Ie ^1;^^' *^# m. ■^ ,fe •t; n (^ DAVIES' REW ^RlTHi^ElJC 2J 1 ^™ "'^'■'^niliiuirir UC-NRLF r^v ^a>?, $B E7fl flbD Wi:^ 51^ o3 3ol)n-5treet, And sold by the Principal i<{)Gheliers \\\n^m\ Ihe llDited Shies. n LIBRARY OF THE University of California. C I /v' C UL A TI N G BRANCH. Keturn in 4«i& week*,' or a week before the end'of/ tjie torm. c5 OD ^ g ft o o H o o c? CR § © <*) o e o a M r-^ g o •0 o o H l-H e o Q ^•^ ' c ( 6 rt fl <» g Ph © a g QQ Q -*-. ,-«-. t ^' r«|l- < H O H» ©la »S Hh* H» .2 a ft w o o «^ «-^ ^•-« is SCHOOL AEITHMETIC. ANALYTICAL A^i D PRA BY CHARLES DA VIES, LL.D. AUTHOU OF A FULL COURSE OF MATHEMATICS, REVISED EDITION, PUBLISHED BY A. S. BARNES & CO., 51 & 53 JOHN STREET. SOLD BY BOOKSKLLERS, GENERALLY, THROUGHOUT THE UNITED STATES. 1858. r'lK ADVERTISEMENT Tub uUeutiou of Teuchcrfc is respectiuUy iuvited to the Kkvibku iLvi ItJNS of §ai}iu' ^rit|mttical Beries FOR SCHOOLS AND ACADEMIES. 1. DA VIES' PRIMARY ARITHMETIC AND TABLE-BOOK. 2. DAVIES' INTELLECTUAL ARITHMETIC. 8. DAVIES' SCHOOL ARITHMETIC. 4. DAVIES' UNIVERSITY ARITHMETIC. 5. DAVIES' PRACTICAL MATHEMATICS. The above Works, by Charles Davies, LL.D., Author ot a Complete Course of Mutliematic.>«, are designed as a full Course of Arithmetical In- !*truction necessary for the practical duties of business life; and also to prepare the Student for the more advanced Series of Mathematics by the same AutJior. EaUretf Recording to Act of Congress, In the year one tbousftiid •Igbt Luudred and fifty-flve, By CHARLES DAVIES, la Uie (J'erk';! OQloe of llio District Conrt of the United States for the gouthent Dl&trict of New York. PREFACE Arithmetic embraces the science of numbers, together with all ttf lules which are employed in applying the principles of this science to practical purposes. It is the foundation of the exact and mixed sciences, and the first subject, in a well-arranged course of instruc- tion, to which the reasoning powers of the mind are directed. Because of its great practical uses and applications, it has become the guide and daily companion of the mechanic and man of business. Hence, a full and accurate knowledge of Arithmetic is one of the most im- portant elements of a liberal or practical education. Soon after the publication, in 1848, of the last edition of my School Arithmetic, it occurred to ijie that the interests of education might be promoted by preparing a full analysis of the science of mathematics, and explaining in connection the most improved methods of teaching. The results of that undertaking were given to the public under the title of " Logic and Utility of Mathematics, with the best methods of in- struction explained and illustrated." The reception of that work by teachers, and by the public generally, is a strong proof of the deep interest which is felt in any effort, however humble, which may be made to improve our systems of public instruction. In that work a few general principles are laid down to which it is ffupjjosed all the operations in numbers may be referred : 1st. The unit 1 is regarded as the base of every number, and the consideration of it as the first step in the aMalysis of every question relating to numbers. "Zd. Every number is treated as a collection of units, or as made up of sets of such collections, each collection having its own base, which is either I, or some number derived from I. 3^/ The numbers expressing the relation between the different units of a number are called the scale; and the employment of this term enables us to generalize the laws which regulate the formation of numbers. Ath. By employing the term '* fractional units " the same principles arc made applicable to fractional numbers , for, all fractions are but Collections of fractional units, tlicse units having a known relation to 1 rV I'KKKAOE. lu the preparation of the work, two objects have beeu kept con- Btuntly in view: \st. To make it educational ; and, '2,(1. To make it practical. To attain these ends, the following plan has been adopted : 1. To introduce every new idea to the mind of the pupil by a sira- ole question, and then to express that idea in general terms under the form of a definition. 2. When a sufficient number of ideas are thus fixed in the mind, they are combined to form the basis of an analysis ; so that all the principles are developed by analysis in their proper order. 3. An entire system of Mental Arithmetic has been carried forward with the text, by means of a series of connected questions placed at the bottom of each page ; and if these, or their equivalents, are care- fully put by the teacher, the pupil will understand the reasoning in eveiy process, and at the same time cultivate the powers of analysis' and abstraction. 4. The work has been divided into sections, each containing a num- ber of connected principles ; and these sections constitute a series of dependent propositions that make up the entire system of principles and rules which the work develops. Grettt pains have beeu taken to make the work practioax in its general character, by explaining and illustrating the various applica- tions of Arithmetic in the transactions of business, and by connecting as closely as possible, every principle or rule, with all the applicationja which belong to it. I have great pleasure in acknowledging my obligations to many teachers who have favored me with valuable suggestions in regard to the definitions, rules, and meth«)d8 of illustration, in the previous edi- tions. 1 hope they will find the present work free from the defects they have so kindly pointed out. Much more than a general acknowledgment is due to Mr. D. W. Fisu, an al>le and distinguished teacher of Western New York, who has rendered me special and valuable aid in the preparation of this edition. His praciical information and zealous labors have given ad- ditional value to many parts of the work. Flshkill Lawoing, February, 1856. CONTENTS ?IR»T riVK RULES. Definition* 9— JO Notation and Numeration 10 — 22 Addition of Simple Numbers 22 — 30 Applications in Addition 30 — 33 Subtraction of Simple Numbers 33 — 37 Applications in Subtraction 37 — 42 Multiplication of Simple Numbers 42 — 50 Factors 60—53 Applications 63 — 56 Division of Simple Numbers , 56 — 6 1 Fractions 61—64 Long Division 64 — 68 Proof of Multiplication -. 68— «« Contractions in Multiplication 69 — 71 Contractions in Division 71 — 74 Applications in the precodinir Rules 74 — 79 UNITED STATES MC^Ef. United States Money defined 79 Table of United States Money 79 Numeration of United States Money 80 Reduction of United States Money 81 — 83 Addition of United States Money 83 — 85 Subtraction of United States Money 85 — 87 Multiplication of United States Money 87 — 91 Division of United States Money 91 — 93 Applications in the Four Rules. 93 — 96 DENOMINATE NUMBERS. English Money 96— 97 Reduction of Denominate Numbers 97 — 99 Linear Measure 99 — 10 J Cloth Measure 1 1— 102 Lund or Squuie Measure 102 — 1(V4 fl C0NTKNT6. Cubic Measure or Measure of Volume 104 — 106 Wine or Liquid Measure. 106—108 Ale or Beer Measure. 108 — 101/ Dry Measure 109—110 Avoirdupois Weight 1 10 — 1 1 1 Troy Weight 111—112 Apothecaries' Weight 112—1 14 Measure of Time 114 — ^116 Circular Measure or Motion 116 Miscellaneous Table 117 Miscellaneous Examples 117 — 1\9 Addition of iJenominate Numbers H9 — 15^ Subtraction of Denominate Numbers 124 — isfe Time between Dates 125 Applications in Addition and Subtraction 126 — 12d, Multiplication of Denominate Numbers 128 — 130 Division of Denominate Numbers 130 — 134 Longitude and Time 134 PKOPERTIE8 OF NUMBKR8. Compt)8ite and Prime Numbers 13.5 — 137 Divisibility of Number^ 137 Greatest Common Divisor 1 37 — 140 Greatest Common Dividend 140 — 142 Cancellation 142—145 OF COMMON FRACTIONS. Definition of, and First Prinriples 14(5 — 149 Of the different kinds of Common Fractions 149 — liiO Six Fundamental Propositions 150 — 154 Reduction of. Common Fractions 154 — 16 1 Addition of Common Fractious 161 — 162 Subtraction of Common Fractions 162 — 164 Multiplication of Common Fractions 164 — 168 Division of Common Fractions 168 — 172 lieduction of Complex Fractions 1 72 Denominate Fractions 1 73— 1 76 Addition and Subtraction of Denominate Fractions 176 — 178 DUODEClMtLS. Defmitions of, &c . 178- ISO Multiplication uf Duudcciiualsi IHO — l.s:.' CONTICNTS. VII DECIMAL rKACTlONS. Definition of Decimal Fractions 1 82 — 183 Decimal Numeration — First Principles 183 — 187 Addition of Decimal Fractions 187 — 19 1 Subtraction of Decimal Fractions , 191 — 193 Multiplication of Decimal Fractions 193 — 195 Divis;on of Decimal Fractions 195 — 197 Applications in the Four Rules 197 — 198 Denominate Decimals 198 Re<3uction of Denominate Decimals 198 — 201 ANALYSIS. Gmeral Principles and Methods 201 — 213 RATIO AND PROPORTION. jlatio defined 213—214 Proportion 214 — 210 Simple and Compound Ratio 216 — 218 Single Rule of Three 218—223 Double Rule of Three 223—228 APPLICATIONS TO BUSINESS. Partnership 228 — 229 Compound Partnership 229 — 231 Percentage 231—234 Stock Commission and Brokerage 234 — 237 Profit and Loss 237—239 Insurance 239- -241 Interest 241—247 Partial Payments 247 — 251 Compound Interest 251 — 253 Discount 253 — 255 Bank Discount 255 — 257 Equalion of Payments 257 — 260 Assessing Taxes 260 — 263 Coins and Currency 263 — 264 Reduction of Currencies 264 — 265 Exchange 265 — 268 Duties 268—271 Aligation Medial 27 1 — 272 Alijiaiiua Alternate 272 — ^276 VIU OO^'TICNTH. INVOLUTION. Definition of, dec 270 EVOLUTION. Definition of, wee when it makes known the properties and relations of numbers ; and an art^ when it applies principles of science to practical pur- poses. 12. A Proposition is something to be done, or demonstrated. 13. An Analysis is an examination of the separate parts of a proposition. 14. An Operation is the act of doing something with numbers. The number obtained by an operation is called a re .suit, or answer. 16. A Rule is a direction for performing an operation, and may be deduced either from an analysis or a demonstration. 16. There are five fundamental processes of Arithmetic : Notation and Numeration, Addition, Subtraction, Multiplica- tion and Division. EXPRESSING NUMBERS. 17. There are three methods of expressing numbers : 1st. By words, or common language ; 2d. By letters, called the Roman method ; 3d. By figures, called the Arabic method. 8. What is a simple number 1 9. What is quantity 1 10. Of what does Science treat ? What is Art 1 11. Of what does Arithmetic treat ! When is it a science ? When ail art 1 12. What is a Proposition T 13. What is an Analysis "? 14. What is an Operation \ What is the number obtained called 1 15. What is a Rule 1 How may it be deduced ? 16. How many fundamental rules are there 1 What are they 1 17. How many methods are there of expressing numbers? Wliat aie they * 11 18. A single thing is called - - - . One. Two. Three Pour. Five. Six. Seven. Eight. Nine. Ten. &iC. Each of the words, one, two, three, four, jive, six, &c., denotes how many things are taken in the collection. NOTATION. 19. Notation is the method of expressing nnmbers either by letters or figures. The method by letters, is called Rar/tan Notation ; the method by figures is called Arabic Notation. ROMAN NOTATION. 20. In the Roman Notation, seven capital letters are used, viz : I, stands ior one ; V, for Jive ; X, ibr ten ; L, tor fifty ; C, hv one hundred; D, ibr five hundred; and M, for 07ie thousand. All other numbers are expressed by combining the letters according to the following ^ NOTATION. BY WORDS. A sing le thing is called - - - One and one more Two and one more Three and one more Four and one ihore Five and one more Six and one more Seven and one more Eight and one more Nine and one more &.C. Ic. ROMAN TABLE. i. . . - One. LXX. - - Seventy. II. - - - Two. LXXX. - Eighty. III.- - - Three. XC. - - Ninety. IV. - - - Four. C. . - - One hundred. V. - - - Five. CC. - - Two hundred. VI - - - Six. CCC. - - Three hundred. VII. - - Seven. cccc - Four hundred. VIII. - - Eight. D. - - - Five hundred. IX.- - - Nine. DC. - - Six hundred. X. - - - Ten. DCC. . - Seven hundred. XX. . - Twenty. DCCC. - Eight hundred. XXX. - - Thirty. DCCCC. - Nine hundred. XL. . - Forty. M. - - ■ One thousand. L. - - - Fifty. MD. - Fifteen hundred LX - Sixty. MM. . . Two thousand. 12 NOTATiUM. Note. — The principles of this Notation arc these 1. Every time a letter is repeated, the number which it denotefc is also repeated. 2. If a letter denoting a less number is written on the right of one denoting a greater ^ their sum will be the number expressed, 3. If a letter denoting a less number is written on the left of one denoting a greater, their difference will be the number ex. pressed. EXAMPLES IN ROMAN NOTATION. Express the following numbers by letters : 1. Eleven. 2. Fifteen. 3. Nineteen. 4. Twenty-nine. 6. Thirty-five. 6. Forty-seven. 7. Ninety-nine. 8. One hundred and sixty. 9 Four hundred and tbrty-one. 10. Five hundred and sixty-nine. 11. One thousand one hundred and six. \ 2. Two thousand and twenty-five. 13. Six hundred and ninety-nine. 14. One thousand nine hundred and twenty-five. 15. Two thousand fiix hundred and eighty. 16. Four thousand nine hundred and sixty-five. 17. Two thousand seven hundred and ninety -one. 18. One thousand nine hundred and sixteen. 19. Two thousand six hundred and forty-one. 20. One thousand three hundred and forty-two. 19. What is Notation { What is the method by letters called 1 What is the method by figures called 1 20. How many letters are used in the Roman notation 1 Which are they 1 What does each stand for 1 Note. — What takes place when a letter is repeated 1 If a letter de- noting a less number be placed on the right of one denoting a greater, now are they read 1 If the letter denoting the less number be written on the left, how are they read ^ 21. What is Arabic Notation 1 How many figures are used I What do they form 1 Name the figures. How many things does 1 express ? How many things does 2 express ] How many units in 3 ] In 4 ! In 6 ] In 9 1 In 8 ? What does express ^ What are the other figures calkd I ^OTAT'UN. It) ARABIC NOTATION. 21. Arabic Notation is the method of expressing uumberg by figures. Ten figures are used, and they Ibrm the alpiiaJbet of Ute Arabic Notation. Thev are, called zero, cipher, or Naught, 1 - . One. 2 ■ - - Two. 3 - - Three. ■ 4 - - Four. 6 - - Five. 6 - - Six. 7 - - Seven. 8 - - Eight. 9 - - Nine. 1 expresses a single thing, or the unit oi' a number. two things - - or two units, three things - - or three units, four things - - or four units, five things - • or five units, six things - - or six uiiits. seven things - - or seven units, eight things - - or eight units nine thinjjs - - or nine units. The cipher, 0, is used to denote the absence of a thing Thus, to express that there are no apples in a basket, we write, the number of apples is 0. The nine other figures are called significant figures, or Digits. 22. We have no single figure for the number ten. We therefore combine the figures already known. This we do by writing on the right hand of 1, thus : 10, which is read ten. This 10 is equal to ten of the units expressed by 1. It is, however, but a si?igle ten, and may be regarded as a unit the value of which is ten times as great as the unit I. It is called a unit of the second order. 22. Have we a separate character for ten I How do we express ten 1 To how many units I is ten equal 1 May we consider it a single unit' Of wiiai order :' 14 NOTATION. 23. When two figures are written by th 3 side of each other, the one on the rigiit is in the 'place of units, and the other in the flace of tens, or of units oj the second Qrder. Each unit of the second order is equal to ten units of the first order. When units simply are named, units of the first order a^t always meant. Two tens, or two units of the second order, are written 26 Three tens, or three units of the second order, are written 30 Four tens, or four units of the second order, are written 40 Five tens, or five units of the second order, are written 50 Six tens, or six units of tlie second order, are written 60 Seven tens, or seven units of the second order, are written 70 Eight tens or eight units of the second order, are written 80 Nine tens, or nine units of the second order, are written 90 These figures are read, twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety. The intermediate numbers between 10 and 20, between 20 and 30, &c., may be readily expressed by considering their tens and units. For example, the number twelve is made up of one ten and two units. It must therefore be written by setting 1 in the place of tens, and 2 in the place of units . thus, ...... 12 Eighteen has 1 ten and 8 units, and is written - - 18 Twenty-five has 2 tens and 5 units, and is written - 25 Thirty-seven has 3 tens and 7 units, and is written - 37 Fifly-four has 5 tens and 4 units, and is written - - 54 Hence, any number greater than nine, and less than one hundred, may be expressed by two figures. 24. In order to express ten units of the second order, or one hundred, we form a new combination. It is done thus, - - - 100 by writing two ciphers on the right of 1. This number is read, one hundred. 23. When two figures are written by the side of each other, what \» the place on the right called ! The place on the left ! When units simply are named, what units are meant 1 How many units of the second order in 20 ! In 30 \ In 40 1 In 50 \ In 60 ! In 70 1 lu 80 ? In 90 I Of what is the number 12 made up I Also 18, 2o, 37 54 1 What numbers may be expressed by two figures 1 NOTATION. 15 Now this one hundred expresses 10 units of the second m-der, or 100 units of the first order. The one hundred is but an individual hundred, and, in this light, may be regarded as a unit of the tJdrd order. We can now express any number less than one thousand. For example, in the number three hundred and Beventy-five, there are 5 units, 7 tens, and 3 hundreds, g * -^ Write, therefore, 5 units of the tirst order, 7 units of the J | § second order, and 3 of the third ] and read from the 3 7 5 right, units^ tensj hundreds. In the number eight hundred and ninety-nine, there ^ ^ ^ are 9 units of the first order, 9 of the second, and 8 of a g "S the third ; and is read, units, tens, hundreds. 8 9 9 In the number four hundred and six, there are 6 units « „; 2 of the first order, of the second, and 4 of the third. 5 § S The right liand figure always expresses units of the first order; the second, units of the second order; and the third, units of the third order. 25. To express ten units of the third order, or one thous and, we form a new combination by writing three ciphers on the right of 1 ; thus, - - - 1000 Now, this is but one single thousand, and may be regarded as a miit of the fourth order. Thus, we may form as many orders of units as we please : a single unit of the first order is expressed by I, a unit of the second order by 1 and ; thus, 10, a unit of the third order by 1 and two O's ; 100, a unit of the fourth order by 1 and three O's ; 1000, a unit of the fifth order by 1 and four O's ; 10000 ; and so on, ibr units of higher orders : 24. How do you write one hundred 1 To how many units of the second order is it equal 1 To how many of the first order 1 May it be considered a single unit ! Of what order is it ! How many units of the third order in 200 ! In 300 ! In 400? In 5001 In 6001 Of what is the number 375 composed 1 The number 899 \ The number 406 ! What numbers may be expressed by three figures ■ Wha» urdcl uf units will'each figure express • 1 6 NOTATION. 26, Therefore, 1st. The same jig \ ire expresses different units according to the place which it occupies : 2d. Units of the first order occupy the place on the right ; units of the second order, the second place ; units of the third order, the third place ; and so on for places still to the left : 3d. Ten units of the first order make one of the second ; ten of the second, one of the third ; ten of the third, one of the fourth ; and so on for the higher orders : 4th. When figures are written hy the side of each other, ten units in any one place make one unit of the pdace next to the left. EXAMPLES IN WRITING THE ORDERS OF UNITS. 1. Write 3 tens. 2. Write 8 units of the second order. 3. Write 9 units of the first order. 4. Write 4 units of the first order, 5 of the second, 6 of the third, and 8 of the fourth. 5. Write 9 units of the fifth order, none of the fourth, 8 of the third, 7 of the second, and 6 of the first. Ans. 90876. 6. Write one unit of the sixth order, 5 of the fiftli, 4 of the fourth, 9 of the third, 7 of the second, and of the first Ans. 7. Write 4 units of the eleventh order. 8. Write forty units of the second order. 9. Write 60 units of the third order, with four of the 2d, and 5 of the first. 10. Write 6 units of tl>e 4th order, with 8 of the 3d, 4 of the 1st. 25. To what are ten snits of the third order equal 1 How do you write it 1 How is a single unit of the fir,si order written 1 How do you write a unit of the second order 1 One of the third 1 One of the fourth! One of the fifth ! 26. On what does the unit of a figure depend 1 What is the unit Oi the first place on the right 1 What is the unit of the second place 1 What is the unit of the third place 1 Of the fourth ? Of the fifth 1 Sixth 1 How many units of the first order make one of the second ] How many of the second one of the third 1 How many of the third one of the fourtii, Ac. When figures are written by the side of each other, how many units of any place make one utjit of the' place next to the NUMERATION. 17 11. Write 9 units of tiie 5th order, of the 4th, 8 of the Sd, 1 of the 2d, and 3 of the 1st. 12. Write 7 units of the 6th order, 8 of the 5th, of the 4tli, 5 of the 3d, 7 of the 2d, and 1 of the 11th. 13. Write 9 units of tlie 7th order, of the 6th, 2 of the 5th, 3 of the 4th, 9 of the 3d. 2 of the 2d, and 9 of the 1st. 14. Write 8 units of the 8th order, 6 of the 7th, 9 of tho 0th, 8 of the 5th, 1 of the 4th, of the 3d, 2 of the 2d, and 8 of the 1st. 15. Write 1 unit of the 9th order, 6 of the 8th, 9 of the 7th, 7 of the 6th, 6 of the 5th, 5 of the 4th, 4 of the 3d, 3 of the 2d, and 2 of the 1st. 16. Write 8 units of the 10th order, of the 9th, of the 8th, of the 7th, 9 of the 6th, 8 of the 5th, of the 4th, 3 of the 3d, 2 of the 2d, and of the 1st. 17. Write 7 units of the ninth order, with 6 of the 7th, 9 of the third, 8 of the 2d, and 9 of the 1st. 18. Write 6 units of 8th order, with 9 of the 6th, 4 of the 6th, 2 of the 3d, and 1 of the 1st. 19. Write 14 units of the 12th order, with 9 of the 10th, 6 of the 8th, 7 of the 6th, 6 of the 5th, 6 of the 3d, and 3 of the first. 20. Write 13 units of the 13th order, 8 of the 12th, 7 of the 9th, 6 of the 8th, 9 of the 7 th, 7 of the 6th, 3 of the 4th, and 9 of the first. 21. Write 9 units of the 18th order, 7 of the 16th, 4 of the I5th, 8 of the 12th, 3 of the 11th, 2 of the 10th, 1 of the 9th, of the e^h, 6 of the 7th, 2 of the third, and 1 of the 1st. NUMERATION. 27. Numeration is the art of reading correctly any num- Der expressed by figures or letters. The pupil has already been taught to read all numbers from jne to one thousand. The Numeration Table will teach him to read any number whatever ; or, to express numbers in words. 27. What is Numeration 1 What is the unit of the first period ' What is the unit of the second 1 Of the third 1 Of the fourth I Of the fifth 1 Sixth"? Seventh I Eiglith 1 Give the rule fur reading iiuuiheii. 18 NUMiaiATION. NUMERATION TABLE. Glh Period. 5th I'eriod. 4th Period. Quadrillions. Trillions. Billions. 3d Period. 2d Peri< d. 1st PenotL Millions. Thousands. Units. ^ O 3 5 7 2 0, S3 • • S CO gl • ^ .2 3 S-c 8 9 1 4 2 8 3 2 o ^ •-=! '5 « .2 ffiHPQ 3, 8 4 2, ^.2 o 3 CD •- •73 4 3, 4 8, 4 5, 4 9, 2, 0, 7 6, 3 6, 4 1, 6 8, 5 w 2-^ • EI! <^ ^ 02 3 5 §^ 8 2, 1 2 3, 0, 2 1 0, 0, 2 8 9, 3 4 1 2 7 2 9 7, 3 1 9i 5 c 6 7 5 7 9 2 3 1 2 1 2 2 Notes. — 1. Numbers expressed by more than three figures are written and read by periods, as shown in the above tabje. 2. Each period always contains three figures, except the last which may contain either one, two, or three figures. 3. The unit of the first, or right-hand period, is 1 .; of the second period, 1 thousand ; of the 3d, 1 million ; of the fourth, 1 bil- lion ] and so, for periods, still to the left. 4. To quadrillions succeed quintillions, sextillions, seplilhouE, 5. The pupil should be required to commit, thfj'oughly, the names of the periods, so as to repeat them in their regular (irder fiom left to riglit, as "well as from right to left. NL'MEItA'l'tOM. li» RULE FOR READING NUMBERS. I. Divide the number into periods of three figures each^ bee/ in /I in;/ at the riyltt hand. II. Name the order of each figure^ beginning at the right hand III. Then, beginning at the left hand, read each period iw if it stood alone, naming its unit. EXAMPLES IN READING NUMBER.S. 28. Lei the pupil point ofi'aiid read the tbllowiiig numbei-s — then write them in words. 1. 67 7. 6124076 13. 804321049 2. 125 8. 8073405 14. 90067236708 3. 6256 9. 261^40123 15. 870432697082 4. 4697 10. 9602316 16. 1704291672301 5. 23697 11. 87000032 17. 3409672103604 6. 412304 12. 1987004086 18. 49701342641714 19. 8760218760541 23. 9080620359704567 20. 904326170365 24. 9806071234560078 21. 30267821040291 25. 30621890367081263 22. 90762038 0467026 26. 350673123051672607 Note. — Let each of the above examples, after being written on tiie black board, be analyzed as a clays exercise ; thus : Ex. 1. How many tens in 67 ? How many units over ? 2. In 125, how many hundreds in the hundreds place? How many tens in the teiis place ? How many units in the units place ? How many tens in the number ? 