CO - 1 i ' x' 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 103 T38n 1877