306 
 
 AVERAGING ACCOUNTS. 
 
 2. Find the balance of the following Acct., and when due. 
 Dr. A. B. in account with C. D. Cr. 
 
 1860. 
 
 
 
 1860. 
 
 
 
 Aug. ii. 
 
 For Mdse., 
 
 $160.00 
 
 Sept. 2. 
 
 By Sundries, 
 
 $75-oo 
 
 Sept. 5. 
 
 " " 
 
 240.00 
 
 Oct. 10. 
 
 " your Note on 30 d., 
 
 100.00 
 
 Oct. 20. 
 
 " i Horse, 
 
 175-00 
 
 Nov. i. 
 
 " Cash 
 
 110.00 
 
 Dr. 
 
 OPERATION. 
 
 Cr. 
 
 Due. 
 
 Days. 
 
 Items. 
 
 Products. 
 
 Due. 
 
 Days. 
 
 Items. 
 
 Products. 
 
 Aug. II. 
 
 
 
 $160 
 
 0000 
 
 Sept. 2. 
 
 22 
 
 $75 
 
 1650 
 
 Sept. 5. 
 
 25 
 
 240 
 
 6000 
 
 Nov. ) 
 
 93 
 
 100 
 
 9300 
 
 Ot. 20. 
 
 70 
 
 175 
 
 12250 
 
 Nov. i. 
 
 82 
 
 no 
 
 9020 
 
 
 
 575 
 
 18250 
 
 
 
 285 
 
 19970 
 
 
 
 ^?1 
 
 1720-4-290=6 d. (nearly), av. tune. 
 
 18250 
 
 
 Bal=$2go 
 
 Aug. ii 6d.=Aug. 5,bal.isdue. 
 
 1720 
 
 SUGGESTION. In this example, the greater sum of items and the 
 greater sum of products are on opposite bides ; hence, the average 
 time must be subtracted from the standard date. 
 
 3. A man bought a cottage for $2 7 50, payable in i year; 
 in 3 mos. he paid $500, and 3 mos. later $750: to what 
 extension is he entitled on the balance ? 
 
 SOLUTION. The sum of the product is 9000 ; and the balance of 
 the debt $1500. Now 9OOO-7-i5oo=:6. Ans. 6 mcs. after the maturity 
 of the debt. (Note 3.) 
 
 4. A merchant sold a bill of $4220 worth of goods on 
 8 mos.; 2 months after the customer paid him $720, one 
 month later $850, and 2 months later $1000: how long 
 should the balance in equity remain unpaid ? 
 
 5. "What is the balance of the following account, and 
 when is it due ? 
 
 Dr. HENRY SWIFT in Acct. with HOMER MORGAN. Cr. 
 
 \ 1865. 
 
 
 
 1865. 
 
 
 
 March 10. 
 
 For Sundries, 
 
 $2 SO 
 
 ! April i. 
 
 By Bal. of Acct., 
 
 Sno 
 
 JApril 15. 
 
 " Flour on 60 d., 
 
 420 
 
 iMav 21. 
 
 " Dft, onsod.. 
 
 300 
 
 'June 20. 
 
 " Mdse. on 30 d., 
 
 600 
 
 July i. 
 
 " Cash, 
 
 560'
 
 RATIO. 
 
 476. Ratio is the relation which one number has to 
 another with respect to magnitude. 
 
 The Terms of a ratio are the numbers compared. 
 They are often called a couplet. 
 
 477. Ratio is commonly expressed by a colon (:) placed between 
 the two numbers compared. Thus, the ratio of 6 to 3 is written 6 :^, 
 
 478. The first term is called the antecedent, the second the 
 consequent. 
 
 The comparison is made by considering what multiple or part the 
 antecedent is of the consequent. 
 
 NOTES. i. The sign of ratio (:) is derived from the sign of 
 division (-=-), the horizontal line being dropped. 
 
 2. The terms are so called from the order of their position. They 
 must be of the same kind or denomination; otherwise they cannot 
 be compared. 
 
 479. Ratio is measured by & fraction, the numerator of which is 
 the antecedent, and the denominator the consequent; or what is the 
 same thing, by dividing the antecedent by the consequent. 
 
 480. Th value of a ratio is the value of the fraction by which 
 it is measured. Thus, comparing 6 with 2, we say the ratio of 6 : 2 
 is 5, or 3. That is, the former has a magnitude which contains tho 
 latter 3 times ; therefore the value of the ratio 6 : 2 is 3. 
 
 481. A Simple Hatio is one which has but two 
 
 terms ; as, 8 : 4. 
 
 A Compound Hatio is the product of two or more 
 simple ratios. 
 
 Thus : 4 : 2 \ are simple But 4x9:2x3 
 
 9:3) ratios. is a compound ratio. 
 
 NOTE. The nature of compound ratios is the same as that of 
 simple ratios. They are so called to denote their origin, and are 
 usually expressed by writing the corresponding terms of the simple 
 
 ratios one under another, as above. 
 
 \ 
 
 476. What is ratio? The numbers compared called ? 477. How is ratio com- 
 monly expressed ? The first term called ? The second ? Note. Why ? 479. How 
 is ratio measured ? The value of a ratio ? 481. Simple ratio? Compound?
 
 308 RATIO. 
 
 482. Ratio is also distinguished as direct and inverse or reciprocal. 
 
 A direct ratio is one which arises from dividing the antecedent by 
 the consequent. 
 
 An inverse ratio is one which arises from dividing the consequent 
 by the antecedent, and is the same as the ratio of the reciprocals 
 of the two numbers compared. (Art. 106.) Thus, the direct ratio 
 of 4 to i2=-,*2, or ; the inverse ratio of 4 to I2= i 4 i , or 3. It is the 
 same as the ratio of their reciprocals, \ to -f^. , 
 
 483. 1'he ratio between two fractions having a common denomi- 
 nator is the same as the ratio of their numeratort. Thus, the ratio 
 of : 3 is the same as 6 : 3. 
 
 In finding the ratio of two fractions which have different denomi- 
 nators, they should be reduced to a common denominator; then take 
 the ratio of their numerators. (Art. 153.) 
 
 484. I n finding the ratio between two compound numbers, they 
 must be reduced to the same denomination. 
 
 NOTE. Finding the ratio between two numbers is the same in 
 principle as finding what part one is of the other, the number 
 denoting the part being the antecedent. Thus, 2 is J of 4=^ ; and 
 the ratio of 2 to 4 is f =|. (Art. 173.) 
 
 485. Since ratios are measured by fractions whose 
 numerators are the antecedents, and denominators the 
 consequents, it follows that operations have the same effect 
 upon the terms of a ratio as upon the terms of a fraction. 
 (Art. 144.) That is, 
 
 1. Multiplying the antecedent multiplies the ratio; and 
 dividing the antecedent divides the ratio. 
 
 2. Multiplying the consequent divides the ratio; and 
 dividing the consequent multiplies the ratio. 
 
 3. Multiplying or .dividing loth the antecedent and con- 
 sequent by the same number, does not alter the value of the 
 ratio. 
 
 482. Direct? Inverse? 483. The ratio of two fractions having a common 
 denominator ? 484. In finding the ratio of two compound numbers, what must 
 be done? 485. What effect do operations on the terms of a ratio have? Multi- 
 plying the antecedent ? Dividing it ? Multiplying the consequent ? Dividing it ? 
 What effect has multiplying or dividing both the antecedent nd consequent by 
 the same number ?
 
 PBOPORTIOtf. 309 
 
 What are the ratios of the following couplets : 
 
 1. 12:4. 4. 6 : 24. 7. 5 : ios. 6d. 
 
 2. 28:7. 5- 8:40. 8. 10 y. : 6ft. 3 in. 
 3.36:12. -6.9:51. 9. 25 g. : 2 qt. i pt. 
 
 10. Reduce the ratio of 14 to 35 to the lowest terms? 
 SOLUTION. 14 : 35 equals ^ ; and i*=f, or 2 : 5. (Art. 146.) 
 Reduce the following ratios to the lowest terms ? 
 
 11. 154:28. -T3. 73:5ii./> 15. 238:1428. 
 
 12. 39:165. 14. 113:1017. , 16. 576:1728. 
 
 17. Reduce the ratio :| to the lowest integral terms? 
 |=iJ. and $ =f. Now iJ : J is the same as 4 : 5. (Art. 483.) 
 
 18. Multiply the ratio of 21 : 7 by 4 : 8. Ans. 84 : 56, or i. 
 
 19. What is the value of 5:8x4: 10x7:9? 
 
 PROPORTION. 
 
 486. Proportion is an equality of ratios. 
 
 487. Every proportion must have at least four terms , 
 for, the equality is between two or more ratios, and each 
 ratio has tiuo terms, an antecedent and a consequent. 
 
 A proportion may, however, be formed from three num- 
 bers; for, one of the numbers may be repeated, so as to 
 form two terms. 
 
 488. Proportion is denoted in two ways ; by a double 
 colon ( : : ), and by the sign of equality ( = ), placed between 
 the ratios. Thus, each of the expressions 4 : 2 : : 6 : 3, and 
 4 : 2 = 6 : 3 indicates a proportion ; for, f=|. 
 
 The former is read, " 4 is to 2 as 6 to 3," or " 4 is the 
 same part of 2, that 6 is of 3." The latter is read, " t he- 
 ratio of 4 to 2 equals the ratio of 6 to 3." 
 
 NOTE. The sign (::) is derived from the sign (=), the points 
 being the extremities of the parallel lines. 
 
 4 S6. What is proportion ? 4 S 7 . How many terms has every proportion ? Can 
 three numbers form a proportion ? How ?
 
 tit PROPORTION. 
 
 489. The four numbers which form a proportion, are 
 called proportionals. The first and last arc the extremes, 
 the other two the means. 
 
 When a proportion has but three numbers, the second 
 term is called a mean proportional between the other two. 
 
 490. If four numbers are proportional, the product of 
 the extremes is equal to the product of the means. Hence, 
 
 491. The relation of the four terms of a proportion to 
 each other is such, that if any three of them are given, the 
 other or missing term may be found. 
 
 492. To find the Missing Term of a Proportion, the other 
 three Terms being given. 
 
 1. Let 6 be the first term of a proportion, 3 and 10 the two 
 means; the other extreme equals 3x10-5-6=5; for, the product 
 of the means=the product of the extremes; and the product of two 
 factors divided by one of them, gives the other. (Art. 93.) 
 
 2. Let 3, 10 and 5 be the last three terms of a proportion, the first 
 term equals 3 x 10-5-5=30-5-5 or 6. 
 
 3. Let 6 and 5 be the extremes of a proportion, and 3 one of the 
 means ; the other mean equals 6 x 5-5-3=30-5-3 or 10. 
 
 4. If 6 and 5 are the extremes, and 10 one of the means, the other 
 mean equals 6 x 5-5-10=30-5-10 or 3. Hence, the 
 
 KTJLE. I. If one of the extremes and the two means are 
 given, divide the product of the means by the given extreme. 
 
 II. If one of the means and the two extremes are given, 
 divide the product of the extremes ly the given mean. 
 
 Find the missing term in the following proportions : 
 
 1. 52:13: : 62 : . 5. 4 rods : 1 1 ft. : : 18 men : . 
 
 2. 15:90:: '.72. 6. 24yd. : 3 yd. :: :$i2. 
 
 3. 60 : : : 100 : 33^. 7. 20 gal. : : : $40 : $8. 
 
 4. : 25 : : : . 8. : 40 Ib. : : 2 : 8s. 
 
 489. What are the four numbers forming a proportion called? 491. If three 
 terms of a proportion arc given, what is true of the fourth ? If the two means 
 and one extreme arc given, how find the other extreme f If the two extremes 
 and one mean are given, how find the other mean ? 
 
 -f i
 
 1MPL1 
 
 493. Simple Proportion is an equality of two 
 simple ratios. 
 
 Simple Proportion is applied chiefly to the solu- 
 tion of problems having three terms given to find & fourth, 
 of which the third shall be the same multiple or part, as 
 i\\Q first is of the second. 
 
 494. To solve Problems by Simple Proportion. 
 
 1. If 5 baskets of peaches cost $10, what will 3 baskets cost? 
 
 ANALYSIS. The question assumes STATEMENT. 
 
 that 3 baskets can be bought at the 5 bas. : 3 bas. : : $10 rAns. 
 same rate as 5 baskets ; therefore 3 
 
 5 bas. lias the same ratio to 3 bas. 5)^30 
 
 as the cost of 5 bas. has to the cost of *7 . 
 
 3 bas. That is, 5 bas. : 3 bas. : : $10 : 
 
 cost of 3 bas. We have then the two means and one extreme of a 
 proportion to find the fourth term, or other extreme. (Art. 492.) 
 Now the product of the means $10 x 3 =$30 ; and $30-1-5 (the other 
 extreme) =$6, the cost of 3 baskets. Hence, the 
 
 RULE. I. Take that number for the third term, which is 
 the same kind as the answer. 
 
 II. When the answer is to be larger than the third term, 
 place the larger of the other tivo numbers for the second 
 term; but ivhen less, place the smaller for the second term, 
 and the other for the first. 
 
 III. Multiply the second and third terms together, and 
 divide the product by the first; the quotient will be the 
 fourth term or answer. (Arts. 490, 491.) 
 
 PROOF. If the product of the first and fourth terms 
 equals that of the second and third, the answer is right. 
 
 NOTES. i The arrangement of the given terms in the form of a 
 proportion is called " Stating the question." 
 
 2. After stating the question, the factors common to the first and 
 second, or to the first and third terms, should be cancelled. 
 
 493. Simple Proportion ? 4 >4. How solve problems by Simple Proportion ?
 
 312 SIMPLE PBOPORTION. 
 
 3. If the first and second terms contain different denominations, 
 they must be reduced to the same. If the third term is a compound 
 number, it must be reduced to the lowest denomination it contains. 
 
 495. REASONS. i. The reason for placing that number for the 
 third term, which is the same kind as the answer, and the other two 
 numbers for the first and second, is because money has a ratio to 
 money, but not to the other two numbers ; and ihe^ther two Humbert 
 have a ratio to each other, but not to money. 
 
 2. Of the two like numbers, the smaller is taken for the second 
 term, and the larye.r for the first, because 3 baskets being less than 
 5 baskets, will cost less ; consequently, the answer or fourth term 
 must be less than the third, the cost of 5 baskets. 
 
 3. If it were required to find the cost of a quantity greater than 
 that whose cost is given, the answer would be greater than the third 
 term ; consequently the greater of the two similar numbers must 
 then be taken for the second term, and the less for the first. 
 
 4. The reason for multiplying the second and third terms together 
 and dividing the product by the first, is because the product of tht 
 means divided by one of the extremes, gives the other extreme of 
 ansicer. (Arts. 93, 492.) 
 
 2. If 9 yards of cloth cost $54, what will 23 yards cost ? 
 Statement. 9 yd. : 23 yd. : : $54 : Ans. 
 
 And ($54 x 23)-5-9=$i38, the Ans. 
 
 By Cancellation. Since 9 yds. cost OPERATION. 
 
 $54, i yd. will cost i of $54, or $\ A ; $^ X 2 3 = Ans. 
 
 and 23 yds. will cost $y * 23=-^-^ 6, 54 x 23 _ 
 
 9 -r $138, A.THS. 
 
 = $138, Ans. 
 Proof. 9 yd. : 23 yd. : : $54 : $138 ; for 9 x 138=23 x 54. 
 
 3. If 7 barrels of flour cost $56, what will 20 barrels cost? 
 
 4. What cost 75 bushels of wheat, if 15 bushels cost $33? 
 
 5. "What cost 150 sheep, if 17 sheep cost $51 ? 
 
 6. If 5 Ib. 8 oz. of honey cost $1.65, Avhat will 20 Ib. cost ? 
 
 7. Paid i, 153. 6d. for 6 pounds of tea: what must be 
 paid for a chest containing 65 Ib. 8 oz. ? 
 
 495. Why take the number which is of the same kind as the answer for 
 the third term, and the other two for the first and second? When place tin- 
 larger of the other two numbers for the second? Why? When the smaller? 
 Why? Why does the product of the second and third terms divided by the first 
 pive the answer? What is the arrangement of the terms in the form of a propor- 
 tion called ? If the first and second terms contain different denominations, how 
 proceed ? If the third is a compound number, how ?
 
 SIMPLE PROPORTION. 313 
 
 SIMPLE PROPORTION BY ANALYSIS. 
 
 496. The chief difficulty experienced by the pupil in Simple 
 Proportion, lies in " stating the question." This difficulty arises 
 from a wxnt of familiarity with the relation of numbers. He will be 
 assisted by analyzing the examples before attempting to state them. 
 
 8. If 7 bats cost $42, how much will 12 hats cost? 
 ANALYSIS. i hat is i S3vcnth of 7 hats ; therefore i hat will 
 
 cost i seventh as much as 7 hats ; and \- of $42 is $6. Again, 12 hats 
 will cost 12 times as much as i hat, and 12 times $6 are $72. There- 
 fore, 12 hats will cost 72. 
 
 Or, 7 hats are ?-, of 12 hats; therefore the cost of 7 hats is -, T 2 the 
 cost of 12 hats. But 7 hats cost $42 ; hence $42 are -^ the cost of 
 12 hats. Now, if -^ of a number arc $42, |\, of that number is J 7 - of 
 842, which is $6 ; and jf are 12 times 6, or 72. 
 
 By Proportion. 7 h. : 12 h. : : $42 : Ans. That is, 
 7 h. are the same part of 12 h. as $42 arc of the cost of 12 hats. 
 
 497. Solve the following examples both by analysis avid 
 proportion. 
 
 9. If ii men can cradle 33 acres of grain in i day, how 
 many acres can 45 men cradle in the same time ? 
 
 10. When mackerel are $150 for 12 barrels, what must 
 I pay for 75 barrels ? 
 
 11. How far will a railroad car go in 12 hours, if ii 
 goes at the rate of 15 miles in 40 minutes ? 
 
 12. A bankrupt owes $3500, his assets are $1800: how 
 much will a creditor receive whose claim is $560 ? 
 
 13. At the rate of 18 barrels for $63, what will 235 bar- 
 rels of apples cost ? 
 
 14. If $250 earns 817^ interest in i year, how much 
 will $1900 earn in the same time? 
 
 15. If a car wheel turns round 6 times in 33 yards, how 
 many times will it turn round in going 7 miles ? 
 
 16. If a clerk can lav up 81500 in i- year, how long will 
 It take him to lay up $5000 ? 
 
 17. If 6 men can hoe a field of corn m 20 hours, ho\? 
 long will it take 15 men to hoe it? 
 
 14
 
 314 SIMPLE P 11 OP OUT I ON. 
 
 1 8. An engineer found it would take 75 men 220 days 
 to build a fort; the general commanding required it to he 
 built in 15 days: how many men must the engineer 
 employ to complete it in the required time ? 
 
 19. If 5 oz. of silk can be spun into a thread 100 rods 
 long, what weight of silk is required to spin a thread that 
 will reach the moon, 240000 miles distant ? 
 
 20. How many horses will it take to consume a scaffold 
 of hay in 40 days, if 1 2 horses can consume it in 90 days ? 
 
 21. If | acre of land costs $15, what will 25^ acres cost ? 
 
 22. If f of a ton of iron costs f , what will ^- of a ton cost ? 
 
 23. If io Ib. sugar cost $if, what will 30^ Ib. cost ? 
 
 24. If | of a chest of tea costs $35.50, what will 15^ 
 chests cost ? 
 
 25. What will 48! tons of hay cost, if 1 2^ tons cost $126^ ? 
 
 26. If ^ of a ship is worth $16250!, what is T 3 g- of it worth ? 
 
 27. What will it cost me for a saddle horse to go 100 
 miles if I pay at the rate of 37-^ cts. for 3 miles ? 
 
 28. What must be the length of a slate that is 10 in. 
 wide, to contain a square foot ? 
 
 29. How many yards of carpeting f yard wide will it 
 take to cover a floor 15 ft. long and 12 feet wide ? 
 
 30. If a man's pulse beats 68 times a minute, how mam 
 times will it beat in 24 hours ? 
 
 31. If Halley's comet moves 2 45' in u hours, how far 
 will it move in 30 days ? 
 
 32. If a pole 10 ft. high cast a shadow i\ ft. long, how 
 high is a flag-staff whose shadow is 60 ft. long ? 
 ""JijTAt the rate of 3 oranges for 7 apples, how many 
 oranges can be bought with 150 apples? 
 
 34. If an ocean steamer runs 1250 miles in 3 days 8 h., 
 how far will she run in 8 days? 
 
 35. If 12 men can harvest a field of wheat in n days, 
 how many men are required to harvest it in 4 days ? 
 
 36. The length of a croquet-ground is 45 feet ; and its 
 width is to its length as 2 to 3 : what is its width ?
 
 SiilPLE PROPORTION. ?15 
 
 37. A man's annual income from U. S. 6s is $1350 when 
 gold is 1 1 2 \ : what was it when gold was 1 60 ? 
 
 38. George has 10 minutes start in a foot-race; and 
 runs 20 rods a minute : how long will it take Henry, who 
 runs 28 rods a minute, to overtake him? 
 
 39. If the driving wheel of a locomotive makes 227 
 revolutions in going 206 rods 6 ft., how many revolutions 
 will it make in running 18 miles 240 rods? 
 
 40. If 3 Ibs. of coffee cost $1.20, and 10 Ibs. of coffee are 
 worth 6 Ibs. of tea,, what will 60 Ibs. of tea cost ? 
 
 3TT"37can chop a cord of wood in 4 hours, and B in 6 
 hours : how long will it take both to chop a cord ? 
 
 42. A reservoir has 3 hydrants; the first will empty it 
 in 8 hours, the second in 10; the third in 12 hours: if all 
 run together, how long will it take to empty it ? 
 
 43. A man and wife drank a keg of ale in 18 d. ; it would 
 last the man 30 d. : how long would it last the woman ? 
 
 44. A fox is 100 rods before a hound, but the hound 
 runs 20 rods while the fox runs 18 rods: how far must 
 the hound run before he catches the fox ? 
 
 45. A cistern holding 3600 gallons has a supply and a 
 discharge pipe ; the former runs 45 gallons an hour, the 
 latter 33 gallons : how long will it take to fill the cistern, 
 when both are running? 
 
 46. A clerk who engaged to work for 8500 a year, com- 
 menced at 12 o'clock Jan. ist, 1869, and left at noon, the 
 2ist of May following: how much ought he to receive? 
 
 47. A church clock is set at 12 o'clk. Saturday night; 
 Tuesday noon it had gained 3 min. : what will be the true 
 time, when it strikes 9 the following Sunday morning? 
 
 48. A market-woman bought 100 eggs at 2 for a cent, 
 and another 100 at 3 for a cent; if she sells them at the 
 rate of 5 for 2 cents, what will she make or lose ? 
 
 49. Two persons being 336 miles apart, start at the same 
 time, and meet in 6 days, one traveling 6 miles a day 
 faster than the other: how far did each travel?
 
 COMPOUND PROPORTION. 
 
 !2 ) 
 
 > : : 1 2 : 2, is a compound proportion. 
 
 * O J 
 
 498. Compound Proportion is an equality of 
 a jompound and a simple ratio. Thus, 
 
 4 : 2 
 
 9 
 It is read, " The ratio of 4 into 9 is to 2 into 3 as 12 to 2." 
 
 i. If 4 men saw 20 cords of wood in 5 days, how many 
 cords can 1 2 men saw in 3 days ? 
 
 ANALYSIS. The answer is to STATEMENT. 
 
 be in cords ; we therefore make 4 m. : 1 2 m. ) 4 , 
 
 20 c. the third term. The other 5 d. : 3 d. J ' 
 given numbers occur in pairs, two 20 C. X 12 X 3 =: 720 C. 
 of a kind ; as, 4 men and 12 men, 4x5 = 20 
 
 5 days and 3 days. We arrange 72OC.-J-2O 36 C. Ans. 
 theso pairs in ratios, as we should 
 
 in simple proportion, if the answer depended on each pair alone. 
 That is, since 12 ra. will saw more than 4 men, we take ~2 for the 
 second term and 4 for the first. Again, since 12 men will saw lens 
 in 3 days than in 5 days, we take 3 for the second term and 5 for the 
 first. Finally, dividing the product of the second and third terms 
 20 c. x 12 x 3=720 c. by the product of the first terms 4 x 5=20, wo 
 have 36 cords for the answer. Hence, the 
 
 EULE. I. Make that number tho third term which is of 
 the same kind as tho answer. 
 
 II. Taka the other numbers in pairs of the same kind, 
 and arrange them as if the answer depended on each couplet, 
 as in simple proportion. (Art. 494.) 
 
 III. Multiply the second and third terms together, and 
 divide the product by the product of the first terms, cancelling 
 the factors common to the first and second, or to the first 
 and third terms. Tlie quotient will be the answer. 
 
 PROOF. If the product of the first and fourth terms 
 equals that of the second and third terms, the work is ri'jht. 
 