3. In 6256, how many thousands in the thousands place ? How many hundreds in the hundreds place ? How many tens in the tens place ? How many units in the units place? 4. How many thousands in the number 4697 ? How manj hundreds? How many tens ? How many units? 5. How many thousands in the number 23697 ? How many hundreds ? How many tens ? How many uiuts ? 6. How many hundreds of thousands in 412304? How many ten thousands ? How many thousands ? How many hundreds ? How many tens ? How many units ? US. Name the unit!^ uf each order in ewduyh Ml In 10^ In IC In W^ Jiivo tht rule for wfitinii niuiihcrsi 20 NUMKIiATlON. KULE Foil WRlTliNG NUMBERS OR NOTAllON. I. Begin at the lejt hand and write each period in order ^ as if it were a period of units. II. When the number in any period, except the left kana period, is expressed by less than three figures, prefix 07ie or two ciphers ; and when a vacant j^criod occurs, Jill it with ciphers, EXAMPLES IN NOTATION. 29. Express the following numbers in figures : 1. One hundred and five. 2. Three hundred and two. 3. Five hundred and nineteen. 4. One thousand and four. 5. Eight thousand seven hundred and one. 6. Eorty thousand four hundred and six. 7. Fifty-eight thousand and sixty-one. 8. Ninety-nine thousand nine hundred and ninety-nine. 9. Four hundred and six thousands and forty-nine. 10. Six hundred and forty-one thousand, seven hundred and twenty-one. 11. One rnilhon, four hundred and twenty-one thousands, six hundred and two. 12. Nine millions, six hundred and twenty-one thousands, and sixteen. 13. Ninety -four milhons, eight hundred and seven thous- ands, four hundred and nine. 14. Four billions, three hundred and six thousands, nine hundred and nine. 15. Forty-nine billions, nine hundred and forty-nine thous- ands, and sixty-five. 16. Nine hundred and ninety billions, nine hundred and ninety-nine millions, nine hundred and ninety thousands, nine hundred and ninety-nine. 17. Four hundred and nine billions, two hundred and nine lliousands, one hundred and six. 18. Six hundred and forty-five billio.is, two hundred and Bixtytnine miUions, eight luindicd and lilty-nine thousaiid>^ nine huutlrenmbers for addition, figures expressing the same unit must be placed in the same column. 206. When is a cipher prefixed to a number 1 When prefixed to a decimal, does it increase the numerator ? Does it increase the denomi- nator f \V hat efiiect then has it on the value of the fraction ? What do .2 become by prefixing a cipher 1 By prefixing two ciphers ! By prefixing three ? What do .07 become by prefixing a cipher ! By pre- fixing two ? B}' prefixing three 1 By prefixing four 1 207. What parts of unity may be added together 1 How do yon set down the numbers for addition I How will the decimal points fall I How do you then odd 1 How many decimal places du you pohit oil in ll;t tuin ' 188 A.DDrrioN ok The addition of decimals is then made in the same manner as that of whole numbers. I. Find the sum of 37.04, 704.3, and .0376. OPERATION. Place the decimal points in the same column: 37.04 this brings units of the same value in the same 704.3 column : then add as in whole numbers : hence, .0376 741.3776 Rule. — I. Set down the numbers to he added so that figures of the same unit value shall stand in the same column. II. Add as in simple nur/ibers, and point off in the sum^ from the right hand^ as many places for decimals as are equal to the (jreaiest number of places in any of the numbers added. Proof. — The same as 'n simple numbers. EXAMPLES. 1. Add 4.035, 763.196, 445.3741, and 91.3754 together. 2. Add 365.103113, .76012, 1.34976, .3549, and 61.11 together. 3. 67.407 + 97.0044-4-1- .6 + . 06-1- .3 4. .0007-f-1.0436-|-.4-h.05 + .047 5. .0049 + 47.0426 + 37.0410 + 360.0039 = 444.0924. 6. What is the sum of 27, 14, 49, 126, 999, .469, and .2614] 7. Add 15, 100, 67, 1, 5, 33, .467, and 24.6 together. 8. What is the sum of 99, 99, 31, .25, 60.102, .29, and 100.347 \ 9. Add together .7509, .0074, 69.8408, and .6109. 10. Required the sum of twenty-nine and 3 tenths, four hundred and sixty-five, and two hundred and twenty-one thousandths. 1 1 . Required the sum of two hundred dollars one dime three cents and 9 mills, four hundred and forty dollars nine mills, and one dollar one dime and one mill. 12. W^hat is the sum of one-tenth, one hundredth, aufl one theusaudth ? UICOTMAL FKAUTTONS. W* 13. What is the sum of 4, and 6 ten-thousandths'? 14. Required, in dollars and decimals, the sum of one dollar one dime one cent one mill, six dollars three mills, four dol- lars eight cents, nine dollars six mills, one hundred dollars six dimes, nine dimes one mill, and eight dollars six cents. 15. What is the sum of 4 dollars 6 cents, 9 dollars 3 mills, 14 dollars 3 dimes 9 cents 1 mill, 104 dollars 9 dimes 9 cents 9 mills, 999 dollars 9 dimes 1 mill, 4 mills, 6 mills, and 1 mill? 16. If you sell one piece of cloth for |i4,25, another for $5,075, and another for $7,0025, how much do you get for ain 17. What is the amount of $151,7, $70,602, $4,06, and $807,26591 18. A man received at one time $13,25 ; at another $8,4 ; at another $23,051 ; at another $6 ; and at another $0,75 : how much did he receive in all ? 19. Find the sum of twenty-five hundredths, three hundred and sixty-five thousandths, six tenths, and nine millionths. 20. What is the sum of twenty- three millions and ten, one thousand, four hundred thousandths, twenty-seven, nineteen millionths, seven and five tenths 1 21. What is the sum of six millionths, four ten-thousandths, 19 hundred thousandths, sixteen hundredths, and four tenths'? 22. If a piece of cloth cost four dollars and six mills, eight pounds of coffee twenty-six cents, and a piece of muslin three dollars seven dimes and twelve mills, what will be the cost of them all ? 23. If a yoke of oxen cost one hundred dollars nine dimes and nine mills, a pair of horses two hundred and fifty dollars five dimes and fifteen mills, and a sleigh sixty-five dollars eleven dimes and thirty-nine mills, what will be their entire coat ? 24. Find the sum of the following numbers : Sixty-nine thousand and sixty-nine thousandths, forty-seven hundred and forty-seven thousandths, eighty-five and eighty-five hun- dredths, six hundred and forty-nine and six hundred aii'V ibrty-iiine leii-tliousandtlis ? 190 BUBTli ACTION OP SUBT-KACTION OF DECIMALS. 20b Subtraction of Decimal Fractions is the operation of finding the ditlerence between two decimal numbers. I. From 3.275 to take .0879. Note. — Iii this example a cipher is annexed operation to the minuend to make the number of decimal 3.2750 places equal to the number in the subtrahend. This .0879 does not alter the value of the minuend (Art. 205) : "^ilE^^T hence, ^'^^^^ Rule. — 1. Write the less mi?nber under the greater^ so thai Jig u res of the same unit value shall fall in the same column. II. Subtract as in simple numbers^ and point off the deci- mal places in the remainder, as in addition. Proof. — Same as in simple numbers. EXAMPLES. 1. From 3295 take 0879. 2. From 291.10001 take 41.375. 3. From 10.000001 take .1111 IL 4. From 396 take 8 ten-thousandth8. 5. From 1 take one thousandth. 6. From 6378 take one-tenth. 7. From 365.0075 take 3 millionths. 8. From 21.004 take 97 ten-thousandths. 9. From 260.4709 take 47 ten-millionths. 10. From 10.0302 take 19 millionths. 11. From 2.01 take 6 ten-thousandths. 12. From thirty- five thousands take thirty-five thousandths. 15. From 4262.0246 take 23.41653. 14. From 346.523120 take 219.691245943. 16. From 64.075 take .195326. lb. What is the difierence between 107 and .0007 1 17. What is the difierence between i.^ and .3735 ? 18. From 96.71 take 96.709. 208. What is subtraction of decimal fractions'? How do you set down the numbers for subtraction 1 How do you then subtriict ! How jnaiiy Jeciuiul p!a«;t!s do yo\.x p<»liit oHiu the reni:.imltr^ DECIMAL VI MULTIPLIGA.T10N OF DECK 209. To multiply one decimal by another. I. Multiply 3.05 by 4.102. OPERAnoN. Analysis. — If we change both factors to vul- s^-VOS. — 3.05 gar fraciioiis, the product of the numerator will 4 i o 2 _ ,i iaq be the same as that of the decimal numbers, and Tooo — _: the number of decimal places will be equal to the 610 number of ciphers in the two denominators : 305 hence, 12 20 12.51110 Rule. — Maltiphj as in simple numbers^ and point off in the product^ from the right hand^ as many figures for decimals US there are decimal places in both factors; and if there be not so many in the product, siqrply the deficiency by prefixing ciphers. EXAMPLES. 1. Multiply 3.049 by .012. 2. Multiply 365.491 by .001. 3. Multiply 496.0135 by 1.49.6. 4. Multiply one and one milliontb by one thousandth. 5. Multiply' one hundred and forty-seven millionths by one millionth, 6. Multiply three hundred, and twenty-seven hundredths by 31. 7. Multiply 31.00467 by 10.03962. 8. What is the product of five-tenths by five-tenths ? 9. What is the product of five-tenths by five-thousandths'? 10. Multiply 596.04 by 0.00004. 11. Multiply 38049.079 by 0.00008. 12 What will 6.29 weeks' b-^ard come to at 2,75 dollars per week ? 13. What will 61 pounds of sugar come to at $0,234 per pound 209. After multiplying, how many decimal places will you point off in the product \ When there are not so many in the product, what de ytto do"? Give the rule for thr- multiplication of deci turds. 192 UONTKACTlOi^b. 14. If 12.836 dollars are paid for one barrel of flour, what »vill .354 barrels cost 1 15. What are the contents of a board, .06 feet long and .06 wide ? 16. Multiply 49000 by .0049. 17. Bought 1234 oranges for 4.6 cents apiece : how much iid they cost? 18. What will 375.6 pounds of coffee cost at .125 dollars per pound ? 19. If I buy 36.251 pounds of indigo at $0,029 per pound, what will it come to ? 20. Multiply $89.3421001 by .00000^8. 21. Multiply 1341.45 by .007. 22. What arethecontentsof alot whichis .004 miles long and .004 miles wide? 23. Multiply .007853 by .035. 24. What is the product of $26.000375 multiplied by* .00007 ? CONTRACTIONS. 210. W^hen a decimal number is to be multiplied by 10, 100, lOOOj &c., the multiplication may be made by removing the decimal point as many places to the right hand as there are ciphers in the multiplier, and if there be not so many figures on the right of the decimal point, supply the deficieiicy by annexing ciphers. Thus, 6.79 multiplied by Also, 370.036 multiplied by '10 1 67.9 100 679. - 1000 ' = ^ 6790. 10000 67900. 100000^ 679000. 10 1 3700.36 100 37003.6 1000 ► =: • 370036. 10000 3700360. 100000 I 3 7003600. 210. How do you multiply a decimal number by 10, 100, 1000, Ac. ! If there are not as many decimal figures as there are ciphers in the uiultiplier, what do you do ^ DECIMAL FRACTIONS. 103 DIVISION OF DECIMAL FRACTIONS. 211. Division of Decimal Fractions is similar to that of limple numbers. 1. Let it be reqmred to divide 1.38483 by 60.21. Analysis. — The dividend must be equal opkration. the product of the divisor and quotient, 60.21)1.38483(23 (Art. 61); and hence must contain as 1.2042 many decimal places as both of them ; 1 80r*V therefore, There must he as many decimal places in the quotient as the decimal places in the divi- Ans. .023 dend exceed those in the divisor : hence, E.ULE. — Divide as in simple nnmbers, and point off in the quotient^ from the right hand, as many places for dechnals as the decimal J) taces i?i the dividend exceed those in the divisor ; and if there are not so many, supply the deficiency by prefiic- ing ciphers, EXAMPLES. 1. Divide 2.3421 by 2.11. 2. Divide 12.8256f by 3.01. 3. Divide 33.66431 by 1.01. 4. Divide .010001 by .01. 5. Divide 8.2470 by .002. 6. Divide 94.0056 by .08. 7. What is the quotient of 37.57602, divided by 3 ; by .3 , by .03 ; by .003 ; by .0003 ? 8. What is the quotient of 129 75896, divided by 8 ; by .08 ; by .008 ; by .0008 ; by .00008 X 9. What is the quotient of 187.29900, divided by 9 ; by .9 ; by .09 ; by .009 ; by .0009 ; by .00009 ? 10. What is the quotient of 764.2043244, divided by 6 ; by .06 ; by .006 ; by .0006 ; by .00006 ; by .000006? Note. — 1. When there are more decimal places in the divisor than in the dividend, annex ciphers to the dividend and make the decimal places equal ; all the figures of the quotient will then bs whole numbers. 211. How does the number of decimal places in the dividend c(im- pare with thai in the divi.sor and (juotient ! How do you determine the nural)er of decimal places in the quotient "! • If the divisor contains four places and the vlividend six, how many in the quotient ! If the divisor contains three places and the dividend five, liow many in the quotient ? Give the rule for the division of decimals. 13 lU DIVISION OF EXAMPLES. 1 Divide 4397.-1 by 3.49. Note. — We annex one to trie dividend. Had it contained no decimal place we should have annexed two. OP K RATION. 3.49)4397.40(1260 34 9 ~907 698 2094 2094 Ans. 1260. 2. Divide 2194.02194 by .100001. 3. Divide 9811.0047 by .325947. 4. Divide .1 by .0001. | 5. Divide 10 by .15. 6. Divide 6 by .6 ; by .06 ; by .006 ; by .2 ; by 3 by by .003; by .5; by .05; by .005. Note. — 2. When it is necessary to continue the division farther than the figures of the dividend will allow, we annex ciphers, and consider them as decimal places of the dividend. When the division docs not terminate, we annex the plus sign to show that it may be continued ; thus .2 divided by .3 = .666-f. EXAMPLES. 1. Diwde 4.25 by 1.25. Anaia'sts. — In this example we annex one 0, and then the decimal places m the dividend will exceed those in the divisor by 1. OPERATION. 1.25)4.25(3.4 3.75 500 500 Ans. 3.4. 4 Divide 580.4 by 375. 5. Divide 94.0369 by 81.032. 2. Divide .2 by .6. 3. Divide 37.4 by 4.5. Note. — 3. When any decimal number is to be divided by 10, 100. 1000, &c., the division is made by removing the decimal point as tmwy places to the left as there are O'.s in the divisor ; and if there be not hO many tigures on the left of the decimal point the deficiency is supplied by prefixing ciphers. 27.69 divided bv 10 ] r 2.769 100 1 .2769 1000 f-' .02769 10000 J .002769 642.89 divided by ILC IMAL KKACTIONS. fio 1 r 64.289 100 6.4289 1000 > = < .64289 10000 .064289 100000 .0064289 196 QUESTIONS IN THB PRECEDING RULES. 1 If I divide .6 dollars among 94 men, how much will eaoh leceive ? 2. 1 gave 28 dollars to 267 persons : how much apiece ? Divide 6.35 by .425. What is the quotient of 136.2678 divided by 2.25 1 Divide a dollar into 12 equal parts. Divide .25 ol'3.26 into .034 of 3.04 equal parts. How many times will .35 of 35 be contained in .024 01241 8. At .75 dollars a bushel, how many bushels of rye can be bought for 141 dollars'? 9. Bought 12 and 15 thousandths bushels of potatoes for 33 hundredths dollars a bushel, and paid in oats at 22 hun- dredths of a dollar a bushel : how many bushels of oats did it take? 10. Bought 53.1 yards of cloth for 42 dollars : how much was it a yard ? 11. Divide 125 by .1045. 12. Divide one millionth by one billionth. 13. A merchant sold 4 parcels of cloth, the first contained 127 and 3 thousandths yards ; the 2d, 6 and 3 tenths yards ; the 3d, 4 and one hundredth yards ; the 4th, 90 and one millionth yards : how many yards did he sell in all ? 14. A merchant buys three chests of tea, the first contains 60 and one thousandth pounds ; the second, 39 and one ten thousandth pounds ; the third, 26 and one tenth pounds : how much did he buy in all ? Note. — 1. If there are more decimal places in the divisor than in the divitlend, what do you do ! M'hat will the figures of the quotient then be ! 2. How do you continue the division after you have hrout'^ht down all the llgures of the dividend I What sign do you place after the quo- tient ! What does it show 1 / 3. How do you divide a decimal fraction by 10, 100, 1000, &c. ! 196 DIVISION Ot- is. What is the sum of ^20 and three hundredths ; S4 and one-tenth, $6 and one thousandth, and $18 and one hundredth ? 16. A puts in trade $504,342 ; B puts in $350.1965 ; C puts in $100.11; D puts in 899.334; and E puts in 19001.32 : what is the whole amount put in 1 17. B has $936, and A has $1,3 dimes and 1 mill : how much more money has B than A 1 18. A merchant buys 37.5 yards of cloth, at one dollar twenty-five cents per yard : how much does the whole come to ? 19. If 12 men had each $339 one dime 9 cents and 3 mills, what would be the total amount of their money ? 20. A farmer sells to a merchant 13.12 cords of wood at $4,25 per cord, and 13 bushels of wheat at $1,06 per bushel : he is to t^ke in payment 13 yards of broadcloth at $4,07 per yard, and the remainder in cash : how much money did he receive 1 21. If one man can remove 5.91 cubic yards oi" earth in a day, how much could nineteen men remove ? 22. What is the cost of 8.3 yards of cloth at $5,47 per yard ? 23. If a man earns one dollar and one mill per day, how much will he earn in a year of 313 working days ? 24. What will be the cost of 375 thousandths of a cord of wood, at $2 per cord ? 25. A man leaves an estate of $1473.194 to be equally divided among 12 heirs : what is each one's portion ? 26. If flour is $9,25 a barrel, how many barrels can I buy for $1637,25 ? 27. Bought 26 yards of cloth at $4,37^ a yard, and paid for it in flour at $7,25 a barrel : how much flour will pay for the cloth 1 28. How much molasses at 22^ cents a gallon must be given for 46 bushels of oats at 45 cents a bushel ? 29. How many days work at $1,25 a day must be givea for 6 cords of wood, worth $4,12J a cord ? 30. What will 36.48 yards of cloth cost, if 14.25 yard cost $21,375? 31. If you can buy 13.25Z6. of cofl^ee for $2,50, liow much can you buy for $325,50 'i DECIMAL FKAUTIONS. 197 212. To change a common to a decimal fraction. The value of a fraction is the quotient of the numerator, divided by the denominator (Art. 148). 1. Reduce f to a decimal. If we place a decimal point alter the 5, and then operation. write any number of O's, after it, the value of the 8)5.000 numerator will not be changed (Art. 205). g25 If, then, we divide by the denominator, the quo- tient will be the decimal number : hence, Rule. — Annex decimal ciphers to the numerator, and then divide by the denomitmtor, pointing off as in division of decimals. EXAMPLES. 1. Reduce flf to its equivalent decimal. OPKRATION. 125)635(5.08 We here use two ciphers, and therefore point (325 off two decimal places in the quotient. Reduce the following fractions to decimals : 1. Reduce f to a decimal. 2. Reduce ^-^ to a decimal. 3. Reduce ^ to a decimal. 4. Reduce \ and x\2Q ' 5. Reduce^V li'^"^ToVo• 6. Reduce ^ and yyg 5-. 7 RpflllPP 3 14 9 5 7 12 3 ft T?pr1npp 8 137.5 3265 o. iteauce g, g^^^-, 4T21- 9. Reduce ^ to a decimal. 1000 1000 10. Reduce ^ to a decimal. 11. Reduce ^^. 12. Reduce ^-q. 13. Reduce j^}^. 14. Reduce y^i,j. 15. Reduce y^j^. 16. Reduce ^V^- 17. Reduce y^w^J- 18. Reduce f^^^. 213. A decimal fraction may be changed to the form of a vulgar fraction by simply writing its denominatoi (Art. 202). 212. How do you change a vulgar to a decimal fraction ( 21il. How do you change a decimal to the form of a vulgar fraction ? 198 DE&'UMINAilC DKOIMALS. EXAMPLES. 1. What vulgar fraction is equal to .04 ? 2. What vulgar fraction is equal to 3.067 ? 3. What vulgar fraction is equal to 8.275? 4. What vulgar fraction is equal to .00049 1 DENOMINATE DECIMALS 214. A denominate decimal is one in which the unit of the fraction is a denominate number. Thus, .5 of a pound, .6 of a shilling, .7 of a yard, . 3. Reduce .056 poles to the decimal of an acre. 4. Reduce 14 minutes to the decimal of a day. 5. Reduce 21 pints to the decimal of a peck.. 6. Reduce 3 hours to the decimal of a day. 7. Reduce 375678 feet to the decimal of a mile. 8. Reduce 36 yards to the decimal of a rod. 9. Reduce .5 quarts to the decimal of a barrel. 10. Reduce .7 of an ounce, avoirdupois, to the decimal of a hundred. 214. What is a denominate decima. ' 215. How do you change a denominate number to a denominate docimul 1 DEWOMINATF DECIMAI.S. , 199 CASE II. 216. To find the value of a decimal in integets of a less denomination I . I, Find the value of .890625 bushels. OPERATION. ,,,.,. , :, . , ^ . / • . .890625 Analysis. — Multiplying the decimal by 4, (since 4 . pecks make a bushel), we have 3.5025 peeks. Mul- . liphing the new decimal by 8, [Ance Squaris make 3.562500 a peck), we have 4.5 quarts. Then, multiplying 8 this last decimal by 2, (since 2 pints make a quart), 4 500000 we have 1 pint : hence, ' o Ans. 3pk. 4qts. \pt. 1.000000 Rule. — I. Multiply the decimal by that number which vrill reduce it to the next less denomination, pointing of as in mMltiplicati'on of decimal froA^lions. II. Multiply the decimal part of the product as befwe ; and so continue to do until the decimal is reduced to the required denominations. The integers at the left form the ansiver EkAMPLES. 1. A/VTiat is the value of 002084/e compared with each other 1 If you compare by their difference, how do you find it 1 If you compare by the quotient, how do you regard one of the numbers ''- What is the ratio 1 222. From how many terms is a ratio derived ] What is the first tcnn called 1 What is the second called 1 Which is the standard 1 'VIA How limy tlie ratio of two nurubcis be expressed ? Ilinv re^j*! ' 214 RATIO AND PROPORTION. 224. If two couplets have the same ratio, their terms are said to be proportional : the couplets 3 : 12 and r : 4 have the same ratio 4 ; hence, the terms are proportional, and are written, 3 : 12 : : 1 : 4 by simply placing a double colon between the couplets. The terms are read 3 is to 12 as 1 is to 4, and taken together, they are called a proportion : hence, A proportion is a comparison of the terms of two equal ratios.^ 224. If two couplets have the same ratio, what is said of the tenns ( Hov/ are they written 1 How read 1 What is a proportion ? * Some authors, of high authority, make the consequent the stand- ard and divide the antecedent by it to determine the ratio of the couplet. The ratio 3 : 12 is the same as that of 1:4 by both methods ; for, if the antecedent be made the standard, the ratio is 4 ; if the conse- quent be made the standard, the ratio is one-fourth. The question is, which method should be adopted 1 The unit 1 is the number from which all other numbers are derived, and by which they are measured. The question is, how do we most readily apprehend and express the relation between 1 and 4 \ Ask a child, and he will answer, " the dif- ference is 3 " But when you ask him, " how many I's are there in 4 1" he will answer, " 4." using 1 as the standard. Thus, we begin to teach by using the standard 1 : that is, by dividing 4 by 1. Now, the relation between 3 and 12 is the same as that between 1 and 4 ; if then, we divide 4 by 1, we must also divide 12 by 3. Do we, indeed, clearly apprehend the ratio of 3 to 12, until we have referred to 1 as a standard ! Is the mind satisfied until it has clearly perceived that the ratio of 3 to 12 is the same as that of 1 to 4 ? In the Rule of Three we always look for the result in the 4th term Now, if we wish to tind the ratio of 3 to 12, by referring to 1 as a stand ard, we have 3 : 12 : : 1 : ratio, which brings the result in the right place. But if we define ratio to be the antecedent divided by the consequent, we should have 3 : 12 : : ratio : 1, wliicb w(juKl brinji tbe ratio, or icqnvcd number, in the .'3d jilacc. RATIO AND PROPORTION. 215 What are the ratios of the proportions, 3 9 : 12 36? 2 : 10 : 12 60? 4 2 : 8 4? 9 1 : 90 10? 225. The 1st and 4th terms of a proportion are caJl^rl the extremes : the 2d and 3d terms, the means. Thus, in tb*» pr»i portion, 3 : 12 : : 6 : 24 Since (Art. 224), 3 and 24 are the extremes, and 12 and 6 the means: 12_24 3"~"6"' we shall have, by reducing to a common denominator, 12x6 24x3 ~3x6~"6^' But since the fractions are equal, an(J have the same deno- minators, their numerators must be equal, viz ; 12x6 = 24x3; that is, In any proportion^ the product of the extremes is equal to the product of the means. Thus, in the proportions, 1 : 6 : : 2 : 12; we have 1x12= 2x6; 4 : 12 : : 8 : 24 ; " " 4x24=12x8. 226. Since, in any proportion, the product of the extremes \& equal to the product of the means, it follows that, In all cases, the numerical value of a (juantity is the number of time? which that quantity contains an assumed standard, called its unit oj mea» ure. If we would find that numerical value, in its right place, we musl 3ay» standard : quantity : : 1 : numerical value : but if we take the other method, we have quantity standard : numerical value : 1. which brings the iiumerioiil value in the wrotiir place. 2^16 RATIO AND PKOPOiJTION. Isti^ If the product jof the means be divided by one of ike extremes, the quotient will he the other extreme. Thus, in the pioportion 3 : 12 : : 6 : 24, we have 3x24 = 12x6; then, if 72, the product of the means, be divided by one o* the extremes, 3, the quotient will be the other extreme, 24: or, if the product be divided by 24, the quotient will be 3. 2d. If the product of the extremes be divided by either of the means, the quotient will be the other mean. Thus, if 3 X 24 irr 12 X 6:= 72 be divided by 12, the quotient will be 6 ; or if it be divided by 6, the quotient will be 12. EXAMPLES. 1. The first three terms of a proportion are 3, 9 and 12 : "what is the fourth term ] 2; The first three terms of a proportion are 4, 16 and 15 : what is the 4th term 1 3. The first, second, and fourth terms of a proportion are 6, 12 and 24 : what is the third term ? 4. The second, third, and fourth terms of a proportion are 9, 6 and 24 : what is the first term.^ 5. The first, second and fourth terms are 9, 18 and 48 ; what is the third term? 227. Simple and Compound Ratio. The ratio of two single numbers is called a Simple Ratio^ and the proportion which arises from the equality of two such ratios, a Simjjle Proportion. 225. Which are th.; extremes of a proportion 1. Which the means 1 What is the product of the extremes equal to 1 226. If the product of the means be divided by one of the extremes, what will the quotient be 1 If the product of the means be divided by either extreme, what will the quotient be ? 227. What is a simple ratio ? What is the proportion called whicli comes from the equality of two simple ratios 1 What is a conipoiuul ratio ' What is u compound pruporliuu ' eiMl'LE QUESTIO 1. If 12 apples be equally dividea aiiiUliy 4 boys, how many will each have ? Analysis. — Since 12 apples are to be divided equally between 4 boys, one boy will have as many apples as 4 is contained timeh In 12, which is 3 ; therefore, if 12 apples be equally divided be- tween 4 boys, each will have 3 apples. 2. If 24 peaches be equally divided among 6 boys, how many will each have ? How many times is 6 contained in 3. A man has 32 mil&i to walk, and can travel 4 miles an : our, how many hours will it take him ? 4. How many yards of cloth, at 3 dollars a yard, can you buy for 24 dollars ? Analysis. — Since the cloth is 3 dollars a yard, you can buy as many yards as 3 is contained times in 24, which is 8 : therefore, you can buy 8 yards. 5. How many oranges at 6 cents apiece can you buy for i 2 cents ? 6. How many pine-apples at 12 cents apiece can you buy .or 132 cents'? 7. A farmer pays 28 dollars for 7 sheep : how much is that apiece? Analysis. — Since 7 sheep cost 28 dollars, one sheep will cost ass many dollars as 7 is contained times in 28, which is 4 ; therefore, each sheep will cost 4 dollars. 8 If 12 yards of muslin cost 96 cents, how much does 1 yard cost] 9. How many lead pencils could you buy for 42 cents, if they cost 6 cents apiece ? 10. How many oranges could you buy for 72 cents, if they cost 6 cents apiece ? 1 1 . A trader wishes to pack 64 hats in boxes, and can put but 8 hats in a box : how many boxes does he want 1 12. If a man can build 7 rods of fence in a day, how long will it take him to build 77 rods ? 13. If a man pays t>G dollars for seven yards of cloth, how much i« that a yard ? 68 DIVISION. 14. Twelve men receive 108 dollars for doing a piece of work : how much does each one receive '? 15. A merchant has 144 dollars with which he is going to buy cloth at 12 dollars a yard ; how many yards can he pur- chase ? 16. James is to learn forty-two verses of Scripture in a week : how much must he learn each day ? 17. How many times is 4 contained in 50, and how maii) over 'i PRINCIPLKS AND EXAMPLES- 60. 1. Let it be required to divide 86 by 2. Set down the number to be divided and write the other number on the left, drawing a curved line between them. Now there are 8 tens and 6 units to be divided by 2. We say, 2 in 8. 4 timeS; which being tens, we write it in the tens place. We then say, 2 in 6, 3 times, which being units, are written in the units' place. The result, which is called a quotient, is fcheie- lore, 4 tens and 3 units, or 43. OPERATION. n3 > 86 43 quotie't. OPVl'HATIOJI. ;^)729 84^ 2. Let it be required to divide 729 by 3. Analysis. — We say, 3 in 7, 2 times and 1 over. Set down the 2, which are hundreds, under the 7. But of the 7 hundreds there is i hundred, or 10 tens, not yet divided. We pat the 10 tens with the 2 tens, making 12 tens, and then say, 3 in 12, 4 times, and write the 4 of the quotient in the tens' place ; then say, 3 in 9, 3 times. The quotient, therefore, is 243. 3. Let it be required to divide 466 by 8. Analysis. — We first divide the 46 tens by 8, giving a quotient of 5 tens, and 6 tens over. These 6 tens are equal to 60 units, to which we add the 6 in the units' place. We then say, 8 in 66, 8 times and 2 over ; hence, the quotient is 58, and 2 over, wliich we call a remainder. This remainder is written after the last quotient figure, and the 8 placed under it ; the quotient is read, 68 and 2 divided by 8. OPERATION. 8)466 08-2 remain. 58| quotient. CO Ex. 1.- tens or uaitsj 1 ■When you divide 8 tens by 2, is the unit of the quotient When G unite are divided by %, what is the unit? SIMPIJi NUMBERS. 59 Analysis. — In the first example, 86 is divided into 2 equal parts, and the quotient 43 is one of the parts. If one of the equal parts be multiplied by the number of parts 2, the product wilMae 86, the number divided. In the third example. 466 is divided into 8 equal parts, and two units remain that are not divided. If one of the equal parts. 5vS, be multiplied by the number of parts, 8, and the remainder 2 be added to the product, the result will be equal to 466, the numbej divided. 6 1 . Division is the operation of finding from two numbers a third, which multiplied, by the first, will produce the second. The first number, or number by which we divide, is called the divisor. The second number, or number to be divided, is called the dividend. The third number, or result, is called the gtiotie^it. The quotient shows how many times the dividend contains the divisor. II" anything is left after division, it is called a remainder. 62. There are three parts in every division, and sometimes four: 1st, the dividend; 2d, the divisor; 3d, the quotient; and 4th, the remainder. There are three signs used to denote division ; they are the following : 18-^4 expresses that 18 is to be divided by 4. Jj^ expresses that 18 is to be divided by 4. 4)18 expresses that 18 is to be divided by 4. When the last sign is used, if the divisor does not excejd 12, we draw a line beneath, and set the quotient under it. Il the divisor exceeds 12, we draw a curved line on the right of the dividend, and set the quotient at the right. 2. — When the seven hundreds are divided by 3. what is the unit of the quotient 1 To hovy many tens is the undivided hundred equal 1 When the 12 tens are divided by 3, vs-hat is the unit of the quotient 1 When the 9 units are divided by 3, what is the quotient '! 3. — How is the division of the remainder expressed 1 Read the quotient. If there be a remainder after division, how must it be written ! 61. What is division] What is the number to be divided called? What is the number called by which we divide I What is the answci called? What is the number called which is leftl 62. How many parts are there in division 1 Name them. IIow many siyns ure there in divijiimi '' Makf and name tlieiii 00 SHORT DIVISION. SHORT DIVISION. « 63. Short Division is the operation of dividing when the work is performed mentally, and the results only written down. It is limited to the cases in which the divisors do not exceed 12. Let it be required to divide 30456 by 8. A.NAi-Ysis. — We first fiay, 8 in 3 we cannot. Then, operation. 8 in 30; 3 times and 6 over ; then, 8 in 64, 8 times ; 8)30456 then 8 in 5, times j then. 8 in 56, 7 times : hence, — ^S07~ Rule I. — Writs the divisor on the hft of the dividend. Beginning at iht left, divide each Jig are of the dividend hy the divisor, and set each quotient figure under its dividend. II. If there is a. remainder, after ony diviaion, annex to it ^Jie next figure of the dividend, and divide as before. III. If any dividend is less than the divisor, write for th, quotient figure and annex the next figure of the dividend, fo a new dividend. iV. If there is a remainder, after dividing the last figurb^ eet the divisor under it, and annex the result to the quotient Proof. — Multiply the divisor by the quotient, and to the product add the remainder, when there is one ; if the wck is right the result will be equal to the dividend EXAMPLES. • (1) (2.) (3.) ,4) . 3)9369 4)73684 ;)673420 6)325467 Ans. 3123 "18421 T3~4684 T37577| 3 4 5 6 Proof 9369 73684 673420 825^67 5. Divide 86434 by 2. 6. Divide 416710 by 4. 7. Divide 64140 by 5. 8. Divide 278943 by 6. 9. Divide 95040522 by 6. 10. Divide 75890496 by 8. 11. Divide 6794108 by 3. 2. Divide 21090^131 by 9. 13. Divide 2345678964 uy 6 14. Divide 570196382 by 12. 15. Divide 67897634 by 9. 16. Divide 75436298 by 12. 17. Divide 674189904 by 9. 18. Divide 1404967214 by U 19. Divide 27478041 by 10. 20 Divid.' 1671 ^'1329 by 12 FJtACTlONS. 61 21. A man sold his farm for 6756 dollars, and divided the amount equally between his wife and children : how much did each receive i 22. There are 576 persons in a train of 12 cars : how many are there in each car 1 23. If a township of land containing 2304 acres be equally divided between 8 persons, how many acres will each have 1 24. If it takes 5 bushels of wheat to make a barrel of flour, how many barrels can be made from 65890 bushels ? 25. Twelve things make a dozen : how many dozens are there in 2167284 ? 26. Eleven persons are all of the same age, and the sum of their ages is 968 years : what is the age of each ? 27. How many barrels of flour at 7 dollars a barrel can be bought fbr 609463 dollars 1 28. An estate worth 2943 dollars, is to be divided equally between a lather, mother, 3 daughters and 4 sons : what is the portion of each ? 29. A county contains 207360 acres of land lying in 9 town- ships of equal extent : how many acres in a township ? 30. If 11 cities contain the same number of inhabitants, and the whole number is equal to 3800247 : how many will there be in each ? FRACTIONS. 64. 1. If any number or thing be divided into two equal parts, one of the parts is called one-half, which is written thus ; ^. 2. If any number is divided into three equal parts, one of the parts is called one-third, which is written thus ; J ; two of the parts are called two-thirds, and written thus ; f. 3. If any number is divided into four equal parts, one of the parts is called one-fourth, which is written thus ; \ ; two of the parts are called two-fourths, and are written thus ; ^ ; three of them are called three-fourths, and written |- ; and similar ntimes are given to the equal parts into wdiich any number may be divided. 03. What is short division 1 How is it generally pcrfomicil' (Jivt thf ruif. H(»w tlo viru pr(jv<; i-horl (]ivi.>-i()i» 1 62 FKACTIONS. 4. If a number "s divided into five equal parts, what is one of the parts called ? Two of them ? Three of them ] Four ol" them ? 5. If a number is divided into 7 equal parts, what is one of the parts called ? What is one of the parts called when it is divided mto 8 equal parts ? When it is divided into 9 equal parts ? When it is divided into 10 ? When it is divided into 11 1 When it is divided into 12 ? 6. What is one-half of 2 ? of 4 ? of 6 ? of 8 ? of 10 ? of 12 ? of 14? of 16? of 18? 7. What is one-third of 3 ? What is two-thirds of 3 ? ANA.LYSIS. — Two-thirds of three are two times one-third of three. One-third of three is 1 ; therefore, two-thirds of three are two times 1, or 2. Let every question be analyzed in the same manner. Wliat is one-third of 6 ? 2 thirds of 6 ? One-third of 9 ? 2 thuds of 9 ? One-third of 12 ? two- thirds of 12 ? 8. What is one-fourth of 4 ? 2 fourths of 4 ? 3 fourths of 4 ? What is oiie-fourth of 8 ? 2 fourths of 8 ? 3 fourths of 8 ? What is onc-fburth of 12 ? 2 fourths of 12 ? 3 fourths of 12 ? One- fourth of 16 ? 2 fourths of 16 ? 3 fourths'? 9. What is one-seventh of 7 ? What is 2 sevenths of 7 ? 5 sevenths ? 6 sevenths? What is one-seventh of 14 ? 3 sev- enths ? 5 sevenths ? 6 sevenths ? What is one-seventh of 21 1 of 28 ? of 35 ? 10. W^hat is one-eighth of 8 ? of 16 ? of 24 ? of 32 ? of 40 ? of 56 ? 11. What is one-ninth of 9 ? 2 ninths? 7 ninths? 6 ninths? 5 ninths? 4 ninths? What is one-ninth of 18? of 27? of 64? of 72? of 90? of 108? 12. How many halves of 1 are there in 2? Analysis. — There are twice as many halves in 2 as there are in 1 , There are two halves in 1 ; therefore, there are 2 times 2 halves in 2, or 4 halves. 13. How many halves of 1 are there in 3 ? In 4 ? In 5? In 6? In 8? In 10? In 12? 14. How many thirds are there in 1 ? How many thirds of 1 in 2 ? In 3 ? In 4 ? In 5 ? In 6 ? In 9 ? In 12 ? 1 5. How many fourths are there in 1 ? How many f.»urths i;f 1 in 2? In 4? In 6? In 10 ? In 12? FKAOTIONS. 6S 16. How many fifths are there in 1 ? How many fifths oi 1 are there lu 2 V In 3 ? In 6 ? In 7 ? In 11? In 12 ? 17. How many sixths are there in 2 and one-sixth i In 3 and 4 sixths ? In and 2 sixths 1 In 8 and 6 sixths 1 18. How many sevenths of 1 are there in 2 ? In 4 and 3 sevenths how many i How many in 5 and 5 sevenths ? In and 6 sevenths ? 1 9. How many eighths of 1 are there in 2 ? How many in 2 and 3 eighths 'I In 2 and 5 eighths ? In 2 and 7 eighths ? In 3 ? In 3 and 4 eighths ? In 9 ? In 9 and 5 eighths'? In 10^ In 10 and 7 eighths? 20. How many twelfths of 1 are there in 2"? In 2 and 4 twelfths how many ] How many in 4 and 9 twelfths? How many in 5 and 10 twelfths ? In 6 and 9 twelfths 1 In 10 and 11 twelfths? 21. What is the product of 12 multiplied by 3 and one- half, (which is written 3^) ? Analysis. — Twelve is to be taken 3 and one-half times (Art. 45). Twelve taken i times is 6; and 12 taken three times is 36 j therefore, 12 taken 3i times is 42. 22. What is the product of 10 multiphed by 6 J 1 23. What is the product of 12 multiplied by 3^ ? 24. What is the product of 8 multiplied by 4^ ? 25. What will 9 barrels ol' sugar cost at 2| dollars a bairei ? Analysis. — Nine barrels of sugar will cost nine times as ={ a 'i The divisor, 13, is contained in 70 l^j/uoy^040 hundreds, 5 hundreds times and some- ^ " thing over. To find how much over, 5 5 /nultiply 13 by 5 hundreds and subtract 5 2 vn the 5 tens, and consider the 55 *^ >ens a new partial dividend. 65. What is long division ! Docs it embrace the rase o*" short divi- ion ! What is a partial dividend ^ SIMPLE NUMBERS. 66 Then, 13 is contained in 55 tens, 4 tens times and something over. Multiply 13 by 4 tens and subtract the product, 52, from 65, and to tlie remainder 3 tens bring down the 9 units, and con- sider the 39 units a new partial dividend. Then, 13 is contained in 39, 3 times. Multiply 13 by 3, and subtract the product 39 from 39, and we find that nothing remains. 66. Proof. — Each product that has arisen from multiply- ng the divisor by a flj^ure of the quotient, is a partial product, and the sum of these products is the product of the divisor and quotient (Art. 51, Note). Each product iias been taken, separately, from the dividend, and nothing remains. But, taking each product away in succession, leaves the same re- mainder as would be left if their sura were taken away at once. Hence, the number 543, when multiplied by the divisor, gives a product equal to the dividend : therefore, 543 is the quotient (Art. 61) : hence, to prove division, Multiply the divisor hy the quotient and add in the remaiyi- der, if any. If the work is right, the result will be the same as the dividend. 67. Let.)it be required to divide 2756 by 26. We first say, 26 in 27 once, and place 1 in operation. the quotient. Multiplying by 1, subtracting, 26)2756(106 and bringing down the 5, we have 15 lor the 26 first partial dividend. We then say, 26 in 15, times, and place the in the quotient. We 156 156 is contained in 156, 6 times. If any one of the partial dividends is less than the divisor, write for the quotient figure, and then bring down the next figure, forming a new partial dividend. Hence, for Long Division, we have the following Rule. — 1. Write the divisor on the left of the dividend. II. Note the fewest figures of the dividend, au the left^ that will contain the divisor, and set the quotient figure at the right. 66. What is a partial product 1 W'hat is the sum of all the partial products equal to ( How do you prove division ' 67. ^^'hat do you do if any partial dividend is less than the divisor * What is the rule for long division 1 5 6(5 T.Oyr; T>TVTSTON III. Myltiply the divisor by the quntiefit figitrre, subtract the 'product from the first 'partial dividend^ and to the re- mainder annex tlic next figure of the dividend^ forming a second, partial divideiid. IV. Find in tlie same manner the second and. svcceeding 'figures of the quotient, till all the figures of the dividend are brought down. NoTK 1. — There are five operations in Long Division. 1st. To write down the numbers : 2d. Divide, or find how many times : 3d. Multiply : 4th. Subtract: 5th. Bring down, to form the partial dividends. 2. The product of a quotient figure by the divisor must never be larger than the corresponding partial dividend : if it is, the quotient figure is too large and must be diminished. 3. When any one of the remainders is greater than the divisor, the quotient figure is too small and must be inerea.>^ed. 4. The unit of any quotient figure is the same as that of the partial dividejid from which it is obtained. The pupil should always name Lhe unit of every quotient figure. EXAMPLES. 1. Divide 7574 by 54. OPERATION. 64)7574(140 54 2\i 216 11 00 14 Remainder. 2. Divide 67289 by 261, OPERATION. 201)67289(257 522_ "1508 1305 ^039 1827 212 Remainder PROOF. 140 Quotient. 54 Divisor. "560" 700 7560 14 Remainder 7574 Dividend. PROOF. 261 Divisor. 257 (Quotient 1827 1305 522 2~r2 Remainder 67281) Dividend. blMPLK ^UMISKKS. 67 3. Divide 119836687 by 39407. OPERATION. 39407)119836687(304 118221 PROOF 39407 Diviaor. 3041 duotienl 161568 157628 39407 39407 39407 157628 118221 119836687 Dividend. 4. Divide 7210473 by 37. 9. Divide 62015735 by 78. 5. Divide 147735 by 45. 10. Divide 14420946 by 74. 6. Divide 937387 by 54, 11 Divide 295470 by 90. 7. Divide 145260 by 108. 12. Divide 1874774 by 162. 8. Divide 79165238 by 238. 13. Divide 435780 by 216. 14. Divide 203812983 by 5049. 15. Divide 20195411808 by 3012. 16. Divide 74855092410 by 949998. 17. Divide 47254149 by 4674. 18. Divide 119184669 by 38473. 19. Divide 280208122081 by 912314. 20. Divide 293839455936 bv 8405. 21. Divide 4637064283 by 57606. 22. Divide 352107193214 by 210472. 23. Divide 558001172606176724 by 2708630425. 24. Divide 1714347149347 by 57143. 25. Divide 6754371495671594 by 678957. 26. Divide 71900715708 by 37149. 27. Divide 571943007145 by 37149. 28. Divide 671493471549375 by 47143 29. Divide 571943007645 by 37149. 30. Divide 171493715947143 by 57007. 31. Divide 121932631112635269 by 987654321. Notes. — 1. How many operations are there in long division ? Nani€ them. 2. If a partial product is greater than the partial dividend, what loet it indicate ' What do you do ! 3. What do you do when any one of the remainders is greater thMl the divi.sor ] 4. What is the unit of any figure of the quotient 1 When the divisoi is contained in simple units, wliat will be the unit of the quotient figure? When it is contained in tens, what will be the unit of the quotient figure 1 When it is contained in hundreds ^ In thousands ! t)i> i^OiNG DIVISION. 68. PRINCIPLES RESULTING FROM DIVISION. Notes. — 1st. When the divisor is 1, the quotient will be equal to tlje dividend. 2d. When the divisor is equal to the dividend, the quotient will be 1. 3d. When the divisor is less than the dividend, the quotient will be greater than 1. The quotient will be as many times greater than 1, as the dividend is times greater than the divisor. 4th. When the divisor is greater than the dividend, the quotient will be less than 1. The quotient will be such a part of I, as the dividend is of the divisor. PROOF OF MULTIPLICATION. 69. Division is the reverse of multiplication, and the)' prove each other. The dividend, in division, corresponds to the product in multiplication, and the divisor and quotient to the multiplicand and multiplier, which are factors of the pro- duct : hence, If the product of two numbers he divided by the multipli- cand., the quotient will be the multiplier ; or, if it be divided by the multiplier., the quotient will be the multiplicand. EXAMPLES. 3679 Multiplicand. 36 179)1203033 327 Multiplier. 11037 25753 9933 7358 7358 11037 25753 1203033 Product. 25753 2. The multiplicand is 61835720, and the product 8162315040 : what is the multiplier? 3. The multiplier is 270000 ; now if the product be 1315170000000, what will be the multiplicand? 4. The product is 68959488, the multiplier 96: what is tlie multiplicand ? 5. The multiplier is 1440, the product 10264849920 what is the multiplicand ? 6. The product is 6242102428164, the multiplicand 6795634 : what is the multiplier ? tX>NTK ACTIONS IN MX^LTIPLICATION. Oil CONTRACTIONS IN MULTIPLICATION. 70. To multiply by 25. 1. Multiply 275 by 25. Analysis. — If we annex two ciphers to the mnl- operation. tiplicand, we multiply it by 100 (Art. 55) : this 4)27500 product is 4 times too great; for the multiplier is fift75~ but one-fourth of 100 ; hence, to multiply by 25, Annex two ciphers to the multiplicand and divide tJie residt dy 4:. EXAMPLES. 1. Multiply 127 by 25. I 3. Multiply 87504 by 25. 2. Multiply 4269 by 25. | 4. Multiply 704963 by 25. 71. TomuUiply hy \2\. 1. Multiply 326 by 12i , Analysis. — Since 12^ is one-eighth of 100. operation. Annex tivo ciphers to the multiplicand and di- 8)32600 vide the result hy H. ~^40Vy EXAMPLES. 1. Multiply 284 by 121 2. Multiply 376 by 121. 3. Multiply 4740 by 12-^. 4. Multiply 70424 by \2^. 72. To multiply by 331 1. Multiply 675 by 33^. Analysis. — Annexing two ciphers to the mul- operation. tiplicand, multiplies it by 100 : but the multiplier 3)67500 is but one-third of 1 00 : lienee. Annex two ciphers and divide the result by 3. 22500 EXAMPLES. 1. Multiply 889626 by 331. 2. Multipl'y 740362 by 331 3. Multiply 5337756 by 33f 4. MultipK 2221086 by 33^. 68. When the divisor is I, what, is the quotient! When the divisor is equal to the dividend,- what is the quotient I When the divisor is less than the dividend, how does the quotient compare with 1 I When the di- visor is greater than the dividend, how does the quotient compare with I ! 09. If a product l)o divided by one of the lactors. what i.s the quotient ! 70 OONTK ACTION i? IN MULTU'LIOATION. 73. To multiply by 125. 1, Multiply 375 by 125. Analysis. — Annexing three ciphers to the mul- operation. tiplicand, miiltipiie.s it by 1000: but 125 i^ but 8)375000 one-€iN"!RA(TriONS IN DIVISION. 71 CONTRACTIONS IN DIVISION. 75. Contractions in Division are short nnethods of finding the quotient, when the divisors are composite numbers. CASE I. 76. When the divisor is a cow^posite number. 1. Let it be required to divide 1407 dollars equally among 21 men. Here the factors of the divisor are 7 and 3. Anai.vsis. — Let the 1407 dollars be first divided into 7 equal piles. operation. Each pile will contain 201 dollars. 7)1407 Lei each pile be now divided into 3 -^^ ^^^ quotient equal parts. Lack part will contain ^ 67 dollars, and the number of parts 67 quotient sought will be Jil : hence the foilowing Rule. — Divide the dividend by one of the factors of the divisor ; then divide the quotient, thus arising, by a second factor, and so on, till every factor has been used as a divisor : the last qaoiient will be the answer. EXAMPLES. Divide the following numbers by the factors ; 1. 1260 by 12 = 3x4. | 6. 55728 by 4 x9 x4=144. 2. 18576 by 48 = 4x12. 3. 9576 by 72 = 9x8. 4. 19296 by 96-12x8. 6. 92880 by 2 X 2 X 3 X 2 X 2. 7. 57888 by 4x2x2x2. 8. 154368 by 3x2x2. Note. — It often happens that there are remainders after some of tlie divisions. How are we to find the true remainder ? 74. — 1. What is the rule for dividing by 25 ? 2 What is the rule for dividing by 12^^? 3. What is the rule for dividing by 33^ ? 4. What is the rule for dividing by 125 ? 7"^. What are contractions in division ? What is a compoiiite num- ber ' 7ft. W^hat is the rule for division when the divisor is a composite number ^ ' 72 UONTRACTIONS 77. Let it be required to divid© 751 grapes into 16 equal parts. 4X4=:16 ^ 4)187 . . . .3 first remainder. ^ 46 .... 3x4=.12 1 5 true rem. Ans. i&j^. Note. — The factors of the divisor 16, are 4 and 4. Analysis. — If 751 grapes be divided by 4, there will be i87 bunches, each containing 4 grapes, and 3 grapes over. The unit of 187 is 07ie bunch ; that is, a unit 4 times as great as 1 grape. If we divide 187 bunches by 4, we shall have 46 piles, each containing 4 bunches, and 3 bunches over : here, again, the unit of the quotient is 4 times as great as the unit of the dividend. If, now we wish to find tlie number of gra])es not included in the 46 piles, we have 3 bunches with 4 grapes in a bunch, and 3 grapes besides : hence, 4x3 = 12 grapes , and adding 3 grapes, we have a remainder, 15 grapes ; therefore, to find the remainder, in units of the given dividend : I. Multiply the last retnainder by the last divisor hut one, and add in the preceding remainder : II. Multiply this result by the next preceding divisoT", and add in the remainder, and so on, till you reach the unit of the dividend. EXAMPLES. 1. Let it be required to divide 43720 by 45. r 3)43720 45=3x5x3 \ 5)r4o73 .l=ilstrem. 1x5 + 3 = 8; I 3 )2914 . 3 = 2d rem. Sx2>-\-\^2b ^ 971 . l=:3d rem. 25 true rem. Divide the following numbers by the factors, for the divisors : 2. 956789 by 7x8 = 56. 3. 4870029 by 8x9 = 72. 4. 674201 by 10x11 = 110. 6. 445767 by 12x12 = 144. 6. 1913578 b> 7x2x3 = 42 7. 146187 by"3x5x7 = 105 8. 26964 by5x2xll=llC 9. 93696 by3x 7x11 = 231 T Give the rule for the remainder IN DIVISION. /3 VASK II. 78. When the divisor is 10, 100, 1000, ^c. Analysis. — Since any number is made up of units, tens, hun- dreds &c. (Art. 28), the number of tcTis in any dividend will denote how many times it contains 1 ten, and the units will be the renuiinder. The hundreds will denote how many times the divi- dend contains 1 hundred, and the tens and unils will be the remain- der ; and similarly when the divisor is 1000, 10000. &c. ; hence, Cut off from the right hand as mani) figures as there are ciphers in the divisor — the figures at the left will be the quotient, and those at he right, the renminder. EXAMPLES. 