 498. What is Compound Proportion 1 Explain the first example. The rule. 
 Proof. Note. How proceed when the first and second terms contain different 
 denominations ? When the third does ? How else may questions ill Compound 
 Proportion he solved 1
 
 COMPOUND PROPORTION. 
 
 317 
 
 NOTES. i. The terms of each couplet in the compound ratio mu,.^ 
 be reduced to the same denomination, and the third term to the lowest 
 denomination contained in it, as in Simple Proportion. 
 
 2. In Compound Proportion, all the terms are given in couplets or 
 pairs of the srtme kind, except one. This is called the odd term, or 
 demand, and is always the same kind as the answer. 
 
 3. It should be observed that it is not the ratio of 4 to 2, nor of 9 to 
 3 alone that equals the ratio of 12 to 2 ; for, 4-4-2 = 2 and 9-7-3=3, 
 while 12-5-2=6. But it is the ratio compounded of 4 x 9 to 2x3, 
 which equals the ratio of 12 to 2. Thus, (4 x g)-^-(2 x 3)=6 ; and 
 i2-=-2=6. (Art. 498.) 
 
 4. Compound Proportion was formerly called " Double Rule of 
 Three." 
 
 499. Problems in Compound Proportion may also be solved by 
 Analysis and Simple Proportion. Take the preceding example : 
 
 By Analysis. If 4 men saw 20 cords in 5 d., i m. will saw ^ of 
 20 c., which is 5 c., and 12 m. will saw 12 times 5 c., or 60 cords, in 
 the same time. Again, if 12 men saw 60 c. in 5 days, in i d. they 
 will saw ^ of 60 c , or 12 cords, and in 3 d. 3 times 12 c., or 36 cords, 
 the answer required. 
 
 By Simple Proportion. 4 m. : 12 m. : : 20 c. : the cords 12 m. will 
 saw in 5 d. ; and 20 c. x 12-1-4=60 c. in 5 days. Again, 5 d. : 3 d. : : 
 60 c. : the cords 12 men will saw in 3 d. And 60 c. x 3-7-5=36 cords, 
 the same as before. 
 
 2. If 4 men earn 219 in 30 days, working 10 hours a 
 day, how much can 9 men earn in 40 clays, working 
 8 hours a day? 
 
 By Cancellation. 
 
 9m. 3 
 
 5, 15, S& d. ' 
 IBh. 
 
 STATEMENT. 
 
 
 4m.: 9 m. 
 30 d. : 40 d. 
 10 h. : 8 h. 
 
 r : : $219 : Ans. 
 
 (219x9x4 
 
 OX 8) -T- 
 
 = $525, Ans. 
 
 5 
 
 Bh. 4 
 
 x 4 x 3 = 
 
 3. If 6 men can mow 28 acres in 2 days, how long will 
 it take 7 men to mow 42 acres ? 
 
 4. If 8 horses can plow 32 acres in' 6 days, how many 
 hcrses will it take to plow 24 acres in 4 days ? 
 
 5. If the board of a family of 8 persons amounts to $300 
 in 15 weeks, how long will 81000 board 12 persons?
 
 318 COMPOUND PHOPOBTIOX. 
 
 6. "What will be the cost of 28 boxes of candles con tain - 
 'ng 20 pounds apiece, if 7 boxes containing 15 pounds 
 ipiece can be bought for $23.75 ? 
 
 7. If the interest of $300 for 10 months is $20, what 
 \vill be the interest of $1000 for 15 months? 
 
 8. If a man walks 180 miles in 6 days, at 10 h. each, 
 how many miles can he walk in 15 days, at 8 h. each ? 
 
 9. If it costs $160 to pave a sidewalk 4 ft. wide and 
 40 ft, long, what will it cost to pave one 6 ft. wide and 
 125 ft. long? 
 
 10. If it requires 800 yards of cloth yd. wide to supply 
 100 men, how many yards that is -g- wide will it require to 
 clothe 1500 men? 
 
 n. If 75 men can build a wall 50 ft. long, 8 ft. high, and 
 3 ft. thick, in 10 days, how long will it take 100 men to 
 build a wall 150 ft. long, 10 ft. high, and 4 ft. thick ? 
 
 12. If it costs $56 to transport 7 tons of goods 1 10 miles, 
 how much will it cost to transport 40 tons 500 miles ? 
 
 1 3. If 30 Ib. of cotton will make 3 pieces of muslin 42 yds. 
 long and f yd. wide, how many pounds will it take to 
 make 50 pieces, each containing 35 yards i yd. wide ? 
 
 14. If the interest of $600, at 7^, is $35 for 10 months, 
 what will be the int. of $2500, at 6 r / c , for 5 months? 
 
 15. If 9 men, working 10 hours per day, can make 18 
 sofas in 30 days, how many sofas can 50 men make in 
 90 days, working 8 hours per day ? 
 
 1 6. If it takes 9000 bricks 8 in. long and 4 in. wide to 
 pave a court-yard 50 ft. long by 40 ft. wide, how many 
 tiles 10 in. square will be required to lay a hall-floor 75 ft. 
 long by 8 ft. wide ? 
 
 17. If in 8 days 15 sugar maples, each running 12 quarts 
 of sap per day, make 10 boxes of sugar, each weighing 
 6 Ib., how many boxes weighing 10 Ib. apiece, will a maple 
 grove containing 300 trees, make in 36 days, each tree 
 running 16 quarts per day? Ans. 720 boxes.
 
 PAJlTiTlVil PKOPOIITIOX. 319 
 
 PARTITIVE PROPORTION. 
 
 500. Partitive Proportion is dividing a number 
 into two or more parts having a given ratio to each other. 
 
 501. To divide a Number into two OP more parts which 
 shall be proportional to given numbers. 
 
 1. A and B found a purse of money containing $35, 
 which they agree to divide between them in the ratio of 2 
 to 3 : how many dollars will each have ? 
 
 By Proportion. The sum of the STATEMENT. 
 
 proportional parts is to each separate 5 : 2 : : $35 : A's S. 
 
 part as the number to bo divided is to r . , . . *- - : J$'g g 
 
 each man's share. That is, 5 (2 + 3) 
 
 is to 2 as $35 to A's share. Again, 5 ('$35 X 2 ) -^ 5 =!f> 1 4 A S S. 
 is to 3 as $35 to B's share. Hence, the ($35 X 3) -r- 5 = $2 1 B's S. 
 
 EULE. I. Take the number to be divided for the third 
 term ; each proportional part successively for the second 
 1 1' ///?; and their sum for the first. 
 
 II. The product of the second and third terms of each 
 proportion, divided by the first, will be the corresponding 
 part required. 
 
 By Analysis. Since A had 2 parts and B 3, both had 2 + 3, or 5 
 parts. Hence, A will have I and B f of the money. Now of $35 
 $14, and | of $35 = $21. Hence, the 
 
 RULE. Divide the given number by the sum of the pro- 
 portional numbers, and multiply the quotient by each one's 
 proportional part. 
 
 2. It is required to divide 78 into three parts which 
 shall be to each other as 3, 4, and 6. Ans. 18, 24, 36. 
 
 3. A man having 200 sheep, wished to divide them 
 into three flocks which should be to each other as 2, 3, 
 and 5 : how many will each flock contain ? 
 
 4. A miller had 250 bushels of provender composed of 
 oats, peas, and corn in the proportion of 3, 4, and 5^-: how 
 many bushels Avere there of each kind ? 
 
 tan. How divide a number into parts having u irlvtm rti>. >
 
 320 PARTNERSHIP. 
 
 5. A father divided 497 acres of land among his four 
 sons in proportion to their ages, which were as 2, 3, 4, and 
 5 : how many acres did each receive ? 
 
 502. The principles of Partitive Proportion are appli- 
 cable to those classes of problems commonly arranged 
 under the heads of Partnership, Bankruptcy, General 
 Average, etc., in which a given number is to be divided 
 into parts having a given ratio to each other. 
 
 ."? ' 
 
 PAETj^EESHIP. 
 
 503. Partnership is the association of two or more 
 persons in business for their common profit. 
 
 It is of two kinds ; Simple and Compound. 
 
 504. Simple Partnership is that in which the 
 capital of each partner is employed for the same time. 
 
 505. Compound Partner sh fp is that in which 
 the capital is employed for unequal times. 
 
 NOTE. The association is called a firm, Jiousc, or company ; and 
 the persons associated arc termed partners. 
 
 506. The Capital is the money or property employed 
 in the business. 
 
 The Profits arc the gains shared among the partners, 
 and are called dividends. 
 
 NOTES. i. The profits are divided as the partners may agroo. 
 Other things being equal, when the capital is employed for the .<."///,> 
 time, it is customary to divide the profits according to the amount 
 of capital each one furnishes. 
 
 2. When the capital is employed for unequal times, the profits are 
 usually divided according to the amount of capital each furn i 
 and the time it is employed. 
 
 502. To what arc Ihe principles of Partitive Proportion applicable ? 503. \Vliat 
 Is Partnership? Of how many kinds? 504. Simple Partnership? 505. Com- 
 pound? Note. What is the association called? 506. What is the capital ? Th 
 profits ? What called f
 
 PARTNERSHIP. 32i 
 
 PRORLEM I. 
 
 507. To find each Partner's Share of the Profit or Loss, 
 when divided according to their capital. 
 
 1. A and B entered into partnership; the former furnish- 
 ing 8648, the latter $1080, and agreed to divide the profit 
 according to their capital. They made $432 : what was 
 each one's share of the profit ? 
 
 ANALYSIS. The capital equals $648 + $io8o=$i728. 
 
 $1728 (capital) : $648 (A's cap.) : : $432 (profit) : A's share, or $162. 
 
 $1728 " : $1080 (B's cap ):: $432 " : B's share, or $270. 
 
 Or thus: The profit $432-r-$i728 (the cap.) =.2 5 ; that is, the 
 profit is 25 fe of the capital. Therefore each man's share of the 
 profit is 25$ of his capital. Now $648 x .25 = $i62, A's share; and 
 $1080 x .25=8270, B's share. Hence, the 
 
 RULE. The whole capital is to each partner's capital, as 
 the whole profit or loss to each partner's share of the profit 
 or loss. 
 
 Or, Find what per cent the profit or loss is of the whole 
 capital, and multiply each man's capital by it. (Art. 339.) 
 
 2. A and B form a partnership, A furnishing $1200, 
 and B 81500; they lose 8500: what is each one's share of 
 the loss ? 
 
 3. A and B buy a saw-mill, A advancing $3000, and 
 B $4500; they rent it for $850 a year: what should each 
 receive ? 
 
 4. The net gains of A, B, and C for a year are $12500 ; 
 A furnishes $15000, B Si2'ooo, and C $10000: how should 
 the p*rofit be divided ? 
 
 5. Three persons entering into a speculation, made 
 $15300, which they divided in the ratio of 2, 3, and 4: 
 how much did each receive? 
 
 6. A, B, and C hire a pasture for $320 a year; A put in 
 80, B 1 20, and C 200 sheep : how much ought each man 
 to pay ? 
 
 507. How find each partner's profit or loss, when their capital is employed the 
 same time ?
 
 3'2 PARTNERSHIP. 
 
 PROBLEM II. 
 
 508. To find each Partner's Share of the Profit or Loss, 
 when divided according to capital and time. 
 
 7. A and B enter into partnership ; A furnishes $400 
 for 8 months, and B $600 for 4 months; they gain $350: 
 what is each one's share of the profit ? 
 
 ANALYSIS. In this case the profit of the partners depends on two 
 conditions, viz. : the amount of capital each furnishes, and the time 
 it is employed. 
 
 But the use of $400 for 8 months equals that of 8 times $400, or 
 $3200, for I m. ; and $600 for 4 in. equals 4 times $600, or $2400, for 
 i m. The respective capitals, then, are equivalent to $2400 and 
 $3200, each employed for I m. Now, as A furnished $3200, and 
 B $2400, the whole capital equals $3200 + $2400=35600. Therefore, 
 
 $5600 : $3200 : : $350 (profit) : A's share, or $200. 
 $5600 : $2400 : : $350 " : B's share, or $150. 
 
 Or thus: The gain $350 -4- $5600 (the cap.) = .06^, or f>\%. 
 Therefore, $3200 x .06} = $200, A's share; and $2400 x .06; = $150, 
 B's share. Hence, the 
 
 RULE. Multiply each partner's capital by the time it is 
 employed. Consider these products as their respective capi- 
 tals, and proceed as in the last problem. 
 
 Or, Find what per cent the profit or loss is of the whole 
 capital, and multiply each man's capital by it. (Art, 339.) 
 
 NOTE. The object of multiplying each partner's capital by the 
 time it is employed is, to reduce their respective capitals to equivalents 
 for the same time. 
 
 8. A, B, and C form a partnership; A furnishing $500 
 for 9 m., B $700 for i year, and C $400 for 15 months; 
 they lose 8600 : what is each man's share of the loss ? 
 
 9. Two men hire a pasture for $50 ; one put in 20 horses 
 for 12 weeks, the other 25 horses for 10 weeks: how much 
 shoiild each pay ? 
 
 508. How find each one's share, when their capital is employed for unequal 
 time* ? .Y<?te , Why multiply each one's capital by the time it is employed ?
 
 BANKRUPTCY. 323 
 
 to A. B, C, and D commenced business Jan. ist, 1870,- 
 when A .furnished $iooo;/ March ist, B put in $1200; 
 July ist, C put in $1500; and Sept. ist, D put in $2000; 
 during the year they made $1450 : how much should each 
 receive ? 
 
 n. A store insured at the; Howard Insurance Co. for 
 $3000; in the Continental, $4500; in the American, 
 &6ooo, was damaged by fire to the amount of $6750: 
 what share of the loss should each company pay ? 
 
 REMARK. The loss should be averaged in proportion to the risk 
 assumed by each company. 
 
 12. A quartermaster paid $2500 for the transportation 
 of provisions ; A carried 150 barrels 40 miles, B 1 70 barrels 
 60 miles, C 210 barrels 75 miles, and D 250 barrels 100 
 miles : how much did he pay each ? 
 
 13. A, B, and C, formed a partnership, and cleared\ 
 $12000; A put in $8000 for 4 m., and then added $2000 
 for 6 m. ; B put in $16000 for 3 m., and then withdrawing 
 half his capital, continued the remainder 5 m. longer; 
 C put in $13500 for 7 m. : how divide the profit. 
 
 BANKRUPTCY. 
 
 509. Bankruptcy is inability to pay indebtedness. 
 NOTE. A person unable to pay his debts is said to be insolvent, 
 
 and is called a bankrupt. 
 
 510. The Assets of a bankrupt are the property in 
 his possession. 
 
 The Liabilities are his debts. 
 
 511. The Net Proceeds are the assets less the ex- 
 pense of settlement. They are divided among the creditors 
 according to their claims. 
 
 509. What ia bankruptcy? 510. What arc assets? Liabilities? 511. Net pro- 
 ceeds? 512. How find each creditor's dividend, wbeu the liabilities ad the ui 
 proceeds are given?
 
 324 ALLIGATION. 
 
 o 
 
 512. To find each Creditor's Dividend, the Liabilities and 
 Net Proceeds being given. 
 
 i. A merchant failed, owing B 81260, $1800, and 
 D 81940; his assets were $1735, and the expenses of 
 settling $435 : how much did each creditor receive ? 
 
 ANALYSIS. The liabilities are $1260 + $1800 + $1940=15000; and 
 the uet proceeds $1735 $435=81300. Now, 
 
 $5000 : $1260 : : $1300 : B's divideud, or $327.60. 
 $5000 : $1800: .$1300: C's " or $468.00. 
 $5000 : $1940 : : 81300 : D's " or $504 40. 
 Or thus: The net proceeds $1300 -=-$5000= 26, or 26%, the rate 
 he is able to yay. (Art. 339.) Now $1260 x .26 = $327.60 B's; 
 1800 x .26=0468 C's ; $1940 x .26=50440 D's. Hence, the 
 
 RULE, T/ic wliolo liabilities arc to each creditor's claim, 
 as the net proceeds to each creditor's dividend. 
 
 Or, Find u'hat per cent the net proceeds are of the 
 liabilities, and multiply each creditor's claim by it. 
 
 2. A bankrupt owes A $6300, B 84500, and D 83200; 
 his assets are $5250, and the expenses of settling 1500: 
 how much will cacli creditor receive ? 
 
 3. A. B. & Co. Avcnt into bankruptcy, owing 48400, 
 and having $13200 assets; the expense of settling was 
 $i 100. What did D receive on $8240 ? 
 
 ALLIGATION. 
 
 513. Alligation is of two kinds, Medial vx\& Alternate. 
 
 All if/at ion Medial is the method of finding the 
 mean value of mixtures. 
 
 Allif/ation Alternate is the method of finding 
 the proportional parts of mixtures having- a given value. 
 
 NOTES. i. Ailigntion, from the Latin alligo, to tie, or bind 
 together, is so called from the manner of connecting the ingredients 
 by curve lin^s in some of the operations. 
 
 2. Alternate, Latin nlternntus, by turns, refers to the manner of 
 connecting the prices above the moan pries with those bdow. 
 
 3. The term mediftl is from the Latin median, middle or average. 
 
 513. What i? alligation medial ? Alternate- 't
 
 ALLIGATION. 32 
 
 ALLIGATION MEDIAL. 
 
 514. To find the Mean Value of a mixture, the Price and 
 Quantity of each ingredient being given. 
 
 1. Mixed 50 Ibs. of tea at 90 cts., 60 Ibs. at $1.10, and 
 80 Ibs. at $1.25 : what is the mean value of the mixture ? 
 
 ANALYSIS The total value of the first $.90 X 50= $45.00 
 
 kind .90 x so=$45.oo, the second $i.iox 60 $1. 10x60= $66.00 
 
 = $6600, the third $1.25 x 80= $100.00 ; $1.25 x8o=$ioo.oo 
 
 therefore the total value of the rnixture=: 190 ) $21 i.oo 
 
 $211.00. But the quantity raixed^igolbs. 4<no~& , 
 Now, if 190 Ibs. are worth $211, i Ib. is 
 
 worth TTT of $211, or $i.n + . Therefore the mean value of the 
 mixture is $1.11 4- per pound. Hence, the 
 
 KULE. Divide the value of the whole mixture ~by the sum 
 of the ingredients mixed. 
 
 NOTE. If an ingredient costs nothing, as water, sand, etc., its 
 value is o ; but the quantity itself must ba added to the other 
 ingredients. 
 
 2. If I mix 3 kinds of sugar worth 12, 15, and 20 cts. a 
 pound, what is a pound of the mixture worth ? 
 
 3. A farmer mixed 30 bu. of corn, at $1.25, with 25 bu. 
 of oats, at 60 cts., and 10 bu. of peas, at 95 cts. : what was 
 the average value of the mixture ? 
 
 4. A grocer mixed 5 gal. molasses, worth 80 cts. a gal., 
 and 107 gallons of water, with a hogshead of cider, at 20 
 cts. : what was the average worth of the mixture ? 
 
 5. A goldsmith mixed 12 oz. of gold 22 carats fine with 
 8 oz. 20 carats, and 7 oz. 18 carats fine: what was the 
 average fineness of the composition ? 
 
 6. A milkman bought 40 gallons of new milk, at 4 cts. 
 a quart, and 60 gallons of skiriimed milk at 2 cts. a quart, 
 which he mixed with 1 2 gallons of Avater, and sold tho 
 whole at 6 cts. a quart : required his profit ? 
 
 Note. Why is this rule called alligation ? Why alternate ? Wh.it is the import 
 of medial? 514. How find the mean value of a mixture, when the price and 
 quantity are giver. ?
 
 326 ALLIGATION ALTEENATE. 
 
 ALLIGATION ALTERNATE. 
 PROBLEM I. 
 
 515. To find the Proportional Parts of a Mixture, the 
 Mean Price and the Price of each ingredient being given. 
 
 7. A grocer desired to mix 4 kinds of tea worth 33., 8s., 
 i is., and 1 28. a pound, so that the mixture should be worth 
 93. a pound : in what proportion must they be taken ? 
 
 ANALYSIS. To equalize the gain and loss, we OPERATION. 
 
 compare the prices in pairs, one being above and 3 
 
 the other below the mean price, and, for conve- 
 nience, connect them by curve lines. Taking the 
 
 first and fourth ; on i Ib. at 33. the gain is 6s. ; j 2 ^ 6 
 
 on i Ib. at I2s. the loss is 33., which we place op- 
 posite the 12 and 3. Therefore, it takes i Ib. at 33. to balance the 
 loss on 2 Ib. at 123., and the proportional parts of this couplet are 
 as i to 2, or as 3 to 6. But 3 and 6 are the differences between the 
 mean price and that of the teas compared, taken inversely. 
 
 Again, i Ib. at 8s. gains is., and i Ib. at us. loses 2s., which we 
 place opposite the u and 8. Therefore, it takes 2 Ib. at 8s. to bal- 
 ance the loss on i Ib. at us., and the proportional parts of this 
 couplet are as 2 to I. But 2 and i are the differences between the 
 mean price and that of the teas compared, taken inversely. The 
 parts are 3 Ibs. at 33., 2 Ibs. at 8s., i Ib. at us., 6 Ibs. at 123. 
 
 If we compare ihc first and third, the second and fourth, the pro- 
 portional parts will be 2 Ibs. at 33., 3 Ibs. at 8s., 6 Ibs. at us., and 
 i Ib. at 128. Hence, the 
 
 KULE. I. Write the prices of the ingredients in a column, 
 with the mean price on the left, and taking them in pairs, 
 one less and the other greater than the mean price, connect 
 them by a curve line. 
 
 II. Place the difference between the mean price and that 
 of each ingredient opposite the price with which it is com- 
 pared. The sum of the differences standing opposite each 
 price is the proportional part of that ingredient. 
 
 515. How find the proportional parts of a mixture, when the mean price and. 
 the price of each Ingredient are given ?
 
 ALLIGATION ALTERNATE. 327 
 
 REM. Since the results show the proportional parts to be taken, 
 it follows if each is multiplied or divided by the same number, an 
 endless variety of answers may be obtained. 
 
 20 METHOD. Since the mean price is gs. a pound, 
 
 )l Ib. at 33. gains 6s. ; hence, to gain is. takes lb.=. Ib. 
 i Ib. at 8s. gains is. ; hence, to gain is. takes i lb.= Ib. 
 1 i Ib. at us. loses 2s. ; hence, to lose is. takes % lb.=| Ib. 
 ' i Ib. at I2S. loses 33. ; hence, to lose is. takes 5 Ib.rrf Ib. 
 Reducing these results to a common denominator, and using the 
 numerators, the proportional parts are i Ib. at 33., 6 Ibs. at 8s., 
 3 Ibs. at us., and 2 Ibs. at 123. Hence, the 
 
 RULE. Take the given prices in pairs, one greater, the 
 other less than the mean price, and find liow much of each 
 article is required to gain or lose a unit of the mean price, 
 setting the result on the right of the corresponding price. 
 
 Reduce the results to a common denominator, and the 
 numerators will be the proportional parts required. 
 
 NOTES. i. If there are three ingredients, compare the price of 
 the one which is greater or less than the mean price with each of 
 the others, and take the sum of the two numbers opposite this price. 
 
 2. The reason for considering the ingredients in pairs, one above, 
 and the other below the mean price, is that the loss on one may be 
 counterbalanced by the gain on another. 
 
 [For Canfield's Method, see Key to New Practical.] 
 
 8. A miller bought wheat at $1.60, $2.10, and $2.25 per 
 bushel respectively, and made a mixture worth $^a bushel: 
 how much of each did he buy? 
 
 9. A refiner wished to mix 4 parcels of gold 15, 18, 21, 
 and 22 carats fine, so that the mixture might be 20 carats 
 fine : what quantity of each must he take ? 
 
 10. A grocer has three kinds of spices worth 32, 40, and 
 45 cts. a pound : in what proportion must they be mixed, 
 that the mixture may be worth 38 cts. a pound? 
 
 n. A grocer mixed 4 kinds of butter worth 20 cts., 
 27 cts., 35 cts., and 40 cts. a pound respectively, and sold 
 the mixture at 42 cts. a pound, whereby he made 10 cts, 
 a pound : how much of a kind did he mix ?
 
 328 ALLIGATION ALTERNATE. 
 
 PROBLEM II. 
 
 516. When one ingredient, the Price of each, and the Mean 
 Price of the Mixture are given, to find the other ingredients. 
 
 12. How much sugar worth 12, 14, and 21 cts. a pound 
 must be mixed with 12 Ibs. at 23 cts. that the mixture 
 may be worth 18 cts. a pound? 
 