1. Divide 49763 by lO. 2. Divide-7641200 by 100. 3. Divide 496321 by 1000. 4. Divide 64978 by 10000. 79. Whe7i there arc ciphers on the right of the divisor, I. Let it be required to divide 673889 by 700. Analysis. — We may regard the operation. divisor as a composite number, of 7100)673189 which the factors are 7 and 100. T^ - We first divide by 100 by .^trikinij -^^ ' ' ^ remains. off the 89, and then find thai 7 is _l2?. ^^"^ remain. contained in the remaining tigures, Ans. 961^^. 96 times, with a remainder of 1 : this remainder we multiply by 100, and then add 89, forming the true remainder 189 ■: to the quotient 96 we annex 189 divided by 700, for the entire quotient : hence, the following Rule. — I. Cut off the ciphers by a line, and cut off the same number of figures from the right of tJie divide?id. II. Divide the remaining figures of the dividend by the remaining figures of the divisor, and annex to the remciin' der, if there be one, the figures cut off from the dividend : this will form the true remainder. EXAMPLES. 1. Divide 8749632 by 37000. 78. How do you divide when the divisor is 1 with ciphers annexed ■ •Give the reason of the rule ! 79. How do you divide when there are ciphers on the light of the divisor ! How do you iorm I he true remahider '! 74 AP PI -I CATION'S. a7|000)8749|632(236 74 134 Ans. 23611 1? 2 111 239 222 "17 Divide the following numbers 2. 986327 by 210000. 3. 870000 by 6000. 4. 30599503 by 400700. 5. 5714364900 by 36500 6. 18490700 by 73000. 7. 70807149 by 31500. APrLICATIONS. 80. Abstractly, the object of division ia to find from two given numbers a third, which, multiplied by the first, will produce the second. Practically, it has three objects : 1. Knowing the number of things and their entire cost, to find the price of a single thing : 2. Knowing the entire cost of a number of things and the price of a single thing, to find the number of things : 3. To divide any number of things intb a given immbei of equal parts. For these cases, we havp from the previous principles (page 57), the fijllowing RULES. I. Divide the entire cost by the rium^ier of the things : the quotient will be the price of a single thing. II. Divide the entire cost by the ])rice of a single thing : the quotient will he the nutnher of things. III. Divide the whole number of things by the nwmher of parts into which they are to be divided : the quotient will be the number i?i each fart. QUESTIONS INVOLVING THE PREVIOUS RULES. 1. Mr. Jones died, leaving an estate worth 4500 dollars, to be divided equally between 3 daughters and 2 sons : what wag the share of each ? 80 What is the object of division, abstractly 1 How many objects has It, practically ! Nau)e the three objects. Give the rules for the three ca.se8 AFPLIUATIOMS. 76 2. What number must be multiplied by 124 to produce 40796? 3. The sum of 19125 dollars is to be distributed equally among a certain number oi' men, each to receive 425 dollars : hoAv many men are to receive the money ? 4. A merchant has 5100 pounds of tea, and wishes to pack it in 60 chests : how much must he put in each chest ? 5. The product ol' two numbers is 51679680, and one of the factors is 615 : what is the other factor 1 6. Bourrht 156 barrels of flour for 1092 dollars, and sold the same for 9 dollars per barrel : how much did 1 gain '? 7. Mr. James has 14 calves worth 4 dollars each, 40 sheep worth 3 dollars each ; he gives them all for a horse worth 150 dollars : does he make or lose by the bargain ? 8. Mr. VYilsou sells 4 tons of hay at 12 dollars per ton.. 80 bushels of wheat at 1 dollar per bushel, and takes in payment a horse worth 65 dollars, a wagon worth 40 dollars, and the rest in cash : how much money did he receive ? 9. How many pounds of coflee, worth 12 cents a pound, must be given for 368 pounds of sugar, worth 9 cents a pound ? 10. The distance around the earth is computed to be about 25000 miles : how long would it take a man to travel that distance, supposing him to travel at the rate of 35 miles a day ? 11. If 600 barrels of flour cost 4800 dollars, what will 2172 barrels cost '? 12. If the remainder is 17, the quotient 610, and the divi- dend 45767, what is the divisor 1 13. The salary of the President of the United States la 25000 dollars a year : how much can he spend daily and feave of his salary 4925 dollars at the end of the year ? 14. A farmer purchased a farm for which he paid 18050 dollars. He sold 50 acres for 60 dollars an acre, and the re- mainder stood him in 50 dollars an acre : how much land did he purchase ? 15. There are 31173 verses in the Bible: how many verses must be read each day, that it may be read through in a year] 16. A farmer wishes to exchange 250 bushels of oats at 42 cents a bushel, for flour at 7 dollars per barrel ; how many barrels will he receive ? 70 AlTLlliATlONS 17. The owner of an estate 9o1q 125? 8. In ^400, how many cents'? How many mills? 9. In $375, how many cents ? How many mills? 10. How many mills in ^4 ? In |;6 ? In $10,14 cents? 11. How many mills in ^40,36 cents 8 mills'? 12. How many mills in •1>71,45 cents 3 mills? 86. To reduce from a less unit to a greater. 1. How many dollars, cents and mills in 26417 mills'^ ANALVfiis. — We hist divide the mills by 10, operation giving 2641 cents and 7 Jiiills over; we then 10)2641j7 divide the cents by 100, giving 26 dollars, and 100^26141 41 cents over: hence, the answer is 26 dollars ' ' 41 cents and 7 mills: therefore, Ans. ^26,417 I. To reduce mills to cents : cut off the right hand figure II. To reduce cents to dollars : cut off the two right hand figures : and, III. To reduce mills to dollars : cut off the three right hand figures. EXAMPLES. 1. How many dollars cents and mills are there in 67897 mills ? 2. Set down 104 dollars 69 cents and 8 mills. 3. Set down 4096 dollars 4 cents and 2 mills. 4. Set down 100 dollars 1 cent and 1 mill. 5. Write down 4 dollars and 6 mills. 6. Write down 109 dollars and 1 mill. 7. Write down 65 cents and 2 mills. 8. Write down 2 mills. 9. Reduce 1607 mills, to dollars cents and mills. 10. lleduce 170464 mills, to dollars cents and mills. 11. Reduce 6674416 mills, to dollars cents and mills. 12. Reduce 94780900 mills, to dollars cents and mills. 13. Reduce 74164210 mills, to dollars cents and mills. 86. How do you change mills into cents ! How do you change cent* into dollars' How do you change mills lo dollars \ rNTTFD STATES 87. One nii/iiber is said to be an when it is contained in that other an exact nuiliuui Ultimes. Thus ; 50 cents, 25 cents, &c., are aliquot parts of a dollar : so also 2 months, 3 months, 4 months and 6 months are ali- quot parts of a year. The parts of a dollar are sometimes expressed fractionally, as in the following TABLE OF ALIQUOT PARTS $; =100 cents. ^ of a dollar^: 50 cents. J of a dollar ==33^ cents. ^ of a dollars: 25 cents. i of a. dollars 20 cents. •i- of a dollar :=: 121 cents. y\j of a dollars 10 cents. Tfi 2^0 of a dollar = '4 5 cents. ^ of a cent = 5 mills. ADDITION OF UNITED STATES MONEY. and 3^ cents for 6 1 . Charles gives 9^ cents for a top quills : how much do they all cost him ? 2. John gives $1,37^ lor a pair of shoes, 25 cents for a penknife, and 12^ cents for a pencil : how much does he pay ibr all I Analysis. — We observe that half a cent is equal to 5 mills. We then place the mills, cents and dol- lars in separate columns. We then add as in simple numbers. 3. James gives 50 cents for a dozen oranges, 12^ cents for a dozen apples, and 30 cents for a pound of raisins : how much for all ? OPERATION. $1,3/5 ,25 ,125 §1750 OPZRATION. $0,50 ,125 ,30 $0,925 88. Hence, for the addition of United States money, we have the following Rule. — I. Set down the numbers so that units (^ the same value shall fall in the same column. 87. What is an aliquot part 1 How many cents in a dollar I hi half a dollnr'? h) a lliird of a t'ollar ■? in a li)uith uf a tln.'l.'ir'! 84 AP PLICATIONS IN II. Add up the several coluinns as in simple nwm^ers^ and place the separating point in the sum directly undet that in the columns. Pkoof. — The same as in simple numbers EXAMPLES. 1. Add $67,214, $10,049, |6,041, $0,271, togethoi. (1.) (2.) (3.) $ cts. m. $ cts. m. $ cts. m. 67,214 59,316 81,053 10,049 87,425 67.412 6,041 48,872 95,376 , 0,271 56.708 87.064 $83,575 $330,905 APPLICATIONS. 1. A grocer purchased a box of candles for 6 dollars 89 cents : a box of cheese for 25 dollars 4 cents and 3 mills ; a keg of raisins for 1 dollar 12^ cents, (or 12 cents and 5 mills ;) and a cask of wine lor 40 dollars 37 cents 8 mills : what did the whole cost him % 2. A farmer purchased a cow for which he paid 30 dollars and 4 mills; a horse for which he paid 104 dollars 60 cents and 1 mill ; a wagon for which he paid ^b dollars and 9 mills : how much did the whole cost ? 3. Mr. Jones sold farmer Sykes 6 chests of tea for §75,641 ; 9 yards of broadcloth for $27,41 ; a plow for -^9,75 ; and a harness for $19,674 : what was the amount of the bill? 4. A grocer sold Mrs. WiUiams 18 hams ibr $26,497 ; a bag of coffee for $17,419 ; a chest of tea for $27,047 ; and a firkin of butter for $28,147 : what was the amount of her biin 5. A father bought a suit of clothes for each of his four boys; the suit of the eldest cost $15,167 ; of the second, $13,407 ; of the third, 12,75 ; and of the youngest, $11,047 ; how much did he pay in all ? 88. How do you set down the numbers for addition 1 How do you (idd up the columns 1 How do you place the sf-puratinif point ' How it! you j-Tove ;wldiliuii \ 1 UNITKI) STATES MOJi^FY. 85 6. A father has six children ; to the first two he gives each $375,416 ; to each of the second two, 'i5287,DO ; to each of the third two, $259,004 : how much did he give to them air^ 7. A man is indebted to A, $630,49 ; to B, $25 ; to C, 87^ cents; to D, 4 mills : how much does he owe i 8. Bought 1 gallon of molasses at 28 cents per gallon ; a Balf pound of tea for 78 cents ; a piece of flannel for 12 dol- lars 6 cents and 3 mills ; a plow for 8 dollars, 1 cent and 1 mill ; and a^pair of shoes lor 1 dollar and 20 cents : what did the whole cost ? 9. Bought 6 pounds of coffee for 1 dollar 12| cents ; a wash-tub for 75 cents 6 mills ; a tray for 26 cents 9 mills ; a broom 9^-' Zl cents ; a box of soap for 2 dollars 65 cents 7 miU* , -c cheese for 2 dollars 87-^ cents ; what is the whole amount " 10. "'•^^a'^ :s the entire cost of the following articles, viz. : 2 gallons f"^ molasses, 57 cents; half a pound of tea, 37^ cents ; 2 yards of broadcloth, |3,37^ cents; 8 yards of flan- nel, $9,875 ; two skeins of silk, 12^ cents, and 4 sticks of twist, 8A cents ? SUBTRACTION OF UNITED STATES MONEY. 1 . John gives ^ cents lor a pencil, and 5 cents for a top how much more does he give for the pencil than top? 2. A man buys a cow for $26,37, and a calf for $4,50 : how much more does he pay for the cow than calf? OPERATION. Note. — We set down the numbers as in addition, $26.37 and then subtract them as in simple numlers. 4,50 $21,87 89. Hence, for subtraction of United States money, we have the following Rule. — I. Write the leas number under the greater so that units of the same value shall fall in the same column. 89. How do you set down the numbers for subtraction 1 How do you subtract them 1 Where do you place the separating pjiut iu the ivuiiiijilcr ^ How do you provr i^uStr.irtinu ! 86 BUHTRACTK^N (»K II. Subtract as in simijle nvmbcrs^ and place the separating point in the remainder directly under that in the columns. Proof. — Tlie same as in simple numbers. EXAMPLES. (1.) (2.) From $204,679 From $8976,400 Take 98,714 Take 610.098 Remainder "|l05!^965 Remainder $8366^2 (3.) (4.) (5.) 1620,000 $327,001 $2349 19,0 21 2,090 29,33 $600;979 $324,911 $2319,67 6. What is the difference between $6 and 1 mill? Between $9,75 and 8 mills? Between 75 cents and 6 mills? Between $87,354 and 9 mills ? 7. From $107,003 take $0,479. 8. From $875,043 take $704,987. 9. From $904,273 take $859,896. APPLICATIONS. 1. A man's income is $3000 a year ; he spends $187,50 : how much does he lay up ? 2. A man purchased a yoke of oxen for $78, and a cow for $26,003 : how much more did he pay ibr the oxen than for the cow 1 3. A man buys a horse for $97,50, and gives a huxidred dollai bill : how much ought he to receive back 1 4. How much must be added to $60,039 to make the sum $1005,40? 5. A man sold his house for $3005, this sum being $98,039 more than he gave for it : what. did it cost him? 6. A man bought a pair of oxen for $100, and sold thcrn again for $75,37^ : did he make or lose by the bargain, and how much ? 7. A man starts on a journey with $100 ; he spends $67,57 : how much has he left? 8. How much iiiuFt you add to $40,173 U. make $100? UNITED STATES MONET. 87 9. A man purchased a pair of horses for f$450> but finding )ne of them injured, the seller agreed to deduct $106,325 : what had he to pay ? 10. A farmer had a horse worth $147,49, and traded him for a colt worth but $35,048 : how much should he receive in money ? 11. My house is worth $8975,034 ; my barn $695,879: what is the difierence of their values'? 12. What is the diilerence between nine hundred and sixty- nine dollars eighty cents and 1 mill, and thirty-six dollars ninety-nine cents and 9 mills ? MULTIPLICATION OF UNITED STATES MONEY. 1. John gives 3 cents apiece for 6 oranges : how much do they cost him ? 2. John ,buys 6 pairs of stockings, for which he pays 25 cents a pair : how much do they cost him ? 3. A farmer sells 8 sheep for $1,25 each: how much does he receive ibr them ? OPERATION. Analysis. — We multiply the cost of one sheep by $1,25 the number of sheep, and the product is the entire 8 ^^^' $10,00 90. Hence, for the multiplication of United States money by by an abstract number, we have the following Rule. — I. Write the money for the multiplicand, and the abstract number for (he tnaltijAier. II. Multiply as in simple numbers, and the product ivill be the answer in the lowest deyiomination of the midtv- plica nd. * III. Reduce the prroduct to dollars, cents and mills. Proof. — Same as in simple numbers. EXAMPLES. 1. Multiply 385 dollars, 28 cents and 2 mills, by 8. operation. (2.) $385,282 $475,87 8 9 Product $3082,250 Product $42S2,83 ^3 MUT.TIPTJCATIOX OF 3. What will 55 yards of cloth come to at 37 cents pei yard ? 4. What will 300 bushels of wheat come to at $1,25 per* bushel ? 5. What will 85 pounds of tea come to at I dollar Z7^ cents per pound ? 6. What will a firkin of butter containing 90 pounds ccme to at 25-J- cents per pound ? 7. What is the cost of a cask of wine containing 29 gal- lons, at 2 dollars and 75 cents per gallon ? 8. A bale of cloth contains 95 pieces, costing 40 dollars 37J cents each : what is the cost of the whole bale 1 9. W4iat is the cost of 300 hats at 3 dollars and 25 centfl apiece ? 10. What is the cost of 9704 oranges at 3^ cents apiece 1 OPERATION. Note. — We know that the product of two nura- 9704 bars contains tiie same number of uniis, whichever 034 be used as the multiplier (Art. 48). Hence, we 4 8 52~ may multiply 9704 by 3^ if we assign the proper „ ^ unit (1 cent) to the product. 8339,64 11. What will be the cost of 356 sheep at 3^ dollars a head I 12. What will be the cost of 47 barrels of apples at 1| dollars per barrel ? 13. What is the cost of a box of oranges containing 450, at 2-^ cents apiece ? 14. What is the cost of 307 yards of linen at 68^ cents per yard ? 15. What will be the co^t of 65 bushels of oats at 331 cents a bushel ? Analysis. — If the price were 1 dollar a bushel, operation, the cost would be as many dollars as there are 3)65,000 bushels. But the cost is 33i cents=i of a dollar: g2^r666^ hence, the cost will be as many dollars as 3 is con- ' ^ tained times in 65 = 21 dollars, and 2 dollars over, which is re- 90. How do you multiply United States money 1 What will he the denomination of the product^ How will you then reduce it to dollars 'jnd rents!? How do yoji prove nmHiiili<'Mion1 UWiniD STATE© MOiN-KY. 89 duced to cents by adding two ciphers, and to mills by adding three ; then, dividing the cents and mills by 3, we have the entire cost : hence, 91. To find the cost, when the price is an aliquot part of a dollar. Take such a part of the mimher which denotes the commo- dity , as the price is of 1 dollar. EXAMPLES. 1. What would be the cost of 345 pounds of tea at 50 cents a pound ? 2. What would 675 bushels of apples cost at 25 cents a bushel ? 3. If 1 pound of butter cost 12^ cents, what will 4 firkins cost, each weighing 56 pounds'? 4. At 20 cents a yard, what will 42 yards of cloth cost ? 5. At 33^ cents a gallon, what will 136 gallons of mo- lasses cost 1 OPERATION. 6. What will 1276 yds. 4)$1276 cost at 1 dollar a yard. of cloth cost at $1,25 a 319 cost at 25 cts. a yard, y^^^^ • $1595 cost at $1,25 a yard. 7. What would be the cost of 318 hats at ^1,121 apiece ? 8. What will 2479 bushels of wheat come to at $1,50 a bushel 1 9. At $1,33^ a foot, what will it cost to dig a well 78 feet deep? 1 0. What will be the cost of 936 feet of lumber at 3 dollars a hundred ? Analysis. — At 3 dollars a foot the cost would be operation. 936X3 = 2808 dollars; but as 3 dollars is the price 936 of 100 feet, it follows that 2808 dollars is 100 times 3 ilie cost of the lumber : therefore, if we divide ^^^^ ^^ '.808 dollars by 1 00 {which we do by cutting oflT two 5^^o,uo t»f the right hand figures (Art. 73), we shall obtain the cost. Note. — Had the price been so much per thousand, we should have divided by 1000, or sut off three of the right hand figures* ttence. 91. How do you fmd tlu? coft of several thitigs when the price is an aliquot part of u diAlarl 4 tfO' MULllPLICATiON Of 92. To find the cost of articles sold by the 100 or 1000 : Multiply the quantity by the jrrice ; and if the price be by the 100, cut off t-wo figures on the right hand of the product ; if by the 1000, cut off three, and the remaining figures will be the answer in the same denmnination as Ih/e price, which if cents or imlls, may be reduced to dollars. EXAMPLES. 1. What will 4280 bricks cost at $5 per 1000? 2. What will 2673 feet of timber cost at $2,25 per 100 ? 3. What will be the cost of 576 feet of boards at $10,62 per 1000? 4. What is the' value of 1200 feet of lathing at 7 dollars T 1000? 5. David Trusty, Bought of Peter Bigtree, 2462 feet of boards at $7, per 1000. 4520 " " 9,50 600 " scantling " 11,37 960 " timber « 15, 1464 " lathing ,75 per 100 1012 " plank " 1,25 Received Payment, Peter Bigtree, 6. What is the cost of 1684 pounds of hay at $10,50 pei ton! Analysis. — Since there are operation. 2000/6. in a ton, the cost of 2)10,50 1000/6 will be half as much as —^^ -^^ ^^ ^^^q^ for 1 ton: viz. $5.2o, or 525 iaqa cents. Multiply this by the ^"^^ number of pounds (1684), and $8,84100 Ans. cut off three places from the right, in addition to the two places before cut off for cents : henoo, 93. To find the cost of articles sold by the ton : Multiply one-half the price of a ton by the number of pounds, and cut off three figures from the right hand of the pi'oduct. The remaining figures will be the ansic&r in the same denomination as the price of a ton. 92. Huw do yuu find the co^t uf articles siAdhy tiio 100 or KMK) 7 UNriKD STATES MONEY. 91 EXAMPLES. 1. "WTiat will 3426 pounds of plaster cost at $3,4^ per ton 1 2. What will be the cost of" the transportation of" 6742 pounds of iron firom Buflalo to New York, at $7 per ton ? J. What will be the cost of 840 pounds of hay at $9,50 per ton'.' at $12 ? at $15,84? at $10,361 at $18,75? DIVISION OF UNITED STATES MONEY. 94. To divide a number expressed in dollars, cents or mills, into any number of equal parts. Rule. — I. Reduce the dividend to cents or mills, if necessary, II. Divide as in simple numbers, and the quotient will be the answer in ike lowest denomination of the dividend: this may be reduced to dollars^ cents, and mills. Proof. — Same as in division of simple numbers. Note. — The sign -f is added in the examples, to show that there is a remainder, and that the division may be continued. EXAMPLES. 1. Divide $4,624 by 4 ; also, $87,256 by 5. OPERATION. OPERATION. 4)$4,624 5)$87,256 $1,156 $17,451^ 2. Divide $37 by 8. Analvsis. — In this example we first reduce the oPERATinN. $37 to mills by annexing three ciphers. The quo- 8)$37,000 tient will then be mills, and can be reduced to dol- ^~X625 lars and cents, as before. ' 3. Divide 856,16 by 16. 4. Divide S495,704 by 129. 5. Divide $12 into 200 equal parts. 6. Divide $4 00 into 600 equal parts. 7 Divide $857 into 51 equal parts. 8. Divide $6578,95 into 157 equal parts. 93. How do you find the cost of articles sold by the ton 1 94. What is the rule for division of United States money ! Haw do you prove division 1 How do you indi<:aie ihr.t the divisjion maybe continue*]? y^ DlVIbJON OF 95. The quantity, and the cost of a quantity given, to find the price of unity (Art. 80). Divide the cost hy ike quantity, 9. Bought 9 pounds of tea for $5,85 ; what was the price per pound ? 10. Paid|;29,68 for 14 barrels of apples: what was the price per barrel % 11. If 27 bushels of potatoes cost $10,125, what is the price of a bushel % 12. If a man receive $29,25 for a rriunth's work, how much is that a day, allowing 26 working days to the month 1 13. A produce dealer bought 3 barrels of eggs, each con- taining 150 dozens, for which he paid $63 : how much did he pay a dozen ? 14. A man bought a piece of cloth containing 72 yards, for which he paid $252 : what did he pay per yard \ 15. If $600 be equally divided among 26 persons, what will be each one's share ? 16. Divide $18000 in 40 equal parts : what is the value of each part % 17. Divide $3769,25 into 50 equal parts : what is one part? 18. A farmer purchased a farm containing 725 acres, for which he paid $18306,25 : what did it cost him per acre % 19. A merchant buys 15 bales of goods at auction, for which he pays $1000 : what do they cost him per bale ? 20. A drover pays $1250 for 500 sheep ; what shall he sell them lor apiece, that he may neither make nor lose by the bargain % 21. The dairy of a farmer produces $600, and he has 25 cows : how much does he make by each cow ? 22. A farmer receives $840 for the wool of 1400 sheep : how much does each sheep produce him ? 23. A merchant buys a piece of goods containing 105 yards, for which he pays $262,50 ; he wishes to sell it so as to make $52,50 : how much must he ask per yard ? 96. When the price of unity and the cost of a quantity are given, to find the quantity (Art. 80). Note. — The divipor and dividend must both be redueca to tlie lowest iiiul niinicd in cither bc.'btc (liviE. — Mnltiply the highest denomination hy the vnifs of the scale which connect it ivilh the next lower., and add to the product the vnita of that denomination : proceed in the same manner ihrovyh all the dcno)ninatiun.v^ till the unit is brought to the required denomination. 105. To reduce from a lower unit to a higher. 1. Reduce 3138 farthings to pounds. OPERATION. Analysis. — Since 4 farthings 4)3138 make a pennv, we first divide by 4. . Since 12 pence make a shilling, we i^'.^" '^J^'^' ^^"^• next divide by 12. Since 20 shil- 2|0)6;5 - - Ad. rem. linss make a pound, we next divide 3 - - - 5^? rem by"'20, and find that 3138/a/-.=£3 .„— -T-.rT, 4^ 9 far Hence, to reduce from a lower to a higher denomination, we have the following Rule. — I. Divide the given number by the units of the scale 104. How do you reduce from a higher to a lower unit 1 105. How do you reduce from a lower to a higher unit 1 What will be the unit of any remainder 1 How do you prove reduction '' UKNOMINATE NUMBERS. 99 which connect it with the next higher denomination, and set down the remainder, if there be one, II. Divide the quotient thus obtained by the units of th( scale ivhich connect it with the next higher denornijiaiion, and set down the remainder. III. Proceed in the same way to the required denomination^ and the last quotient, with the several remainders annexed^ will be the answer. Note. — Every remainder will be of the same denomination as its dividend. Proof. — After a number has been reduced from a hif^ber denomination to a lower, by the first rule, let it be reduced back by the second ; and alter a number has been reduced from a lower denomination to a hipfher, by the second rule, let it be reduced back by the first rule. If the work is right, the results will agree. EXAMPLES. 1. Keduce £15 7^. 6d. to pence. OPERATION. PROOF. 15 12)36'JQ — ^FQ)30i7 . . . 6d. rem. 307 15 ... 7s. rem. 12 3G90 Ans. £15 75 6c/. 2. In £31 8.S'. 9d. ofar., how many farthings? Also proof 3. In £87 14s. 8^(1., how many farthings'? Also proof. 4. In £407 196'. life/., how many farthings'? Also proof. 5. In 80 guineas, how many pounds '? 6. In 1549/(/r , how many pounds, shillings and pence ? 7. In 6169 pence, how many pounds ? LINEAR MEASURE. 106. This measure is used to measure distances, lengths, breadths, heights aud depths, &c 106. For what is Linear Measure used 1 What are its denominations 1 Repeat the tahle. What is a fathom] What is a hand] What are the units of the scale. 100 KEDUCTIOiV OF TABLE, 12 inches make 3 feet - - - 51 yards or 161 feet - 40 rods - . - - 8 furlongs or 320 rods - 3 miles - - - - 691 statute miles (nearly) or ) 60 geographical miles, ) 360 degrees, ... in. ft. yd. 12 = 1 36 =3 =1 198 = 16^ = 5^ 7920 = 660 = 220 63360 = 5280 = 1760 foot, yard, rod, perch, or pole furlong, - mile, league, - degree of | ^^ 1 1 1 1 1 1 1 the equator, a circum'nce of the earth. rd. marked Ji. yd, rd. fur. inl. [] fun mi. = 1 = 40 = 320 Notes. A fathom is a length of six feet. = 1 = 8=1 and is generally used to measure the depth of water. 2. A band in 4 inches, used to measure the height of horses. 3. The units of the scale, in passing from inches to feet, are 12 ; in passing from feet to yards, 3 ; from yards to rods, 5-^ ; from rods to furlongs, 40 ; and from furlongs to miles, 8. 1. How many inches in 5 feet? In 10 feet 1 In 16 feet? 2. How many yards in 36 hetl In 54 feef? In 96 ? 3. How many feet in 144 inches'? In 96 inches? In 48 ? 4. How many furlongs in 3 miles 1 In 6 miles ? In 8 ? EXAMPLES. 1. How many inches 6rd. iyd. 2ft. '^in. in 2. In 1365 inches, how many rods? OPERATION. Grd. 4.1/d. 2ft. 9m. OPERATION. 12)1365 3)113 feet 9in. 3 34 37 yards. 3 5^)37 yards 2ft. 11)74 6rd. 8halfyds. = 42/d 113 feet. 12 Ans. Qrd. Ayd. 2ft. 9in. 1365 inches. DKNOMINATE NUMBERS. 101 Note. — When we reduce rods to yards, we multiply by the scale 5^ ; that is, we take 6 rods 6 and one-half times. When we reduce yards to rods, we divide by 5^, which is done by reducing the dividend and divisor to halves : the remainder is 8 half-yards, equal to 4 yards. 3. In 59mi. Ifur, 38rc?., how many feet? 4. In 115188 rods, how many miles'? 5. In 719mi. 16/-^. Qyd., how many feet] 6. In 118°, how many miles? 7. In -54° 45mi. Ifur. 20rd. 4:i/d. 2ft. 10m., how many inches ? 8. In 481401716 inches, how many degrees, &c. ? CLOTH MEASURE. 107. Cloth measure is used for measuring all kinds of cloth, ribbons, and other things sold by the yard. TABLE. 2\ inches, in I. make 1 nail, marked na* 4 nails - 1 quarter of a yard, qr. 3 quarters - 1 Ell Flemish," E.FL 4 quarters - 1 yard. - yd. 5 quarters - 1 Ell Enghsh, - E.E, in. na» qr. E.FL yd. E.E. ^ = 1 9 =4 = 1 27 =12 = 3 r= 1 36 = 16 = 4 — ^i = 1 45 = 20 = 5 — ^ = ii = 1 Note. — The units of the scale, in this and 5. measure, are 2^, 4, 3, 4 1. In 9 inches, how many nails ? Plow many nails in 1 yard ? In 2 yard Is? In 6? In 8 ? . 2. In 4 yards, how many quarters ? How many quarters in 8 yards ? In 7 how many 1 3. How many quarters in 12 nails ? In 16 nails? In 20 nails? In 36] In 40? ^07. For what is cloth measure used 1 What are its deuominatiuna \ Rf'Ueat the table. What are the units oi the scales \ 102 KLpLO'ilON OF liXAiMPLKS. 1. How many nails- are there in 35yd. Sgr^Sna. ] OPERATION. o5i/d, 3qr. ona. 4 143 quarters. 4 2. In 575 nails, how many yards ? OPERATION. 4)575 4) J 43 3na. 35 dqr. Ana. 35yd. 3qr. ona. 575 nails. In 49 E. E., how many nails? In 51 E. Ft., 2qr. 3na., how many nails? In 3278 naiis, how many yards ? In 340 nails, how many Ells Flemish] 7. In 4311 inches, how many E. E. ? SQUARE MEASURE. 108. Square measure is used in measuring land, or anything in which length and breadth are both considered. 1 Foot. A square is a figure bounded by four equal lines at right angles to each other. Each line is called a side of the square, if each side be one foot, the figure is called a '"' square foot. 1 yard = 3 feet If the sides of the square be each one yard, the square is called a square yard. In the large square there are nine small squares, the sides of which are each one foot. Therefore, the square yard contains 9 square leet. " The number of small squares that is contained in any large square is always equal to the product of two of the sides of I he large square. As in the figure, 3 x 3 = 9 square feet. The number of square incites contained in a square foot is equal to 1 2 X 1 2 ~ 1 44. 108. For what is Square Measure used 1 What is a square 1 If each side be one foot, what is it called ? If each side be a yard, what is it called 1 How many square foct does the square yard contain ? How IS the number of small squares contained in a large square found] Repeat tlie table. What are the units of the scale 1 CO II DENOMINATE 144 square inches, sq in. 9 square feet 30A square yards 40 square rods or perches 4 roods - 540 acres - fi- square yard, Sq yd square rod or perch, P. rood, - R. acre, - A. 1 square mile. M. Sq. in. 144 ' 1296 39204 1568160 6272640 Sq.ft. 1 9 2721 10890 Sq.yd. = 1 = 301 = 1210 = 43560 = 4840 = 1 40 160 R. = 1. Note. — The units of the scale are 144, 9, 30-^, 40, and 4. 106 1. How many square inches in 2 square feet ? How many square feet in 3 square yards ] How many in 6 ? In 8 ? 2. How many perches in 1 rood ? In 3 roods ? How many roods in 4 acres ? In 8 ] In 1 2 ? 3. How many perches in an acre? How many in 2 acres! How many square yards in 8 1 square feet ? SURVEYORS' MEASURE. 109. The Surveyor's or Gunter's chain is generally used in Burveying land. It is 4 poles or 66 feet in length, and ia divided into 100 links. TABLE. ^tVo ii^ches make 1 link, marked - - I. 1 chain, - - - c. 1 mile, - - - mi. 16 square rods or perches, P. 1 acre, - - ' A. Note — 1. Land is generally estimated in square miles, acres, roods, and square rods or perches. 2, The unitss of the scale are 7j^g-, 4, 80, 1, and 10. 4 rods or 66ft. 80 chains - 1 square chain - 10 square chains 109. What chain is used in land surveyinof 1 What is its length ? How is it divided I Repeat the table. In what is land generally esti- mated ? What are the units of the scale 1 104 RKDUCTK^N OF 1. How many rods m 1 chain ? How many in 4 ? In 5 f 2. How many chains in 1 mile ] In 2 miles 1 In 3 ] 3. How many perches in 1 square chain 1 In 4 ? In 6 ? 4. How many square chains in 2 acres ? How many perches in 3 acres ? In 5 ? In 6 1 EXAMPLES. 1 . How many perches in 32if. 25 A. Sli. 19P ? OPERATION. 'S2M. 25 A, 3E. 19P. 640 20505 acres. 4 82023 roods. 40 2. How many square miles, &c., in 3280939jP. ? OPERATION. 40 )328093 9 4 )8202 3 i9P. 640)20505 SB. 32 25^1. Ans. 32M. 25 A. ZJR. 19P. 3280939 perches. 3. In 19^. 2Ii. 31 P., how many square rods ? 4. In 175 square chains, how many square feet ? 5. In 37456 square inches, how many square feet ? 6. In 14972 perches, how many acres? 7. In 3674139 perches, how many square miles? 8. Mr. Wilson's farm contains 104^1. 3B. and 19P. ; he paid for it at the rate of 75 cents a perch : what did it cost? 9. The ibur walls oi' a rooni are each 25 feet in length and 9 feet in height and the ceiling is 25 feet square : how much will it cost to plaster it at 9 cents a square yard ? CUBIC MEASURE. 110. Cubic measure is used for measuring stone, timber earth, and such other things as have the three dimensions, length, breadth, and thickness. TABLE. 1728 cubic inches, Cu. in. make 1 cubic foot, Cu. ft. 27 cubic feet, - - - 1 cubic yard, Cu. yd 40 feet of round ^^ ) j |on - - T 50 fieet of hewn timber, ) 42 cubic feet - - - 1 ton of shipping, T. 16 cubic feet - " 1 cm d foot, - C. ft 8 cord feet or > _ j^^^j . ^ 125 «uhifi feet, \ II ill III III '^ DENOMINATE NUMUEK8. 105 Note. — 1. A cord of wood is a pile 4 feet wide, 4 feet high, and 8 feel loiig. , 2. A cord foot is 1 foot in length of the pile which makes a cord. 3. A CUBE is a figure bounded by six equal squares, called faces ; the sides of the squares are called edges. 4. A cubic foot is a cube, each of whose faces is a square foot; ts edges are each 1 foot. 5. A cubic yard is a cube, each of whose edges is 1 yard. '^ 6. The base of a cube is the face >. on which it stands. If the edge of r* the cube is one yard, it will contain ii 3X3 = 9 square feet j therefore, 9 ^ cubic feet can be placed on the base, n and hence, if the figure were 1 foot thick, it would contain 9 cubic feet; 3 feet=l yard. if it were 2 feet thick it would contain 2 tiers of cubes, or 18 cubic feet ; if it were 3 feet thick, it would contain 27 cubic feet ; hence, The contents of a figure of this form are found by miilti- l^lymg the length, breadth, and thickness together. 7. A ton of round timber^ when square, is supposed to produce 40 cubic feet ; hence, one-fifth is lost by squaring. 1. In 1 cubic foot, how many cubic inches ? How many in 2 1 In 3 ? 2. In 1 cubic yard, how many cubic feet ? How many in 2 '? In 41 In 6 ? 3. How many cord feet in 3 cords of wood ? In 5 ? In 6 ? 4. How many cubic feet in 2 cords 1 In half a cord, how many 1 How many in a quarter of a cordl 5. How many cubic yards in 54 cubic feet ? In 81 ? 6. In 120 feet of round timber, how many tons'? 7. How many tons of shipping in 84 cubic feet 1 In 168 ? 8. How many cords of wood in 64 cord lieeil In 96 ? In 128 ] 9. How many cubic feet in a stone 8 feet long, 3 feet wide and 2 feet thick 1 110. For what is cubic measure used ] What are its denominations ? >What is a cord of wooiH What is a cord foot 1 What is a cube? What is a cubic foot 1 What is a cubic yard 1 How many cubic feet in a cubic yard 1 What are the contents of a solid equal to ? Repeat the table. What are the units of the scale % 106 REDUCTION OF EXAMPLES. 1. In 15r?/. yd. 18c?/. ft. \^Kn. in., how many cubic inches ? cu. yt 15 27 113 31 OPERATION. I. cu.ft. CU. in, 18 16 423 X 1728 = 730960 cw.m, 2. In 730960 cubic inch OS, how many cubic yards, &c. 1 OPERATION. 1728 )730960 cu. in. "27)423" cu. ft. 15 cu, yd. cu. yd. cu.ft. cu. in. Ans. 15 18 16 3. How many small blocks 1 inch on each edge can be sawed out of a cube 7 feet on each edge, allowing no waste for sa\ving ? 4. In 25 cords of wood, how many cord feet 1 How many cubic feet 1 5. How many cords of wood in a pile 28 feet long, 4 feet wide, and 6 feet in height "? 6. In 174 904 cord feet, how many cords ? 7. In 7645900 cubic inches, how many tons of hewn timber ] WINE OR LIQUID MEASURE. 111. Wine measure is used for measuring all liquids except ale, beer, and milk. TABLE. 4 gills, yi, make 1 pint, marked pi. 2 pints 1 quart. ■ - qt. 4 quarts 1 gallon, gal. 31^ gallons 1 barrel, bar. or bbl. 42 gallons 1 tierce. tier. 63 gallons • 1 hogshead hhd. 2 hogsheads 1 pipe. pi. 2 pipes or 4 hogsheads 1 tun, - tun. 111. What is measurrd by wine or liquid measured What are its denominations ! Repeat the table. What are the units of the scale ? VV'hat is the standard wine gallon ? DKMOMINATE NUM23EKS. 107 gl. pt. qt. gal. b ir. tier. hhd. pi 4 =1 8 =2 = 1 32 =8 = 4 = 1 1008 =252 = 126 = 311 = 1 1344 =336 = 168 = 42 — 1 2016 =504 = 252 = 63 = \\ = 1 4032 =1008 = 504 = 126 = 3 = 2 = 1 8064 =2016 = 1008 = 252 = 6 = 4 = 2 1 NoTE.-^The standard nnit, or gallon of liquid measure, in the United States, contains 231 cubic inches. 1. How many gills in 4 pints'? How many pints in 3 quarts ? In 6 quarts ? In 9 ? In 10 ? 2. How many quarts in 2 gallons ? In 4 gallons ? In 6 gallons? How many pints in 2 gallons'? In 5 ? 3. How many barrels in a hogshead 1 How many in 4 hogsheads 1 In 6 1 4. How many quarts in 3 gallons 1 In 5 gallons? In 20? In a barrel how many ? In a hogshead how many ? EXAMPLES. 1 . In 5 tuns 3 hogs 17 gallons of wine, many gallons ? heads how 2. In 1466 many tuns, 6zc gallons, how . ? OPERATION. 5tuns 3hhd . 17 gal. OPERATION. 63jl466 4 23 63 4)23 5 17 gal. "3 hhd. 76 139 Ans. 5tun. 3hhd. 17 gal 1466 gallons. 3. In 1 2 pipes 1 hogshead and 1 quart of wine, how many pints ? 4. In 10584 quarts of wine, how many tuns ? 5. In 201632 gills, how many tuns? 6. What will be the cost of 3 hogsheads, 1 barrel, 8 gal- lons, and 2 quarts of vinegar, at 4 cents a quart ? 103 REDUOTIOW OF ALE OR BEER MEASURE. 112. Ale or Beer Measure is used for measuring ale, beer, and milk. TABLE. make 1 2 pints, pt. 4 quarts 36 gallons 54 gallons pt. 2 = 8 z= 288 = 482 = qt. 1 4 144 216 quart, gallon, barrel, - hogshead, gal. bar. marked qt. gal. bar. hhd. hhd. 1 36 54 Note. = 1 •1 gallon contains 282 cubic inches. 1. How many pints in 3 quarts? How many in 5 ? 2. How many quarts in 3 gallons? In 4 gallons? In 91 EXAMPLES. 1. How many quarts are there in Vihd. 2bar. 29gal. 3qt. ? OPERATION. 2. In 1271 quarts, how many hogsheads, &c. ? OPERATION. Ahhd. 2har. 29gal. 3qt. H 4)1271 2 4 4 36)317 3qt. 11)8 29gal. 8bar. 4 2bar. 36 / 57 26 Am. 4hhd. 2bar. 2gal. 2qt, 3\7gal. 4 1271.7/s. 3. In 47 bar. 4. In 27 hhd 5. In 55832 6. In 64972 IQgal 4qt., he . 3bar. 26gal. : pints, how mar quarts, how m^ )w many pints ? ^qt., how many pints? ly hogsheads? my barrels ? 112. For what is ale or beer measure usedl What uatioiis ] Rcj>cat the table What are the scaler "i its denuiiii- DENOMINATE NUMBERS. 109 DRY MEASURE. 113. Dry Measure is used in measuring all dry articles, such as ffrain, fruit, salt, coal, &:c. 2 pints, pL. 8 quarts - 4 pecks 36 bushels pt.' 2 16 TABLE. make 1 quart, marked 1 peck, - 1 bushel, 1 chaldron, pk. qt. pk. bu. ch. bu. ch. 64 2304 = 1. = 1 = 36 In 5 ] In 8 ? In 32? In 64? In 8? In 12? How In 40? qt z 1 = 8 =1 z 32 =4 = 1152 = 144 1. How many quarts in 2 pecks ? 2. How many pecks in 24 quarts 1 3. How many pecks in 6 bushels ? many bushels in 16 pecks ? In 32 ? 4. How many bushels in 2 chaldrons ? In 3 ? In 4 ? Note. — The standard bushel of the United States is the Win- chester bushel of England. It is a circular measure. 18-^ inches in diameter and 8 inches deep, and contains 21 50| cubic inches, nearly. 2. A gallon, dry measure, contains 268^ cubic inches. EXAMPLES. 1 . How many quarts are there in Qoch. 20bu. opk. Iqt. ? OPERATION. ^5ch. 20bu. 3pk. Iqt. 36 390 197 "2360 4 9443 7o551 quarts. &c. How many chaldrons, in 75551 quarts ? OPERATION. 8)75551 4)9443 36)2360 65 Iqt. 3pk. 2Qbu. Ans. 65ch. 20bu. opk. Iqt 113. What articles are measured by dry measure 1 What are its denominations 1 Repeat the table. What are the scales 1 What is the standard busliel ! What are the contents of a gallon \ 8 no KKDUCTION OF 3. In 372 bushels, how many pints 1 4. In 6 chaldrons 31 bushels, how many pecks? 5. Ill J 7408 pints, how many bushels? 6. In 4220 pints, how many chaldrons ? AVOIRDUPOIS WEIGHT. 114. By this weight all coarse articles are weighed, such as hay. grain, chandlers' wares, and all metals except gold and silver. TABLE. 16 drams, dr. make 1 ounce, marked oz, 16 ounces - - 1 pound, - lb. 25 pounds - - 1 quarter, - qr, 4 quarters - 1 hundred weight, cwt. 20 hundred weight 1 ton, dr. oz. lb, 16 = 1 256 =16 =1 6400 = 400 = 25 25600 = 1600 = 100 512000 = 32000 = 2000 =80 =20 =1 Notes. — 1. The standard avoirdupois pound is the weight of 27.7015 cubic inches of distilled water. 2. By the old method of weighing, adopted from the English system, 112 pounds were reckoned for a hundred weight. But now, the laws of most of the States, as well as general usage, fix the hundred weight at 100 pounds. 3. The units of the scale, in passing from drams to ounces, are 16; from ounces to pounds, 16; from pounds to quarters, 25; from quarters to hundreds, 4; and from hundreds to tons, 20. 1. In 2oz., how many drams ? In 3 ? In 4 ? In 5 ? 2. In 4/6., how many ounces'? In 3 how many ? In 2 I 3. In Qtqr., how many hundred weight ] In 5qr.'\ 4. In Zc'wt., how many quarters ? How many in Acwt. ? 5. In 60 hundred weight, how many tons ] In 80 ? 114. For what is avoirdupois weijiht used] How is /he table to b* read 1 How can you detennhie, from the second table, the \alue ol any unit in units of the lower denominations ? ., - T. qr. cwt. = 1 = 4 . = 80 = 20 DKNOMINATK NUMUIORS. Ill EXAMPLES. 1. How many pounds are there in 15T. ScwL 3qr. X6lb. ? OPERATION. 15T. 8cwL 3qr. 15/6. _20 SOScwt. 4' 1235 qr. 25 6180 2471 30b90 lb. 5 lb. added. 1 ten added. 2. In 30890 pounds, hew many tons ? OPERATION. 25)30890 4)1235 qr. i5lb. Sqr. 15 T. Scwt, 20)308 cwt 15 T. SctvL 3qr. 15Ib 3. In 5T. Scwt. 3qr. 2Hb. \3oz. 14^/r., how many drams? 4. In 287^. Acwt. Iqr. 21/6., how many ounces'? 5. In 27903G6 drams, how many tons ? G. In 903 136 ounces, how many tons? 7. In 3124446 drams, how many tons? 8. In 93 7'. \3"wt. 3qr. 8lb., how many ounces? 9. In 108910592 drams, how many tons'? 10. What will be the cost of 117\ l7cwL 3qr. 24/6. of hay at half a cent a pound ? How much would that be a ton 1 11. What is the cost of 2T. \3cwt. 3qr. 21/6. of beef at 8 cents a pound ? How much would that be a ton ? TROY WEIGHT. 115. Gold, silver, jewels, and liquors, are weighed by Troy weight. TABLE. 24 grains, gr. make 1 pennyweight, marked pwt. 20 pennyweights - 1 ounce - - - oz. 12 ounces - - 1 pound - • -lb, gr. pwt. oz. lb. 24 = 1 480 =20 =1 5760 =240 =12 =1 1V2 REDUCTION OF Notes. — 1. The standard Troy pound is the weight of 22.794377 cubic inches of distilled water. It is less than the pound avoirdupois. 2. The units of the scale, in passing from grains to penny- weights, are 24 ; from pennyweights to ounces, 20 ; and from ounces to po\inds, 12. 1. How many grains in 2 pennyweights ? In 3 ? In 41 2. How many pennyweights in 48 grains? In 72 1 3. How many ounces in 40 pennyweiglits ? In 60 ? 4. How many ounces in 4 pounds'? In 12 ? In 9 ? In 7 ? 5. How many pounds in 24 ounces 1 In 36 ? In 96 1 EXAMPLES. 1 . How many grains are therein IQlb. lloz. 15j)wt. OPERATION. 16/6. lloz. 12 15pwt. 203 20 ounces. 4075 pennyweights. 24 97817 grains. 2. In 97817 grains, many pounds ? how OPERATION, 24 )07817 20)4075 pwt. 12)2203 02. ■ 16 Z6. 1 6pwt^ lloz. Ans. IQib. lloz. Idj^wt. 11 gT 3. In 25lb. 9oz. 20gr., how many grains ? 4. In 6490 grains, how many pounds 1 6. In 148340 grains, bow many pounds? 6. In 11716. 9oz. I5pwt. ISgr., how many grains? 7. In 8794 pwt., how many pounds 1 8. In 6Ib. 9oz. 21 gr. how many grains 1 9. In lib. loz. lOj^wt. 16«-r., how many grains ? 10. A jewel weighing 2oz. lApwt. I8gr., is sold for half a dollar a grain : what is its value 1 Notes. 1. — What is the standard avoirdupois pound 1 2. — What is a hundred weight by the English method! What is a hundred weight by the United States method ] 3. Name the units of the scale in passing from one denomination to another. 115. What articles are weighed by Troy weight 1 What are its de- nominations ] Repeat the table. W'hat is the standard Troy pound 1 Wliut are the units of the scale, in passiui; from one unit to another 1 DENOMINATE NUMBERS. APOTHECARIES' WEIGHT. 118 116. This weight is used by apothecaries and physicians in mixing their medicines. But medicines are generally sold, in the quantity, by avoirdupois weight. TABLE. 20 grains, gi. make 1 scruple, marked 9. 3 scruples - ■ 1 dram, - - - 5. 8 drams - • 1 ounce, - - - !. 12 ounces - • 1 pound, • - • fc. gr. B 3 5 20 = 1 60 =3 = 1 480 = 24 = 8 =1 6760 = 288 =96 =12 = 1 Notes. — 1. The pound and ouikce are ihe same as the pound and ounce in Troy weight. 2. The units of the scale, in passing from grains to scruples, are 20 ; in passing from scruples to drams, 3 ; from drams to ounces, 8 ; and from ounces to pounds, 12. 1. How many grains in 2 scruples 1 In 3 ? In 4 *? In 6 ? 2. How many scruples in 4 drams 1 In 7 drams 1 In 51 3. How many drams in 5 ounces ? How many ounces in 32 drams? EXAMPLES. 1. How many grains in 9fe 8 5 6 3 2 9 12^r. OPERATION. 9fe 8! 6 3 2 9 12^r. 12 116 ounces. 8 934 scruples. _^ *\^04 drams. 20_ 600^.2 grains. 2. In 56092 grains, how many pounds ? OPERATION. 20)56092 3)2804 9 12^r. 8)934 3 29 12)116! 65 9fe 8! Ans. 9fe 8 5 6 3 2 9 I2gr. lU REDUCTION OF In271fe95 63 J9, how many scruples ? In 94)fe 115 13, how many drams] 8011 scruples, how many pounds? In 9113 drams, how many pounds'? How many grains inl2}fe 9 5 73 29 18^r. ? In 73918 grains, how many pounds ? MEASURE OF TIME. 117. Time is a part of duration. The time in which the earth revolves on its axis is called a day. The time -in which it goes round the sun is 365 days and 6 hours, and is called a year. Time is divided into parts according to the following TABLE. m. hr. da, wk. mo, y^' yr. yr = 1 Notes. — 1. The years are numbered from the beginning of the Christian Era. The year is divided into 12 calendar months, numbered from January : the days are numbered from the begin ninii of the month : hours from 12 at night and 12 at noon. 60 seconds, sec. make 1 minute, marked 60 minutes . 1 hour, 24 hours - . 1 day, - 7 days . 1 week. 4 weeks - . 1 month, iomo. Ida. and Ohrs. or 365da. G/ir. t 1 Julian year. 12 calendar months • 1 year, sec. m. hr. da. wk. CO = 1 3600 = 60 = 1 86400 = 1440 — 24 = 1 * 604800 = 10080 — 168 = 7 = 1 31557600 = 525960 = 8766 = 3651 = 52 Names. No. January, - - 1st. February, - - 2d. March, - - . 3d. April, . - May, - - June, - - . 4lh. . 5th. . 6th. No. days. - - 31 - - 28 - - 31 - - 30 - - 31 - - 30 Names, No. No. days July, - - - 7th. - - 31 August, - - 8th. - - 31 September, - 9Lh. - - 30 October, - . 10th. - - 31 November - lUh. - - 30 December, - 12th. - - 31 DENOMINATE NUMBERS. 115 2. The length of the tropical year is 365d. ohr. 48m. ASscc. n*^arly; but in the examples we shall regard it as 365d. 6hr. 3. Since the length of the year is 365 days and 6 hours, the o(W 6 hoais, by accumulating for 4 years, make 1 day, so that every fourth year contains 366 days. This is called Bissextile or Leap Year The leap years are exactly divisible by 4 : 1852, 1856, 1860, Rre leap years. 4. The additional day, when it occurs, is added to the month of February, so that this month has 29 days in the leap year. Ihirty days hath September, April, June, and November; All the rest have thirty-one, Excepting February, twenty-eight alone. 1. How many seconds in 4 minutes'^ How many in 6? 2. How many hours in 3 days ? How many in 5 ? In 3 ? 3. How many days in 6 w^eeks ? In 8, how many ? 4. How many hours in 1 week ] How many weeks in 4:2da, 1 1. How many seconds in 365da. 6hr. ? OPERATION. 365c/a. 6hr, 24 EXAMPLES, 2. How many days, &c. in 31557600 seconds ? OPERATION. 60)31557600 1466 730 8766 60 525960 X60=:31557600sec. 60)525960 24 )8766 365" 6Ar. Ans. 365da. ijkr. 3. If the length of the year were 365cfa, 237^r. 57m, 39sec how many seconds would there be in 12 years'? 4. In 126230400 seconds, how many years of 365 days'? 5. In 756952018 seconds, how many years of 365 days'? 117. What are the denominations of time? How long is a year? How many days in a common year1 How many days in a Leap year ! How njany calendar months in a year ? Name them, and the number of days in each. How many days has February in the leap year! How J>» \o\i Tfineiiilifr whicfi of ihu nunitlis have !M) dayp, and which DENOMINA'] IER6. MISCELLANEOI 12 units, or things 12 dozen - - - 12 gross, or 144 dozen 20 things - 100 pounds - 196 pounds - 200 pounds - 18 inches - - - 22 inches, nearly 1 4 pounds of iron or lead 21 i stones - - - make 8 pigs i^J^dnf^ 1 great gioss. 1 score. 1 quintal of fish, 1 barrel of flour. 1 barrel of pork. 1 cubit. 1 sacred cubit. 1 stone. 1 pig- 1 Ibther. BOOKS AND PAPER. The terms, folio^ quarto^ octavo^ duodecimo, (fee, indicate the number ol" leaves into which a sheet of paper is folded. A sheet folded in 2 leaves is called a folio. A sheet folded in 4 leaves A sheet folded in 8 leaves A sheet folded in 12 leaves A sheet folded in 16 leaves A sheet Iblded in 18 leaves A sheet folded in 24 leaves A sheet folded in 32 leaves 24 sheets of paper 20 quires 2 reams 6 bundles MISCELLANEOUS EXAMPLES. 1. How many hours in 344tf^. &da. Mhr.l 2. In 6 signs, how many minutes'? 3. In 15 tons of hewn timber, how many cubic inches? 4. In 171360 pence, how many pounds'? 5. In 1720320 drams, how many tons? 6. In 65799 grains of laudanum, how many pounds? 7. In 9739 grains, how many pounds Troy? 8. In 59lb \3pwt. 5gr., how many grains? 9. In £85 85., how many guineas ? U). In 346 ii. F., liow many Ell? English'? " a quarto, or 4to. " an octavo, or 8vo. " a 12mo. " a 16mo. " an 18mo. " a 24mo. " a 32mo. ce 1 quire 1 ream. 1 bundle. 1 bale. 118 KEDITCTTON OF 11. In 'dhhd. ISgal. 2gt., how many half-pints ? 12. In 12T. \5civt. \qr. 19lb. I2dr', how many drams? 13. In 40144896 square inches, how many acres 1 14. In 5760 grains, how many pomids ? 15. In 6 years (of 52 weeks each), o2wk. 6da. 17/*/-., how nany hours? 16 In 811480'', how many signs ? 17 In 2654208 cubic inches, how many cords'? 16. In 18 tons of round timber, how many cubic inches? 19. In 84 chaldrons of coal, how many pecks 1 20. In 302 ells Enghsh, how many yards? 21. In 24:hhd. 18gal. 2(]L of molasses, how many gills? 22. In 76-4. \R. 8P., how many square inches ? 23. In £15 195. 11^. 2>far., how many farthings? 24. In 445577 feet, how many miles ? 25. In 37444325 square inches, how many acres? 26. If the entire surfoce of the earth is found to contain 791300159907840000 square inches, how many square miles are there ? 27. How many times will a wheel 16 feet and 6 inches in circumference, turn round in a distance of 84 miles ? 28. What will 28 rods, 129 square feet of land cost at $12 a square foot ? 29. What will be the cost of a pile of wood 36 feet long, 6 feet high and 4 feet wide, at 50 cents a cord foot ? 30. A man has a journey to perform of 288 miles. He travels the distance in 12 days, travelling 6 hours each day : at what rate does he travel per hour ? 31. How many yards of carpeting 1 yard wide, will carpet a room 18 feet by 20? 32. If the number of inhabitants in the United States is 24 millions, how long will it take a person to count them, counting at the rate of 100 a minute ? 33. A merchant wishes to bottle a cask of wine containing 126 gallons, in bottles containing 1 pint each : how many bottles are necessary ? 34. There is a cube, or square piece of wood, 4 feet each way : how many small cubes of 1 inch each way, can be sawed from it, allowing no waste in sawing ? 35. A merchant wishes to ship 285 bushels of flax-seed in casks containing 7 bushels 2 pecks each : what number of (!.!ifel<* MTO leqiiin"! ^ DKKOMINATE NUMBEKS. 119 36. How many times will the wheel of a car, 10 feet and 6 inches in circumference, turn round in going from Hartlbrd to New Haven, a distance of 34 miles ? 37. How many seconds old is a man who has lived 32 years and 40 days '? 38 There are 15713280 inches in the distance from N't w York to Boston, how many miles ? 39. What will be the cost of 3 loads of hay, each weighing ISctvt. 3qr. 24/6., at 7 mills a pound? AbDITION OF DENOMINATE NUMBERS. 119. Addition of denominate numbers is the operation oi finding a single number equivalent in value to two or more given numbers. Such single number is called the sum. How many pounds, shillings, and pence in X4 8s. 9d.f £27 145. Ik/., and Xl56 175. 