 ANALYSIS. If neither ingredient were limited, 
 the proportional parts would be 3 Ibs. at 12 cts., 
 5 Ibs. at 14 cts., 6 Ibs. at 21 cts., and 4 Ibs. at 
 
 12 v 3 
 
 14 5 
 21 J\ 6 
 
 23 cts. 23 ^ 4 
 
 But the quantity at 23 cts. is limited to 12 Ibs., 
 which is 3 times its difference 4 Ibs. Now the ratio of 12 Ibs. to 
 4 Ibs. is 3. Multiplying each of the proportional parts found by 3 
 the result will be 9 Ibs., 15 Ibs., 18 Ibs., and 12 Ibs. Hence, the 
 
 EULE. Find the proportional parts as if the quantity 
 of neither ingredient were limited. (Art. 515.) 
 
 Multiply the parts thus found oy the ratio of the given 
 ingredient to its proportional part, and the products will 
 be the corresponding ingredients required. 
 
 NOTE. When the quantities of two or more ingredients are given, 
 find the average value of them, and considering their sum as one 
 juantity, proceed as above. (Art. 515.) 
 
 13. How much barley at 40 cts., and corn at 80 cts., 
 must be mixed with . IP. bu. of oats at 30 cts. and 20 bn. 
 of rye at 60 cts., that the mixture may be 55 cts. a bu. ? 
 
 SUGGESTION. The mean value of 10 bu. of oats at 30 cts. an(' 
 20 bu. of rye at 60, is 50 cts. a bu. Ans. 30 bu. barley, and 24 bu. corn. 
 
 14. How much butter at 40, 45, and 50 cts. a pound 
 respectively, must I mix with 30 Ibs. at 65 cts. that the 
 mixture may be worth 60 cts. a pound? 
 
 15. How many quarts of milk, worth 4 and 6 cts. a quart 
 respectively, must be mixed with 50 quarts of water s<. 
 that the mixture may be worth 5 cts. a quart ? 
 
 516. When one ingredient, the price of each and mean price are given, hoM 
 End the other ingredients ?
 
 ALLIGATION ALTERNATE. 329 
 
 PROBLEM III. 
 
 517. To find the Ingredients, the Price of each, the 
 Quantity mixed, and the Mean Price being given. 
 
 1 6. How much water must be mixed with two kinds of 
 Bourbon costing $4 and $6 a gal., to make a mixture of 
 T 5o gal. the mean price of which shall be $3 a gal. ? 
 
 ANALYSIS. The price of the water is o. Disregarding the quantity 
 to be mixed, and proceeding as in Problem I., the proportional part? 
 are 4 g. water, 3 g. at $4, and 3 g. at $6, the sum of which is 4 g. , 
 
 3 g. + 3 g. 10 gallons. 
 
 But the whole mixture is to be 150 gallons. Now the ratio of 
 150 g. to 10 g. equals L p$ , or 15. 
 Multiplying each of the parts previously obtained by 15, we have 
 
 4 gal. x 15=60 gal. water; 3 gal. x 15=45 al. at $4 ; and 3 gal. x 
 15=45 ?al. a t $6- Hence, the 
 
 RULE. Find the proportional parts without regard tc 
 the quantity to be mixed, as in Problem /. 
 
 Multiply each of the proportional parts thus found by 
 the ratio of the given mixture to the sum of these parts, and 
 the several products will be the corresponding ingredients 
 required. 
 
 17. A grocer mixed 100 Ib. of lard worth 6, 8, and 12 cts. 
 a pound, the mean value of the mixture being 10 cts. : 
 how many pounds of each kind did he take ? 
 
 1 8. Having coffees worth 28, 30, 38, and 42 cts. a pound 
 respectively, I wish to mix 200 Ibs. in such proportions 
 that the mean value of the mixture shall be 36 cts. 
 a pound : how many pounds of each kind must I take ? 
 
 19. A grocer wished to mix 4 kinds of petroleum worth 
 40, 45, 50, and 60 cts. a gal. respectively : how much of 
 each kind must he take to make a mixture of 300 gallons, 
 worth 52 cts. a gallon? 
 
 517. How find the in^rcdu-uts. when the price of each, the quantity mixed, 
 nod the mean price a.-c gi'veii?
 
 INVOLUTION. 
 
 518. Involution is finding a power of a number. 
 
 A Poiver is the product of a number multiplied iutc 
 itself. Thus, 2x2 4; 3 x 3 9> etc., 4 and 9 are powers. 
 
 519. Powers are divided into different degrees; as, first, 
 second, third, fourth, etc. The name shows how many 
 times the number is taken as a factor to produce the power. 
 
 520. The First Poiver is the root or number itself. 
 The Second Poiver is the product of a number 
 
 taken twice as a factor, and is called a square. 
 
 The Third Power is the product of a number taken 
 three times as a factor, and is called a cube, etc. 
 
 NOTES. i. The second power is called a square, because the pro- 
 cess of raising a number to the second power is similar to that of 
 finding the area of a square. (Art. 243.) 
 
 2. The third power in like manner is called a cube, because the 
 process of raising a number to the third power is similar to that of 
 finding the contents of a cube. (Art. 249.) 
 
 521. Powers are denoted by a small figure placed above the 
 number on the right, called the index or exponent ; because it shows 
 how many times the number is taken as a factor, to produce the power. 
 
 NOTE. The term index (plural indices), Latin indicere, to proclaim. 
 Exponent is from the Latin exponere, to represent. Thus, 
 
 2 ] = 2, the first power, which is the number itself. 
 
 2' 2 :=2 x 2, the second power, or square. 
 
 2 3 =2 x 2 x 2, the third power, or cube. 
 
 2 4 =2 x 2 x 2 x 2, the fourth power, etc. 
 
 522. The expression 2 4 is read, "2 raised to the fourth power, or 
 the fourth power of 2." 
 
 1. Read the following, g 5 , I2 7 , 25", 245", 381', 4&5 15 , looo 24 . 
 
 2. 6 3 x y 4 , 25" x 48', I40 8 75 s , 256' -^ 97'. 
 
 518. What is involution ? A power . 519. How are powers divided ? 520. What 
 is the fir^t power ? The second ? Third ? Note. Why is the second power called 
 square? Why the third a cube ? 521. How are powers denotdy
 
 INVOLUTION. 331 
 
 3. Express the 4th power of 85. 5. The 7th power of 340. 
 
 4. Express the 5th power of 348. 6. The 8th power of 561. 
 
 623. To raise a Number to any required Power. 
 
 7. What is the 4th power of 3 ? 
 
 ANALYSIS. The fourth, power is the product of a number into 
 itself taken four times as a factor, and 3 x 3 x 3 x 3=81, the answer 
 required. Hence, the 
 
 RULE. Multiply the number into itself, till it is taken 
 as many times as a factor as there are units in the index 
 of the required power. 
 
 NOTES. I. In raising a number to a power, it should be observed 
 that the number of multiplications is always on,e less than the number 
 of times it, is U'jen as a factor; and therefore one less than the 
 number of the index. Thus, 4 3 =4 x 4 x 4, the 4 is taken three times 
 as a factor, but there are only two multiplications. 
 
 2. A decimal fraction is raised to a power by multiplying it into 
 itself, and pointing off as many decimals in each power as there are 
 decimals in the factors employed. Thus, .I 2 =.oi, .2 3 =.oo8, etc. 
 
 3. A common fraction is raised to a power by multiplying each 
 term into itsel f. Thus, (\)*= , 2 6 . 
 
 4. A mixed number should be reduced to an improper fraction, or 
 the fractional part to a decimal; then proceed as above. Thus, (2i) 2 
 =(5)2=**; or 21=2.5 and (2.5)2=6.25. 
 
 5. All powers of I are I ; for i x i x i, etc.=i. 
 
 Compare the square of the following integers and that 
 of their corresponding decimals : 
 
 8. S> 6 > 7> 8 > 9> I0 > 20 > 3> 4> 5> 6o > 7> 8o > 9- 
 9- -5> - 6 > -7> - s > -9> - OI > - 02 > -3> - 4 ? -5> - 6 > -7> -S, .09. 
 
 Raise the following numbers to the powers indicated : 
 
 io. 5 3 - 
 
 13- 4 s - 
 
 16. 2.O3 3 . 
 
 19- I 4 - 
 
 II. 2 6 . 
 
 14. 8 4 . 
 
 17. 4.0003 3 . 
 
 20. - 3 . 
 
 12. I32 3 . 
 
 J 5- 2 5 3 - 
 
 1 8. 400.05 3 . 
 
 21. 2-^ 4 . 
 
 5_'3. How raise a number to a power? Note. In raising a number to a power, 
 how many multiplications are there ? How is a decimal raised to a power ? A 
 common fraction'/ A mixed number? 524. How find the product of two or 
 wore powers of ihe s:imw mimhorV
 
 332 
 
 INVOLUTION. 
 
 524. To find the Product of two or more Powers cT the 
 same Number. 
 
 22. What is the product of 4 3 multiplied by 4 2 
 ANALYSIS. 4 3 =4x4x4, and 4.^=4x4; therefore in the product 
 of 4 3 x 4 4 , 4 is taken 3 + 2, or 5 times as a factor. But 3 and 2 are the 
 given indices ; therefore 4 is taken as many times as a factor as there 
 are units in the indices. Ans. 4". Hence, the 
 
 EULE. Add the indices, and the sum will be the index 
 of the product. 
 
 23. Mult, 2 3 by 2 2 . 25. Mult. 4 3 by 4*. 
 
 24. Mult 3 4 by 3 3 . 26. Mult. 5* by 5'. 
 
 FORMATION OF SQUARES. 
 
 525. To find the Square of a Number in the Terms of its 
 
 Parts. 
 
 i. Find the square of 5 in the terms of the parts 3 and 2. 
 
 ANALYSIS. Let the shaded part of the 
 diagram represent the square of 3 ; each 
 side being divided into 3 inches, its con- 
 tents are equal to 3 x 3, or 9 sq. in. 
 
 The question now is, what additions 
 must be made and how made, to preserve 
 fhcform of this square, and make it equal 
 to the square of 5. 
 
 i st. To preserve the form of the square 
 it is plain equal additions must be made 
 to two adjacent sides ; for, if made on one side, or on opposite sides, 
 the figure will no longer be a square. 
 
 2d. Since 5 is 2 more than 3, it follows that two rows of 3 squares 
 each must be added at the top, and 2 rows on one of the adjacent 
 sides, to make its length and brendth each equal to 5. Now 2 into 3 
 plus 2 into 3 are 12 squares, or twice the product of the two parts 
 2 and 3. 
 
 But the diagram wants 2 times 2 small squares, as represented by 
 the dotted lines, to fill the corner on the right, and 2 times 2 or 4 is 
 the square of the second part. We have then 9 (the sq. of the ist 
 part), 12 (twice the prod, of the two parts 3 and 2), and 4 (the square 
 of the zd part). But 9 + 12 + 4=25, the square required.
 
 EVOLUTION. 333 
 
 Again, if 5 is divided into 4 and i, the square of 4 is 16, twice the 
 prod, of 4 into i is 8, and the square of i is i. But 16 + 8 + 1=25. 
 
 2. Eequired the square of 25 in the terms of 20 and 5. 
 
 ANALYSIS. Multiplying 20 by 20 gives 25 20 + 5 
 
 400 (the square of the ist part); 20x5 25 20 + 5 
 
 plus 20 x 5 gives 200 (twice the prod, of ^2^ AOO 4- 100 
 
 the two parts) ; aud 5 into 5 gives 25 (the Q 3 . 
 
 square of the 2d part). Now 400 + 200 + 25 -- 
 
 =625, or 25 2 . Hence, universally, 625 =400 + 200 + 25 
 
 The square of any number expressed in the terms of its 
 parts, is equal to the square of the first part, plus twice the 
 product of the two parts, plus the square of the second part. 
 
 3. What is the square of 23 in the parts 20 and 3 ? 
 
 4. What is the sqrare of 2%, or 2 + ? Ans. 6. 
 
 5. What is the square of f or + ? Ans. 4. 
 
 EVOLUTION. 
 
 526. Evolution is finding a root of a number. 
 A Hoot is one of the equal factors of a number. 
 
 527. Roots, li-ke powers, are divided into degrees; as, the 
 square, or second root; the cube, or third root; the fourth 
 root, etc. 
 
 528. The Square Root is one of the two equal 
 factors of a number. Thus, 5 x 5 = 25 ; therefore, 5 is the 
 square root Of 25. 
 
 529. The Cube Hoot is one of the three equal 
 factors of a number. Thus, 3x3x3 27; therefore, 3 is 
 the cube root of 27, etc. 
 
 525. To what is the square of a number equal in the terms of its parts? 
 5*6. What is evolution f A root 1 528. Square root ? 529. Cube root ?
 
 334 EVOLUTION. 
 
 530. Roots are denoted in two tuays: ist. By prefixing 
 to the number the character ( \/), called the radical sign, 
 with a figure over it ; as V4, V8. 
 
 2d. By a fractional exponent placed above the number 
 on the right. Thus, Vg, or 9*j denotes the sq. root of 9. 
 
 NOTES. i. The figure over the radical sign (i/) and the denomi- 
 nator of the exponent respectively, denote the name of the root. 
 
 2. In expressing th? square root, it is customary to use simply the 
 radical sign (i/), the 2 being understood. Thus, the expression 
 1/25=5, is read, " the square root of 25 = 5." 
 
 3. The term radical is from the Latin radix, root. The sign (j/) 
 is a corruption of the letter r, the initial of radix. 
 
 531. A Perfect Power is a number whose exact 
 root can be found. 
 
 An Imperfect Power is a number whose exact 
 root can not be found. 
 
 532. A Surd is the root of an imperfect power. 
 Thus, 5 is an imperfect power, and its square root 2.23 + 
 is a surd. 
 
 NOTE. All roots as well as powers of r, are i. 
 Eead the following expressions : 
 i. 1/40. 3. 119*. 5. 1.5'. 7- ^256. 9. V- 
 
 
 2. Vi5- 4- 2433. 6. V29- 8. '1/45-7- 1. = . 
 1 1. Express the cube root of 64 both ways ; the 4th root 
 of 25 ; the jtli root of 81 ; the loth root of 100. 
 
 533. To find how many figures the Square of a Number 
 contains. 
 
 ist. Take i and 9, the least and greatest integer that can 
 be expressed by one figure; also 10 and 99, the least and 
 greatest that can be expressed by two integral figures, etc. 
 Squaring these numbers, we have for 
 
 TheEoots: i, 9, 10, 99, 100, 999, etc. 
 
 The Sq-uares: i, 81, 100, 9801, 10000, 998001, etc. 
 
 530. How are roots denoted? 531. A perfect power? Imperfect? 532. A 
 rord ? 5j3. How many figures has the square of a number ?
 
 SQUARE ROOT. 335 
 
 2(1. Take . i and .9, the least and greatest decimals that 
 can be expressed by one figure; also .01 and .99, the least 
 and greatest that can be expressed by two decimal figures, 
 etc. Squaring these, we have for 
 
 The Roots: .1, .9, .01, .99, .001, .999, etc. 
 
 The Squares: .01, .81, .0001, .9801, .000001, .998001, etc. 
 
 By inspecting these roots and squares, we discover that 
 
 Tlie square of the number contains twice as many figures 
 as its root, or twice as many less one. 
 
 534. To find how many figures the Square Hoot of a 
 Number contain*. 
 
 Divide the number into periods of two figures each, 
 placing a dot over units' place, another over hundreds, etc. 
 The root will have as many figures as there are periods. 
 
 REMARK. Since the square of a number consisting of tens and 
 units, is equal to the square of the tens, etc., when a number has two 
 periods, it follows that the left hand period must contain the squart 
 of the tens or first figure of the root. (Art. 525.) 
 
 EXTRACTION OF THE SQUARE ROOT. 
 535. To extract the Square Root of a Number. 
 
 i. A man wishes to lay out a garden in the form of a 
 square, which shall contain 625 sq. yards : what will be 
 the length of one side ? 
 
 ANALYSIS. Since 625 contains two periods, its root OPEHATION. 
 will have two figures, and the left hand period con- (etc 
 
 tains the square of the tens' figure. (Art. 534, Hem.) 
 
 But the greatest square of 6 is 4, and the root of 4 
 is 2, which we place on the right for the tens' figure 45) 22 5 
 of the root. Now the square of 2 tens or 20 is 400, ?_?5_ 
 
 and 625400=225. Hence. 225 is twice the product of 
 the tens' figure of the roct into the units, plus the square of the 
 units. (Art. 525.) 
 
 534. How many figures has the square root of a number ?
 
 336 
 
 EXTRACTION OF SQUABE BOOT. 
 
 20 yds. 5 yds. 
 
 But one of the factors of this product is 2 times 20 or 40 ; there- 
 fore the other factor must be 225 divided by 40 ; aud 225-7-405, 
 the factor required. (Art. 93.) Taking 2 times 20 into 5=200 
 (i. e., twice the product of the tens into the units' figure of the root) 
 from 225, leaves 25 for the square of the units' figure, the square 
 root of which is 5 Hence 25 is the square root of 625, and is there- 
 fore the length of one side of the garden. 
 
 2d ANALYSIS. Let the shaded part of 
 the diagram be the square of 2 tens, the 
 first figure of the root ; then 20 x 20, or 
 400 sq. yds., will be its contents. Sub- 
 tracting the contents from the given area, 
 we have 625400=225 sq. yds. to be added 
 to it. To preserve its form, the addition 
 must be made equally to two adjacent 
 sides. The question now is, what is the 
 width of the addition. 
 
 20 yds. 5 yds. 
 
 Since the length of the plot is 20 yds., adding a strip i yard wide 
 to two sides will take 20 + 20 or 40 sq. yds. Now if 40 sq. yds. will 
 add a strip i yard wide to the plot, 225 sq. yds. -will add a strip as 
 many yds. wide as 40 is contained times in 225 ; and 40 is contained 
 in 225, 5 times and 25 over. 
 
 That is, since the addition is to bo mado on two sides, we double 
 the root or length of one side for a trial divisor, and find it is con- 
 tained in 225, 5 times, which Bhows the width of the addition to be 
 5 yards. 
 
 Now the length of each side addition being 20 yds., and the widtL 
 5 yds., the area of both equals 20 x 5 + 20 x 5, or 40 x 5 = 200 sq. yards. 
 But there is a vacancy at the upper corner on the right, whose length 
 and breadth are 5 yds. each ; hence its area=5 x 5, or 25 sq. yards 
 and 200 sq. yd. + 25 sq. yd. =225 sq. yd. For the sake of finding the 
 area of the two side additions and that of the corner at the same 
 time, we place the quotient 5 on the right of the root already found, 
 and also on the right of the trial divisor to complete it. Multiplying 
 the divisor thus completed by 5, the figure last placed in the root, 
 we have 45x5 = 225 sq. yds. Subtracting this product from the 
 lividend, nothing remains. Therefore, etc. Hence, the 
 
 535. What is the first step in extracting the square root of a numher ? The 
 second? Third? Fourth? How proved? Note. If the trial divisor is not con- 
 tained in the dividend, what must be done? if there is a remainder after the 
 root of the la<t period is found, what? now many decimals docs the root of a 
 decimal fraction have ?
 
 EXTRACTION OF SQUARE BOOT. 33? 
 
 KULE. I. Divide the number into periods of two figures 
 each, putting a dot over units, then over every second figure 
 towards the left in whole numbers, and towards the right 
 in decimals. 
 
 II. Find the greatest square in the left hand period, and 
 place its root on the right. Subtract this square from the 
 period, and to the right of the remainder bring down the 
 next period for a dividend. 
 
 III. Double the part of the root thus found for a tria't 
 divisor; and finding how many times it is contained in the 
 dividend, excepting the right hand figure* annex the quotient 
 both to the root and to the divisor. 
 
 IV. Multiply tJie divisor thus increased by this last 
 figure placed in the root, subtract the product, and bring 
 down the next period. 
 
 V. Double the right hand figure of the last divisor, and 
 proceed as before, till the root of all the periods is found. 
 
 PROOF. Multiply the root into itself. (Art 528.) 
 
 NOTES. i. If the trial divisor is not contained in the dividend, 
 annex a cipher both to the root and to the divisor, and bring down 
 the next period. 
 
 -2r Since the product of the trial divisor into the quotient figure 
 cannot exceed the dividend, allowance must be made for carrying, if 
 the product of this figure into itself exceeds 9. 
 
 . -ylt sometimes happens that the remainder is larger than the 
 divisor ; but it does not necessarily follow from this that the figure 
 in the root is too small, as in simple division. 
 
 4. If there is a remainder after the root of the last period is found, 
 annex periods of ciphers, and proceed as before. The figures of the 
 root thus obtained will be decimals. 
 
 5. The square root of a decimal fraction is found in the same way 
 as that of a whole number; and the root will have as many decimal 
 figures as there are periods of decinu 1; in the given number. 
 
 6. The left hand period in whole numbers may have but one figure; 
 luit in decimals, each period must have two figures. (Art. 533.} 
 Hence, if the number lias but one decimal figure, or an odd number 
 of decimals, a cipher must be annexed to complete the period. 
 
 536. REASONS. I. Dividing the number into periods of two 
 figures each, shows how many figures the root will contain, and 
 15
 
 338 EXTRACTION OF SQUARE ROOT. 
 
 enables us to find its first figure. For, the left hand period contains 
 the square of this figure, and from the square the root is easily 
 found. (Art. 534, Bern.) 
 
 2. Subtracting the square of the first figure of the root from the 
 left hand period, shows what is left for the other figures of the root. 
 
 3. The object of doubling the first figure of the root, and dividing 
 the remainder by it as a trial divisor, is to find the next figure of the 
 root. The remainder contains tioice th-3 product of the tens into the 
 units ; consequently, dividing this product by double the tens' factor, 
 the quotient will be the other factor or units' figure of the root. 
 
 4. Or, referring to the diagram, it is doubled because the remainder 
 must be added to two sides, to preserve iheform of the square. 
 
 5. The right hand figure of the dividend is excepted, to counter- 
 balance the omission of the cipher, which properly belongs on tho 
 right of the trial divisor. 
 
 6. The quotient figure is placed in the root ; it is also annexed to 
 the trial divisor to complete it. The divisor thus completed is 
 multiplied by the second figure of the root to find the contents of the 
 additions thus made. 
 
 The reasons for the steps in obtaining other figures of the root 
 may bo shown in a similar manner. 
 
 2. What is the square root of 381.0304? .4s. 19.52. 
 
 SUGGESTION. Since the number contains decimals, we begin at 
 the units' place, and counting both ways, have four periods ; as, 
 381.0304. The root will therefore have 4 figures. But there are two 
 periods of decimals ; hence we point off two decimals in the root. 
 
 3. What is the square root of 1012036 ? Ans. 1006. 
 
 4. What is the square root of 2 ? Ans. 1.41421 +. 
 
 Extract the square root of the following numbers : 
 
 5. 182329. 
 
 ii. .1681. 
 
 17-5- 
 
 23- I9-5364- 
 
 6. 516961. 
 
 12. .725. 
 
 18. 7. 
 
 24. 3283.29. 
 
 7- 595984- 
 
 13. .I26l. 
 
 19. 8. 
 
 2 5 . 8 7 .6 5 . 
 
 8. 3.580. 
 
 14. 2.6752. 
 
 20. 10. 
 
 26. 123456789. 
 
 9. .4096. 
 
 15. 4826.75. 
 
 21. II. 
 
 27. 61723020.96. 
 
 10. .120409. 
 
 1 6. 452-634. 
 
 22. 12. 
 
 28. 9754.60423716. 
 
 536. Why divide the number into periods of two figures each ? Why subtract 
 the square of the first figure from the period ? Why double the first figure of the 
 root for a trial divisor? Why omit the right hand figure of the dividend? Why 
 place the quotient figure on the right of the trial divisor? Why multiply the 
 trial divisor thus completed by the figure last placed in the root?
 
 APPLICATIONS OF SQUARE ROOT. 339 
 
 537. To find the Square Root of a Common Fraction. 
 
 I. When the numerator and denominator are both perfect 
 squares, or can be reduced to such, extract the square root 
 of each term separately. 
 
 II. When they are imperfect squares, reduce them to 
 decimals, and proceed as above. 
 
 NOTE. To find the square root of a mixed number, reduce it to 
 an improper fraction, and proceed as before. 
 
 29. "What is the square root of T \? 
 ANALYSIS. -^^, and -1/^=5, Ans. 
 
 30. "What is the square root off? .^^..7745 +, 
 
 31. "What is the square root of if? Ans. 1.1726+. 
 
 Find the square root of the following fractions : 
 
 3 2 - iff- 35- 6 f 3 8 - Iff- 41- !& 
 
 33. f. 36. 13$. 39. f. 42. 27^. 
 
 34- A- 37- i7f- 4o. HI- 43- 
 
 APPLICATIONS. 
 
 538. To find the Side of a Square equal in area to a 
 given Surface. 
 
 1. Find the side of a square farm containing 40 acres. 
 
 ANALYSIS. In i acre there are 160 
 
 , . ,. OPERATION. 
 
 eq. rods, and m 40 acres, 40 times 
 
 !6o, or 6400 sq. rods. The ^6400= 4 A. x 160 = 6400 Sq. r. 
 80 r. Therefore the side of the farm is 4/6400 = 80 r. Ans. 
 80 linear rods. Hence, the 
 
 RULE. I. Extract the square root of the given surface. 
 