10^/.? Analysis. — We write the units of the same operation. Qanie in the same column. Add the column £, g, cl, of pence ; then 30 pence are equal to 2 siiil- 4 8 9 lings and 6 pence : write down the 6, carrying oj 14 11 the tM'o to the shillings. Find the sum or' the ^Tp ^- ^ p. shillings, which is 41 ; that is. 2 pounds and 1 shilling over. Write down Is. ; then, carrying Xl89 Is. (JcL the 2 to flie column of pounds, we find the Bum to be £189 l5. 6d. Note. — In simple numbers, the number of units of the scale, at any place, is always 10. Hence, we carry 1 for every 10. In denominate numbers, the scale varies. The number of units, in pafssing from pence to shillings, is 12; hence, we carry one for every 12. In passing from shillingstopounds.it is 20 ; hence, we carry one for every' 20. In passing from one denomination to another, we carry 1 for so many units as are contained in the scale at that place. Hence, for the addition of denominate numbers, we have the following Rule. — I. Set down the riumbers so that units of Ji^ same name shall stand in the same column ; II. Add as in simple numheis, and carry from one de- nomination to another according to the scale. Proof. — The same as in simple numbers. 119. What is addition of denominate numbers'? How do you set down the immbers for addition 1 How do you adJI How do you pi uve addlliuu 1 L5S0 ADDITION OK , EXAMPLES. , ( 1) ( ;2.) ( S.) £ s. d. £ s. d. £ .. d 173 13 5 705 17 31 104 18 9. 87 17 7| 354 17 2f 404 17 8. 11 10-- 14 4 75 18 7^ 175 17 3J 467 25 17 8-i- 87 19 7J 12 7| 597 10 10 lOl 52 22 18 5 373 18 3 18 6 5 TROY WEIGHT. (4.) (5.) lb. OS. JTWt. gr. ( lb oz. ^z/;^. gr. KM IOC 1 10 19 20 171 6 13 14 43k I 6 5 391 11 9 12 8C 1 3 2 1 230 6 6 13 ^ 9 94 7 3 18 ) 11 10 23 42 10 15 20 1 8 9 ARIES' WEK 31 21 APOTHEC GtHT. 1 (6.) (7.) (8.) % 5 5 9 ^^. ! 3 9 gr^ 3 9 gf 24 7 2 1 1€ » 11 2 1 17 3 2 15 17 11 7 2 1£ 1 7 4 2 14 1 13 36 6 5 7 r 4 1 19 2 2 11 15 9 7 1 12 2 5 2 11 7 17 9 3 4 1 « 1 10 1 2 16 >IRDUP01S WEIGHT.' 5 2 14 AVC (9-) (10 .) AVt . qr. lb. 02. dr. T. cwt. qr . lb. oz. 14 2 14 9 15 12 1 10 10 13 2 20 1 15 71 8 2 6 9 3 6 7 3 83 19 3 15 5 10 18 12 11 36 7 20 14 7 3 2 3 2 47 11 2 2 11 6 1 19 8 1 63 5 2 19 7 4 3 15 5 12 13 1 14 9 12 2 13 9 7 5 10 DEMOMlNATJi NUilBiaiS. 121 11. A merchant bought 4 barrels of potash of the following wei|;hts, viz. : 1st, Zcwt. 2qr. Olb. \2oz. Mr. ; 2d, ^civt. \qr, 21/6. 4.0Z. ; 3d, Acwt. ; 4th, ^cwt. Oqr. 2lb. l5oz, IGdr '. what was the entire weight of the four barrels ? LONG MEASURI (12 X. mi. fur 16 2 7 rd. yd. ft, 39 9 2 (13.) r^Z. ^cZ. ft. in. 16 9 2 11 327' 1 2 20 7 1 12 11 1 9 87 1 15 6 1 18 14 7 1 1 1 1 2 2 EASIJR 19 15 2 1 CLOTH M E. (14.) E. Fl. qr, na. 126 4 4 (15.) yd. qr. 4 3 na. 2 ^. 11. qr. na. m. 128 5 1 3 65 3 1 5 4 1 20 3 1 2 72 1 ^3 6 1 19 1 4 1 157 2 3 25 2 2 15 3 1 2 LAND OR SQUARE MEASURE. (17.) Sq.yd. Sq.ft. 97 4 Sq. in. 104 M. 2 (18.) A. R. P. Sq. yd. 60-3 37 25 22 3 27 6 375 2 25 21 105 8 2 7 450 1 31 20 37 7 127 11 30 25 19 19. There are 4 fields, the 1st contains 12^. 2R. 38P. ; ihe 2d, 4A. IR. 26P. ; the 3d, 85^. OR. 19P. ; and the 4th, 51 A. IR. 2 P. : how many acres in the four fields'? CUBIC MEASURE. (20.) (21.) (22.) Cu, yd. Cu.ft. Cu. in. C. S.ft. C. Cwdfeet 65 25 1129 16 127 87 9 37 26 132 17 12 26 7 50 1 1064 18 119 16 6 22 19 17 37 104 19 6 122 AIJDITION OF WINE OR LIQUID MEASURE. ... (?^-) (^^-^ kha. gal. qU pt. tun. pi. hhd. gal. qt 127 65 3 2 14 2 1 27 3 12 60 2 3 15 1 2 25 2 450 29 1 4 2 1 27 1 2J L 2 3 5 1 62 3 14 39 1 2 MEAJ 7 1 2 21 2 DRY >URE. ch. 27 (25.) hu. pk. qt.pt. 25 3 7 1 ch. 141 (26.) hu. pk. qt.pt. 36 3 7 2 69 21 2 6 3 21 32 2 4 1 2 12 7 1 85 9 10 3 5 9 18 2 TIME. 10 4 4 13 yr. (27.) mo. wk. da. hr. wk. da. (28.) hr. m. sec. 4 11 3 6 20 8 8 14 55 57 3 10 2 5 21 10 7 23 57 49 5 8 1 4 19 20 6 14 42 01 101 9 3 7 23 6 5 23 19 59 55 8 4 6 17 CIRCULAR MEASUR 2 2 20 45 48 E OR M OTION. 8. (29.) o / ti s. (30.) O / ft 5 17 36 29 6 29 27 49 7 25 41 21 8 18 29 16 8 15 16 09 7 09 04 58 Note. — Since 12 signs make a circumference of a circle, we write down only the excess over exact 12's. APPLICATIONS IN AIDITION. 1. Add 46lh. 9oz. 15/;w;^ legr., 87lb. lOoz. 6pwt. 14^r., lOUM. lOa-ir. lOpwt. lOyr., and C)UO opwt. G//r loncther. DENOMINATE NUMUJCltS. 123 2. "What is the weight of forty-six pounds, eight ounces, thirteen pennyweights, fourteen grains ; ninety-seven pounds, three ounces ; and one hundred pounds, five ounces, ten pen- nyweights and thirteen grains ? 3. Add the following together: 29 T. 16ctoL Iqr. Ulb, VZoz. Mr., \Scwt. Sqr. lib., 50 T. 3qr. 4:0Z., and 2T. Iqr. Udr. 4. What is the weight of 39 7". lOcwt. 2qr. 2lb. 15o2. 12dr^^ \lcwt. Qlb., 12civt. 3qr., and 2qr, 8lb. 9dr. ? 5. What is the sura of the following : SUA. 2R. 39P. 20e«^. ft. 136.vy. in., 16^. IR. 20P. lOsq. ft., SB. 36P. and 4: A. IE. 16P. ? 6. What is the solid content of 64ton SSft. 800m., 9ion 1200m., 25ft. 700m., and 95lon Sift. 1500m. 7. Add together, 96bu. Spk. 2qt. Ipt., 466m. Spk. \qt. Ipt., 2pk. \qt. Ipt. and 2Sbu. Spk. 4qt. Ipt. 8. What is the area of the four following pieces of land ; the first containing 20^. SB. 15P. 250sq.ft. llQsq. in. ; the second, 19 A IB. S9P. ; the third, 2B. lOP. 60sq.ft. ; and the fourth, 5A. 6P. 50sq. in. 1 9. A farmer raised from one field S7bu. \ph. Sqt. of wheat ; from a second, 416m. 2/>'A-. 5qt. of barley ; from a third, 356/^. Ijik. Sqt. of rye ; from a fourth, 436m. Spk. \qt. of oats ; how much grain did he raise in all ? 10. A grocer received an invoice of Ahhd. of sugar; the first weighed llcwt. 15/6. ; the second, ]2cwt. Sqr. 15/6. ; the third, 9ciot. Iqr. 16/6. ; the fourth, 12cm;/. Iqr. : how much did the four weigh ? 11. A lady purchased 32 «/o?5. 3$'r6\ of sheeting ; Slpds. Iqr. of shirting ; li^/ds 2qrs. of linen ; and (5yds. 2qrs. of cambric: what was the whole number of yards purchased ? 12. Purchased a silver teapot weighing 2302. \lpwt. llgr, ; a sugar bowl, weighing 802. ISpiut. 19 gr. ; a cream pitcher, weighing 5oz. llgr. : what was the weight of the whole? . 13. A stage goes one day, 87?7i. 6far. 24r{/. ; the next, 75w Sfiir. llrd.) the third, SOm. Ifar. \Qrd. ; the fourth, 78 w. 5fii.r. : how far does it go in the four days ? 14. Bought thr^e pieces of land ; the first contained 17 uc?'€s IB. S5rd. ; the second, 36 acres 2B. 21rd. ; and the tliiid. Ajide, 125. The sun apparently goes round the earth once in 24 hours. This time is called a day. Hence, in 24 hours, the sun apparently passes over 360° of longitude ; and in 1 hour over 360°-^-24=: 15°. 126. Since the sun, in passing over 15° of longitude, re- quires 1 hour or 60' of time, 1° v^'ill require 60'-^15=z:4 minutes of time ; and V of longitude will be equal to one sixtieth of 4' which is 4" : hence, 15° of longitude require 1 hour. 1° of longitude requires 4 minutes. 1 ' ol' longitude requires 4 seconds. Hence, we see that, 1. If the degrees (f longitude he multiplied by 4, the irro- duct will be the corresponding time in minutes. 2. If ilie minutes m longitude be ?nultiplied by 4, the j^ro- duel will be the corresponding time in seconds. 127. When the sun is on the meridian of any place, it ia 12 o'clock, or noon, at that place. Now, as the sun apparently goes from east to west, at the instant of noon, it will be past noon for all places at the east, and before noon for all places at the west. If then, we find the difierence of time between two places and know the exact time at one of them, the corresponding time at the other will be found by adding their difierence, if that other be east, or by subtracting it lowest. 124. How is the circumference of the earth supposed to be divided ] 125. How does the sun appear to move ! What is a day 1 How far does the sun appear to move in 1 hour \ 126. How do you reduce degrees of longitude to timel How do you reduce minutes of longitude to time "^ 127. What is the hour when the sun is on the meridian ■; When the sun is on the meridian of any place, how will the time be for all places east ! How for all places west ! If you have the difl'erence of time how do you find the time'' Longitude and timk. 135 1. The longitude of New York is 74° 1' west, and that of Philadelphia 75° 10' west: what is the difference of longi- tude and what their difference of timel 2. At 12 M. at Philadelphia, what is the time at New York ? 3. At 1 2 M. at New York, what is the time at Philadelphia ? 4. The longitude of Cincinnati, Ohio, is 84° 24' Avest : what is the diiierence of time between New York and Cin- cinnati ? t). What is the time at Cincinnati, when it is 12 o'clock at New York 1 6. The longitude of New Orleans is 89° 2' west : what time is it at New Orleans, when it is 12 M. at New York ? 7. The meridian from which the longitudes are reckoned passes through the Greenwich Observatory, London : hence, the longitude of that place is : what is the difference of time between Greenwich and New York 1 8. What is the time at Greenwich, when it is 12 M. at New York ? 9. The longitude of St. Louis is 90° 15' west : what is the time at St. Louis, when it i^ 3k. 25' P. M. at New York ? 10. The longitude of Boston is 71° 4' west, and that of New Orleans 89° 2' west : what is the time at New Orleans when it is 7 o'clock 12' A. M. at Boston ? 11. The longitude of Chicago, Illinois, is 87° 30' west • what is the time at Chicago, when it is 12 M. at New York ? PROPERTIES OF NUMBERS. COMPOSITE AND PRIME NUMBERS. 128. An Integer, or whole number, is a unit ©r a collection of units. 129. One number is said to be divisible by another, when the quotient arising from the division is a whc^e number. The division is then said to be exact. Note — Since every number is divisible by itself and 1, the term divisible will be applied to such numbers orzZi/, as have other divisors. 128. What is an uiteger 1 lob PKOPEKTIES OF NUMBEKS. 130. Every divisible number is called a comjoosite nunibej* (Art. 54), and any divisor is called di factor: thus, 6 is a com- posite number, and the factors are 2 and 3. i31. Every number which is not divisible is called a prime number: thus, 1, 2, 3, 5, 7, 11, &c. are prime rmmbers. 132. Every prime number is divisible by itself and 1 ; but since these divisors are common to all numbers, they are not called /ac^OTA-. 133. Every factor of a number is either prime or compo- site : and since any composite factor may be again divided, it follows that. Any number is equal to the product of all its prim^e factors. For example, 12 = 6x2 ; but 6 is a composite number, of which the factors are 2 and 3 ; hence, 12=2x3x2; also, 20 = 10x2=5x2x2. Hence, to find the prime factors of any number, Divide the number by any pri^e number that will exactly divide it : then divide the quotient by any prime number that will exactly divide it^ and so on^ till a quotient is found ivhich is a prime number ; the several divisors and the last quotient will be the prime factors of the given number. Note. — It is most convenient, in practice, to use the least prime number, which is a divisor. 1. What are the prime factors of 42 ] Analysis. — Two being the least divisor that is a prime number, we divide by it, giv- ing the quotient 21, which we again divide 3)21 by 3, giving 7 : hence, 2. 3 and 7 are the Y prime factors. 2X3X7 = 42. 129. When is one number divisible by another 1 By what is every number divisible 1 Is 1 called a divisor 1 130. What is a composite number 1 What is a factor 1 r.U. What is a prime number \ 132. By what divisors is every prime number divided 1 133. To what product is every number equal 1 Give the rule fox finding the prime factors of a numler. What number is it nwst conve- nient to use as a divisor 1 OPERATION. 2)42 PiilME FACTOKS. 137 What are the prime fabtors of the following numbers t 1. Of the number 9? 2. Of the number 15 1 3. Of the number 24? 4. Of the number 16 ] 6. Of the number 18? 6. Of the number 32? 7. Of the number 48? 8. Of the number 56? 9. Of the number 63 ? 10. Of the number 76? Note. — The prime factors, when the number is small, may generally be seen by inspection. The teacher can easily multiply the examples. 134. When there are several numbers whose prime factors are to be found, I^ind the prime factors of each and then select those factors vahich are common to all the numbers. 11. What are the prime factors common to 6, 9 and 24 ? 12. What are the prime factors common to 21, 63 and 84 ? 13. What are the prime factors common to 21, 63 and 106? 14. What are the common ffictors of 28, 42 and 70? 15. What are the prime factors of 84, 126 and 210 ? 16. What are the prime factors of 210, 315 and 525? 135. DIVISIBILITY OF NUMBERS. 1. 2 is the only even number which is prime. 2. 2 divides every even number and no odd number. 3. 3 divides any number when the sum of its figures is di visible by 3. 4. 4 divides any number when the number expressed by the two right hand figures is divisible by 4. 5. 5 divides every number which ends in or 5. 6. 6 divides any even number which is divisible by 3. 7. 10 divides any number ending in 0. GREATEST COMMON DIVISOR. 136. The greatest common divisor of two or more num- bers, is the greatest number which will divide each of them, separately, without a remainder. Thus, 6 is the greatest common divisor of 12 and 18. 134. IIovv do you fnul the prime factors of two or moie umubersi 138 COMMON DIVISOK. Note. — Since 1 divides every number, it is not reckoned among the common divisors. 137. If two numbers have no common divisor, they are called prime with respect to each other. 138. Since a factor of a number always divides it, it fol- lows that the greatest common divisor of two or more rmm- bers, is simply the greatest factor common to these numbers. Hence, to find the greatest common divisor of two or more numbers, I. Resolve each number into its p^'ime factors. II. The jyroduct of the factors common to each result will he the greatest comrrwn divisor. EXAMPLES. 1. What is the greatest common divisor of 24 and 30 ? Analysis. — There are four prime opkration. factors in 24, and 3 in 30 : the factors 24 = 2x2x2x3 2 and 3 are common: hence, 6 is the 30 = 2x3x5 greatest common divisor. 2x3 = 6 com. divisor 2. What is the greatest common divisor of 9 and 18 1 3. What is the greatest common divisor of 6, 12 and 301 4. What is the greatest common divisor of 15, 25 and 30 ? 5. What is the greatest common divisor of 12, 18 and 72 ? 6. What is the greatest common divisor of 25, 35 and 70 ? 7. What is the greatest common divisor of 28, 42 and 70 1 8. What is the greatest common divisor of 84, 126 and 210 1 139. When the numbers are large, another method of find- ing their greatest common divisor is used, which depends on the following principles : 135. What even number is prime 1 What numbeis will 2 divide 1 What numbers will 3 divide 1 What numbers will 4 divide ? 51 6 ^ 101 136. What is the greatest common divisor of two or more numbers ] 137. When are two numbers said to be prime with respect to each other ] 138. What is the greatest factor of two numbers'! How do you find the greatest common divisor of two or more numbers ? PliOFERTIES OF NUMBERS. Iti9 ^. Any nuviber which tvill divide tico numbers separately, will divide their sum; else, tve shaidd have a 04 4- ^7 — -51 ivhole number equal to a proper fraction. " "~ 2. Any number which ivill divide two numbers separately, ivill divide their difference; and any number which will divide their difftfr- 51 — 27 =z 24 ence and one of the numbers, ivill divide the other ; else, tue should have a whole number equal to a 'proper fraction. 1. What is the greatest common divisor of 27 and 51 ? Divide 51 by 27 ; the quotient is 1 and the remainder 24; then divide the preceding divisor 27 by the^^^re- mainder 24: the quotien is 1 and there- t)n\K\i^ mainder 3; then divide the preceding '^IS divisor 24 by the remainder 3 ; the quo- ^' tient is 8 and the remainder 0. 24)27(1 iVow, since 3 divides the difference 3, 24 and also 24, it will divide 27, by principle ~3T^478 2d ; and since 3 divides the remainder 24, ^ , ^ and 27. it will also divide 51 ; hence, it is "* a common divisor of 27 and 51 ; and since it is the greatest com- mon factor, it it their greatest common divisor. Since the above reasoning is as applicable lo any other two numbers as to 27 and 5], wc liave the following rule : Divide the greater number by the less, and then divide the jyreceding divisor by the remainder, and so on, till nothing re- mains : the last divisor will be the greatest common divisor, EXAMPLES. 1. What is the greatest common divisor of 216 and 408 ? 2. Find the greatest common divisor of 408 and 740. 3. Find the greatest common divisor of 315 and 810. 4. Find the greatest common divisor of 4410 and 5670. 5. Find the greatest common divisor of 3471 and 1869. 6. Find the greatest common divisor of 1584 and 2772 ? Note. — If it be required to find the greatest common di\isor of more than two numbers, first find the greatest common divisor of 139. When the numbers are large, on what principles does the oper- ation of finding the greatest common divisor depend ! What is the rule for thiding it ! 140 COMMON DIVIDEND. two of tliem, then of that common divisor and one of the remaiu ing numbers, and so on for all the numbers : the last commou divisor will be the greatest common divisor of ail the numbers. 7. What is the greatest common divisor of 492, 744, and 1044 ? 8. What is the greatest common divisor of 944, 1488, and 2058 1 9. What is the greatest common divisor of 216, 408, and 740? 10 W^hat is the greatest common divisor of 945, 1560, and 22083 ? LEAST COMMON DIVIDEND. 140. The least common dividend of two or more numbers is the least number which they will separately divide without a remainder.* Notes. — 1. If a dividend is exactly divisible by a divisor, it can be resolved into two factors, one of which is the divisor and the other the quotient. 2. If the divisor be resolved into its prime factors, the cor- responding factor of the dividend may be resolved into the same factors ; hence, the dividend will contain every prime factor of the divisor. 3. The question of finding the least common dividend of several numbers, is therefore reduced to finding a number which shall con- tain all their prime factors and none others. 1. Let it be required to fmd the least common dividend of 6, 8 and 12. Anai.ysis.— We see, from inspec- operation. tion, that the prime factors of 6, are 2^3 2X2X2 2X2X3 2 and 3 ;— of 8 ; 2, 2 and 2 ;— and ©f 12; 2, 2 and 3. b . . . . » xlf.'ij w!\iclj follow ■? COMMON FRAC'no»^S. 149 3d. The numerator sliows Jioiv many fractional units are taken : 4th. The value of every fraction is equal to the quotient arising from dividing the numerator by the denominator : 5th. When the nutnerator is less than the deno7ni?tator, the value of the fraction is less tJian 1. 6th. When the numerator is equal to the denominator, the value of the fraction is equal to 1. 7th. When the numerator is greater than the denominor tor, the value of the fraction is greater than 1. EXAMPLES IN WRITING AND READING FRACTIONS. 1 Read the following fractions ; 5 5 1_6 7 3 _9_ 6 5_ T2' 9' 7 ' T&' 85 50' Tl 7 in each example? How many fractional units are taken in each? 2. Write 12 of the 17 equal parts of 1. 3. If the unit of the fraction is 1, and the fractional unit one-twentieth, express 6 fractional units. Express 12, 18, 16, 30, fractional units. 4. If the fractional unit is one 36th, express 32 fractional units ; 35, 38, 54, 6, 8. 5. If the fractional unit is one-fortieth, express 9 fractional units; 16, 25. 69, 75. DEFINITIONS. 149. A Proper Fraction is one whose numerator is less than the denominator. The following are proper fractions : 111135. 9_85. 2' 35 4> 4' 7' 8' TO' 9' 6* 150. An Improper Fraction is one whose numerator is equal to, or exceeds the denominator. Note. — Such a fraction is called improper because its value equals or exceeds 1. 149. What is a proper fraction 1 Give examples. 150 What is an improjier fraction 1 Wl>y iiniToperl Give exam - 160 riio POSITIONS uv The following are improper fractions : 3 5. 6 8. 9 1_2 1_4 X9 2' 3' 5' 7' 8' 6 ' 7 » 7 ' 151. A Simple Fraction is one whose numerator and de- nominator are both whole numbers. Note. — A simple fraction may be either proper or improper. The following are simple fractions : i.3.58.££6 1 4' 2' 6' 7' 2' 3> 3' 5* 152. A Compound Fraction is a fraction of a fraction, or several fractions connected by the word of. The following are compound fractions : i of i, i of i of i i of 3, 4 of J of 4. 153. A Mixed Number is made up of a whole number and a fraction. The following are mixed numbers : 3i, % 6f, 5f, 6f, 3i. 154. A Complex Fraction is one whose numerator or do- norainator is fractional ; or, in which both are fractional. The following are complex fractions : 4 2 I 45j 5' 19^' f 69f 155. The numerator and denominator of a fraction, taken together, are called the ter7ns of the fraction : hence, every fraction has two terms. fundamental propositions. 156. By multiplying the unit 1, we form all the whole numbers, 161, What is a simple fraction 1 Give examples. May it be proper or improper ^ 152. What is a compound fraction 1 Give examples. 153. What is a mixed number 1 Give examples. 154. What is a complex fraction 1 Give examples. 155. Hov^' many terms has every fraction 1 What are they T 15G. How may all the whole numbers be formed 1 How may the fractional units be formed 1 How many times is one-half less than 1 1 How many thucs is auy fraitiunal unif less than 1 1 COMMON FRACTIONS. 151 2, 3, 4, 5, 6, 7, 8, 9, 10, &c. ; and by dividing the unit 1 by these numbers we orm ail the fractional units, 1 JL 1 I 1 i JL 1 JL &e 2' 3' 4> 5' 6' 7' 8' 9' 10' *^*^" Now, since in 1 unit there are 2 halves, 3 thirds, 4 fourths, 5 filths, 6 sixths, &c., it follows that the fractional unit becomes less as the denominators are increased : hence, The fractional unit is such a part of I, as 1 is of the dcncnninator of the fraction. Thus, 1 is such a part of 1, as 1 is of 2 ; J is such a part of 1, as 1 is of 3 ; i is such a part of 1 as 1 is of 4, ^c. 6lc. 157. Let it be required to multiply ^ by 3. Analysis. — In -| there are 5 fractional operation. units, each of which is ^, and these are to |^ X 3=:-^^==-^ be taken 3 times. But 5 things taken 3 times, gives 15 tilings of the same kind ; that is, 15 sixths ; hence, the product is 3 limes as great as the multiplicand : therefore, we have V Proposition I. — If the numerator of a fraction be mitltiplied by any number^ the fraction will be increased as many times as there are units in the multiplier. EXAMPLES. 1. Multiply I by 8. 2. Multiply I by 5. 3. Multiply 4 by 9. 4. Multiply ^ by 14. 5. Multiply I by 20. 6. Multiply Yf by 25. 158. Let it be required to multiply J by 3. Analysis. — In ^ there are 4 fractional operation. units, each of which is ^. If we divide -i X 3 r:z -4_ — 4 the denominator by 3. we change the frac- tional unit to ^, which is 3 times as great as ^. since the first is contained in 1, 2 times, and the second 6 times. If we take ibis fractional unit 4 times, the result ^. is 3 times as great as | : therefore, we have Proposition II. — If the denominator of a fraction be divi- ded by any multiplier, the value of the fraction will be in- creased as many times as there are u?tits in that multiplier. 167. What is proved hi Proposition I \ 152 PHOi'OSITlO^'S IN EXAMPLES. 4. Multiply If by 2, 4, G, 5. Multiply fi by 2, 6, 7. 6. Multiply i§J by 5, 10. 1. Multiply I by 2, by 4. 2. Multiply if by 2, 4, 8. .3. Multiply j% by 2, 4, 6. 159. Let it be required to divide ^j by 3. Analysis. — In ^j. there are 9 fractional operation. niiits^ each o!" which is ^, and these are o _i_3— »^ir;^. to be divided by 3. But 9 things, divided by 3, sives 3 things of the same kind for a quotient ; hence, the quotient is 3 elevenths, a number one-third as great as ^j ; hence, we have Proposition III. — If the numerator of a fraction he divi- ded by any mimber, ike value of t}i£ fraction will he dimin- isited as many times as there are units in the divisor. JXAMPLKS. 1. Divide fl by 2, by 7. 2. Divide |i| by 56. 3. Divide f^f by 25, by 8. 4. Divide fl-g by 8, 16,10. 160. Let it be required to divide ^ by 3. xVnalysis. — In y\, there are 9 fractional operation, units, each of which is ^. Now, if we ^-^3=j^yx5=^' multiply the denominator by 3 it becomes 33, and the fractional unit becomes ^, which is only \ of ^, be- cause 33 is 3 times as great as 11. if we take this fractional unit 9 times, the result, ^, is exactly ^ of -^^ : hence, we nave Proposition IV. — If the denominator of a fraction be taultiplied by any divisor, the value of the fraction will be diminished as many times as there are units in that divisor. 1. Divide i by 2. 2. Divide l by 7. 3. Divide y\ by 4. EXAMPLES. 4. Divide ^ by 8. 5. Divide^ by 17. 6. Divide ^f^ by 45. 158. What is proved in proposition II. 1 159. What is proved in proposition III.1 ItJO. What is ppDvod in propo.'^ition IV. ! (X>MM:on FRAOl'lONS. 153 161. Let it be required to multiply both terms of the frac- tion I by 4. Analysis. — In |^, the fractional unit is ^, and it operation. is taken 3 times. By multiplying the denominator ■? "^ ^ ■_ 12. by 4, the fractional unit becomes X, the value of * ** * ^ ** which is \ times as great as \ By multiplying the numerator by 4, we increase the number ot fractional units taken, 4 times; that is, we increase the number just as many times as we decrease the 1 Mite ; hence, the value of the fraction is not changed ; there- fore, we liave rRorosiTiON V. — If both terms of a fraction be multiplied by the same number, the value of the fraction will not be changed. EXAMPLES. 1. Multiply the numerator and denominator of | by 7 : this gives y5|-H- 2. Multiply the numerator and denominator of ^ by 3, by 4, by 5, by 6, by 9. 