 NOTE. The root is in linear units of the same name as the given 
 surface. 
 
 2. "What is the side of a square tract of laud containing 
 1 102 acres 80 sq. rods? 
 
 537. How find the square root of a common fraction? Note. Of a mixed 
 number ? 538. How find the side of a square equal to a given surface f
 
 340 APPLICATIONS OF SQUAfiE KOOT. 
 
 3. How many rods of fencing does it require to inclose 
 a square farm which contains 122 acres 30 sq. rods ? 
 
 4. A bought 14161 fruit trees, which he planted so as 
 to form a square : how many trees did he put in a row ? 
 
 5. A general has an army of 56644 men: how many 
 must he place in rank and file to form them into a square ' J . 
 
 539. A Triangle is a figure having three 
 sides and three angles. 
 
 A Hiyht-angled Triangle is one that 
 contains a right angle. (Art. 260.) 
 
 The side opposite the right angle is called the 
 hypothenuse; the other two sides the base and 
 perpendicular. 
 
 Base 
 
 540. The square described on the hypothenuse of a right- 
 angled triangle is equal to the sum of the squares described 
 on the other two sides.* 
 
 541. The truth of this principle may 
 be illustrated thus : Take any right-angled 
 triangle ABC; let the hypothenuse h, 
 be 5 in., the base b, 4 in., and the per- 
 pendicular p, 3 in. It will be seen that the 
 square of h contains 25 sq. in., the square 
 of b 16 sq. in., and the square of p 9 sq. in. 
 Now 25 16 + 9, which accords with the 
 proposition. In like manner it may be 
 shown that the principle is true of all 
 right-angled triangles. Hence, 
 
 542. To find the Hypothenuse, the Base and Perpendicular 
 being given. 
 
 To the square of the base add the square of the per- 
 pendicular, and extract the square root of their sum. 
 
 539. What is a triangle 1 A right-angled triangle ? Draw a right-nngled 
 triangle? The side opposite the right-angle called? The other two sides? 
 540. To what is the square of the hypothennse equal? 541. Illustrate this prin- 
 ciple by a figure? 542. How find the hypothenuse when the base and per- 
 pendicular are given ? 
 
 * Thomson's Geometry, IV, 1 1 ; Euclid, I, 47.
 
 APPLICATIONS OF SQUARE ROOT. 341 
 
 543. To find the Base, the Hypothenuse and Perpendicular 
 
 being given. 
 
 From the square of the hypothenuse take the square 
 of the perpendicular, and extract the square root of the 
 remainder. 
 
 544. To find the Perpendicular, the Hypothenuse and 
 
 Base being given. 
 
 From the square of the hypothenuse take the square 
 of the base, and extract the square root of the remainder. 
 
 NOTE. The pupil should draw figures corresponding with the 
 conditions of the following problems, and indicate the parts given : 
 
 6. The perpendicular height of a flag-staff is 36 ft: 
 what length of line is required to reach from its top to a 
 point in a level surface 48 ft. from its base ? 
 
 SOLUTION. The square of the base .. =48x48=2304 
 " " perpendicular = 36 x 36=1296 
 The square root of their sum =1/3600 =60 ft. Ans. 
 
 7. The hypothenuse of a right-angled triangle is 135 yds., 
 the perpendicular 81 yds. : what is the base ? 
 
 8. One side of a rectangular field is 40 rods, and the 
 distance between its opposite corners 50 rods: what is the 
 length of the other side ? 
 
 9. Two vessels sail fronv the same point, one going 
 due south 360 miles, the other due east 250 miles: how 
 far apart were they then ? 
 
 10. The height of a tree on the bank of a river is 100 ft., 
 and a line stretching from its top to the opposite side is 
 144 ft. : what is the width of the river ? 
 
 n. The side of a square room is 40 feet: what is the 
 distance between its opposite corners on the floor ? 
 
 12. A tree was broken 35 feet from its root, and struck 
 the ground 21 ft. from its base: what was the height of 
 the tree ? 
 
 543. How find the base when the other two sides are given? 544. How find 
 the perpendicular when the other two sides are given ?
 
 342 SIMILAE FIGURES. 
 
 SIMILAR FIGURES. 
 
 545. Similar Figures are those which have the 
 same form, and their like dimensions proportional. 
 
 NOTES. i. All circles, of whatever magnitude, are similar. 
 
 2. All squares, equilateral triangles, and regular polygons are 
 similar. And, universally, 
 
 All rectilinear figures are similar, when their several angles are 
 equal each to each, and their like dimensions proportional. 
 
 3. The like dimensions of circles are their diameters, radii, and 
 circumferences. 
 
 546. The areas of similar figures are to each other as 
 the squares of their like dimensions. And, 
 
 Conversely, the like dimensions of similar figures are to 
 each other as the square roots of their areas. 
 
 13. If one side of a triangle is 12 rods, and its area 
 72 sq. rods, what is the area of a similar triangle, the 
 corresponding side of which is 8 rods ? 
 
 SOLUTION. (12)* : (8) 2 : : 72 : Ans., or 32 sq. rods. 
 
 14. If one side of a triangle containing 36 sq. rods is 
 8 rods, what is the length of a corresponding side of a 
 similar triangle which contains 81 sq. rods? 
 
 SOLUTION. ^36 : j/8i : : 8 : Ans., or 12 rods. 
 
 15. If a pipe 2 inches in diameter will fill a cistern in 
 42 min., in what time will a pipe 7 in. in diameter fill it ? 
 
 547. To find a Mean Proportional between two Numbers. 
 
 1 6. What is the mean proportional between 4 and 16? 
 ANALYSIS. When three numbers are proportional, the product of 
 
 the extremes is equal to the square of the mean. (Arts. 487, 490.) 
 But 4x 16=64 ; an( i 1/64=8. Ans. Hence, the 
 
 RULE. Extract the square root of their product. 
 
 Find the mean proportional between the following : 
 17. 4 and 36. 19. 56 and 72. 21. $$ and -^-. 
 
 *8. 36 and 81. 20. .49 and 6.25. 22. -^ and 
 
 545. What are similar figures ? 546. How arc their areas to each other Y
 
 FOKMATION OF CUBES. 343 
 
 FORMATION OF CUBES. 
 
 548. To find the Cube of a Number in the terms of its parts, 
 i. FiaJ the cube of 32 in the terms of the parts 30 and 2. 
 
 ANALYSIS. 32=30 +2, and 32 3 =( 
 
 32=30 +2 
 "96"=: 
 64= 
 
 IO24=30 2 + 2 x V 3O x 2) + 2 s 
 
 32 = 3O + 2 
 
 3072 = 30 3 + 2 x (30 s x 2) 4- (30 x 2 2 ) 
 
 2048 = (30* X 2) + 2 X v30j< 2 2 ) + 2 8 
 
 32768=30^3 X (30 2 X 2) + 3 X (30 X Z 9 ) + 2 3 . 
 
 Or thus : Let the diagram represent a 
 cube whose side is 30 ft. ; its contents^ 
 the cube of 3 tens, or 2700 cu. ft. 
 
 What additions must be made to this 
 cube, and how made, to preserve its form, 
 and make it equal to the cube of 32. 
 
 ist. To preserve its cubical form, the 
 additions must be equally made on three 
 adjacent sides ; as the top, front, and right. 
 
 2d. Since 32 is 2 more than 30, it follows that this cube must be 
 made 2 ft. longer, 2 ft. wider, and 2 ft. higher, that its length, 
 breadth, and thickness may each be 32 ft. 
 
 But as the side of this cube is 30 ft., the contents of each of these 
 additions must be equal to the square of the tens (3O 2 ) into 2, ihe units, 
 and their sum must be 3 times (3O 2 x 2)=3 x 900 x 2=5400 cu. ft. 
 
 But there are three vacancies along the edges of the cube adjacent 
 to the additions. Each of these vacancies is 30 ft. long, 2 ft. wide, 
 and 2 ft. thick ; hence, the contents of each equals 30 x 2 2 , and the sum 
 of their contents equals 3 times the tens into the square of the units 
 =3 times (30 x 2 2 )=3 x 30 x 4=360 cu. ft. But there is still another 
 vacancy at the junction of the corner additions, whose length, 
 breadth, and thickness are each 2 ft., and whose contents are equal 
 to 2 3 =2x 2x2 = 8. The cube is now complete. Therefore 32*= 
 27000 (3o s ) + 54Oo (3 times 3O 2 x2) + 3&o (3 times 30x2*) + 8 (2 3 )= 
 32768 cu. ft. In like manner it may be shown that, 
 
 548. To what is the cube of a number consisting of tens and units equal?
 
 344 FORMATION OF CUBES. 
 
 The cube of any number consisting of tens and units is 
 equal to the cube of the tens, plus 3 times the square of tlw 
 tens into the units, plus 3 times the tens into the square of 
 the units, plus the cube of the units. 
 
 549. To find how many figures the Cube of a Number contains. 
 
 ist. Take i and 9, also 10 and 99, 100 and 999, etc., the least and 
 
 greatest integers that can be expressed by one, two, three, etc., figures. 
 
 2d. In like manner take .1 and .9, also .01 and .99, etc., the least 
 
 and greatest decimals that can be expressed by one, two, etc., decimal 
 
 figures. Cubing these, we have 
 
 The root land r'=i, - - - - .1 and .I 3 =.i 
 " 9 " 9 3 --729, - - - .9 " .g 3 =.729 
 
 " IO " IO 3 =IOOO, - - - .OI " .OI 3 =.OOOOOI 
 
 99 " 995=970299, - - .99 " .gg 3 -. 970299 
 " 100 " ioo 3 =ioooooo, - - .001 " .ooi 3 =.000000001 
 999 " 999 3 = 997002999, - .999 " -999 3 =. 997002999. 
 By comparing these roots and their cubes, we discover that 
 TJie cube of a number cannot have more than three times 
 as many figures as its root, nor but two less. Hence, 
 
 550. To find how many figures the Cube Hoof, contains. 
 
 Divide the number into periods of three figures each, 
 putting a dot over units, then over every third figure 
 towards the left in whole numbers, and towards the right 
 in decimals, 
 
 REMAKKS. i. Since the cube of a number consisting of tens and 
 units is equal to the cube of the tens, plus 3 times the square of the 
 tens into the units, etc., when a number has two periods, it follows 
 that the left hand period must contain the cube of the tens, or first 
 figure of the root. 
 
 2. The left hand period in whole numbers may be incomplete, 
 having only one or two figures ; but in decimals each period must 
 always have three figures. Hence, if the decimal figures in a given 
 number are less than three, annex cipher* to complete the period. 
 
 How many figures in the cube root of the following : 
 2. 340566. 4. 576.453. 6. 32.7561. 
 
 3- 1467- 5- 5-732I- 7- 456785- 
 
 549. How many figures has the cube of a number?
 
 EXTRACTION OF THE CUBE BOOT. 345 
 
 EXTRACTION OF THE CUBE ROOT. 
 551. To extract the Cube Root of a Number. 
 
 i. A man having 32768 marble blocks, each being a 
 cubic foot, wishes to arrange them into a single cube: 
 what must be its side ? 
 
 ANALYSIS. Since 32768 contains two periods, OPERATION. 
 
 we know its root will have two figures; also that 12768(32 
 
 the left hand period contains the cube of the tens 2 
 or first figure of the root. (Art. 550, Hem.) 
 
 5768 
 
 57 68 
 
 , . 
 
 The greatest cube in 32 is 27, and its root is 3, ' ~ 
 which we place on the right for the tens or first 
 figure of the root. Subtracting its cube from the 
 first period, and bringing down the next period 
 to the right of the remainder, we have 5768, which 
 by the formation of the cube is equal to 3 times the square of the 
 tens' figure into the units, plus 3 times the tens into the square of 
 the units, plus the cube of the units. We therefore place 3 times 
 the square of the tens or 2700, on the left of the dividend for a 
 trial divisor ; and dividing, place the quotient 2 on the right for the 
 units' figure of the root. 
 
 To complete the divisor, we add to it 3 times 30 the tens into 2 
 units=i8o; also 4, the square of the units, making it 2884. Multi- 
 plying the divisor thus completed by the units' figure 2, we have 
 2884 x 2 = 5768, the same as the dividend. Ans. 32 ft. 
 
 Or thus : Let the cube of 30, the tens of the root, be represented 
 by the large cube in the set of cubical blocks.* The remainder 5768, 
 is to be added equally to three adjacent sides of this cube. 
 
 To ascertain the thickness of these side additions, we form a trial 
 divisor by squaring 3, tho first figure of the root, with a cipher 
 annexed, for the area of one side of this cube, and multiply this 
 square by 3 for the three side additions. Now 3O 2 = 30 x 30=900; 
 and 900x3 = 2700, the trial divisor. Dividing 5768 by 2700, the 
 quotient is 2, which shows that the side additions are to be 2 ft. 
 thick, and is placed on the right for the units' figure of the root. 
 
 551. The first step in extracting the cube root? The second? Third? 
 Fourth 1 Fifth ? Note. If the trial divisor is not contained in the dividend, how 
 proceed ? If there is a remainder after the root of the last period is found, how ? 
 
 * Every school in which the cuhe root is taught is presumed to be furnished 
 with a set of Cubical Blocks,
 
 346 EXTRACTION OF THE CUBE BOOT. 
 
 To represent these additions, place the corresponding layers on 
 the top, front, and right of the large cube. But we discover three 
 vacancies along the edges of the large cube, each of which is 30 ft. 
 long, 2 ft. wide, and 2 ft. thick. Filling these vacancies with the 
 corresponding rectangular blocks, we discover another vacancy at 
 the junction of the corners just filled, whose length, breadth, and 
 thickness are each 2 ft. This is filled by the small cube. 
 
 To complete the trial divisor, we add the area of one side of each 
 of the corner additions, viz., 30 x 2 x 3, or 180 sq. ft., also the area of 
 one side of the small cube=2 x 2, or 4 sq. ft. Now 2700+ 180+4= 
 2884. The divisor is now composed of the area of 3 sides of the 
 large cube, plus the area of one side of each of the corner additions, 
 plus the area of one side of the small cube, and is complete. 
 
 Finally, to ascertain the contents of the several additions, we 
 multiply the divisor thus completed by 2, the last figure of the root ; 
 and 2884 x 2=5768. (Art. 249.) Subtracting the product from the 
 dividend, nothing remains. Hence, the 
 
 KULE. I. Divide the number into periods of three figures 
 each, putting a dot over units, then over every third figure 
 towards the left in ivhole numbers, and towards the right 
 in decimals. 
 
 II. Find the greatest cube in the left hand period, and 
 place its root on the right. Subtract its cube from the 
 period, and to the right of the remainder bring down the 
 next period for a dividend. 
 
 III. Multiply the square of the root thus found with a 
 cipher annexed, by three, for a trial divisor; and finding 
 hoiv many times it is contained in the dividend, icrite the 
 quotient for the second figure of the root. 
 
 IV. To complete the trial divisor, add to it three times 
 the product of the root previously found witli a cipher 
 annexed, into the second root figure, also the square of the 
 second root figure. 
 
 V. Multiply the divisor thus completed by the last figure 
 placed in the root. Subtract the product from the divi- 
 dend; and to the right of the remainder bring down the 
 next period for a new dividend. Find a new trial divisor, 
 as before, and thus proceed till the root of the last period is 
 found.
 
 EXTRACTION OF THE CUBE BOOT. 347 
 
 NOTES. i. If a trial divisor is not contained in the dividend, put 
 a cipher in the root, two ciphers on the right of the divisor, and bring 
 down the next period. 
 
 2. If the product of the divisor completed into the figure last 
 placed in the root exceeds the dividend, the root figure is too large. 
 Sometimes the remainder is larger than the divisor completed ; but 
 it does not necessarily follow that the root figure is too small. 
 
 3. When there are three or more periods in the given number, the 
 first, second, and subsequent trial divisors are found in the same 
 manner as when there are only two. That is, disregarding its true 
 local value, wo simply multiply the square of the root already found 
 with a cipher annexed, by 3, etc. 
 
 4. If there IP- a remainder after the root of the last period is found, 
 annex periods of ciphers, and pioceed as before. The root figures 
 thus obtained will be decimals. 
 
 5. The cube root of a decimal fraction is found in the same way as 
 that of a whole number ; and will have as many decimal figures as 
 there are periods of decimals in the number. (Art. 549.) 
 
 552. REASONS. i. Dividing the number into periods shows 
 how many figures the root contains, and enables us to find the first 
 figure of the root. For, the left hand period contains the cube of 
 the first figure of the root. (Art. 550.) 
 
 2. The object of the trial divisor is to find the next figure of the 
 root, or the thickness of the side additions. The root is squared to 
 find the area of one side of the cube whose root is found, the cipher 
 being annexed because the first figure is tens. This square is multi- 
 plied by 3, because the additions are to be made to three sides. 
 
 3. The root previously found is multiplied by this second figure 
 to find the area of a side of one of the vacancies along the edges of 
 the cube already found. This product is multiplied by 3, because 
 there are three of these vacancies ; and the product is placed under 
 the trial divisor as a correction. The object of squaring the second 
 figure of the root is to find the area of one side of the cubical vacancy 
 at the junction of the corner vacancies, and with the other correction 
 this is added to the trial divisor to complete it. 
 
 4. The divisor thus completed is multiplied by the second figuro 
 of the root to find the contents of the several additions now made. 
 
 552. Why divide the number into periods of three figures ? What is the object 
 of a trial divisor ? Why square the root already found ? Why annex n cipher to 
 it? Why multiply this square by 3? Why is the root previously found multi- 
 plied by the second figure of the root? Why multiply this product by 3* Why 
 square the second figure of the root ? Why multiply the divisor whn completed 
 by the second figure of the root*
 
 348 EXTRACTION OF THE CUBE ROOT. 
 
 2. "What is the cube root of 130241.3 ? 
 
 ANALYSIS. Having completed the OPEEATION. 
 
 period of decimals by annexing two 130241.300(50.64 
 
 ciphers, we find the first figure of the I2! . 
 
 root as above. Bringing down the 
 nest period, the dividend is 5241. 75 OOO 5241-3' 
 The trial divisor 7500 is not contained 
 in the dividend; therefore placing a _ 3_ 
 cipher in the root and two on the 7593^ 4554 21 ^ 
 right of the divisor, we bring down 687084 Rem. 
 
 the next period, and proceed as before. 
 
 3. Cube root of 614125 ? 6. Cube root of 3 ? 
 
 4. Cube root of 84.604 ? 7. Cube root of 21.024576 ? 
 
 5. Cube root of 373248 ? 8. Cube root of 17 ? 
 9. "What is the cube root of 705919947264? 
 
 10. What is the cube root of .253395799 ? 
 
 11. What is the side of a cube which contains 628568 
 cu. yards ? 
 
 NOTE. The root is in linear units of the same name as the given 
 contents. 
 
 12. What is the side of a cube equal to a pile of wood 
 40 ft. long, 1 5 ft. wide, and 6 ft. high ? 
 
 13. What is the side of a cubical bin which will hold 
 1000 bu. of corn; allowing 2150.4 cu. in. to a bushel ? 
 
 553. To find the Cube Root of a Common Fraction. 
 
 If the numerator and denominator are perfect cubes, of 
 can be reduced to such, extract the cube root of each. 
 
 Or, reduce the fraction to a decimal, and proceed as before 
 NOTE. Reduce mixed numbers to improper fractions, etc. 
 
 14. What is the cube root of -f^s ? Am. f. 
 
 15. Cube root off? Ans. .75+. 
 1 6. Cube root of jf& ? 18. Cube root of T V 2 \ ? 
 17. Cube root of -ff^fj ? 19. Cube root of 8 if ? 
 
 553. How find the cube root of a common fraction? Note. Of a mixed number?
 
 SIMILAR SOLIDS. 349 
 
 APPLICATIONS. 
 
 554. Similar Solids are those which have the same 
 form, and their like dimensions proportional. 
 
 NOTES. i. The like dimensions of spheres are their diameters, 
 radii, and circumferences; those of cubes are their sides. 
 
 2. The like dimensions of cylinders and cones are their altitudes 
 and the diameters or the circumferences of their bases. 
 
 3. Pyramids are similar, when their bases are similar polygons 
 and their altitudes proportional. 
 
 4. Polyhedrons (i. e., solids included by any number of plane faces) 
 are similar, when they are contained by the same number ot similai 
 polygons, and all their solid angles are equal each, to each. 
 
 555. The contents of similar solids are to each other aa 
 the cubes of their like dimensions; and, 
 
 Conversely, the like dimensions of similar solids are a& 
 the cube roots of their contents. 
 
 1. If the side of a cubical cistern containing 1728 en. in. 
 is 1 2 in., what arc the contents of a similar cistern whose 
 side is 2 ft. Ans. i 3 : 2 3 : : 1728 cu. in. : con., or 13824 en. in. 
 
 2. If the side of a certain mound containing 74088 cu. ft. 
 is 84 ft., what is the side of a similar mound which con- 
 tains 17576 en. ft? 
 
 3. If a globe 4 in. in diameter weighs 9 Ibs , what is the 
 weight ot a globe 8 in. in diameter? Ans. 72 Ibs 
 
 4. If 8 cubic piles of wood, the side of each being 8 ft-., 
 were consolidated into one cubic pile, what would be the 
 length of its side ? 
 
 5. If a pyramid 60 ft. high contains 12500 cu. ft., what 
 are the contents of a similar pyramid whose height is 20 ft.? 
 
 6. If a conical stack of hay 15 ft. high contains 6 tons, 
 what is the weight of a similar stuck whose height is 12 ft.? 
 
 554. What are similar s. lii--? .'u/'V. \Vlmt .ire like dimensions of spheres t 
 Of cylinders and cones f Of pyramids ? 555. What relation have similar solids ?
 
 ARITHMETICAL PROGRESSION. 
 
 556. An Arithmetical Progression is a series 
 of numbers which increase or decrease by a common dif- 
 ference. 
 
 NOTE. The numbers forming the series are called the Terms. 
 The first and. last terms are the Extremes; the intermediate terms 
 the Means. (Arts. 476, 489.) 
 
 557. The Common Difference is the difference 
 between the successive terms. 
 
 558. An Ascending Series is one in which the successive 
 terms increase ; as, 2, 4, 6, 8, 10, etc., the common difference being 2. 
 
 A Descending Series is one in which the successive terms 
 decrease ; as, 15, 12, 9, 6, etc., the common difference being 3. 
 
 559. In arithmetical progression there are five parts or elements 
 to be considered, viz. : the first term, the last term, the number of 
 terms, the common difference, and the sum of all the terms. These 
 parts are so related to each other, that if any three of them are given, 
 the other two may be found. 
 
 560. To find the Last Term,, the First Term, the Number 
 of Terms, and the Common Difference being given. 
 
 i. Required the last term of the ascending series having 
 7 terms, its first term being 3, and its common difference 2. 
 
 ANALYSIS. From the definition, each succeeding term is found 
 by adding the common difference to the preceding. The series is : 
 
 3, 3 + 2, 3 + (2 + 2), 3 + (2 + 2 + 2), 3 + (2 + 2 + 2 + 2), etc. Or 
 3> 3 + 2, 3 + (2x2), 3 + (2x3), 3 + (2x4), etc. That is, 
 
 561. The last term is equal to the first term, increased by 
 the product of the common difference into the number of terms 
 less i. Hence, the 
 
 RULE. Multiply the common difference by the number 
 of terms less i, and add the product to the first term. 
 
 556. What is an arithmetical progression ? Note. The first and last terms 
 called? The intervening ? 5^7. The common difference? 5^8. An ascending 
 pcries? Descending? 560. How And the last term, when the first term, th$ 
 number of terms, and the common difference are given ?
 
 ARITHMETICAL PROGRESSION. 351 
 
 ES. i. In a descending series the product must be subtracted 
 from the first term ; for, in this case each succeeding term is found 
 by subtracting the common difference from the preceding terms. 
 
 2. Any term in a series may be found by the preceding rule. 
 For, the series may be supposed to stop at any term, and that may 
 be considered, for the time, as the last. 
 
 3. If the last term is given, and the first term required, invert the 
 order of the terms, and proceed as abate. 
 
 2. If the first term of an ascending series is 5, the 
 common difference 3, and the number of terms 1 1, what 
 is the last term? Ans. 35. 
 
 3. The first term of a descending series is 35, the com- 
 mon difference 3, and the number of terms 10: what is 
 the last ? 
 
 4. The last term of an ascending series is 77, the number 
 of terms 1 9, and the common difference 3 : what is the 
 first term ? Ans. 23. 
 
 5. What is the amount of $150, at 1% simple interest, 
 for 20 years ? 
 
 562. To find the Number of Terms, the Extremes, and 
 the Common Difference being given. 
 
 6. The extremes of an arithmetical series are 4 and 37, 
 and the common difference 3 : what is the number of terms ? 
 
 ANALYSIS. The last term of a series is equal to the first term 
 increased or diminished by the product of the common difference 
 into the number of terms less i. (Art. 561.) Therefore 374, or 
 33, is the product of the common difference 3, into the number of 
 terms less i. Consequently 33-f- 3 or 1 1, must be the number of terms 
 less i ; and n + i,or 12, is the answer required. (Art. 93.) Hence, the 
 
 RULE. Divide the difference of the extremes by the com- 
 mon difference, and add i to the quotient. 
 
 7. The youngest child of a family is i year, the oldest, 
 21, and the common difference of their ages 2 y. : how 
 many children in the family ? 
 
 562. How find the number of terms, when the extremes and common different 
 (ire given ?
 
 352 ARITHMETICAL PROGRESSION. 
 
 563. To find the Common Difference, the Extremes and 
 
 the Number of Terms being given. 
 
 8. The extremes of a series are 3 and 21, and the number 
 of terms is 10: what is the common difference? 
 
 ANALYSIS. The difference of the extremes 21 3=18, is the pro- 
 duct of the number of terms less i into the common difference, and 
 10 i, or g, is the number of terms less i ; therefore 18-^-9, or 2, is the 
 common difference required. (Art. 93.) Hence, the 
 
 EULE. Divide the difference of the extremes by the 
 number of terms less i. 
 
 9. The ages of 7 sons form an arithmetical series, the 
 youngest being 2, and the eldest 20 years: what is the 
 difference of their ages ? 
 
 564. To find the Sum of all the Terms, the Extremes and 
 
 the Number of Terms being given. 
 
 10. Eequired the sum of the series having 7 terms, the 
 extremes being 3 and 15. 
 
 ANALYSIS. (i.) The series is 3, 5, 7, 9, n, 13, 15. 
 
 (2.) Inverting the same, 15, 13, n, 9, 7. 5, 3. 
 
 (3.) Adding (i.) and (2.), 184-18 + i8~+ 18 + 18 + 18 + 18= twice the sum. 
 
 (4.) Dividing (3.) by 2, 9 + 9 + 9 + 9 + 9 + 9 + 9=63, the sura. 
 
 By inspecting these series, we discover that half the sum of the 
 extremes is equal to the average value of the terms. Hence, the 
 
 RULE. Multiply half the sum of the extremes by the 
 number of terms. 
 
 REMARK. From the preceding illustration we also see that, 
 The sum of the extremes is equal to the sum of any two terms 
 
 equidistant from them ; or, to twice the sum of the middle tern; , if 
 
 the number of terms be odd. 
 
 11. How many strokes does a common clock strike in 
 12 hours? 
 
 563. How find the common difference when the extremes and number of terms 
 are jriven ? 564. How find the sum of all the terms, when the extremes &nd 
 number of terms are given ?
 
 GEOMETRICAL PROGRESSION. 
 
 565. A Geometrical Progression is a series of 
 numbers which increase or decrease by a common ratio. 
 
 NOTE. The series is called Ascending or Descending, according 
 as the terms increase or decrease. (Art. 558.) 
 
 566. I Q Geometrical Progression there are also five parts or 
 elements to be considered, viz. : the first term, the last term, the 
 number of terms, the ratio, and the sum of all the terms. 
 
 567. To find the Last Term, the First Term, the Ratio, 
 and the Number of Terms being given. 
 
 1. Required the last term of an ascending series having 
 6 terms, the first term being 3, and the ratio 2. 
 
 ANALYSIS. From the definition, the series is 
 3> 3 X 2, 3 x (2 x 2), 3 x (2 x 2 x 2), 3 x (2 x 2 x 2 x 2), etc. Or 
 3, 3x2, 3 x 2 2 , 3 x 2 3 , 3 x 2 4 , etc. That is, 
 
 Each successive term is equal to the first term multiplied by the 
 ratio raised to a power whose index is one less than the number 
 of the term. Hence, the 
 
 RULE. Multiply the first term by that power of the ratio 
 whose index is i less than the number of terms. 
 
 NOTES. i. Any term in a series may be found by the preceding rule. 
 For, the series may be supposed to stop at that term. 
 
 2. If the last term is given and the first required, invert the order 
 of the terms, and proceed as above. 
 
 3. The preceding rule is applicable to Compound Interest; the 
 principal being the first term of the series ; the amount of $i for i 
 year the ratio ; and the number of years plus i, the number of terms. 
 
 2. A father promised his son 2 cts. for the first example 
 he solved, 4 cts. for the second, 8 cts. for the third, etc. : 
 what would the son receive for the tenth example ? 
 
 3. What is the amount of $1500 for 5 years, at 6% com- 
 pound interest ? Of $2000 for 6 years, at 7% ? 
 
 565. WTiat is a geometrical progression ? 567. How find the last term, when 
 the flrt term, the ratio, and number of terms are given ?
 
 354 GEOMETKICAL PEOGEESSION". 
 
 568. To find the Sum of all the Terms, the Extremes 
 
 and Ratio being given. 
 
 4. Required the sum of the series whose first and last 
 terms are 2 and 162, and the ratio 3. 
 
 ANALYSIS. Since each succeeding term is found by multiplying 
 the preceding terra by tlie ratio, tlie series is 2, 6, 18, 54, 162. 
 
 (i.) The sum of the series, 21-6 + 18 + 54+162. 
 
 (2.) Multiplying by 3, = 6 + 18 + 54+162 + 486. 
 
 (3.) Subt. (i.) from (2.), 4862=484, or twice the sum. 
 
 Therefore 484-4-2=242, the sum required. But 486, the last term 
 of the second series, is the product of 162 (the last term of the given 
 series) into the ratio 3 ; the difference between this product and the 
 first term is 4862 or 484, and the divisor 2 is the ratio 31. 
 Hence, the 
 
 KULE. Multiply the last term ~by the ratio, and divide 
 the difference between this product and the first term by the 
 ratio less i. 
 
 5. The first term is 4, the ratio 3, and the last term 972 : 
 what is the sum of the terms ? 
 
 6. What sum can be paid by 1 2 instalments ; the first 
 being $i, the second $2, etc., in a geometrical series ? 
 
 569. To find the Sum of a Descending Infinite Series, the 
 
 First Term and Ratio being given. 
 
 REMARK. In a descending infinite series the last term being 
 infinitely small, is regarded as o. Hence, the 
 
 RULE. Divide the first term by the difference between 
 the ratio and i, and the quotient will be the sum required. 
 
 7. What is the- sum of the series f, f, , -fa, continued 
 to infinity, the ratio oeing ? Ans. i}. 
 
 NOTE. The preceding problems in Arithmetical and Geometrical 
 Progressions embrace their ordinary applications. The others involve 
 principles with which the pupil is not supposed to be acquainted. 
 
 5<ss. How find the sum of the terms, when the extremes and ratio are given ? 
 .509. How find the sum of an infinite descending series, when the first term and 
 ratio are given ?
 
 MENSURATION. 
 
 570. Mensuration is the measurement of magni- 
 tude. 
 
 571. Magnitude is that which has one or more of 
 the three dimensions, length, breadth, or thickness; as, 
 lines, surfaces, and solids. 
 
 A Line is that which has length without breadth. 
 A Surface is that which has length and breadth, 
 without thickness. 
 
 A Solid is that which has length, breadth, and. thickness. 
 
 572. A Quadrilateral Figure is one which has 
 four sides and/owr angles. 
 
 NOTES. i. If all its sides are straight lines it is rectilinear. 
 
 2. If all its angles are right angles it is rectangular. 
 
 3. Figures which have more than/ww sides are called Polygons. 
 
 573. Quadrilateral figures are commonly divided into the rect- 
 angle, square, parallelogram, rhombus, rhomboid, trapezium, and 
 trapezoid. 
 
 ]@p~ For the definition of rectangular figures, the square, etc., se 
 Arts. 240, 241. 
 
 574. -A- Rhombus is a quadrilateral 
 which has all its sides equal, and its angles 
 oblique. 
 
 575. A Rhomboid is a quadrilateral 
 in which the opposite sides only are equal, and 
 all its angles are oblique. 
 
 576. The altitude of a quadrilateral figure 
 having two parallel sides is the perpendicular 
 distance between these sides ; as, A, L. 
 
 577. A Trapezium is a quadrilateral which has only two of 
 its sides parallel.* (See next Fig.) 
 
 570. What is Mensuration ? 571. Magnitude ? Line ? Surface ? Solid ? 574. A 
 hombus J 575. Rhomboid ? 576. The altitude ? 577. A trapezium ? 
 * Legendre, Dr. Brewster, Young, Du Morgan, etc.
 
 356 MENSURATION. 
 
 578. A Diagonal is a straight line joining the 
 vertices of two angles, not adjacent to each other'; 
 as, A D. 
 
 579. The common measuring unit of surfaces is 
 a square, whose side is a linear unit of the same 
 name. (Thomson's Geometry, IV. Sch.) 
 
 Jg^~ To find the area of rectangular surfaces, see Art. 280. 
 The rules or formulas of mensuration are derived from Geometry 
 to which their demonstration properly belongs. 
 
 580. To find the Area of an Oblique Parallelogram, Rhombus, 
 or Rhomboid, the Length and Altitude being given. 
 
 Multiply the length by the altitude. 
 
 NOTE. If the area and altitude, or one side are given, the other 
 factor is found by dividing the area by the given factor. (Art. 244.* 
 
 1. What is the area of a rhombus, its length being 
 60 rods, and its altitude 53 rods? Ans. 3i8osq. r. 
 
 2. A rhomboidal garden is 75 yds. long, the perpendicular 
 distance between its sides 48 yds. : what is its area ? 
 
 3. The area of a square field whose side is 120 rods ? 
 
 4. If one side of a rectangular grove containing 80 acres 
 is 1 60 rods, what is the length of the other side ? 
 
 581. To find the Area of a Trapezium, the Altitude and 
 
 Parallel Sides being given. 
 
 Multiply half the sum of the parallel sides by the altitvde. 
 
 5. The altitude of a trapezium is n ft-, and its parallel 
 sides are 16 and 27 ft.: what is its area? Ans. 236.5 sq.ft 
 
 582. To find the Area of a Triangle, the Base and 
 
 Altitude being given. 
 
 Mutely the base by half the altitude. (Art. 539.) 
 
 NOTE. Dividing the area of a triangle by the altitude gives the 
 base. Dividing the area by half the base gives the altitude. 
 
 578. A diagonal V 579. The common measuring null of surface?? 580. How 
 find the area of an oblique parallelogram or rhombus? 583. How find the are* 
 of a trapezium ? 582. Of a triangle ?
 
 MENSURATION. 35? 
 
 6. What is the area of a triangle whose base is 37 ft., 
 and its altitude 19 ft. ? Ans. 351.5 sq. ft. 
 
 7. Sold a triangular garden whose base is 50 yds., and 
 altitude 40 yds., at $2.75 a sq. rod: what did it come to? 
 
 583. To find the Circumference of a Circle, the Diameter 
 
 being given. 
 
 Multiply the diameter by 3.14159. 
 
 E2T For definition of the circle and its parts, see Art. 257. 
 
 8. The diameter of a cistern is 1 2 ft. : what is its 
 circumference? Ans. 37.69908 ft. 
 
 9. The diameter of a circular pond is 65 rods : what is 
 its circumference ? 
 
 584. To find the Diameter of a Circle, the Circum- 
 
 ference being given. 
 
 Divide the circumference by 3.14159. 
 
 10. What is the diameter of a circle whose circum- 
 ference is 150 ft. ? 
 
 1 1. The diameter of a circle 100 rods in circumference ? 
 
 585. To find the Area of a Circle, the Diameter and 
 
 Circumference being given. 
 
 Multiply half the circumference by half the diameter. 
 
 12. Eequired the area of a circle whose diameter is 75 ft. 
 xi3. What is the area of a circle 200 r. in circumference ? 
 
 586. The common measuring unit of solids is a cube, 
 whose sides are squares of the same name. The sides of 
 a cubic inch are square inches, etc. 
 
 jgjp For definition of rectangular solids, and the method of find- 
 ing their contents, see Arts. 246, 247, and 284. 
 
 583. How find the circumference of a circle when the rliamcter is given* 
 ^84. How find the diameter of a circle, when the circumference is given? 
 585. How find the area of a circle ? 586. What is the men* uring unit of solids ?
 
 358 
 
 MEN SITUATION. 
 
 587. A Pyramid is a solid 
 whose base is a triangle, square, or 
 polygon, and whose sides terminate 
 in a point. 
 
 NOTE. This point is called the vertex of 
 the pyramid, and the sides which meet in 
 it are triangles. 
 
 588. A Cone is a solid which 
 has a circle for its base, and termi- 
 nates in a point called the vertex. 
 
 589. A Frustum is the part 
 which is left of a pyramid or cone, 
 after the top is cut off by a plane 
 parallel to the base ; as, a, It, c, d, c. 
 
 590. To find the Contents of a Pyramid or Cone, the 
 
 Base and Altitude being given. 
 
 Multiply the area of the base by % of the altitude. 
 
 XOTH. The contents of a frustum of a pyramid or cone are found 
 by adding the areas of the two ends to the square root of the product of 
 those areas, and multiplying the sum by ^ of the altitude. 
 
 14. What are the contents of a pyramid whose base is 
 22 ft. square, and its altitude 30 ft. Ans. 4840 cu. ft 
 
 15. Of a cone 45 ft. high, whose base is 18 ft. diameter? 
 
 1 6. The altitude of a frustum of a pyramid is 32 ft, 
 the ends are 5 ft. and 3 ft square : what is its solidity ? 
 
 591. A Cylinder is a roller-shaped solid of uniform 
 diameter, whose ends are equal and parallel circles. 
 
 592. To find the Contents of a Cylinder, the Area of the 
 Base and the Length being given. 
 
 Multiply the area of one end by the length. 
 
 17. "What is the solidity of a cylinder 20 ft. long and 
 4 ft in diameter? Ans. 251.3272 cu. ft. 
 
 5*7. What is a pyramid? 588. A cone! 590. How find the contents of each?
 
 MENSUBATION. 359 
 
 593. To find the Convex Surface of a Cylinder, the 
 
 Circumference and length being given. 
 
 Multiply the circumference by the length. 
 
 1 8. Eequired the convex surface of a cylindrical log 
 whose circumference is 18 ft., and length 42 ft.? 
 
 594. A Sphere or Globe is a solid terminated by a 
 curve surface, every part of which is equally distant from 
 a point within, called the center. 
 
 595. To find the Surface of a Sphere, the Circumference 
 
 and Diameter being given. 
 
 Multiply the circumference by the diameter. 
 
 19. Eequired the surface of a 15 inch globe. Ans. 4.91 sq.ft. 
 
 20. Required the surface of the moon, its diameter 
 being 2162 miles. 
 
 596. To find the Solidity of a Sphere, the Surface and 
 
 Diameter being given. 
 
 Multiply the surface by % of the diameter. 
 
 21. "What is the solidity of a 10 inch globe? A. 523.60^ in. 
 
 22. What is the solidity of the earth, its surface being 
 197663000 sq. miles, and its mean diameter 7912 miles? 
 
 597. To find the Contents of a Cask, its length and head 
 
 diameter being given. 
 
 Multiply the square of the mean diameter by the length, 
 and this product by .0034. The result is wine gallons. 
 
 NOTES. i. The dimensions must be expressed in inches. 
 
 2. If the staves are much curved, for the mean diameter add to the 
 head diameter .7 of the difference of the head and bung diameters ; 
 if little curved, add .5 of this difference ; if a medium curve, add .65. 
 
 23. Required the contents of a cask but little curved, 
 whose length is 48 in., its bung diameter 36 in., and its 
 head diameter 34 inches. Ans. 199.92 gal. 
 
 591. What is a cylinder ? 592. How find its contents? 594. What is a sphere 
 or globe ? 593. How find its surface ? 596. Its contents T 597. Contents of a cask T
 
 360 MISCELLANEOUS EXAMPLES. 
 
 MISCELLANEOUS EXAMPLES. 
 
 1. A square piano costs $650, which is 3-fifths the price 
 of a grand piano : what is the price of the latter ? 
 
 2. A man sold his watch for $75, which was 5 -eighths 
 of its cost : what was lost by the transaction ? 
 
 3. Two candidates received 2126 votes, and the victor 
 had 742 majority : how many votes had each ? 
 
 4. A man owning 3 lots of 154,242 and 374 ft. front 
 respectively, erected houses of equal width, and of the 
 greatest possible number of feet : what was the width ? 
 
 5. Three ships start from New York at the same time 
 to go to the West Indies ; one can make a trip in 10 days, 
 another in 12 days, and the other in 16 days: how soon 
 will they all meet in New York ? 
 
 6. Two men start from the same point and travel in 
 opposite directions one goes 33^ m. in 7 h. ; the other 
 27-^ m. in 5 h. : how far apart will they be in 14 h. ? 
 
 7. Divide $200 among A, B and C, giving B twice as 
 much as A, and C 3 1 times as much as B. 
 
 8. How many bushels of oats are required to sow 
 35f acres, allowing i i-J bushels to 5 acres ? 
 
 9. Bought 4-| bbls. of apples at $3$, and paid in wood 
 at $3!: a cord : how many cords did it take ? 
 
 10. A mason worked u| days, and spending f of his 
 earnings, had $20 left: what were his daily wages? 
 
 11. A fruit dealer bought 5250 oranges at $31.25 per M., 
 and retailed them at 4 cents eacli : what did he make or lose ? 
 
 12. Cincinnati is 7 50' 4" west of Baltimore: when 
 noon at the former place, what time is it at the latter ? 
 
 13. A grocer bought 1000 doz. eggs at 12 cts., and sold 
 them at the rate of 20 for 25 cts. : what was his profit ? 
 
 14. Bangor, Me., is 21 13' east of New Orleans: when 
 9 A. M. at Bangor, what is the hour at New Orleans ? 
 
 15. What is the cost of a stock of i ^boards 15 ft. long 
 and i o in. wide, at 1 6 cts. a foot ?
 
 MISCELLANEOUS EXAMPLES. 361 
 
 1 6. A farmer being asked how many cows he hud, 
 replied that he and his neighbor had 27 ; and that f of his 
 number equaled -f-y of his neighbors : how many had each ? 
 
 17. How many sheets of tin 14 by 20 in. are required to 
 cover a roof, each side of which is 25 ft. long and 2 1 ft. wide ? 
 
 1 8. A and B counting their money, found they had 
 $100 ; and that of A's plus $6 equaled f of B's: how 
 much had each ? 
 
 19. How many pickets 4 in. wide, placed 3 in. apart, are 
 required to fence a garden 21 rods long and 14 rods wide ? 
 
 20. Bought a quantity of tea for $768, and sold it for 
 $883.20 : what per cent was the profit ? 
 
 21. What must be the length of a farm which is 80 rods 
 wide to contain 75 acres ? 
 
 22. What must be the height of a pile of wood 36 ft. long 
 and 12 feet wido to contain 27 cords ? 
 
 23. Sold 60 bales of cotton, averaging 425 lb., at 22^ cts., 
 on 9 m., at 1% int. : what shall I receive for the cotton ? 
 
 24. When it is noon at San Francisco, it is 3 h. i m. 
 39 sec. past noon at Washington : what is the difference 
 in the longitude ? 
 
 25. A man bought a city lot 104 by 31 \ ft., at the rate 
 of $22^ for 9 sq. ft. : what did the lot cost him ? 
 
 26. If ^J of a ton of hay cost 3 , what will |- ton cost ? 
 
 27. What sum must be insured on a vessel worth $16500, 
 to recover its value if wrecked, and the premium at 2%? 
 
 28. How many persons can stand in a park 20 rods long 
 and 8 rods wide ; allowing each to occupy 3 sq. ft. ? 
 
 29. A builder erected 4 houses, at the cost of $4284-} 
 each, and sold them so as to make 16% by the operation: 
 what did he get for all ? 
 
 30. A man planted a vineyard containing 16 acres, the 
 vines being 8 ft. apart : what did it cost him, allowing he 
 paid 6 \ cts. for each vine ? 
 
 31. If a perpendicular pole 10 ft. high casts a shadow 
 of 7 ft., what is the height of a tree whose shadow is 54 ft ?
 
 362 MISCELLANEOUS EXAMPLES. 
 
 32. A miller sold a cargc of flour at 20% profit, by which 
 he made $2500: what did he pay for the flour? 
 
 33. Paid $1.25 each for geographies: at what must I 
 mark them to abate 6%, and yet make 20% ? 
 
 34. Sold goods amounting to $1500, | on 4 m, the 
 other on 8 m. ; and got the note discounted at ]%: what 
 were the net proceeds ? 
 
 35. A line drawn from the top of a pole 36 ft. high to 
 the opposite side of a river, is 60 ft long: what is the 
 width of the river ? 
 
 36. A school-room is 48 ft. long, 36 ft. wide, and 1 1 ft. 
 high : what is the length of a line drawn from one corner 
 of the floor to the opposite diagonal corner of the ceiling ? 
 
 37. The debt of a certain city is $212624.70: allowing 
 (>% for .collection, what amount must be raised to cover 
 the debt and commission ? 
 
 38. A publisher sells a book for 62^ cents, and makes 
 20%'. what per cent would he make if he sold it at 75 cents? 
 
 39. What will a bill of exchange for 534, los. cost in 
 dollars and cents, at $4.87! to the sterling ? 
 
 40. Three men took a prize worth $27000, and divided 
 it in the ratio of 2, 3, and 5 : what was the share of each ? 
 
 41. An agent charges 5^ for selling goods, and receives 
 $135.50 commission : what are the net proceeds? 
 
 '42. A, B, and C agreed to harvest a field of corn for 
 $230; A furnished 5 men 4 days, B 6 men 5 days, and 
 C 7 men 6 days: what did each contractor receive? 
 
 43. A and B have the same salary ; A saves of his, 
 but B spending $40 a year more than A, in 5 years was 
 $50 in debt: what was their income, and what did each 
 spend a year ? 
 
 44. If a pipe 6 inches in diameter drain a reservoir in 
 80 hrs., in what time would one 2 ft in diameter drain it ? 
 
 45. A young man starting in life without money, saved 
 $i the first year, $3 the second, $9 the third, and so on, 
 for 12 vears: how much was he then worth?
 
 ANSWERS. 
 
 ADDITION. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Ex. Aus. 
 
 Ex. ANS. 
 
 J. age <$*>. 
 
 14. 2616263 
 
 33. 13800 
 
 Page 29. 
 
 2. 13839 
 3. l82<0 
 
 15- 9539381 
 
 1 6. $4668 
 
 Page 28. 
 
 $33658, all- 
 
 O -J 
 
 4. 20000 
 
 
 34- 159755 
 
 . o o _. ^ 
 
 54. 156 str. 
 
 5- 20438 
 6. 212269 
 
 Page 27. 
 17. 1376 yds. 
 
 35. 848756 
 
 36. 182404 
 
 37. 1039708 
 
 55. 7213 bu. 
 56- 366 d. 
 
 Page 2G. 
 
 1 8. $6332 
 19. 1695 Ibs. 
 
 38. 11485 
 39. 9929 
 
 58. 50529 
 
 CQ. -2674, A. 
 
 i. 2806 
 
 20. 2668 g. 
 
 40. 13720 
 
 y' o I ^ ** 
 
 60. $4.37.4.4, 
 
 2. $1941 
 
 21. 10438 
 
 41- 233331 
 
 *-' vi rO / T 1 r 
 
 3. 25285 Ibs. 
 
 4- 14756 yds. 
 
 22. 8636 
 
 23. 10672 
 
 42. 1328464 
 43. 8237027 
 
 62. $376.02 
 63. $476.19 
 
 5- 98937 r. 
 
 24. 3874 
 
 44. 25148 
 
 
 6. 2051834 ft 
 7. 2460 A. 
 
 25. 15246 
 
 26. 100980 
 
 45. limiio 
 46. 22226420 
 
 6 5 . $475.8 9 
 
 8- 23459 
 
 27. 1207053 
 
 47. $1460 
 
 Page 3O. 
 
 9. 185462 
 
 28. $9193 
 
 48. 1925 y. 
 
 66. $1704.28 
 
 10. 76876 
 
 29. 3998 bn. 
 
 49. $8190 
 
 67. $16988.71 
 
 " 33367 
 
 30. $107601 
 
 50. 6987 Ibs. 
 
 68. $16580.34 
 
 12. 179589 
 
 31. $38058 
 
 5 1 - 93 y rs - 
 
 69. $179403.71 
 
 13. 273070 
 
 32. 2844 '52. $22338 170. $157011.73 
 
 SUBTRACTION. 
 
 Page 35. 
 
 4- 3779 7- 
 
 15. 309617 T. 
 
 Page 37. 
 
 2. 346 
 
 5- J7i9 A. 
 
 16. 209354 A. 
 