3. Multiply each term of Jf by 2, by 3, by 4, by 5, by G. 162. Let it be required to divide the numerator and de- nominator of ^ by 3. Analysis. — In ^, the fractional unit is ^, and operation. is taken 6 times. By dividing the denominator 6 -t-3 2 by 3, the fractional unit becomes i. the value of i5-:-3 which is 3 times as great as ^. By dividing the numerator by 3, we diminish the number of fractional units taken 3 times : that is, we diminish the number just as many times as we increase itie value: hence, the value of the fraction is not changed : therefore, we have Proposition VI. — Jf both terms of a fraction be divided by the same number, the value of the fraction vdll not be clmnged. EXAMPLES. 1. l)ivide both terms of the fraction xe ^Y ^ • ^^ gives 8 -^2_4 Tg-2—6 ^^—^ Arts. ICI. What is proved in proposition V. 1 1G2. Wliat is proved in proposition VI \ c (51 KEDUCTTON OF 2. Divide both terms by 8 : this ^ves ^^^:t|=:J. 3. Divide both terms of the fraction ^^ by 2, by 4, by 8, by 16. 4. Divide both terms of the fraction ^-^^ by 2, by 3, by 1, by 5, by 6, by 10, by 12. REDUCTION OF FRACTIONS. 163. Keduction of Fractions is the operation of changing the fractional unit without altering the value of the fraction. A fraction is in its lowest terms, when the numerator and denominator have no common factor. CASE I. 164. To reduce a fraction to its lowest terms. 1. Reduce ^-^ to its lowest terms. Analysis. — By inspection, i1 is seen that 5 is a common lactor of the numerator and 1st. operation, denominator. Dividing by it, we have |^. 5^ JLo___i4^ We then see that 7 is a common factor of 14 and 35 : dividing by it, we have f . Now, 7 \ i 4 _ 2 there is no factor common to 2 and 5 ; hence, ^^ ^' \ is in it.s Imjoest terms. The areatest common divisor of 70 and 175 2d operation. is 35, (Art. 136) ; if we divide both terms of 35)y'^=|. the fraction by it, we obtain ^. The value of the fraction is not changed in either operation, since the numera- tor and denominator are both divided by the same number (Art. 162) : hence, the following Rl'LE. — Divide the numerator and denominator by any number thai will divide them both without a remainder., and divide the quotient, in the same manner until they have no common factor. Or : Divide the numerator and denominator by their great est common diuisor. 163 What is reduction of fractions i When is a fraction in ite lowest terms T 104. How ()o you reduce a iractiun to its lowest terms '' COMMON FKAOTIONB. 155 -^ EXAMPLES. Reduce the following fractions to their lowest terms. 1. Reduce |f . : 9. Reduce ^|. 2. Reduce if. j 10. Reduce 1^. 3. Reduce fl. 4. Reduce j^, 5. Reduce f|. 6. Reduce ^. 7. Reduce ^5^. 8. Reduce ^. 11. Reduce if 1^. 12. Reduce j^. 13. Reduce ffj. 14. Reduce T^nr- 15. Reduce ff. 16. Reduce fi£^. CASE n. 165. To redvce an improper fraction to its equivalent whole or mixed nuwher. 1. In *^ how many entire units ? Analysis. — Since there are 8 eighths in 1 unit, operation. in '^ there are as many units as 8 is contain- 8)59 ed times in 59, which is 7| times. 7T Hence, the following Rule. — Divide the numerator by the denominator, and the result will be the whole or mixed number. examples. 1. Reduce ^-^ and y to their equivalent whole or mixed numbers. OPERATION. OPERATION. 4)84 9)67 21 7J. 2. Reduce ®^ to a whole or mixed number. 3. In ^-^ yards of cloth, how many yards 1 4. In ^^ of bushels, how many bushels 1 105. How do you reduce an hup roper fraction to a whole or iiiiied lunilier *? 156 KKDUOTION OF 5. If I give ^ of an apple to each one of 15 children, how many apples do I give 1 G. Reduce ffj, \%\^, s^^^ %3^7^2_s ^ to their whole or mixed numbers. 7. If I distribute 878 quarter-apples among a number of boys, how many whole apples do I use ? 8. Reduce %Vt¥. tW' 'ttTO^'. to their whole or mixea numbers. 9. Reduce L±1R^.AJ^, lA^^, -^|-H^, to their whole or mixed numbers. CASE III. 166. To reduce a mixed number to its equivalent improper fraction. 1. Reduce 4| to its equivalent improper fraction. Analysis. — Since in any number operation there are 5 times as many fillhs as av'S — 90 fflVia units, in 4 there will be 5 times 4 fifths, ai a I!f i! or 20 fifths, to which add 4 fifths, and . ^^^ ^^ J r®' we have 24 fifths. gives Y =24 filths. Hence, the following Rule. — Multiply the whole number by the denominator of iKe fraction : to the product add the numerator^ and place the sum over the given denominator. EXAMPLES. 1. Reduce 47^ to its equivalent fraction. 2. In 17|^ yards, how many eighths of a yard 1 3. In 42^^Q rods, how many twentieths of a rod 1 4. Reduce ^'^^^'^ to an improper fraction. 5. How many 112ths in 20 5y*^? 6. In 84i^ days, how many twenty-fourths of a day 1 7. In 15^^ years, how many 365ths of a year 1 8. Reduce 916|-§^ to an improper fraction. 9. Reduce 25^, 156fJ, to their equivalent fractions. ir>G. How do ycm reduce a mixed number to its equivalent inipropei fruttivin 1 COMMON FRA.CT10NS. 157 CASE VI. 167. To reduce a whole number to a fraction havmg a given denominator. 1. Eeduce 6 to a fraction whose denominator shall be 4. Analysis. — Since in 1 unit there are 4 fourths. operation. it follows that in 6 units there are 6 times 4 fourths, 6 X 4 = 24, or 24 fourths: therefore, 6=^ : hence, ^.. KuLE^ — Multiply the whole number and denominator together, and write the product aver the required derimiii- nator. EXAMPLES. 1. Reduce 12 to a fraction whose denominator shall be 9. 2. Reduce 46 to a fraction whose denominator shall be 15. 3. Change 26 to 7ths. 4. Change 178 to 40ths. 5. Reduce 240 to IHths. 6. Change $54 to quarters. 7. Change 96yc?. to quarters. 8. Change 426/6. to I6ths. CASE V. 168. To reduce a compound fraction to a simple one. 1. What is the value jf f of |^ 1 Analysis. — Three-fourths of ^ is 3 times 1 fourth operation. of |; 1 fourth of ^ IS ^ (Art. 160) ; 3 fourths of f is 3 X 5_ 15 3 times ^, or ^ : therefore, | of 4=M • hence, ^ X 7 "28* Rule. — Multiply the numerators together for a new numerator, and the denominators together for a new de- nominator. Note. — If there are mixed numbers, reduce them to their equiv alent improper fractions. EXAMPLES. Reduce the following fractions to simple ones. 1. Reduce i of 1 of f . j 4. Reduce 2\ of ^ of 7. 2. Reduce f of | of f . j 5. Reduce 5 of .J- of \ of 6. U Reduce { of I of ^j.. \ 0. Reduce 6 J of 7^ of 6§ J 158 KEDUCTION OF METHOD BY CANCELLING. 169. The work may often be abridged by cancelling com mon Ikctors in the numerator and denominator (Art. 143). In every operation in fractions, let this be done whenever it is possible. EXAMPLES. 1 . Reduce f of | of f to a simple fraction. Here, -x-X^ = -, or, $ 5 7 7 5 Here. Note. — The divisors are always written on the left of the vertical line, and the dividends on the right. 2. Reduce f of | of ^ to its simplest temis. $ 2 Note. — Besides cancelling the like factors 8 and 8, and 9 and 9, we also cancel the factor 3, common to 15 and 6, and write over them, and at the right, the quotients 5 and 2. 3. Reduce f of | of | of y^JL of ^ to its simplest terms. 4. Reduce j-\2_ of _3_ of _4^ of ^ to its simplest terms. 5. Reduce 3| of f of Pit of 49 to its simplest terras. CASE VI. 170. To reduce fractions of different denominators to fractions having a common denominator. 1. Reduce ^, \ and | to a common denominator. 167. How do you reduce a whole number to a fraction having a given denominator 1 IGy. How do you reduce a compound fraction to a simple one. 169. How is the reduction of compound fractions to simple ones aliriilyt'd l»y raru'cllati'in 'J i COMMuN FRACTIONS. 159 0/»ERATTON 1 x3x5—\5 1st num. 7x2x5 = 70 2d num, 4x3x2 = 24 3d num. 2x3x5 = 30 denom. Analysts. — If both terms of the first fraction be multiplied by 15, the product of the other denomina- tors, it will become ^^. If both terms of the second traction be mul- tiplied by 10, the product of the other denominators, it will become ^ If both terms of the bird be multiplied by 6, the product of the other denominators, t will become ^. In each case, we have multiplied both terms of the fraction by the same number; hence, the value has not been altered (Art. 161) : hence, the following Rule. — Redvce to simjile fractimis when necessary ; then multiply the numerator of each fraction by all the denomi- nators except its own^ for the new numerators, and all the denominators together for a cmnmon denominator. Note. — When the numbers are small the work may be per- formed mentally. Thus. 5> 4? 20 40' 10 40' 13. 40* EXAMPLES. Reduce the fbllowins: fractions to common denominators. 1. Reduce |, |, and \. 2. Reduce f , y^y, and |. 3. Reduce 4, i, and |. 4. Reduce 2^, and A of \. 5. Reduce 5^, f of J, and 4. 6. Reduce 3^ of 1 and f. 7. Reduce J, ^, and 37. 8. Reduce 4, fi, and ^. 9. Reduce 7i ^, 6f 10. Reduce 4^ ^, and 2 J. Note. — We may often shorten the work by multiplying the nu- merator and denominator of each fraction by such a number as will make the denominators the same in all. 10. Reduce \ and J to a common denominator. OPERATION. Analysis. — Multiply both terms of the first by 2=f 3, and both terras of the second by 2. i _2 14. Reduce f, 3|, and f. 11. Reduce \ and ^. 12 Reduce i, ^^, and ^. 13. Reduce f , ^, y\. 15. Reduce 6^, 9^, and 5. 16. Reduce 7f, J, ^ and J. 170. How do you reduce fractions of different denominatora to frac- tious having a connnou denominator 1 When the numbers are •small liuw m;i} llie 'Auik lie jjerforuiedl ■• , 160 KEDUCnON OF CASE vn. 171. To reduce fractions to their least c&minon denominator. The least common denominator is the number which con- tains only the prime factors of the denominators, I. Reduce J, |, and f , to their least common denominator. OPERATION. (12-^3) xl= 4 1st Numerator. 3)3 . 6 . 4 (12-^6) x5:ii:10 2d " 2)1 . 2 . 4 (12-^4) X 3= 9 3d " "~1 \ 1 . 2 3x2x2=12, least com. denom. Therefore, the fractions \, J, and |-, reduced to their least nmon denominator, a Hence, the fbliowinsr common denominator, are ^^, \^, and ^ Rule. — I. Find the least common dividend of the denomu nators (Art. 14:0), which will be the least common denominator of the fractions. 11. Divide the least common denominator by the denomina- f.ors of the given fractions separately, and multiply the nume- rators by the corresponding quotients, and place the products over the least common denominator. Notes. — 1. Before beginning the operation, reduce every frac- tion to a simple fraction and to its lowest terms. 2. The expressions, (12-^3)X1, (12-t6)X5, ( 1 2-^4) X 3, indi- cate that the quotients are to be multiplied by 1, 5, and 3. EXA.MPLES. Reduce the following fractions to their least, common denominator. 2. Reduce f , |, /^. 7. Reduce 31, 4^% 8^. 3. Reduce H-J, 6f, 5^. 8. Reduce 1, f , f , and f. 4. Reduce fj, ^, f . 9. Reduce2iofi 3iof2. 5. Reduce tVo. 4^. f • 10. Reduce f , |, f , and ^V 6. Reduce fl, 3/^, 4. 11. Reduce I f , f , 1, «• 171. What is the least common denominator of several fractious' How do vou reduce fractions to their least conuuoa denominator 1 COMMOM FKA \DDITION OF FRA 172. Addition of Fractions is the operation number of liactional units in two or more fractions. 1. What is the sum of i, |, and J ? Analysis. — The fractional unit is the same m each > traction, viz: i; but the numerators show how many such units are taken (Art. 148) ; hence, the svm of the numerators written over the common denominator, expresses the sutn of the fractions. 2. What is the sum of i and | ? Analysis. — In the first, the fractional unit is ^, in the second it is ^. These units, not being of the same kind, cannot be expressed in the same colJection. But the i=|, and § = f, OPERATION. 1 + 3-1-0 = 9. Ans, 4} OPERATION. 1—3 2—6 2 _4 3—^ +i=i=H in each of which the unit is J : hence, iheir sum is |~1^. Note. — Only units of the same kind, whether fractional or inte- gral, can be expressed in the same collection. From the above analysis, we have the following Rule. — 1. When the fractions have the same denominator^ add the numerators, and place the sum over the comTuon deno- minator. II. When they have not the same denominator, reduce them to a common denominator, and then add as before. Note. — After the addition is performed, reduce every result to its lowest terms. EXAMPLES. 1. Add \, I, f , and f 2. Add! 13 6 oTifl ^, 2. ^. ttl^tJ — — a 1 ul — 3 4 6 13 ^• 3. Addf, i, f, V^andJ 4. Add -/y, t\, t\, and ^ 5. Add A, f^, and ^\. 6. Add 1, f . 1 and ^\. 7. Add I f , |, and ^\. 11 8. Add f , f , 1 and ^^. 9. Add9,f,T-L|,andi. 10". Add 1, f, f , ±, and |. 11. Add j%, f , ^6_, and f . 12. Add 1 |, and |. 13. Add tV> ?> I and i 14. Add t\, f , i, and Z^. l(J2 SUIiTliAOTION 01'' 15. "WTiat is the sum oflQl, 6f, and 4| ? OPERATION. Whole numbers. Fractions. 19 + 6+4 = 29 i+l+4^|4a=l^^. Sum = 29 + lT%V = 30iV5- 173. Note. — When there are mixed numbers, add the whole numbers and fractions separately^ and then add their sums. Find the sums of the following fractions : 16. Add 3i Vy^o, 12A II . , 20. Add 900yL 450|, 75if. 17. Add 16, 9f, 2oJ, li. : 21. Addiof/-j-of i^to^ofl. 18. Add 1 off, I of 9, 14^. 22. Add 17| to | of . 7|. 19. Add 2^-85-, 6-1, and 12if. 23. Add |, 7i and 8f. 24. What is the sum off of 12| of 7|-, and | of 25 ? 25. What is the sum of 3^ of 9f and ^j of 32S| 1 174. 1. What is the sum of ^ and i? Note. — If each of two fractions has operation. 1 for a numerator, the sum of the frac- 1-4-1—6 15 — 11 .,, , ' 1 . xu i- ii • 5 '6—30 I 30 — To* tions will be equal to the sum of their 1 1 i_5+6 n denominators divided by their product. s ""«— 5x6 ~30" 2. What is the sum of ^ and ^ 1 of J and ^ 1 3. What is the sum of j and Jg ? of Jg and j\ ? of j\ and 1? 4. What is the sum of J and y^j "^ ^^ i ^^^^ i • ^^ i and^V^ SUBTRACTION OF FRACTIONS. 175. Subtraction of Fractions is the operation of finding the difference between two fractions. 172. What is addition of fractions? When the fractional unit is the same, what is the sum of the fractions ! What units may he exj)rt'ssed in the same collection ? What is the rule for the addition of fractions 1 173. When there are mixed numbers, how do you add ! 1 74 When two fractions have 1 for a numerator, what is their sum eqvml to ! 175. What is subtraction of fractious 1 COMMON FK ACTIONS. 11)3 1. "WTiat is the difTerence between ^ and 1 1 Analysis. — In Ihis example the fractional unit operation. is i : there are 5 such units in tlie minuend and | — |^^|=:i. 3 in the .subtrahend : their difference is 2 eighths; ^^^^. i^ therefore. 2 is written over the common denomi- * **' nator 8. 2. From if take ^. | 4. From ^ff take\^. 3. From f take f . | 5. From ff| take -jff . 6. What is the difTerence between J and J 1 OPERATION. Analysis. — Reduce both to the same frac- 5 _j o. tional unit ^ : then, there are 10 such units 5^ V in the minuend and 4 in the subtrahend : , q ^~ ^^ , tT""T2— T2— ^• A lis, 1. From the above analysis we have the following Rule. — I. Wlien the fractions have the same denondnatw^ subtract the less numeiator from the yr eater, and place the difference over the common denominator. II. When they have not the same denominator, reduce them to a common denominator^ and then subtract as before. EXAMPLES. Make the following subtractions : 4. From 1, take f^^. 5. Fromiof 12, take jf of J. 6. F'mf ofllof7,takc'f off. 1. From f take f . 2. From f take f . 3. From /^ take y\. 7. From f of f of 1 take f^ off of 1. 8. From f of f of ^, take f of f of f 9. From j\ of fl of i take {^ off 10. What is the difference between 4 i and 2^1 OPERATION. 21— 15 - 90 or, 01_9 6 l5 _ 2^ Aus. 2^5 104 MULTII'LICATION OF 176. Therefore : When there are mixed numbers, chMugc both to improper fractiois and subtract as in Art. 175 ; or, subtract the integral and /^-actional numbers separately^ and write the results. 11. From 84y7_ take 161 | 12. From 246-| take 164^. 13. From 7f take 4J : f =^\ ^nd i = 2T- Note. — Since we cannot take ^ from -^ we operation. borrow 1, or |j, from the minuend, which added 72 —76 to 2j = fy ; then -^ from J^ leaves J^. We must 41 —4 J now carry 1 to the next figure of the subtrahend ^^""73! 5" and proceed as in subtraction of simple numbers. ^^^^- %r* 14. From 16f take 5f. | 16. From 3Gf take 27yV 15. From 26f take 19^. I 17. From 400y5^ take 327|-. 18. From i take ^j. Note. — When the numerators are 1, operation. the difference of the two fractions is 1.— ^1^ r^^l — .^ = .^ equal to the differeuce of the denomina- j^ 1 11-8 _ _3_ tors divided by their product. ^ TT — nxs ~ ss 19. What is the difference between J and \ ? Between I and J5 ? J and f, ? J^ and 3V ? yV ^nd ^V ? io and ^\^ ? MULTIPLICATION OF FRACTIONS. 177. MuLTirLTCATiON of Fractions is the operation of taking one number as many times as there are units in another, when one of the numbers is fractional, or when they are both fractional. 1. If one yard of cloth cost f of a dollar, what will 4 yards cost? Analysis. — Four yards will cost 4 operation. ^imes as much as 1 yard; if 1 yard 5 x 4=^-^ — — =21. costs 5 eighths of a dollar, 4 yards will ^ . ^ cost 4 times 5 eighths of a dollar, which are 20 eighths : there- fore, if 1 yard cost f of a dollar, 4 yards will cost ^ = 2^ dollars. 17C. When there arc mixed numbers, how do you subtract ! Explain ihc case when the fractional part of the subtrahend is the greater 1 177. What is umltiplicatioi: of friicliouii 1 COMMON KBAOTIONS. 165 OPERATION. OR, $ 2 5 : — Multiply tl^ 2d. If we divide the denominator by 4, the fraction will be multiplied by 4 (Prop. 11) : performing the operation, we obtain, I which = 2^ : hence. To multiply a fractior. by a whole number numerator^ or divide the denominator by the multiplier. EXAMPLES. 1. Multiply ^ by 12. 14. Multiply ^V ^Y ^' 2. Multiply li by 7. 5. Multiply fff by 49. 3. Multiply Vs^ by 9. | 6. Multiply ^ by 26. 7. If 1 dollar will buy f of a cord of wood, how much will 15 dollars buy ? 8. At f of a dollar a pound, what will 12 pounds of tea cost ? 9. If a horse eats J of a bushel of oats in a day, how much will 18 horses eat? 10. What will 64 pounds of cheese cost, at ^ of a dollar a pound ? 11. If a man travel |- of a mile an hour, how far will he travel in 16 hours ? 12. At f of a cent a pound, what will 45 pounds of chalk costi 13. If a man receive ^^ of a dollar for 1 day's labor, how much will he receive for 1 5 days ? 14. If a family consume |^ of a barrel of flour in 1 month, how much will they consume in 9 months ? 15. If a person pays \^ of a dollar a month for tobacco, how much does he pay in 18 months'? 181. To multiply a whole number by a fraction. 1. At 15 dollars a ton, what will f of a ton of hay costl Analysis. — 1st. Four-fifths of a ton will cost 4 times as much as 1 fifth of a ton ; if operation. 1 ton cost 15 dollars, 1 fifth will cost -I of 15 (15-^5) x 4rz 12. dollars, or 3 dollars, and ^ will cost 4 times 3 dollars, which are 12 dollars. 180. How do ynu multiply a frdctijn by a whole uumberl Iti(> MULTU'LICATION OK Or : 2d. 4 fifths of a ton will cost 1 fifth of 4 times the cost of 1 ton; 4 times 15 is 60, 15x4-^5=12. and 1 fifth of 60 is 12. .„ 3 Note. — Both operations may he combined in one by the use of the vertical line and can- cellation : hence, 12 Ana. Divide the whole number hy the denominator of the fraction and multiply the quotieiit hy the numerator ; Or : Multiply the whole number by the numerator of the fi'uction and divide the product by the denominator'. EXAMPLES. 1. Multiply 24 by J. 2. Multiply 42 by jf 3. Multiply 105 byf. 4 Multiply 64 by i|. 5. What is the cost of f of a yard of cloth at 8 dollars a yaid ? 6. If an acre of land is valued at 75 dollars, what is ^ of it worth ? 7. If a house is worth 320 dollars, what is y^^ of it worth ? 8. If a man travel 46 miles in a day, how far does he travel in | of a day ? 9. At 18 dollars a ton, what is the cost of y®^ of a ton of hay ? 10. If a man earn 480 dollars in a year, how much does he earn in ^ of a year ? 182. To multiply one fraction by another. 1. If a bushel of corn cost J of a dollar, what will f of a bushel cost 1 OPERATION. Analysis. — 5-sixths of a bushel will cost Jxf = ^|=:|-. I times as much as 1 bushel, or 5 times ^ | 1 sixth as much : i of f is ^, (Art. 180), g ^ and 5 times ^ is ijr=^|- : hence, | 5 $ I 5=|. 181. How Jo you multiply a whole number by a fraction ? C<)MMUM JTiACTlvWS. 167 Multiply the numerators together for a new numerator and the deuoinmators together for a new denominator. Notes. — 1. When the multiplier is less than 1, we do not take the whole of the multiplicand, but only such a part of it els the multiplier is of 1. 2. When the multiplier is a proper fraction, multiplication does not imply increase, as in the multiplication of whole numbers. The product is the same part of the multiplicand which the multi- plier is oM. EXAMPLES. Multiply g ^^yf 2. Multiply T% by 3. Find the pro't off, f, ^. 4. Find the pro't off, ^, ff 5. If silk is worth ^^ of a dollar a yard, what is | of a yard worth ? 6. If I owTi f of a farm and sell J of my share, what part of the whole farm do I sell ? 7. At 3^ of a dollar a pound, what will ^^ of a pound of tea cost ? 8. If a knife cost f of a dollar and a slate f as much, what does the slate cost ? 9. Multiply 51 by \ of f . Note. — Before multiplying, reduce both fractions to the form of simple fractions. — 2J. 5J-^ OPERATION. ; 1 ofl-^A. ¥x 5T 5T- n 1 . l = 'LAns GENERAL EXAMPLES. Mult. lof|.of|by ^. Mult. T^o by i ^^^' 4. Mult. 5 of I of I by 4f 5. Mult. 14 off of 9 by 6f 6. Mult, f of 6 off by f of 4. 3. Mult. J of 3 by i of 15^ 183. When the multiplicand is a whole and the. multi- plier a mixed number. 182. How do you multiply one fraction by another 1 When the multiplier is less than 1, what part of the multiplicand is taken T If the fraction is proper, does multiplication imply mcrease 1 What part is the product of the multiplicand '? 168 . Division of 7. What is the prod^jfct of 48 by 8^ 1 Note.- First mtiltiply 48 by i, which gives 4?*5YI!°^*q 8 ; then by' 8, which gives 384, and the sura j 392 ^o S~qq^ is the product : hence, ^^ ^ ° — ♦^Q^ 392 Multiply first hy the fraction, and then hy the whole number, and add the 'products. 8. MuJt. 67 by 9^. 1 10. Mult. 108 by 12f 9. Mult. 12f by 9. I 11. Mult. 5f by 3^. 12. What is the product of 6J, 2| and | of 12. 13. What will 24 yards of cloth cost at 3f dollars a yard ? 14. What will 6| bushels of wheat cost at 3| dollars a bushel 1 15. A horse eats ^ of J of 12 tons of hay in three months ; how much did he consume % 16. 11 I of I of a dollar buy a bushel of corn, what will T^ of y^ of a bushel cost % 17. What is the cost of 5 J gallons of molasses at 96 J cents a gallon ? 18. What will 7^ dozen candles cost at ^^ of a dollar per dozen ? 19. What must be paid for 175 barrels of flour at 7f dol- lars a barrel 'I 20. If I of f of 2 yards of cloth can be bought for one dol lar, how much can be bought ibr J of 13 J dollars ? 21. What is the cost of 15f cords of wood at 3| dollars a cord? DIVISION OF FRACTIONS. 184. Division of Fractions is the operation of finding a number which multiplied by the divisor will produce the divi- dend, when one or both of the parts are fractional. 185, To divide a fraction by a whole number. 1. If 4 bushels of apples cost f of a dollar, what will 1 bushel cost 1 183. How may you mulfiply when the muttipiicand is a whole and tht multiplier a mixed number 1 184. What is division of fractions ! 185. How do you divide a fraction by a whole number 1 OOALMt)N FKA Analysis. — Since 4 bushels cost f 1 bushel wiJl cost i of | of a dollar, the numerator of the fraction f by 4 f (Art. 159). Multiplying the denominator by 4 will pro- duce the same result (Art. 160) : hence, ^4 = 5-«=§ Divide the numerator or multiply the denominator hy the divisor S Note, — By the use of the vertical line and the principles of cancellation (Art. 143), all operations in division of fractions may be greatly abridged. 9 2=1 EXAMPLES. 1. Divide |f by 6. 2. Divide if by 9. 3. Divide W^ by 1 5. 4. Divide Iffi by 75. 5. Divide jf by 6. ' Divide ^2_ by 12. 6. j^iviue ^^ oy lis. 7. Divide Jf by 20. 8. Divide ^f f by 27. 9. If 6 horses eat ^^ of a ton of hay in 1 month, how much will one horse eat ? 10. If 9 yards of ribbon cost f of a dollar, what will 1 yard cost? 11. If 1 yard of cloth cost 4 dollars, how much can be bought for f of a dollar ? 12. If 5 pounds of coffee cost ^ of a dollar, what will 1 pound cost '? 13. At $6 a barrel, what part of a barrel of flour can be bought for I of a dollar \ 14. li 10 bushels of barley cost 3 J dollars, what will 1 bushel cost'^ Note. — We reduce the mixed number to •~>i._io an improper fraction and divide as in the 3 3 * case of a simple fraction. U)_l_i n — i A 15. If 21 pounds of raisins cost 4J dollars, what will i pound cost ? 16. If 12 men consume 6| pounds of meat in a day, how much does 1 man consume ? 170 DI VISTON i)¥ 186. To divide a ivhole nuinher by a fraction. 1. At |- of a dollar apiece, how many hats can be bought, for 6 dollars % Analysis. — Since \ of a dollar will operation. buy one hat. 6 dollars will buy as many 6-f-|- = 6 X 5-f-4=:7i. hats as \ is contained times in 6; and * ^ as there are 5 times as many fifths as i o whole things in any number, in 6 there 2 ^. 2cla. J of a w*^ek=:J of 7 days = - - 2da, 8kr. J of a Aay =----=---- ohr. Ans. Slwk. Ada. llAr. 3. From -^ of a X take J of a shilling. J of a shillingz=i of j^ of a £=zA- of a £. Then, i-eV-M-^o-fS of a £ = 9*. 8d. 4. From If Z6. Troy weight, take ^z. lb. oz. pwt. gr. \\lh.^\lh. of yo2. = 21o0. = l 9 \oz. =::iof \o ofW. = 80^r.- 3 8 Ans. 1 8 16 16 Rule. — Reduce the given fractions to the same unity and then add or subtract as in simple fractions, after which reduce to integers of a lower denomination : Or : Reduce the fractions separately to integers of lower de- nominations^ and then add or subtract as in denominate num- bers. EXAMPLES. 