 26. $279979 
 
 3- H7 
 
 6. 11574 
 
 J7- 34943 
 
 27. iiiiiu 
 
 4- 3 IO(5 
 
 7- 22359 
 
 1 8. 1235993 
 
 28. 6333333334 
 
 5- 26 3 
 
 8. 27179 
 
 r 9- 3633805 
 
 29. mil 1 1 112 
 
 6. 509 
 
 9. 267642 
 
 20 - 33 2 376 
 
 30. 289753017- 
 
 
 10. 235009 
 
 21. 349629696 
 
 746. 
 
 Page 3d. 
 
 II. 5009009 1 22. $18990 |3I. 270305844* 
 
 ' 53637 
 
 12. 5542809 | 23. $1915 28516. 
 
 2. 35 r - 
 
 13. 2738729 
 
 24. $415026 J32. 226637999- 
 
 3. 67 Ibs. 
 
 14. 51989 Ibs. 
 
 25. $2OOOO5 
 
 876130.
 
 364 
 
 AN8WEBS. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Ex. ANS. i Ex. ANS. 
 
 33. 1990005 
 34. 995500 
 35. 64564 
 
 36. 999001000 
 
 39. 2235 A. 
 
 40. 26530000 
 
 41. 1732 yrs. 
 42. 84 yrs. 
 43. 1642 y. 
 
 44. 413000000 
 45. $3115027 
 46. 8253204 
 
 QUESTIONS FOR REVIEW. 
 
 Page 38. 
 
 7- 3771 
 
 Page 39. 
 
 23. 2198 
 
 
 8- 803559 
 
 
 24. $4730 
 
 I. $1200 
 
 9. $14292 
 
 16. 108888 
 
 25. $522. 
 
 2. $1107 
 
 10. $963 
 
 17. 126950 
 
 ?6. $1802 
 
 3. 2365 sold, 
 
 ii. $12523 
 
 18. 38022 
 
 27. $13970 B's 
 
 1195 left. 
 
 12. 1700 
 
 19. 43898 
 
 $27440 C's 
 
 4. $4331 
 
 13- 4434 
 
 20. 33885 
 
 28. 5673 
 
 5- 447i 
 
 14. 1 1 21 22 
 
 21. 762 
 
 29- $737 
 
 6. 1279 
 
 15. 127680 
 
 22. l68o 
 
 30. $18775 
 
 MULTIPLICATION. 
 
 Page 46. 
 
 2. 10224 
 
 3- 1970S 
 
 4. 64896 
 
 5. 7.61824 
 
 Page 48. 
 
 16. 2923420500 
 
 17. 1572150300 
 
 18. 450029050401 
 
 19. 1924105179680 
 
 20. 86625 Iks. 
 
 21. $5535 
 
 22. 22360 bu. 
 
 23. $222984 
 
 24. 78475 m. 
 
 25. 1351680 ft 
 
 26. 48645 
 
 27. $33626 
 
 28. $2367500 
 
 29- 54075 yds, 
 
 30. $1786005 
 
 31. $242284 
 
 32. 432 m. 
 
 Page 47. 
 
 2- 242735 
 3. 1230710 
 
 4- 3627525 
 5. 20136672 
 
 Page 5O. 
 
 3. 65100 
 
 4. 290496 
 
 5- 509733 
 6. 3263112 
 
 Page 48. 
 
 6. 332671482 
 
 7. 941756556 
 
 8. 1524183620 
 
 9. 2751320848 
 
 10. 131247355 
 
 11. 351410400 
 
 12. 532125480 
 
 13. 1651148750 
 
 7. 14515201.111. 
 
 8. $295008 
 
 9. $197500 
 
 Page SI. 
 
 21. I49IOOO 
 
 22. 3328000 
 
 23. 166092000 
 
 24. 740000 Ibs. 
 
 25. $184000 
 
 26. 2600000 cts. 
 
 27. $1050000 
 
 28. 604800000 t. i 46. 454842 
 
 29. $7350000 47. 1585873 
 
 I5833 l8 55 
 
 30. 40226351915148000 
 
 31. 64003360044100000 
 
 32. 22812553589100000 
 
 Page 52. 
 
 33. 12615335010000000 
 
 34. 312159700930000000 
 
 35. 263200000756000000 
 
 36. 20776000000000 
 
 37. 42918958404000 
 
 38. 370475105000000 
 
 39. 652303302651 
 
 40. 400800840440000 
 42. 17784 
 
 43- 29484 
 
 44- 45975 
 
 45- 107872
 
 ANSWERS. 
 
 365 
 
 SHORT DIVISION. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Paye 59. 
 
 13. 8243 h. 
 
 22. 1280604^ 
 
 34. 9065} t. 
 
 2. 218392 
 
 14. 211 S. 
 
 23. IOOII62 
 
 35. 52483- bar. 
 
 3. 186782 
 
 15. 8 97 8f r. 
 
 24. 74 6 3 6 7rr 
 
 36. 343^ F- 
 
 4. I72258| 
 
 16. $10671 
 
 25. 1200381 . 
 
 37- 6 469f yds. 
 
 5. 1496471 
 
 17. 122 yds. 
 
 26. 2346842 
 
 38. $9405 
 
 6. 662107^ 
 
 18. 14140^ Ibs 
 
 27. 3562695^ 
 
 39. 5812 sq.yd. 
 
 7. 6865867 
 
 19. $8121 
 
 28. 5848142! 
 
 40. 2500 hrs. 
 
 8. 923808 
 
 
 29. 84472327 
 
 41. 70440 b. 
 
 9. 922969! 
 10. 5762314 
 
 Page GO. 
 
 30. 59363694^ 
 31.64519169^ 
 
 42. $i453 r 
 43- 335 18 3 A. 
 
 ii. 60663768-^ 
 
 20. 1067102^ 
 
 32. 3794| w. 
 
 44. 13920! bar. 
 
 12. 680O2 1 033-1^ 
 
 2I - 933539 
 
 33. $6412 
 
 45. 8090 cows 
 
 LONG DIVISION. 
 
 Page 63. 
 
 3- 2312^ 
 4. 4091 
 5- 2076^ 
 
 /\ f\ I 3 
 
 7- 
 8. 
 
 9- 9759if 
 
 10. 1025277 
 
 11. 50872!^ 
 
 -5 
 
 12. 
 
 13- 
 
 14. 1080887^ 
 
 17- 24377 
 18. 454 am. 
 
 Page 64. 
 
 20. 29796^ 
 
 21. 
 
 22. 
 
 23- 
 24. 
 
 25- 
 26. 
 27. 
 28. 
 2 9 . 
 3- 
 
 32. 
 
 33- 
 
 34- 
 35- 
 36. 
 37- 
 
 38. 
 
 I2294ffi 
 
 450 sh. 
 411^8. ft. 
 
 $94 
 
 54<>H* 
 
 $60 2 
 
 cu.ffc. 
 
 Page 65. 
 
 $4000 
 
 39- 
 
 40. 
 
 41. 
 
 42. 
 
 43- 
 
 2. 
 
 3- 
 4- 
 
 5- 
 6. 
 
 7- 
 
 $3797iM 
 
 $8050^^ 
 
 $275 
 
 66. 
 
 17- 
 
 ft ; 19 or. 
 2 3 pounds. 
 12 com. 
 
 17 
 23 
 26 
 
 Page 67. 
 
 9. Given 
 
 10. 
 
 ii- 7974H 
 
 12. 
 
 14. 66042 if|- 
 
 15. 1502085^ 
 
 Page 69. 
 
 24- 2283^ 
 
 25. 4 o6if> 
 
 26. 
 
 27. 
 
 28. 
 
 29. 
 
 3- 
 
 33. 229 horses 
 
 34- 
 
 35- 
 
 36. 60 bales
 
 366 
 
 ANSWERS. 
 
 QUESTIONS FOR REVIEW. 
 
 Page 6'.9. 
 
 i. 146 J's; 
 365 both 
 2. 407 sheep 
 3. 76 years 
 4. 1779 
 
 5- 393 
 6. 83^4 
 
 /. 4,5 JLUUQ 
 
 8. 5118 bu. 
 
 9. 8613 
 
 10. 35 
 ii. 200849 
 
 Page 7O. 
 
 12. $631 gain 
 
 15. i22-r*y bar. 
 1 6. $i537i 
 17. 60 days 
 18. $156 
 1 9- 1 05^ yds. 
 
 20. $2 
 
 a. T5J033: 
 23. 252 sheep 
 24. 41 J bar. 
 
 2 5- HT& 
 26. 2383/7 
 27. 482 books 
 28. 13 cows 
 
 PROBLEMS AND FORMULAS. 
 
 Page 73-6. 
 
 2. 1157 votes 
 4. 1147 votes 
 6. 1 60 rods 
 
 8. 31 miles 
 10. $22680 
 
 12. 60 
 
 14. $2196, ist. 
 $3172, 2d. 
 
 15. 2428, ist 
 3136 2d. 
 16. $104 ch. 
 $146 w. 
 
 17. 45, A's. 
 30, B's. 
 1 8. 269, A's. 
 $231, B's. 
 
 ANALYSIS. 
 
 Page 78. 
 
 i. 15 cows 
 2. 50 Ibs. 
 3. 432 Ibs. 
 4. 60 bu. 
 5- 3 1 Iks. 
 
 6. 378 bu. 
 7. $ 1 80 cloth; 
 
 $10 
 
 8. Si8 
 9. $315 
 10. 1 8 pears 
 
 Page 7.9. 
 
 12. 31 
 
 13- 54 
 14. ii chil. 
 1 6. 12 less, 
 60 greater 
 
 17. Given 
 18. $112 B's. 
 6361 A's. 
 19. 67, ist 
 176, 2d. 
 
 20. 2Il8 
 
 FACTORING. 
 
 Page 88. 
 
 23. 2 and 3 
 24. 2 and 3 
 
 25. 2 and 2 
 26. None. 
 27. 2, 2, and 2 
 28. 2, 3, and 2 
 
 29. 5 and 3 
 30. 2 and 2 
 31. 2 and 2 
 
 32. 5 
 33. 2 and 2 
 34. 2 and 3 
 
 35- !> 2, 3, 5, 7, n, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 
 
 59, 61, 67, 71, 73, 79, 83, 89, 97. 
 
 36. From 100-200 are 101, 103, 107, 109, 113, 127, 131, 137 
 i39> *49> IS 1 * i57 l6 3> l6 7> i73> *79> *8i, 191, 193, 197, 199-
 
 ANSWERS. 
 
 307 
 
 CANCELLATION. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Page 9O. 
 
 8. 28 
 
 13. 90 
 
 18. 36* 
 
 23. i So ch. 
 
 4. if 
 
 9. 18 
 
 14." 78 J 
 
 19- 378 
 
 24. 315 bu. 
 
 5- 8 
 
 10. 15 
 
 15. 84 
 
 20. 30 bar. 
 
 25. 28 yrs. 
 
 6. 3 
 
 II. 12$ 
 
 1 6. 84 
 
 21. sfffc. 
 
 26. 81 1. 
 
 7- 9 
 
 12. 10 
 
 I 7 . 318^ 
 
 22. 46^ bgsj 
 
 COMMON DIVISORS. 
 
 Page 91. 
 
 8. 5 and 3 _ .11.6 
 i Pane 94. 
 
 9. 2 and 4 12. 4 
 
 1 8. one 
 19- 37 
 
 3- 3 
 
 6. 21 13. 12 
 
 2O. 2 
 
 4- 4> 3> 6 
 5- 6 
 
 Page 93. 7- *5 M- 2 
 
 8. 12 15. 12 
 
 21. 2040 
 
 22. 1 8 yd. 
 
 6. 7 
 
 2. 12 9. 3 1 6. 2 
 
 23. 21 each 
 
 7. 10 
 
 3. 24 10. 25 17. 192 
 
 24. 8 A. 
 
 MULTIPLES. 
 
 Page 98. 
 
 6. 600 10. 288 14. 55440 
 
 17. 60 cts. 
 
 3- 48 
 
 7. 480 n. 12852 15. 57600 
 
 1 8. 720 ro's 
 
 4. 84 
 
 8. 330 12. 15120 16. Jooo 
 
 19. 60 Ibs. 
 
 5. 720 
 
 9. 240 13. 73440 
 
 
 FRACTIONS. 
 
 Page 1O5. 
 
 15- H 
 
 29. ^ 
 
 13. 5^ 
 
 27. $2 9 ffff 
 
 2. | 
 
 16. ii 
 
 
 14. 46-^5- 
 
 28. 3516 yr. 
 
 3- f 
 
 17. 
 
 Page 1O5. 
 
 15. iSflf 
 
 
 4.1 
 
 18. | 
 
 2- 37 
 
 1 6. gojfr 
 
 Page 1O6. 
 
 5- f 
 
 19. T^y 
 
 3. 1 6 
 
 17. 22-,-^ 
 
 2. IfA 
 
 6. j 
 
 20. || 
 
 4. 19 
 
 18. 2IOJ 
 
 3- V 
 
 7- ii 
 
 21. | 
 
 5- : 5l 
 
 19. io7 T y T 
 
 4. -^l 1 - 
 
 8- i 
 
 22. | 
 
 6. 12 
 
 20. 38311 
 
 s-^tt 1 
 
 9-1 
 
 23-1 
 
 7. 16 
 
 21. 20| 
 
 6. 2^^ 
 
 10. | 
 
 24-1 
 
 8. I2| 
 
 22. 2449^4 
 
 7. 1 W 1 
 
 ii. Hi 
 
 25- 1 
 
 9- 3 6 fi 
 
 23. lo||4& ; 8. -2^^ 
 
 12. f 
 
 26. 3 
 
 10. 45^ 
 
 24. 29^1 
 
 9. -S^f^S. 
 
 T 3- * 
 
 7-4 
 
 11. 61 
 
 25- ^frlM 
 
 10. ' R / g 
 
 14. 
 
 28. -^ 12. 3^4 
 
 26. 41-A^lb ii. J '2-3-i
 
 ANSWERS. 
 
 Ex. ANS. 
 
 12. 
 
 3-267-2 
 S3 
 
 1 6. 
 
 1 7- I006T3T 
 
 1 8. 2Llbs. 
 
 Ex. ANS. 
 
 9. 28 
 10. |f 
 
 3% 
 
 Ex. 
 
 ANS. 
 
 -m 
 
 Piojye 1O 7. 
 
 3- i 
 
 4- if 
 
 5- f 
 6. f 
 
 7- H 
 
 T f 3 
 
 ! 5- s 
 
 1 6. ?, 
 
 17. i 
 
 20. 
 21.4 
 
 22. |J bll. 
 
 2 3- 6 s ? yds. 
 
 I4 ' 
 
 II , 
 
 T& 
 
 Hf 
 
 iff 
 
 144 
 
 ^60^ 
 650 
 
 TOOCTO- 
 
 T 
 
 Ex. 
 1 6. 
 
 ANK. 
 
 Ex. ANS. 
 
 1O9. 
 
 9- 
 
 I0 . 
 n. 
 
 12. 
 13- 
 
 Page <//<?. 
 
 4- 7 a 5 
 
 r Jj 6 
 
 6. f 
 
 7- f 
 8. -f 
 
 9-1 
 
 10. ff 
 
 11. -ff 
 
 12. f 
 
 13- lit 
 14. iff 
 
 Page 111 
 
 2 - if, A 
 
 3- ff, *f, ** 
 
 4- iff> 
 
 5- -& 5 v> 
 6- 
 
 7. 
 
 8. 
 
 . if 
 
 *f, Hff H, 
 
 Pttye 112. 
 
 17. -Hi*, *f IMS 
 18. f!|, ifi |H 
 
 =o. if, H f 
 
 21- ff,ff,ff . 
 
 22- ff, U 
 
 23. W, ft 
 
 , , Tns QR Q I 
 2 4 1irB T5"S> 7T 
 
 25* 1000, TtfVo", TO 00 
 
 Page 113. 
 
 3. f i , ft 
 
 4- > i i 
 
 5- -if, tt, If 
 
 6- f&, 
 
 7- if, if, - 2 A- 
 
 8- t, Mi 
 
 n 25 7 
 y. TTJ' To", T 
 
 O I 7 S 
 "> 20~ 
 
 304 154 
 26<f, 21T6 
 
 1 * "JTT, IjV, 1HJ, 
 
 T o _SJ> 140 
 I2 - ffif, 2TC> 
 
 TlO, 2Tff 
 
 13- W, i _ 
 
 15. 45,' w, H 
 
 17. tHf,mtmi 
 
 T Q 32 53 4" 
 10 " 81), TO, 89
 
 ANSWEBS. 
 
 369 
 
 ADDITION OF FRACTIONS. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Page lid. 
 
 8, 9. Given 
 
 10. 
 
 ii. 
 
 12. 
 13- 
 
 14. 
 
 '5- 
 1 6. 
 
 17. 320 
 
 1 8. 75 if! 
 19- 554-BtJ 
 
 22. I3TV 
 
 23- 
 
 2 4 . 
 25. 
 
 26. 51 J y. 
 
 27. $ 4 8 9 | 
 
 29- 
 
 30. 
 
 SUBTRACTION OF FRACTIONS. 
 
 Page 118. 
 
 10. fj 
 
 11. 2 
 
 12. I T 7 ^ 
 !4- H 
 
 16. 3tt* 
 
 19. 
 
 20. 32-! Ibs. 
 
 21. 87^ A. 
 
 Page 119. 
 
 27. 102-f^- 
 
 34- 53 
 
 
 28. 
 
 35- I74H 
 
 22. Given 
 
 29. Given 
 
 36. iu-& 
 
 23- 3 8 i 
 
 30. i 
 
 37. 40-! gaJ 
 
 24. 37i 
 
 3 1 * TTZ 
 
 38. $29$ 
 
 25- 43* 
 
 32. 44 
 
 40- i-H- 
 
 26 ' 6 7* ' 33- 5l 
 
 41- ifl 
 
 MULTIPLICATION OF FRACTIONS. 
 
 JFVeflre /^J, 
 
 3- 2j 
 
 4- 7^r 
 
 6- 3571 
 
 7- 583i 
 
 8. 36 
 
 9. 277! 
 
 u. 2566]- 
 
 '12. 
 
 13- 2.75 
 
 14. $6of 
 
 15. $231! 
 
 16. $1218^ 
 
 1 7 . $8o 4 f 
 
 1 8. $1592^ 
 
 20. 
 
 21. $S4l8 
 
 Page 123. 
 
 3- 24! 
 4. 26^ 
 
 5- 336 
 6. 406 
 
 7- 542$ 
 
 10. $6429!- 
 
 12. 1407 
 
 14. 1256^ 
 
 15. 6i2j 
 
 16. 1009! 
 
 17. 21721-^ 
 
 1 8. 70003! 
 
 Page 124. 
 
 3-1 
 
 4- i 
 
 5- 6| 
 6. |4 
 
 7-1 
 8. | 
 
 10. ii 
 
 II. 
 
 15- 3 
 
 1 6. 78^ 
 
 1 7. 9 cts. 
 
 1 8. $36 
 
 20. 703! 
 
 2L 19531 
 
 22. 15352^ 
 
 24- i 
 
 2 5- iV 
 J'wfire ^^5. 
 
 4- 149^ 
 
 5- 37 ii 
 
 6. 600 
 
 7. 3986ff 
 
 8. 12377^5 
 9- I7317H 
 
 n. u|J 
 12. $5of 
 13- 433 i -eta 
 
 15. 750 tin, 
 3000 cop. 
 
 16. $1918! 
 
 X 7- 44839^
 
 370 
 
 ANSWERS. 
 
 DIVISION OF FRACTIONS. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Ex. Ass. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Page 126. 
 
 28. 68f 
 
 Page ISO. 
 
 O* ^ o 
 
 II. -jVj 
 
 * T 
 
 30. uSiVjr 
 
 2.2|f 
 
 4. i7f^ 
 5. 15 times 
 
 12. 50^ bu. 
 
 13. 245 I T? f. 
 
 5- <T 7 / 
 
 6- * 4 y 
 
 7- "z 2 ? 
 
 3 '** 
 32. $126^ 
 
 33. i|- 
 
 4- I ^ I ff 
 5-4H 
 
 X 
 
 15. $103^ 
 
 8 4 
 e " T2T 
 
 9- TT7 
 
 Page 128. 
 
 2. I26f 
 
 7- 2|^ 
 
 9 I-20-J- 
 
 10. 50 -J Ibs. 
 1 1. 304 A. 
 
 
 I0 *ffSr 
 
 3.672 
 
 9- A 
 
 12. 10 lots 
 
 I 2O 
 
 II. -^j^y 
 
 4-35 
 5-375 
 6. 1479 
 
 10. 4^ 
 
 " i! 
 12.4 
 
 13- 42-iV 4 j J- 
 i4.^ 
 
 I2O 
 
 14- rH^r 
 
 7- 447f 
 
 
 15. 273 
 
 1 8. T 4 j 
 
 
 15. A bar. 
 
 8. 1645^ 
 
 !"> JT^ 
 
 400 
 
 J 9- M 
 
 v/ *fr U 
 
 1 6. S-nnr 
 
 9. 21630 
 
 l6 - 5-H 
 
 Page 132. 
 
 20. 80 days 
 
 
 10. 56727^ 
 
 17. 3^. 
 
 i. $43! err. 
 
 21. $41^ 
 
 1 8. $7f 
 
 12.135 
 
 1 8. 7 fl- 
 
 2. $619^0. 
 
 22. 23!-! 
 
 
 J 3- 477 
 
 ip. 6 
 
 $19^5 er. 
 
 23- IIU 
 
 Page 12 7. 
 
 14. 266 
 
 20. 140^ r. 
 
 3.5o| 
 
 24- T% 
 
 19. 6-^y 
 
 15. 804 
 
 2I< ioff^s.r. 
 
 4- 3 2 f A. 
 
 25. 50 vests 
 
 2. 5-Ui 
 
 17- & 
 
 23- f^f 
 
 5- II 
 
 26. 44^ c - 
 
 21. if 
 
 18. 8 
 
 24. ij- 
 
 6. 3fJ 
 
 27. 12-^ hr. 
 
 22. l|f 
 
 19. I2^| 
 
 2 ^C 2 Y^Vrf 
 
 7. $J sold ; 
 
 28. i6J 
 
 23- lo'oi 
 
 2O. I if 
 
 27. T 4 >A 
 
 il own ; 
 
 29. 60 bu. 
 
 24. 12-zVff 
 
 22. 9 
 
 
 $24783 
 
 30. 244! yd. 
 
 2 r j r 4 81 
 
 23- 3^ 
 
 Prtgre 131. 
 
 8. 86^| A. 
 
 31. 2 
 
 26. 15^^ 
 
 24. 3lr 
 
 i. 5 J mo. 
 
 9. itfr 
 
 ->-> 400 
 3* T089 
 
 *7- 20^5 25. 4^ 
 
 2. 8| Ibs. 
 
 10. f| 
 
 33' 3384 
 
 FRACTIONAL RELATIONS OF 
 
 J'af/c 1.54. 
 
 3- i A 
 
 4- TT 
 
 5- I 
 
 6. | 
 
 8. A 
 
 10. f *wk. 
 
 n. 
 
 12. 
 
 J- i 
 
 NUMBERS. 
 
 Page 135. 
 
 15- $252 
 1 6.
 
 ANSWERS. 
 
 371 
 
 Ex. 
 I?- 
 
 20. 
 
 22. 
 24. 
 
 27- 
 28. 
 
 30. 
 32. 
 
 AKS. 
 
 $132 
 
 Ex. ANB. 
 
 V 
 
 iJl 
 
 33- if 
 
 34- rV 
 
 35- W 
 
 136, 
 
 37- i 
 
 38. I 
 
 39- i 
 
 40. I 
 
 42- 
 
 44- ^ 
 
 Ex. ANS. 
 
 45- A 
 
 46. ff Ib. 
 
 47. 2 ft. 
 
 48. A Ib. 
 49- T* 
 
 3- 84 
 
 4- 9f 
 
 Ex. ANB. 
 
 6. 138 
 
 7. 1017! 
 
 8. 729^ 
 
 9. 2283$ 
 
 10. 2283* 
 
 11. 270 
 
 12. $14720 
 13- 33 2 50 
 
 16. i/ 7 
 
 17- 4 
 
 19. 62-J 
 
 Ex. ANS. 
 
 20. 
 
 22. 288 
 
 25. $3 each 
 
 27. n| t. 
 
 28. 100 cts. 
 8 cigars 
 
 32. 7 times 
 34- 3i 
 
 REDUCTION OF DECIMALS. 
 
 Page 142. 
 
 Page 144. 
 
 JPrtf/e 145. 
 
 Page 14fi. 
 
 8. 4-7 
 
 3- AV 
 
 i. Given 
 
 19. Given 
 
 9. 21.06 
 
 4- TV? 
 
 2. -25 
 
 20. -3333 + 
 
 10. 84.45 
 
 5- 1 
 
 3- -4 
 
 21. .8333 + 
 
 ii. 93.009 
 
 
 4- -75 
 
 22. .2857 + 
 
 12. 7.045 
 
 i 
 
 5- -8 
 
 23. .4444 + 
 
 13. 10.00508 
 
 a IS 
 
 6. .625 
 
 24. .2727 + 
 
 14. 46.0007 
 
 TO onj 
 
 7- -25 
 
 25. .6428 + 
 
 15. 80.000364 
 
 9- TTfftf 
 
 8. .875 
 
 26. .604166 + 
 
 17. .06; .063; 
 
 I0 ?Wtr 
 
 9. .8 
 
 27. .5466 + 
 
 .0109 
 
 II I 0000 
 
 10. .95 
 
 28. 75.6 
 
 18. .3055.00021; 
 
 1 2. 'J 28 0~0" 
 
 n. .6 
 
 2 9 . 136.875 
 
 .000095 
 
 ^ 3- 'J'S'OiJTT 
 
 12. .0875 
 
 30. 26l.68 
 
 19. .004; .0108; 
 .46; .000065; 
 
 14- 20000 
 IP S QI7 I 
 
 13. .02 
 
 14. .000375 
 
 31. 346.8133 + 
 32. 465.0025 
 
 .0001045 
 
 I 5- ^30"ffO-ff 
 
 15. .0078125 
 
 33. 523.00390625 
 
 20. 69.004; 
 
 l ^' TTT^Tnrtr 
 
 16. .00875 
 
 34. 740.01375 
 
 10.0075 ; 
 
 I 7- tW^ff 17. .01125 
 
 35. 956.0078125 
 
 160.000006 
 
 1 8. ff$$ff ! 
 
 
 ADDITION OF DECIMALS. 
 
 Page 147. 
 
 \, 2. Given 
 3. 881.6217 
 4. 139.26168 
 
 6. 118.792 
 7. 892.688 
 8. 2.76231 
 
 9- 92.00537 
 10. 37.417 
 ii. 2.3948 
 
 12. 23.25553 
 
 Page 148. 
 
 13- 575-729i5 
 14. 53.1 bn. 
 
 ^5- 75-97 lbs -
 
 372 
 
 A N s \v i; n s . 
 
 SUBTRACTION OF DECIMALS. 
 
 Ex. ANS. 
 
 Ex. ANS. ' Ex. ANS. 
 
 Ex. ANS. 
 
 Page 14ff. 
 
 Page 149. 
 
 12. .8969755 
 
 19. 44.9955 
 
 i. Given 6. 7.831 
 
 13. .5496933 
 
 2O. .000098 
 
 2. 7.831 
 
 7. 6.60249 
 
 14. .876543211 
 
 21. $443.825 
 
 3. 6.60249 
 
 8- 17-3675 
 
 15. .01235679 
 
 22. 139.83 A. 
 
 4- 17.3675 
 
 9- 77-94794 
 
 1 6. .099 
 
 23. 99.063 
 
 5. 17.94794 
 
 10. 78.569966 
 
 17. .00999 
 
 24. 70.333 m - 
 
 
 ii. 2.896216 
 
 1 8. 99.999 
 
 io72.407m 
 
 MULTIPLICATION OF DECIMALS. 
 
 Page 15O. 
 
 15. 43*. 25 Ibs. 
 
 26. .OOOO00252 
 
 3- -I453 
 
 16. $222.9375 
 
 27. .OOI 
 
 4. .000151473 
 
 17. 469.0625 bu. 
 
 28. I [ooooi 
 
 5. .0000016872 
 
 1 8. $10639.75 
 
 29. .OOOOOOOOOO- 
 
 6. 21800.6 
 
 19. .0126 
 
 31. 3205.05 
 
 7. .012041505 
 
 20. $686.71875 
 
 32. 8003.56 
 
 8. 20.08591442 
 
 
 33- 2.43 
 
 9. 318.0424 
 
 Page 151. 
 
 34- 5-8 
 
 10. 721.36 
 
 21. .0025 
 
 35- 5 
 
 ii. .00004368 
 
 22. .OOOOO4 
 
 36. $50 
 
 12. 1.50175036 
 
 23. .OOOOOO49 
 
 37. $600 
 
 13. .000721236 
 
 24. .000000603 
 
 38. 27.625 bu. 
 
 14. .O20007 
 
 25. .OOOOOO3 
 
 39. 65.625 m. 
 
 DIVISION OF DECIMALS. 
 
 
 8. .13+ | 16. 10 
 
 24. 27 stoves 
 
 Page 153. 
 
 9- .7^5 + 
 
 17. .01 
 
 26. 4-3753 
 
 2. .007 
 
 10. .9768 
 
 1 8. .000002 
 
 27. .063845 
 
 3- 4 
 
 ii. .00675 
 
 19. .0005 
 
 28. .0000253 
 
 4. 600 
 
 T2. .0000576 
 
 20. 50 
 
 29. .0000005 
 
 5- .4154 + 
 
 13- -015 
 
 21. .OOOOOO27. 
 
 30. $.0005 
 
 6. 30.153 + 
 
 14. 625.5 
 
 22. 46 coats 
 
 31. $.0475 
 
 7. 2.4142+ i 15. 10 
 
 23. 44.409 + r. 
 
 
 ADDITION OF U. S. MONEY. 
 
 Page 158. 
 
 i. Given 
 2, $48.625 
 
 3- $780.25 
 4. $200.09 
 5- $224.53 
 6. $761.4785 
 
 7. 817.28 
 8. $69.1925 
 9. $13131.72 
 
 Page 169. 
 
 10. $54-50 
 ii. $120.48 
 
 12. $76.82
 
 ANSWERS. 
 
 373 
 
 SUBTRACTION OF U. S. MONEY. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Ex. Aus. 
 
 Page 159. 
 
 2. $533-105 
 3. $604.625 
 4. $524.50 
 
 5- $585-25 
 
 Page 16O. 
 
 6. $948.33 
 
 7. $171.9625 
 8. $411.075 
 9- $-955 
 
 ]. $74275 
 ii. $494-945 
 12. $28.75 
 
 MULTIPLICATION OF U. S. MONEY. 
 
 Page 16O. 
 
 Page 161. 
 
 12. $.00056 
 
 1 8. $45 
 
 2. $747-5 
 
 7. $100.9125 
 
 | 13- $5 
 
 19. $80.4375 
 
 3. $1569.24 
 
 8. $14.124375 
 
 j 14. $.001011 
 
 20. $1890 
 
 4. $290.625 
 
 9. $67.8375 
 
 15. $6.5625 
 
 21. $548.625 
 
 5. $60.165 
 
 10. $310.596255 
 
 1 6. $58.905 
 
 22. $178.75 
 
 6. $459.25 
 
 ii. $4630.70025 
 
 17. $56.25 
 
 23. $14208 
 
 DIVISION OF 
 
 U. S. MONEY. 
 
 Page 162. 
 
 4. $2.75 
 
 9. .01 
 
 14. $2.965 + 
 
 
 5- $2-921 
 
 10. .1 
 
 15. 925.405 + 
 
 i. Given 
 
 6. 100 
 
 II. IOO 
 
 16. 293.039 + 
 
 2. $1.964 
 
 7. 361.455 + 
 
 12. IOOOO 
 
 17. $.0625 
 
 3. $.05 
 
 8. 4000 
 
 13. 6OOOOO 
 
 18. $.0547 + 
 
 COUN1 
 
 Page 163. 
 
 i. $13958.38 bal. 
 2. $159857.16 bal. 
 
 r I N G - R O O M 
 Page 165. 
 
 2. $206.83 
 3. $4367.125 
 
 EXERCISE 
 
 4. $3183.07$ 
 Page 166. 
 
 5. $71.15 
 
 :s. . 
 
 6. $5201.70 
 7. $717.77 
 8 - $395-37 
 
 ANALYSIS. 
 
 Page 167. !i 3 . $114.06$ 
 
 20. 6132 A. 
 
 10. $31! 
 
 4- $789^25 
 
 14. $47.82 ami 
 
 21. $478.125 1. 
 
 ii. $291! 
 
 5. $4312.50 
 
 15. $3007750 
 
 2. $66J 
 
 12. $35 
 
 6. $43l 
 
 16. 946. i+lb. 
 
 3- $65 
 
 13- $27! 
 
 7. $892* 
 
 17. 7897 Ib. s'd; 
 
 4- $22 
 
 14. $107$ 
 
 8. $0.33^ 
 
 149 Ib. av. 
 
 5. $409$ 
 
 15. $115$ 
 
 9- $5- 26 4 
 
 1 8. 36.45 bu. 
 
 6. $12$ 
 
 
 10. $31.50 
 
 
 7 . $! 5 
 
 Page 169 
 
 ii. 102 bibles 
 
 Page 16 S. 
 
 8. $208$ 
 
 17. $210 
 
 j2. 148^ vests i 19. $247 
 
 9- $58if 
 
 18, $ 547 j
 
 374 
 
 ANSWERS. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 I 9 . $576 
 20. $168 
 
 21. $240 
 
 23. 13.66 yds. 
 24. 1125 Ibs. 
 
 25. 1800 coc's 
 26. 2690 pine's 
 
 Page 17 O. 
 
 2. $78.177 
 
 3. $240.625 
 4. $789.625 
 5. $1169.83-! 
 6. $1203.93! 
 
 7. $40.4875 
 
 8. $22.31} 
 
 9. $100.2472 
 10. $4.51 
 
 II. 814.18-} 
 
 12. $15.975 
 
 REDUCTIO N. 
 
 fage WO. 
 
 Page 192. 
 
 55- *'2-75 
 
 3. 614 far. 
 
 30. 495782 in. 
 
 56. 790 gi. 
 
 4. 39618 far. 
 
 31. 126720 in. 
 
 57. 603 qts. 
 
 6. 8, IDS. 7<1. 
 
 32. $1480 
 
 58. 5713 qts. 
 
 7. 128. gel. 2 far. 
 
 33. $1556.10 
 
 59. 34616 gi. 
 
 8. 41, 53. 2cl. 2 far. 
 
 34. 456 eighths 
 
 60. 10834 gal. i pt 
 
 9. 1 66 1 yds. 
 
 35. 2608 sixteenths 
 
 6 1. 1 68 bottles 
 
 (o. 225 
 
 36. i 44 J yds. 
 
 62. $126.00 
 
 ii. 4440 pwts. 
 
 37. 123! yds. 
 
 63. 2612530 sec. 
 
 
 38. 22 vests 
 
 64. 176010 min. 
 
 Page 191. 
 
 39. $16.25 profit 
 
 6 5- 3 I 55 6 9 2 9-7 sec. 
 
 12. 7865 pWtS. 
 
 40. 1 60 r. 256 sq. ft. 
 
 66. gw. 6d.2oh.i5m. 
 
 13. 80 Ibs. 7 oz. 
 
 41. 5 A. 405. r. 2os.y. 
 
 67. 639 y. 225 d. 5 h. 
 
 14. 7 Ibs. i oz. iSpwt. 
 
 42. 6976414} sq. ft. 
 
 68. 19 y. 164 d. 6 h. 
 
 1 8 grs. 
 
 43. 10240000 sq. r. 
 
 20 min. 
 
 15. 1491 rings 
 
 44. 10 A. 1 08 sq. r. 
 
 69. $212.625 
 
 1 6. $27 
 
 45. $22875 profit 
 
 70. 13 29' 21" 
 
 17. 2653 oz. 
 
 46. 3983040 en. in. 
 
 71. 365.7 17' 
 
 1 8. 6721944 oz. 
 
 47. 32000 cu. ft. 
 
 72. 855631" 
 
 19. 2 t. 725 11). 
 
 48. 10 c. ft. 985 c. in. 
 
 73. 1296000" 
 
 2O. II t. 1141 Ib. I2OZ 
 
 49. 64 C. 86 cu. ft. 
 
 74. 163^ dozen 
 
 21. $700.55 
 
 50. 5259 qts. 
 
 75. 1 500 eggs 
 
 22. $280 profit 
 
 51. 24051 pts. 
 
 76. 694! gross 
 
 23. 24048 drams 
 
 
 77. 9360 pens 
 
 24. 7lb.6oz. idr. 2sc. 
 
 Page 19.'}. 
 
 78. 67 Ibs. 
 
 26. 1928495 ft. 
 
 52. 6641 bu. 
 
 79. 1800 sheets 
 
 28. 7 in. 79 r. i^ ft. 
 
 53. 254 bu. 2 p..3 q. 
 
 80. 4i6f quires 
 
 29. 443440 rods 
 
 54. $412.08 
 
 
 APPLICATIONS OF WEIGHTS AND MEASURES. 
 
 Page 194. Page 19fi. 7. 2 sq. ft 10. 192 bulbs 
 
 i. Given 4-72 rods 8. 1200 bu. n. 3630 gr. v. 
 
 2. i\ A. 5. $46500 pr. $300 $190.575 & 
 
 3. 165 ft, 6. 4116 s. ft 9. Given 12. z\ A.
 
 ANSWERS. 
 
 375 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Page 196. 
 
 23. 147 b. ft. 
 24. $278.10 
 
 35. $112 
 3 6. $337.68 
 
 47- i795ff g- 
 
 14. $324 
 
 
 37. 86751 br. 
 
 Page 2O1. 
 
 15. $8.88 
 16. $402.27! 
 17. $22.50 
 18. $45*139 
 
 Page 198. 
 
 28. $14.85 
 29. $23.40 
 
 38. $3600 
 
 Page 2OO. 
 
 40. 56 yds. 
 
 48. 56^ in. 
 
 49- 5 7-85 7 +b. 
 50. 42 1 Ac- ft 
 
 51. 9-9551 in- 
 
 
 *2O 5P4.R 
 
 41. 44^ yds. 
 
 52. 3750 Ibs. 
 
 Page 197. 
 
 Page 199. 
 
 42. 12^ yds. 
 43. 320 sods 
 
 C ^, <tp7T^TTT 
 J -J ^ / L o 
 
 56. 9y 5 ^ Ibs. 
 
 21. i8fb.fi; 
 
 32. 2-J-^f cords 
 
 44. 48 yds. 
 
 57. 87! rings 
 
 $1.40 val. 
 
 33. i6| tons 
 
 45. 640 tiles 
 
 58. $72-f| 
 
 22. 12^ b. ft. 
 
 34. $133.20 
 
 46. 1 3^ rolls 
 
 59. $805^1 
 
 DENOMINATE FRACTIONS. 
 
 Page 202. 
 
 Page 2O4. 
 
 30. |^| wk. 
 
 41. 17.28 grs. 
 
 2. $ qt. 
 
 1 6. 133. 4d. 
 
 31. | C. 
 
 42. 5, i2S. 6d. 
 
 
 17. 3 pk. 6 qt. 
 
 32- ff 
 
 0.4032 far. 
 
 3- "JT u 
 
 1 8. 1250 Ibs. 
 
 
 
 4. -fe br. 
 
 19. 10 oz. 10 p. 
 
 Page 2O5. 
 
 Page 2O6. 
 
 5- 3f z - 
 
 20. 3 fur. 13 r. i 
 
 34. 2S. 6d.o.4 + 
 
 44. .5423 + lb. 
 
 6. ^ s. in. 
 
 yd. 2 ft. 6 in. 
 
 far. 
 
 45. .005 ton 
 
 
 J 
 
 21. 146 s.r. 2os.y. 
 
 35. 10 oz. iSp. 
 
 46. .45539 + m. 
 
 
 i s. ft. 72 s.in. 
 
 22.56 grs. 
 
 47. .04 lb. 
 
 Page 2O3. 
 
 22. 112 CU. ft 
 
 36. 2 furlongs 
 
 4 8. .5 bl. 
 
 g. Sy^ 
 
 24. gal. 
 
 32 rods. 
 
 49. .409+ r. 
 
 Q T 
 
 25. i-i|? lb. 
 
 37. 2 qt. i pt. 
 
 50. .25 
 
 10. Tt^oz. 
 
 26. fl bu. 
 
 38. 39^36" 
 39. i d. i6b.29 
 
 51. .48 bl. 
 52. 1.44 r. 
 
 ii- ritfgal- 
 
 27. ^ ff ton 
 
 m. 1 6.8 s. 
 
 53. .166+ wk. 
 
 12. -f-SV m - 
 
 28. ff sq. yd. 
 
 40. 505 lb. 1.92 
 
 54. .28182 + 
 
 '3-trfwt. 
 
 2 9- tWl 
 
 oz. 
 
 55. .41666 + 0. 
 
 METRIC NOTATION AND NUMERATION. 
 
 Page 212. 
 
 5370.9845 dekameters; 
 537.09845 hektometers ; 
 53.709845 kilometers; 
 537098.45 decimeters. 
 
 2. 450.5108 dekagrams; 
 450510.8 centigrams; 
 45.05108 hektograms ; 
 4.505108 kilograms.
 
 37G 
 
 ANSWERS. 
 
 REDUCTION OF METRIC WEIGHTS AND MEASURES. 
 
 Ex. Ass. 
 
 Ex. Aus. 
 
 Ex. ANS. 
 
 rage 213. 
 
 i. Given 
 2. 437500 sq. m. 
 3. 867000 grams 
 4. 26442 liters 
 
 5. 256100 sq. m. 
 6. 8652000011. dm. 
 7. 4256250 grams 
 8. Given. 
 9. 65.2254 hektars. 
 
 10. 0.087 kilos, 
 u. 1.48235 kg. 
 12. 39.2675 kl. 
 
 APPLICATIONS OF METRIC WEIGHTS AND MEASURES 
 
 Page 215. 
 
 8. Given 
 
 15. 72.492+ kl. 
 
 3. 39.14631 m. 
 
 9. 148.87775 A. 
 
 16. 143.223+ kg. 
 
 4. 1 9.8 134 gals. 
 
 10. 4237.92 cu. ft. 
 
 17. 6000.06+ s. m. 
 
 5. 15.89 bu. 
 
 12. 58.2934- m. 
 
 18. 16.378+ beet 
 
 6. 4.2324 oz. 
 
 13- 6236.959+ kg. 
 
 19. 410.748+ c. m. 
 
 7. 303.68365 Ibs. 
 
 14. 236.5 85 + liters. 
 
 20. 27958.715 +c.m 
 
 COMPOUND ADDITION. 
 
 Page 217. 
 
 9. 1 96 bu. 2 pk. 7 q. 
 
 14. 64 A. 7 s.r. i o-J s.y 
 
 3. 11, xos. od. 2 f. 
 
 10. 6 C. 80 cu. ft. 
 
 17. 15 cwt. 50 Ib. 4oz 
 
 4. z6T.3Cwt.83lb. 
 
 ii. 98 bu. 3 pk. 2 q. 
 
 1 8. 2 pk. 4 qt. 
 
 3 oz. 
 
 
 19. goz. ipwt. logr. 
 
 5. 45 bu. o pk. 2 qt. 
 
 Page 218. 
 
 20. 3 s. 6 d. 3.2 far. 
 
 6. 74$ yds. 
 
 12. Given 
 
 21. 2 A. 52 sq r. 
 
 7. 1093 Ib. 5 oz. 
 
 13. 23 wk. i d. 17 h. 
 
 22. 2 C. 8 1 en- ft 
 
 8. 55 gal. 2 qt. 
 
 58 m. 
 
 1209! cu. in 
 
 COMPOUND SUBTRACTION. 
 
 Page 219. 
 
 10. 3 m. 5 fur. 38 r. 
 
 4. 150 days 
 
 
 5 yd. o ft. i in. 
 
 5. 224 days 
 
 2. i fur. 39 r. i yd. 2^ ft. 
 
 2." 9 s. 9 d. 
 
 6. 10 1 days 
 
 3. 7 Ib. 4 oz. 1 7 p. 9 grs. 
 4. 7 T. 8 cwt. 26 Ib. 
 
 '3. 2pk.sqti.2p. 
 
 Page 223. 
 
 5. 8 gal. i qt. i pt. 2 gi. 
 
 4- 4^ pt- 
 5. i jib. 2 oz. 2 p. 
 
 10. 86 19' 24" 
 
 o / // 
 
 
 6. ii 4. 4 sq. r. 
 
 11. 7 24' 7 
 
 OO ' 
 
 Page 22O. 
 
 7. 31.25 Ibs. 
 
 12. 10 2 
 
 13. 5 6' 46" 
 
 6. 159 A. 12 sq. r. 
 
 Page 221-2. 
 
 14. 15 4' 16" 
 
 222^ sq. ft. 
 7. 46 cu. ft. 1689 cu. in. 
 
 3. 67 y. gm. 22 d. 
 
 15- 19 35' 
 16. 56 u' 
 
 8. 145 A. 21 sq. r, 
 
 2. 113 davs 
 
 17. 70 29' 
 
 9. 46$ yd. 1 3. 74 days 
 
 18. 10 19' 38"
 
 ANSWERS. 
 
 377 
 
 COMPOUND MULTIPLICATION. 
 
 Ex. AN-. 
 
 Ex. ANS. 
 
 Ex. Ass. 
 
 Page 224. 
 
 2. 98T.i7Cwt 28 Ib. 
 3. 151, 153. 9fd. 
 4- 33oz.ispw.iog. 
 
 Page 225. 
 
 7. 331 gal. 2 qt 
 
 8. 22 C. 91 cu. ft 
 9. 23, 153. 3^(1. 
 10. 562m. 4 fn. 24 r. 
 ii. 1937 bu. i pk. 
 12. 22 C. 57 cu. ft. 
 13. 6 1 T. 844 Ibs. 
 14. 1307 r. 8 qr. 8 s. 
 
 15- 161 37' 30" 
 16. 431 h. 15 m. 
 !? J 33 sq. r. 21 sq. 
 yd. sq. ft 
 18. 73 T. 1492 Ibs. 
 19. 1571 bu. 2 p. 4 q. 
 20. 1 946 gal 3 q. i p. 
 
 COMPOUND DIVISION. 
 
 Page 227. 
 
 3. 4 fur. 8 r. 2 yd. 2^ ft 
 4- 6 gal. 3 qt o pt 3! gi. 
 
 5. 25 bu. o pk. i qt I pt. 
 
 6. 15 A. 1 06 sq. r. 5 sq. yd. 
 sq. ft 
 
 9- 
 10. 
 
 26f spoons 
 8800 rails 
 1 049^ times 
 12 books 
 
 11. 2 A. 64 sq. r. 
 
 12. 6 bu. i pk. i qt. 
 
 COMPARISON OF TIME AND LONGITUDE. 
 
 Page 228. 
 
 Given. 
 
 43 m. 32.13+ s. 
 
 1 1 o'c. 1 2 m. 4 s. 
 
 1 2 o'c. 3 701. 8.43. 
 49 m. 20 s. 
 
 6. i o'c. 30 m. E. ; 
 10 o'c. 30 m. W. 
 
 7. 54 m. 19.8 sec. 
 
 8. 13 m. 44.86+3. 
 
 Page 229. 
 
 9. Given 
 
 PERCENTAGE. 
 
 10. 6 9' 
 
 11. 6 45' 5" 
 
 12. 45 24' 45" 
 
 13. 56 m. 49+ sec, 
 
 14. 12 o'clk. 34 m. 
 47 sec. 
 
 13 
 14 
 '5 
 Page 234. 
 
 3. $24.21 
 
 4. 10.8 bu. 
 
 5. 22.6 yds. 
 
 6. 8.64 oxen 
 
 7. 8.25 yds. 
 
 8. $11.584 
 
 Page 232. 
 
 5; - 2 s; -75; - 2 ; 4; -60; .80 
 .10; .70; .90; .05; .35; .12; .06 
 
 .66f; .i6|; .125; .625; .875; .58$; .91$ 
 
 9. 100.8 Ibs. 
 10. 63.14 T. 
 IT. 52.5 men 
 12*. 145.656 m. 
 
 13. $857.785 
 
 14. .106 
 15- 1-25 1- 
 
 1 6. 1.38 k. 
 
 17. 840 Ibs. 
 
 18. 81000 
 
 19. 78.75 bu. 
 
 20. .35 Ibs. 
 
 21. 9.765 gals. 
 
 22. 840 men 
 
 24. $215 
 
 25. 157.2 Ibs. 
 
 26. 32.25 m. 
 
 27. 116 1. 
 
 28. 580 sheep 
 
 29. 156 bu. 
 
 30. $925
 
 378 
 
 ANSWERS. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 .ji. 576 cu. ft. 
 
 14. $10200 
 
 Page 238. 
 
 19. 3000 m. 
 
 32. 2, is. 
 
 15- $9750 
 
 4. 250 bu. 
 
 20. 360000 p. 
 
 
 
 5. $362.50 
 
 
 Page 236. 
 
 Page 237. 
 
 6. 1 80 tons 
 
 Page 24O. 
 
 3. 1150 s. C.; 
 
 2-r -7 I 7 
 * o^r/^ 
 
 7- 45 
 
 5. 809 
 
 880 s. D. 
 
 3- 25$ 
 
 8. 600 
 
 6. 1820 
 
 4. $2300.91 
 
 4. 33$$ 
 
 9. 360 fr. 
 
 7. Given 
 
 5. $2850 
 
 5. 15% 
 
 10. 8000 
 
 8. f 
 
 6. 504 bales 
 
 6. t)\% 
 
 II. $200 
 
 9- A B's 
 
 7. 2018.75 gal. 
 
 7- S's/o 
 
 12. $i6666f 
 
 10. 5175 m. 
 
 8. 1312 m. 
 
 8. 10% 
 
 13. $17600 
 
 ii. $3648 
 
 9. $470-40 
 
 9. 80^ 
 
 14. 54.4 yds. 
 
 12. 6875 P- 
 
 10. 129 t. 
 
 10. 10% 
 
 15- $50 
 
 13- $3350 
 
 ii. $1410.50 
 
 II. 23-^% 
 
 16. 100 
 
 14. 228 sheep 
 
 12. $228 
 
 12. 42^ 
 
 17. $5000 
 
 1 5. 400 pupils 
 
 13. $4823.33 J 
 
 X 3- 3fj/ 
 
 18. $1920 
 
 1 6. 5800 m. 
 
 COMMISSION AND BROKERAGE. 
 
 Page 242. 
 
 3. $31.1431$; 
 4. $366.225 com.; 
 $10902.225 paid 
 
 Page 243. 
 
 5. $4310.145 
 6. $238.66-}; 
 8. \% 
 9- 5% 
 10. \% 
 
 Page 244. 
 
 12. $9000 
 
 13. $4212 sales; 
 $4001.40 net p. 
 14. $1500 col.; 
 $1432.50 paid 
 15. $9000 sale ; 
 $8865 rec'd 
 17. $5100 
 1 8. $6834.87 2 sales 
 $170.872 com. 
 
 Page 245, 6. 
 
 19. $15488.89 + 
 
 20. $26635.294 + 
 
 22. $3010.75 
 
 23. $2419.23 + 
 
 24. #49261.083 +in.; 
 $738.916+ com. 
 
 25. $13687.50 g. a.; 
 $360.75 ch.; 
 $13326.75 net p. 
 
 26. $12127.8? net p. 
 
 PROFIT AND LOSS. 
 
 Page 247-9. 
 
 ii. $453.60 
 
 19. $12810 
 
 3. $22.20 
 
 12. $2942.50 
 
 21. 28f% 
 
 4. $22 
 
 13. $9520 
 
 22. 11^% 
 
 5. $.0875 
 
 14. 1 1 cts. 
 
 23. 6o< 
 
 6. $16.65 
 
 15. $i a pair 
 
 '25. 100% 
 
 7. $1.40 
 
 1 6. $2.77! each 
 
 26. 50^ 
 
 10. $2925 A.; 
 
 17. $4.90 a yd. 
 
 27. 66$% 
 
 $2075 B. 
 
 1 8. $4331.25 
 
 28. 40^
 
 A X S W E E S . 
 
 379 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Page 25O. 
 
 Page 252. 
 
 14. $7 cost; 
 
 29. 50% 
 
 51. $.736 marked p. 
 
 $8.05 sell'g pr. 
 
 30. Given 
 
 52. $34.78+ m. pr. 
 
 15. $2 cost; 
 
 31- s% 
 
 53. $4.80 m. pr. 
 
 $1.75 sell'g pr. 
 
 32. 5$ 
 
 54. $i.i6f m. pr. 
 
 
 33- 33i# 
 
 55. $75 each 
 
 Page 254. 
 
 34. $12.60 pr.; 
 
 56. $5.75 per yd. 
 
 1 6. $5200 sales; 
 
 7H# ' 
 
 
 $4940 net 
 
 36. $9750 
 
 Page 253. 
 
 17. $24000 sales 
 
 
 i. $2.691 prof. 
 
 18, $1.25 cost 
 
 Page 251. 
 
 2. $6356.72^ 
 
 19. $5250 cost; 
 
 39. $i cost per Ib. 
 
 3. $3487.575 sh.; 
 
 $1750 gain 
 
 40. 822500 cost; 
 
 $850.84 ging. ; 
 
 20. .7894^ per gal.; 
 
 $30000 sell'g pr. 
 
 $4338.415 both 
 
 . 1 48 1 gain per g.; 
 
 41. $17500 spend; 
 
 4. $22 loss 
 
 $49.73 wh. cost; 
 
 $2 1 ooo am t. sales 
 
 5. $77812.50 cost 
 
 $9.33 wh. gain 
 
 42. $12000 A's inv. ; 
 
 6. $897.45 net pr. 
 
 21. $10 a barrel 
 
 $9375 B'sinv. 
 
 7- 38f# 
 
 22. $51311! cost; 
 
 43. 8250000 cost; 
 
 8 - 5 6 ^ prof.; 
 
 $6413! gain 
 
 $275000 amt. s. 
 
 $3240 prof. 
 
 23. $157 98.963 sale; 
 
 44. $.02 cost; 
 
 9. i6f% loss 
 
 $55 2 -9 6 3 com. 
 
 $025 selling pr. 
 
 10. .052 or ST%-#C.; 
 
 24. 3*&% 
 
 45. Given 
 
 $1766.835 net p. 
 
 2 5- 5Tn&# loss 
 
 46. $10061. 71} 
 
 ii. 25$ loss 
 
 26. $60 m. p. 
 
 47- $13319-672 
 
 12. 125$ 
 
 27. (>\% loss 
 
 48. 22 cost 
 
 13. $9300 cost; 
 
 28. 15 cts. 
 
 49. 3 cents 
 
 $11 1 60 sell'g pr. 
 
 29. lo-j^ib loss 
 
 INTEREST. 
 
 Page 258. 
 
 14. $680.28 
 
 27. $99.96 | 6. $73.96 
 
 3- $2.54 
 
 15. $81.44 + 
 
 28. $24.00 7. $64.18 
 
 4. $1.676 
 
 16. $2875.792 
 
 29. $163.842 ; I 8. $26.49 
 
 5. $1.224 
 
 17. $362.315 
 
 $2740.652 ! 9. $1.65 
 
 6. $4-523 
 
 18. $2759.962 
 
 
 10. $29.933 
 
 7- $43-356 
 
 19. $5032.083 
 
 Page 259. 
 
 1 1 . ^60.04 
 
 8. $110.77 
 
 -20. $122.40 
 
 2. $24.44 I2 $30.52 
 
 9- $33-42 
 
 21. $226.69-^ 
 
 3- $6.252 
 
 13. #2450.80 
 
 10. $23.364 
 
 22. $45.053 
 
 
 14. $20819.80 
 
 11. $25.577 
 
 23. $216.489 
 
 Page 2(iO. 
 
 
 12. $94.35 
 
 24. $493.20 
 
 4. 4.82 
 
 Page 261. 
 
 13. $17,91 ^25. $3923.47 
 
 5. $6.75 
 
 3. $2.8435
 
 8U 
 
 ANSWERS. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Ex. AKS. 
 
 Ex. ANS. 
 
 4. $2.192 
 
 4- <>% 
 
 7. i4f jrs. 
 
 2. $235.85 
 
 5- $11-413 
 
 5- 8# 
 
 8..i6f yrs. 
 
 3- $327-36 
 
 6. $13.89 
 
 6- 9&% 
 
 9. i8| yr-s. 
 
 4. $8928.57 
 
 7. $1285.33 7- *\% 
 
 
 5. $892.86 
 
 8. $13.876 8. s%\ % 
 
 2. $i666f 
 
 6. $5582.142 
 
 9- $4363-044 
 
 9- 7$ 
 
 3. $3000 
 
 
 10. $H4.i6|- 
 
 10. 6$ 
 
 
 Page 268, .<>. 
 
 ii. $185.18 
 
 
 Page 264. 
 
 i. Given 
 
 12. $81.358 
 
 Page 263. 
 
 4- $2333$ 
 
 2. 81519.71 
 
 
 3- 5f .Y-> or 5 
 
 5. $2500 
 
 3- *5 38-63 
 
 Page 262. 
 
 y.8m.i7d. 
 
 6. $40000 
 
 5. $270.19 
 
 2. 1% 
 
 4. 9 m. 2 d. 
 
 7. $10000 
 
 6. 8388.23 
 
 3- 1% 
 
 5. y.,or3m. 
 
 8. $25000 
 
 7. $516,32 
 
 COMPOUND INTEREST. 
 
 Page 273, 4. 
 
 5. 8200.63 
 
 8. $551.58 
 
 13. $21825.26 
 
 i. Given 
 
 6. 82165.713 
 
 9. $1377-41 
 
 14. $29849.56 
 
 2. $112.52 
 
 
 10. $1543-65 
 
 15- $37/04.95 
 
 3. $161.63 
 
 Page 275. 
 
 ii. $8104.25 
 
 1 6. $3298.77 
 
 4. $164.61 
 
 7. Given 
 
 12. $32564.58 
 
 - 
 
 DISCOUNT. 
 
 Page 277. 
 
 7. Given 
 
 I 3 . $528.33 
 
 6. 8' 17.*$^ i 
 
 i. Given 
 
 8. 886.57 
 
 14. 846,73 for. 
 
 7- $735-7 
 
 2. $283.47 + 
 
 9. 8101.892 
 
 X 
 
 8. Given 
 
 3. $462.96 
 
 10. 891.35 dis; 
 
 Page 279. 
 
 9. 8768.44 -^ 
 
 4- $1213-59 
 
 82 283.65 p. 
 
 3. $718.685 
 
 10. $15787*4. 
 
 5. $2336.45 
 
 ii. $3775.264 
 
 4. 8989.50 
 
 ii. $2226.23 
 
 6. 84464.28 
 
 12. $20.377 
 
 5. $1721.875 
 
 12. $7503.83 
 
 STOCKS AND BONDS. 
 
 Page 282. 
 
 8. $12880 
 
 12. $10852.37$ 
 
 Page 284. 
 
 2. $243 
 
 
 13. $19293.75 
 
 20. 40 bonds. 
 
 3- $525 
 
 Page 2S3. 
 
 14. $672 cur. 
 
 21. $12000 
 
 
 9. $10497.50 
 
 1 6. 15% 
 
 23. $44166$ 
 
 5. 8400 
 
 10. $6363.61 
 
 17. 6\% 
 
 24. 835625 
 
 6. $476 
 
 ii. $6336.11 
 
 1 8. 6%% 
 
 25. $56000
 
 ANSWERS. 
 
 381 
 
 EXCHANGE. 
 
 Ex. Asa 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Page 287. 
 
 2. $2049 
 3 . $3488/80 
 4. $4130.647 
 5. $203 
 
 8. $1219.51 
 
 Page 288. 
 o. $1491.053 
 
 10. $2617.801 
 II. $3751.95 
 12. $3750 
 
 Page 289. 
 
 1,2. Given. 
 3. $1858.80! 
 4. $4866.50 
 
 Page 29 O. 
 
 6. $56125.12 + 
 7. Given. 
 8. 5 1 1, 15 s. 
 4 d. 3.2 far. 
 9. 774, 7 s. 
 10 d. i. 28 far. 
 10. 1026, 8s. 
 7.2 d. 
 
 1 1. .1542,1 is. 
 
 6d. 
 
 Page 291. 
 
 12. Given 
 13. $675.68 
 14. Given 
 15. 13050.00 f. 
 1 6. 16474.50 f. 
 
 INSURANCE. 
 
 Page 293. 
 
 i. Given 
 2. $6.65 
 
 3- $38-40 
 4. $84.375 
 
 5. $10.70 
 6. $25.00 
 7. $197.60 
 8. $875 
 9. $1569.625 
 
 Page 294. 
 
 10. Given 
 ii. $15747.423 
 
 12. $27806.122 + 
 
 *3- $37105.263 + 
 
 i. Given 
 
 2. $125.00 
 
 3- $2437.50. 
 4. $5000 
 5. $45000 gr. 
 
 Page 297. 
 
 3. $1003.75, C'stax 
 $1250, D's "' 
 $1375, E's " 
 $925, Fs " 
 
 TAXES. 
 
 4. 2% rate; 
 $159.75, A's tax 
 
 5. z% rate; 
 $450, G's tax 
 
 6. $309.75 H's tax 
 
 7. $% rate; 
 $170, A's tax 
 
 9- 
 10. 
 
 ii. $11052.63 + 
 
 Page 299. 
 
 2. $4062.50 
 3. $1063.314 
 
 DUTIES. 
 4. $1999.20 
 6. $945 v 
 7. $2296.80 
 
 Page 3OO. 
 
 2. $Il8.25 
 3. $425.75 
 
 EQUATION OF PAYMENTS. 
 
 Page :t(>2. 
 
 2. 4 m. from J. 20, or Oct. 20 
 
 3. 8 ra. 18 d. 
 
 4. Mch. 10+17 d.=Mch. 27 
 
 5. June i, or in 3 in. 
 
 AVERAGING 
 Page 3OO. 
 
 1-3. Given 
 4. 7 m. extension ; 
 Bal., $1650 
 
 Page 803. 
 
 6. Given 
 
 8. July 15+51 d. Sept. 4 
 
 9. $1483.25 ami; 
 
 42.9 d. av. time ; due Nov. 1 7 
 
 ACCOUNTS. 
 
 5. $300 btil. debts; 
 22320 bal. prod.; 
 74 d. av. time 
 Due May 23.
 
 38S 
 
 A N 6 W E B S . 
 
 SIMPLE PROPORTION. 
 
 Ex. AN*. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Ex. Ana. 
 
 Page 312. 
 
 14. 8133 
 
 26. 85223^ 
 
 38. 25 min. 
 
 ?. 8160 
 
 15. 2 240 times 
 
 27. 812.50 
 
 39. 6600 rev. 
 
 O 
 
 4- 8165 
 
 1 6. 5 years 
 17. 8 hours 
 
 28. 14.4 in. 
 29. 26f yds. 
 
 40. 840 
 41. 2\ hrs. 
 
 6 &6 
 
 
 30. 97920 t. 
 
 42. 3 7 ? y hrs. 
 
 7. 19, 7s.6|d. 
 
 Page 314. 
 
 18. 1 100 men 
 
 31. 180 
 32. 80 ft. 
 
 43. 45 days 
 44. 1000 rods 
 
 
 19. 240000 Ib. 
 
 33. 64^ orang. 
 
 45. 300 hrs. 
 
 -P<jre 3 13. 
 
 20. 27 horses 
 
 34. 3000 mi. 
 
 46. 8191.78 
 
 9- i35 A. 
 
 21. 8612 
 
 35. 33 men 
 
 47. 8m. 518. 
 
 10. 8937^ 
 
 22. l 
 
 36. 30 ft. 
 
 before 9 
 
 ii. 270 miles 
 
 23. 85.06^ 
 
 
 48. 3^ cts. loss 
 
 12. 8288 
 
 24. 8880.40 
 
 Page 315. 
 
 49. 1 50 m. less; 
 
 13. 8822^ 
 
 25. 849 -5 2 i 
 
 37. $1920 
 
 1 86 m. gr. 
 
 COMPOUND PROPORTION. 
 
 Page 317. 
 
 Page 3 IS. 
 
 9- $75 
 
 13. 750 Ibs. 
 
 3. 2$ days 
 
 6. Si26.66f 
 
 10. 10285^ yd. 
 
 14. 862.50 
 
 4. 9 horses 
 
 7. $100 
 
 ii. 37-Jdays 
 
 15. 240 sofas 
 
 5. 331 weeks 
 
 8. 360 miles. 
 
 12. &I454A 
 
 1 6. 864 tiles 
 
 PARTITIVE 
 Page 319. 
 
 40 s. ; 60 s. ; 100 s. 
 
 PROPORTION. 
 
 4. 60 bu. oats; Sob. p.; nob. c. 
 5- 7i; !o6|; 142; 177! A. 
 
 2. $222$, A's loss; 
 $277^, B's " 
 
 3. 8340, A's share ; 
 8510, B's " 
 
 PARTNERSHIP .Page 321-3. 
 
 896B's;$i6oC's n. 81500, H'd.; 
 8. 8i42.8q4,A'ssh. 82250, Cont.; 
 83000, Am. 
 
 12. 8263.38^] |,A's; 
 
 4. $5 06 7 1|, A's g'n; 
 
 -H, C's 
 
 5. $3400; $5100; 
 and 86800 
 
 6. 864, A's share ; 
 
 Page 324. 
 
 2. 81687.50, A's sh.; 
 
 $266.66f, B's " 
 
 8 190.47!!, C's" 
 
 9. 824.48!!, one ; 
 $25.51^, other 
 10. 8424^, A's sh.; 
 8424^, B's u 
 C's 
 
 , D'S " 
 
 13. 8402 1 -f ||, A's; 
 $3846!!!, B's; 
 
 BANKRUPTCY. 
 
 81205.35^, B's sb.; 
 8857.14!, D's 
 
 3. 82060, D rec'd ; 
 ^12100 net proc.
 
 AN S WEES. 
 
 383 
 
 ALLIGATION. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Page 323. 
 
 2. I5| Cts. 
 
 9. 2 p. at 15 ; 
 i p. at 1 8 ; 
 
 Page 328. 
 
 1 8. 54-j^lb. 280; 
 iS-^j- Ib. 30 ; 
 
 3- 95A cts. 
 
 2 p. at 2 1 ; 
 
 14. 3-J Ibs. ea. 
 
 54iT -Ib. 3" 5 
 
 4-93? ct s - 
 5. 2o carats 
 6. $15.68 
 
 Page 327. 
 
 8 T. <5 bu i st * 
 
 5 p. at 22 
 10. 9lb.at32 c; 
 61b. at 40; 
 6 Ib. at 45 
 ii. 81b.at2oc; 
 
 15. 5oq.at4C.; 
 300 q. at 6 
 
 Page 329. 
 
 17. 2olb. at6c.; 
 
 72-frlb. 42 
 For other ans. 
 see Key. 
 
 53i g- 45 5 
 
 40 bu. 2d ; 
 40 bu. 3d 
 
 5 Ib. at 35 ; 
 1 2 Ib. at 40 
 
 20 Ib. at 8 ; 
 60 Ib. at 12 
 
 140 g. 60 
 
 INVOLUTION .Page 331. 
 
 8. 25; 36; 49; 64; 81 ; 100 ; 400; 900; i6oa; 2500; 3600, 
 
 4900; 6400; 8100. 
 
 9. .25; .36; .49; .64; .81; .0001; .0004; .0009; .0016; .0025; 
 
 .0036; .0049; .0064; .0081. 
 
 10. 125 
 
 11. 64 
 
 12. 2299968 
 
 13. 1024 
 
 14. 4096 
 IS- IS625 
 
 16. 8.365427 
 
 17. 64.014401080027 
 
 1 8. 64024003.000125 
 
 21. 
 
 EXTRACTION OF THE SQUARE ROOT. 
 
 Page 338. 12. .8514 + 
 
 20. 3.16 + 
 
 28. 98.7654 
 
 37. 4-1683 
 
 5- 4 2 7 
 
 13- -355 + 
 
 21. 3.316 + 
 
 
 38- i 
 
 6.719 (14.1.635 + 
 
 22.3.46 + 
 
 Page 339 
 
 ^o. 24 
 
 7.772 
 8. 1.892 + 
 9. .64 
 
 15.69.47 + 
 16. 21.275 + 
 17. 2.236 + 
 
 23. 4.42 
 
 24- 57-3 
 25. 9.36 + 
 
 33- -745 + 
 34. .866 + 
 
 40. .8545 + 
 41. i.or8 + 
 
 10. .347 
 
 18. 2.64 + 
 
 26. 1 1 ii i. ii + 
 
 35. 2.529 + 
 
 42. -V- 
 
 ii. .41 ; 19. 2.828 + 
 
 27. 7856.4 
 
 36. 3.63 + 
 
 43- 
 
 Page 339-41. 
 
 2. 420 rods. 
 3- 559- 2 8 r. 
 
 4. 119 trees 
 
 5. 238 men 
 
 7. 1 08 yds. 
 
 8. 30 rods 
 
 APPLICATIONS. 
 
 9. 438.294- m. 
 10. 103.61 + ft. 
 ii. 56.56+ ft. 
 12. 75.81+ ft 
 
 1 6. Given 
 17. 12 
 
 18. 54 
 19. 63.49 + 
 
 Page 342. 
 
 20. 1.75 
 
 15. 3^ min. | 22. &
 
 AK8TTEBS. 
 
 EXTRACTION OF THE CUBE ROOT. 
 
 Ex. AKS. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Ex. ANS. 
 
 Page X4S. 
 
 1-2. Given 
 
 3- 85 
 
 4. 4-38 + 
 5- 72 
 6. 1.44 + 
 
 7. 2.76 
 
 8. 2.57 + 
 9. 8904 
 10. .632 + 
 ii. 85.6+ yds. 
 12. 15.3+ ft. 
 
 13. 129.07 -fin 
 14. Given. 
 16. .7464- 
 
 !*.. 
 
 18. A 
 19. 4-3 + 
 
 Page 34ft. 
 
 i. Given 
 2. 52 ft. 
 4. 1 6 ft. 
 5. 462.96 + c.f. 
 6. 3.072 tons 
 
 ARITHMETICAL PROGRESSION .Page 351. 
 
 I- IS 
 3- 8 
 
 5. $360 
 7. ii children 
 
 Page 3~2. 
 
 9- 3 FS. 
 
 10. Given 
 ii. 78 strokes 
 
 GEOMETRICAL 
 
 Page 353. 
 
 1. 96 
 
 2. $10.24 
 
 2007.3383664; 
 
 3001.460703698 
 
 242 
 
 PROGRESSION. 
 
 Page 354. 
 
 5. 1456 
 
 6. $4095 
 
 MENSURATION .Page 35<i-9. 
 
 i. Given 
 
 9. 204.20335 r. 
 
 15. 3817.03185 cu. ft 
 
 2. 3600 s. y. 
 
 10. 47 . 74 6-mm ft- 
 
 I 6. 522f CU. ft. 
 
 3. 1 44008.1'. 
 
 TT ? T 5\ ^ T 4-8 i 
 A O O 14159 * 
 
 1 8. 756 sq. ft. 
 
 4. 80 rods 
 
 12. 4417.8609375 S. ft 
 
 20. 14684558.207965. m. 
 
 7. $90.90-^ 
 
 13. 3183.1 s. r. 
 
 22. 260651609333^00. m. 
 
 MISCELLANEOUS EXAMPLES .Page 3GO-2. 
 
 I. $1083! 
 
 
 $45.93^ pr. 
 
 21. 150 rods 
 
 36. 6 1 ft 
 
 2. $120 COSt 
 
 5 
 
 12. 12 O'cl. 31 
 
 22. 
 
 !ft. 
 
 37.1226196.489 
 
 $45 loss 
 
 
 m. 2o-f T s. 
 
 23. $6038.72 
 
 38. 44% 
 
 3. 692 less: 
 
 
 13. $120 cost; 
 
 24. 45 24' 45" 
 
 39. $2607.291 
 
 1434 great 
 
 $30 profit 
 
 25. $8125 
 
 40. $5400. ist. ; 
 
 4. 22 ft. 
 
 
 14. 35 m. 8 s. 
 
 26. 3 f| 
 
 $8100, 2d. ; 
 
 5. 240 days 
 
 
 past 7 A.M. 
 
 27. $16836.734 
 
 $13500, 3d 
 
 6. 143^ miles 
 
 15. $24 
 
 28. 14520 per. 
 
 41. 82574.50 
 
 7. $20, A's; 
 
 
 1 6. 3 farmer's; 
 
 29. $19878.92 
 
 42. $50, A; 
 
 $40, B's ; 
 
 
 24 neigh b's 
 
 30. $680.625 
 
 $75, B; 
 
 $140, C's 
 
 
 17. 540 sheets 
 
 31. 77* ft. 
 
 $105, C 
 
 8- 79-591 ku 
 
 . 
 
 1 8. $40, A's; 
 
 32. $12500 
 
 43. Si 20 each 
 
 9. 5.26 T 2 j C. 
 
 
 $60, B's 
 
 33- * 
 
 1.59 
 
 $130, B 
 
 10. $3 
 
 
 19. 1980 pick. 
 
 34. $1446.625 
 
 44. 5 hrs. 
 
 ii. $164.06$ 
 
 c. 
 
 20. 15^ i 35. 48 ft, 
 
 45. $265720
 
 o
 
 UNIVERSITY OF CALIFORNIA AT LOS ANGELES 
 
 THE UNIVERSITY LIBRARY 
 This book is DUE on the last date stamped below 
 
 . _ 
 
 JUKI 01957 
 
 JUL2 193 
 
 Form L-9 
 20m-l, '42(8518) 
 
 7 
 
 UNIVEHSTTY OF CALIFORNIA 
 
 AT
 
 fa'&A- 
 
 ~t< 
 
 UC SOUTHERN REGIONAL LIBRARY FACILITY 
 
 A 000 937 537 9 
 
 */ 
 
 t 
 
 i-^c^L />te 
 &sQ>tsri,i 
 
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 T38n 
 
 1877