5. Add \\ miles, ^q furlongs, and 30 rods. 6. Add f of a yard, f of a foot, and |^ of a mile. 7. Add f of a cwt., ^^ of a /6., 13o«., J of a cwt. and 6/6 8. From J of a day take f of a second. 9. From f of a rod take f of an inch. 10. From y^g of a hogshead take ^ of a quart. 11. From \oz. take \'pwt. 12. From ^cwt. take \^Jb. 12 178 DDODECIMALS. 13. Mr. Merchant bought of farmer Jones 22 J bushels of wheat at one time, 19y^^ bushels at another, and 3o^ at an- other : how much did he buy in all 1 14. Add ^ of a ton and y^^ of a cwt. 15. Mr. Warren pursued a bear for three successive days; the first day he travelled 2«| miles ; the second 33^^ miles ; the third 29 Jy miles, when he overtook him : how larhad he travelled 1 16. Add 6^ days and 52^^^ minutes. 17. Add ^cwL, 8^lb., and 3j%lb. 18. A tailor bouirht 3 pieces of cloth, containing respect- ively, i8f yards, 2 If Ells Flemish, and 16| Ells English: how many yards in all I 19. Bought 3 kinds of cloth ; the first contained ^ of 3 of J of I yards ; the second, j of J of 5 yards ; and the third, | of I of ^ yards : how much in them all ? 20. Add l\cwt. 17f/6. and 7joz. 2 1 . From f of an oz. take J of a pwt. 22. Take -^ of a day and ^ of | of f of an hour from 3|^ weeks. 23. A man is GJ miles from home, and travels 4mi. I fur. '^24rd, , when he is overtaken by a storm : how far is he then from home ? 24. A man sold J^ of his farm at one time, ^ at another, and y2y at another : what part had he left 1 25. From i^ of a £ take f of a shilUng, 26. From Ijoz. take ^pwt, 27. From 8^cwL take ^^yh. 28. From 3^lb. Troy weight, take ^z. 29. From 1^ rods take f of an inch. 30. From f f fe take y^^ ! . DUODECIMALS. 197. If the unit 1 foot be divided into 12 equal parts, each part IS called an inch or prune, and marked '. If an inch be divided into 12 equal parts, each part is called a second, and marked ". If a second be d-ividod. in DU0UECIMAL6. 179 equal parts, each part is called a third, and marked '" ; and so on for divisions still smaller. This division of the foot gives 1 ' inch or prime - - - - - = tV ®^ ^ ^^^^' \" second is j^ of y'^ - - - = jj^- of a foot. V" third is yL of ^^2 of ^ - - =T^ of a foot. Note. — The marks ', ", '", &c., which denote the fractionat itnitSy are called indices. TABLE. 12'" make 1" second. 12" " 1' inch or prime. 12' " 1 foot. Hence : Duodecimals are denominate fractions, in which the primary unit is I foot, and 12 the acule of division. Note. — Duodecimals are chiefly used in measuring surfaces and solith. ADDITION AND SUBTRACTION. 198. The units of duodecimals are reduced, added, and subtracted, like those of other denominate numbers. The 8qr. 2^lb. to the decimal of a ton. 9. Reduce 5 A. SR. 2lsq. rd. to the denomination of acres. 10. Reduce 11 pounds to the decimal of a ton. 11. Reduce 3da. 12^5^6*. to the decimal of a week. 12. Reduce lAbu. ofqt. to the decimal of a chaldron, 13. Reduce 77n. Ifur. \r. to the denomination of miles. 217. How do you reduce a compound denomiaate nuuiber to decimal ' ANALYSIS. 201 ANALYSIS. 218. An analysis of a proposition is an examination of its separate parts, and their connections with each other. The sohjtion of a question, by analysis, consists in an exami- nation of its elements and of the relations which exist between these elements. "We determine the elements and the rela- tions which exist between them, in each case, by examining the nature of the question. In analyzing, we reason from a given uumber to its unity and then from this unit to the required number. EXAMPLES. 1. If 9 bushels of wheat cost 18 dollars, what will 27 bushels cost ? Analysis. — One bushel of wheat will cost one ninth as much as 9 bushels. Since 9 bushels cost 18. dollars, 1 bushel will cost ^ of 18 dollars, or 2 dollars ; 27 bushels will cost 27 times as much as 1 bushel: that is. 27 times \ of 18 dollars, or 54 dollars: cost 54 dollars. OPERATION. M^ 1 27 ^^, -f X;rX — .= $54; Or, 19 1 IS 3 54 Ans, Note. — 1. We indicate the operations to be performed, and then cancel the equal factors (Art. 141). 219. Although the currency of the United States is ex- pressed in dollars cents and mills, still in most of the States the dollar (always valued at 100 cents), is reckoned in shil- lings and pence ; thus, In the New England States, in Indiana, Illinois, Missouri, Vir- ginia, Kentucky, Tennessee, Mississippi and Texas, the dollar is reckoned at 6 shillings : In New York, Ohio and Michigan, at 8 shillings : In New Jersey, Pennsylvania, Delaware and Mary- land, at Is. 6d : In South Carolina, and Georgia, at 45. 8d. : In Canada and Nova Scotia, at 5 shillings. 218. What is an analysis ] In what does the solution of a question by analysis consist ? How do we determine the elements and theii relations ! How do we rea&ou in analyzing 1 202 ANALYSTS. Note. — In many of the States the retail price of articles is given in shillings and pence, and the result, or cost, required in doilarB and cents. 2. What will 12 yards of cloth cost, at 5 shillings a yard, New York currency ? Analysis. — Since 1 yard cost 5 shillings 12 yards will cost 12 times 5 shillings, or 60 shillings : and as 8 shillings make 1 dollar. New York cuiiency, there will be as many dollars as 8 is contain ed times in 60=$7i. OPERATION. 6X12-H8 = $7,50; Or, 5 15=:i«:^|;7,50. $7,50. Note. — The fractional part of a dollar may always be reduced to cents and mills by annexing two or three ciphers to the nume- rator and dividing by the denominator ; or, which is more conve- nient in practice, annex the ciphers to the dividend and continue the division. 3. What will be the cost of 66 bushels of oats at 3&'. 3d. a bushel, New York currency ? 4 $ 00 ^ 13 OPERATION. 4 4 91 Or, 4_ 91 22,75. ^22,75 Ans. Note. — When the pence is an aliquot part of a shilling- the price may be reduced to an improper fraction, which will be the multiplier: thus. 35. 3d. = 3^s.= ^s. Or: the shillings and ])ence may be reduced to pence : thus, 'Ss. 3d. = 39d., in which case the product will be pence, and must be divided by 96, the number of pence in 1 dollar : hence, 220. To find the cost of articles in dollars and cents. 219. In what is the currency of the States expressed 1 the currency of the States often reckoned ! 230. How do you find the cost of a commoditv In what if ANALYSIS. Wii Multiply Ihe commodity by the price and divide the p-oduct by the value of a dollar reduced io the sam.e denominational unit. 4. What will 18 yards of satinet cost at 3s. 9c?. a yard Pennsylvania currency 1 u OPERA! 'ION. u' Q 1$ S0 n t Or, 4$ ^9. $9 Ans Note. — The ahove rule will apply to the currency in any of ihe States. In the last example the multiplier is "^s. 9(i. = 3i5 = 1^.9. or 45rf. The divisor is 75. Qd. = 1U.= ^s.^dOd. 5. What will 7^/6. of tea cost at 65. Sd. a pound, Ncm Eng'und currency ? OPERATION. it U ' Or, $0 10 25 3 ! 25 = 25 r= $8,333-1- 5.333-|-^w«. 6. What will be the cost of '[20yds. of cotton cloth at 1*. 6d. a yard, Georgia currency ] 7. What will be the cost in New York currency? 8. What will be the cost in New England currency ? 9. What will be the cost of 15 bushels of potatoes at 3s 6fl?., New York currency ? 10. What will it cost to build 148 feet of w^all at Is. 8d.. per loot, N. Y. currency ? 11. What will a load of M'heat, containing 46 J bushels, come to at lO.s. 8d. a bushel, N. Y. currency ] 12. What will 7 yards of Irish linen cost at 3s. 4c/. a yard, Penn. currency ? 13. How many pounds of butter at Is. 4.d. a pound must be given for 12 gallons of molasses at 2.v. 8d. a gallon ? 204 AWALYISIS. OPERATION. I 12 , 12 - $ \ $ ' H\n ^ ^ I ^ 0^' \2Ub, I 24/6. Note. — The same rule applies in the last example as in the preceding ones, except that the divisor is the price of tlie article received in payment, reduced to the same unit as the price of the article bought. 14. What will be the cost of \2cwt. of sugar at \^d. per lb. N. Y. currency ? Note. — Reduce the cwts. to Ihs. by multiplying by 4 and then by 25. Then multiply by the price -per pound, and then divide by the value of a dollar in the required currency, reduced to the same denomination as the price. $112,50 15. What will be the cost of 9 hogsheads of molasses at Is. 3c?. per quart, N. E. currency '\ 16. How many days work at 75. 6r/. a day must be given ibr 12 bushels of apples at 3s. 9^. a bushel? 17. Farmer A exchanged 35 bushels of barle} , worth 65. 4:d., with farmer B for rye worth 7 shillings a bushel : how many bushels of rye did farmer A receive ? 18. Bought the following bill of goods of Mr. Merchant: what did the whole amount to, JST. Y. currency 1 12^ yards of cambric at Is. ^d. per yard. 8 " ribbon " 2s. &d. 21 " calico •' Is. M. 6 *' alpaca " 5s. dd. " 4 gallons molasses ** 3s. 5d. per gallon. 2^ pounds tea " 6s. 6d. per pound. 30 . " sugar " 9d. " " 19. If f of a yard of cloth cost $3,20, what will || of a yard cost 1 Analysis. — Since 5 eighths of a yard of cloth costs $3,20. 1 eighth of a yard will cost ^ of $3,20 ; and 1 yard, or 8 eighths, will cost 8 times as much, or | of $3,20 ; ^| of a yard will cost \^ as much as 1 yard, or {^ of | of $3,20 =$4.80. 160 J ANALYSIS. '"205 OPERATION. 3 ftii'irtl'GO $$;^0X^XyX~ = S4.8O. Or, ^ 10 10 $4,80. 20. If 3j pounds of tea cost 3^ dollars, what will 9 poundg cost 1 Note. — Reduce the mixed numbers to improper fractions, and then apply the same mode of reatsoning as in tlie preceding ex- ample. 21. What will 8i cords of wood cost, if 2|- cords cost 7-1 dollars ? 22. If 6 men can build a boat in 120 days, how long will it take 24 men to build it ? Analysis. — Since 6 men can build a boat in 120 days, it will take 1 man 6 times 120 days, or 720 days, and 24 men can build it in ^ of the time that 1 man will require to build it, or -jlj- of 6 times 120, which is 30. OPERATION. 120x6-^24=30 days. Or, . Ans. 30 da^s. 23. If 7 men can dig a ditch in, 21 days, how many men will be required to dig it in 3 days ? 24. In what time will 12 horses consume a bin of oats, that will last 21 horses 6|- weeks ? 25. A merchant bought a number of bales of velvet, each containing 129i|^ yards, at the rate of 7 dollars for 5 yards, and sold them at the rate of 1 1 dollars for 7 yards ; and gained 200 dollars by the bargain : how many bales were there ? Analysis. — Since he paid 7 dollars for 5 yards, for 1 yard he paid ^ of $7 or I of 1 dollar ; and since he received 11 dollars for 7 yards, for 1 yard he received \ oi 11 dollars or ^ of 1 dollar. He gained on 1 yard the difference between ^ and ^=-^ of a dol- lar. Since his whole gain was 200 dollars, he had as many yards as the gain on one yard is contaiiied times in his whole gain, or as ^ is contained times in 200. And there were as many bales as 129^. (the number of yards in one bale), is contained times in the whole number of yards ^<^ ; which gives i) bale*. 200 ANALYSIS. $000 *^00 OPERATION. 129^\—^^, number ofyardsin a ba.e : ^ ^^^^^^ ^ 2004-3^5 =—^, "^'hol*^ number of yards : i00 Ui^o o_^3|p_o ^ 9 bales. ^^.. | 9 bales, 2G. Suppose a number of bales of cloth, each contairiiuy 133^ yards, to be bought at the rate of 12 yards for 1 1 doJ lars, and sold at the rate of 8 yards for 7 dollars, and the loss in trade to be ^100 : how many bales are there? 27. If a piece of cloth 9 feet long and 3 feet wide, contain 3 square yards ; how long must be a piece of cloth ihat is 2| feet wide be, to contain the same number of yards? 28. A can mow an acre of grass in 4 hours, B in 6 hours, and C in 8 hours. How many days, working 9 hours a day, would they require to mow 39 acres ? Analysis. — Since A can mow an acre in 4 hours, B in 6 hours, and C in 8 hours, A can mow \ of an acre, B ^ of an acre, and C ^ of an acre in 1 hour, Together they can mow i+i+i=M of an acre in 1 hour. And since they can mow 13 twenty-fourths of an acre in 1 hour, they can mow 1 twenty-fourth of an acre in ^ of 1 hour; and 1 acre, or f|, in 24 times ^==f | of 1 hour: and to mow 39 acres, they will require 39 times f |=r \2/ hours, which reduced to days of 9 hours each, gives 8 days. OPERATION. 8 $ U $0 1 n — X T =^ ^ days. Or 1 ^ Ans. u n 8 days. 29. A can do a piece of work in 4 days, and B can do the same in 6 days ; in what time can they both do the work if they labor together ? 30. If 6 men can do a piece of work in 10 days, how long will it take 5 men to do it \ Analysis. — If 6 men can do a piece of work in 10 day.'^, 1 man will require 6 times as long, or 60 days to do tiie t^ame work. Five men will require but onc-fifLh as long as one man. or bO-f-'O = 12 days. 10x6-^5=12 days. ANALYSIS. OPERATION. u $ 6 Ans. 12 days 207 31. Three men together can perform a piece of work in 9 days. A alone can do it in 18 days, B in 27 days ; in what time can C do it alone 1 32. A and B can build a wall on one side of a square piece of ground in 3 days ; A and C in 4 days ; B and C in G days : what time will they require, working together, to complete the wall enclosing the square I 33. Three men hire a pasture, lor M'hich they pay 66 dol- lars. The first puts in 2 horses 3 weeks ; the second 6 horses for 2^ weeks ; the third 9 horses for 1^ weeks : how much ought each to pay 1 Analysis. — The pasturage of 2 horses for 3 weeks, would be the same as the pasturage of 1 horse 2 times 3 weeks, or 6 weeks: that of six horf^es 2^ weeks, the same as for 1 horse 6 times 2-| weeks, or 15 weeks ; and that of 9 horses \^ weeks, the same as 1 horse tor 9 times 1-|^ weeks, or 12 weeks. The three persons had an equivalent for the pasturage of 1 horse for 6 + 1 5 + 1 2 = 33 weeks ] therefore, the first must pay ^, the second ^, and the third \^ of 66 dollars. OPERATION. 3 x2i=:6; then $66x^=$12. 1st. 2^X6 = 15; " $66xifrz:$30. 2d. 11x9 = 12; " $66x^1=: $24. 3d. 34. Two persons, A and B. enter into partnership, and gain $175. A puts in 75 dollars for 4 months, and B puts in 100 dollars for 6 months : what is each one's share of the gain ? 35. Three men engage to build a house for 580 dollars. The first one employed 4 hands, the second 5 hands, and the third 7 hands. The first man's hands worked three times as many days as the third, and the second man's hands tM'ice as many days as the third man's hands : how much must each receive 1 208 .ANALYSIS. 36. If 8 students spend $192 in 6 months, how much will 12 students spend in 20 months ? Analysis. — Since 8 students spend $192, one student will spend J of $192, in 6 months; in 1 month 1 student will spend ^ of \ of $192= $4. Twelve students will spend, in 1 month, 12 times as much as 1 student, and in 20 months they will spend 20 times as much as in 1 month. OPERATION. 24 1 1 I^ 20 ^^^^ 1 ^6^^^T^T=^^^^ m 20 48 $960. Ans, 37. If 6 men can build a waJl 80 feet long, 6 feet wide, and 4 feet high, in 15 days, in what time can 18 men build one 240 feet long, 8 feet wide, and 6 leet high ? Analysis. — Since it takes 6 men 15 days to build a wall, it will take 1 man 6 times io days, or 90 days, to build the same wall. To build a wall 1 foot long, will require ^ as long as to build one 80 feet long; to build one 1 foot wide, \ as long as to build one 4 feet wide ; and to build one 1 foot high, \ as long as to build one 6 feet high. 18 men can build the same wall in ^ of the time that one man can build it: but to build one 240 feet long, will take them 240 times as long as to build one 1 foot in length; to build one 8 feet wide, 8 times as long as to build one 1 foot wide, and to build one 6 feet high, 6 times as long as to build one 1 foot high. OPERATION. % 2 15x0 1 1 1 1 ^^0 $ '- X "T-r X —; X T X ~r X z X~X~ — oU. 1 $0 ^ $ 1$ I I 1 $ 15 $ 2 Ans. I 30 days. 38. If 96Z^5. of bread be sufficient to serve 5 men 12 days, how many days will 6716. serve 19 men? ANALYSIS. 39. If a man travef 220 miles hours a day, in how many days wil travelling 16 hours a day i 40. If a family of 12 persons consume a certain quantity of provisions in 6 days, how long will the same provisions last a I'amily of 8 persons ? 41. If 9 men pay $135 for 5 weeks' board, how much must 8 men pay for 4 weeks' board I 42. If 10 bushels of wheat are equal to 40 bushels of corn, and 28 bushels of corn to 56 pounds of butter, and 39 pounds of butter to 1 cord of wood ; how much wheat is 12 cords of wood worth ? Analysis. — Since 10 bushels of wheat are worth 40 bushels of corn, 1 bushel of corn is worth ^ of 10 bushels of wheat, or \ of a bushel ; 28 bushels are worth 28 limes ^ of a bushel of wheat, or 7 bushels : since 28 bushels of corn, or 7 bushels of wheat are worth 56 pounds of butter. 1 pound of butter is worth ^ of 7=^ of a bushel of wheat, and 39 pounds are worth 30 times as much as 1 pound, or 39x-^=^ bushels of wheat; and since 39 pounds of butter, or ^ bushels of wheat are worth 1 cord of wood, 12 cords are worth 12 times as much, or 12X^ = ^8"^ Dushels. OPEKATION. 3 ^ .. 4 2 10 n 39 n Note. — Always commence analyzing from the term which is of the same name or kind as the required answer. 43. If 35 women can do as much work as 20 boys, and 16 boys can do as much as 7 men : how many women can do the work of 1 8 men ? 44. If 36 shillings in New York are equal to 27 shillings in Massachusetts, and 24 shillings in Massachusetts are equal tx) 30 shilHngs in Pennsylvania, and 45 shillings in Pennsyl- vania are equal to 28 shillings in Georgia ; how many shil- lini^s in Georgia are equal to 72 shillings in New York ? 14 210 PROMISCUOUS EXAMPLES PROMISCUOUS EXAMPLES IN ANALYSIS. 1. How many sheep at 4 dollars a head must I give fcr 6 cows, worth 1 2 dollars apiece ? 2. If 7 yards of cloth cost ^^49, what will 16 yards cost ? 3. If 36 men cun build a house in 16 days, how long will it take 12 men to build it ? 4. If 3 pounds of butter cost 71 shillings, what will 12 pounds cost ? 0. If 5i bushels of potatoes cost S2|, how much will 1 2^ bushels cost ? 6. How many barrels of apples, worth 12 shillings a barrel, will pay for 16 yards of cloth, worth 9s. 6d. a yard ? 7. If 31i gallons of molasses are worth $9f, what are 5 J gallons worth"? 8. What is the value of 24j bushels of corn, at 5s. Id. a bushel, New York currency ? 9. How much rye, at 8s. 3d. per bushel, must be given for 40 gallons of whisky, worth 2s. dd. a gallon? 10. If it* take 44 yards of carpeting, that is li yards wide, to cover a floor, how many yards of J yards wide, will it take to cover the same floor] 11. Li' a piece of wall paper, 14 yards long and 1^ feet wide, will cover a certain piece of wall, how long must an- other piece be, that is 2 feet wide, to cover the same wall ? 12. If 5 men spend $200 in 160 days, how long will $300 last 12 men at the same rate ? 13. If 1 acre of land cost J of f of f of $50, what will 3 J acres cost ? 14. Three carpenters can finish a house in 2 months ; two of them can do it in 2i months : how long will it take tho third to do it alone 1 15. Three persons bought 2 barrels of flour lor 15 dollars The first one ate from them 2 months, the second 3 months and the third 7 months : how much should each pay ? 16. What quantity of beer will serve 4 persons 18-J days if 6 persons drink 7^ gallons in 4 days ? IN ANALYSIS. 211 17. If 9 persons use If pounds of tea in a month, how much will 10 persons use in a year ? 18. If ^ of f of a gallon of wine cost f of a dollar, what will 5J gallons cost ? 19. How many yards of carpeting, If yards wide, will it take to cover a floor that is 4f yards wide and 6 and three- fifths yards long ? 20. Three persons bought a hogshead of sugar containing 413 pounds. The first paid $2^ as often as the second paid 831, and as often as the third paid $4 : what was each one's share of the sugar ? 21. A, wath the assistance of B, can build a wall 2 feet wide, 3 feet high, and 30 feet long, in 4 days ; but with the assistance of C, they can do it in 3^ days : in how many days can C do it alone 1 22. If two persons engage in a business, where one advances $875, and the other $625, and they gain $300, what is each one's share 1 23. A person purchased ^ of a vessel, and divided it into 5 equal shares, and sold each of those shares for $1200 : what was the value of the whole vessel t 24. How many yards of paper, j of a yard wide, will be sufficient to paper a room 1 yards square and 3 yards high 1 25. What M'ill be the cost of 4.6lbii. of coffee, New Jersey currency, if 9lbs. cost 27 shillings ? 26. Wliat will be the cost of 3 barrels of sugar, each weigh- ing 2cwl. at lOd. per pound, Illinois currency? 27. If 12 men reap 80 acres in 6 days, in how many days will 25 men reap 200 acres ? 28. If 4 men are paid 24 dollars for 3 days' labor, how many men may be employed 16 days for $96 ? 29. If $25 wall supply a family with flour at $7,50 a bar- rel for 2|- months, how long would $45 last the same family when flour is worth $6,75 per barrel 1 30. A wall to be built to the height of 27 feet, was raised to the height of 9 feet by 12 men in 6 days : how many men must be employed to finish tlie wall in 4 days at the same rute of work in'' l 212 PKOMISCUOU^ EXAMPLES. 31. A, B and C, sent a drove of hogs to market, of which A owned lOo, B 75, and C 120. On the way 60 died : how many must each lose '? 32. Three men, A, B and C, agree to do a piece of work, for which they are to receive $315, A works 8 days, 10^ hours a day ; B 9-J days, 8 hours a day ; and C, 4 days, 12 hours a day : what is each one's share ? 33. If 10 barrels of apples will pay for 5 cords of wood, and 12 cords of wood for 4 tons of hay, how many barrels of apples will pay for 9 tons of hay ? 34. Out of a cistern that is |- full is drawn 140 gallons, "when it is found to be | full : how much does it hold ? 35. If .7 of a gallon of wine cost $2,25, what will .25 of a gallon cost? 36. If it take 5.1 yards of cloth, 1.25 yards wide, to make a gentleman's cloak, how much surge, ^ yards wide, will be required to hue it ? 37 A and B have the same income. A saves i of his annually ; but B, by spending '1>200 a year more than A, at the end of 5 years finds himself $160 in debt : what is their income ? 38. A father gave his yomiger son $420, which was J of what he gave to his elder son ; and 3 times the elder son's portion was ^ the value of the father's estate : what was the value of the estate ? 39. Divide $176,40 among 3 persons, so that the first shall have twic^ as much as the second, and the third three times as much as the first : w^hat is each one's share 1 40. A gentleman having a purse of money, gave ^ of it foi a span of horses ; j of |^ of the remainder for a carriage ; when he found that he had but $100 left : how much was in his purse before any was taken out ? 41. A merchant tailor bought a number of pieces of cloth, each containing 25^^ yards, at the rate of 3 yards for 4 dol- lars, and sold them at the rate of 5 yards for 13 dollars, and gained by the operation 96 dollars : how many pieces did he buy i .EATIO AND PKOPORTION. 213 RATIO AND PROPORTION. 221. Two numbers having the same unit, may be com- pared in two ways : Ist. By considering how much one is greater or less than the other, which is shown by their difference ; and, 2d. By considering how many times one is contained in the other, which is shown by their quotient. In comparing two numbers, one with the other, by means of their difference, the less is always taken from the greater. In comparing two numbers, one with the other, by means of their quotient, one of them must be regarded as a standard which measures the other, and the quotient which arises by dividing by the standard, is called the ratio. 222. Every ratio is derived from two terms: the first is called the antecede?it, and the second the consequent ; and the two, taken together, are called a couplet. The antecedent will be regarded as the standard. If the numbers 3 and 12 be compared by then- difierence, the result of the comparison yill be 9 ; for, 12 exceeds 3 by 9. If they are compared by means of their quotient, the result will be 4; for, 3 is contained in 12, 4 times: that is, 3 measuring 12, gives 4. 223. The ratio of one number to another is expressed in two ways : \st. By a colon ; thus, 3 : 12 ; and is read, 3 is to 12 ; or, 3 measuring 12. i2 2d. In a fractional form, as — ; or, 3 measuring 12. 221. In how many ways may two numbers, having the same unit, &e compared with each other 1 If you compare by their difference, how do you find it ^ If you compare by the quotient, how do you regard one of the numbers "^ What is the ratio 1 222. From how many terms is a ratio derived ? WTiat is the first tf rm called ? "Wliat is tlie second called 1 Which is the standard 1 *'.43 How may the ratio of two iituiibcrs be expresi^ed ^ Hinv rcuJ ' 214 RATIO AND PBOPOETION. 224. If two couplets have the same ratio, their tenns are Bdid to be proportional : the couplets 3 : 12 and 1 : 4 have the same ratio 4 ; hence, the terms are proportional, and are written, 3 : 12 : : 1 : 4 by simply placing a double colon between the couplets. The terms are read 3 is to 12 as 1 is to 4, and taken together, they are called a lyroportion : hence, A proportion is a comparison of the terms of two equal ratios.* 224. If two couplets have the same ratio, what is said of the tenns .' Hov/ are they written 1 How read 1 What is a proportion ? * Some authors, of high authority, make the consequent the stand- ard and divide the antecedent by it to determine the ratio of the couplet. The ratio 3 : 12 is the same as that of 1:4 by both methods ; for, if the antecedent be made the standard, the ratio is 4 ; if the conse- quent be made the standard, the ratio is one;fourth. The question is, which method should be adopted I The unit 1 is the number from w^iich all other numbers are derived, and by which they are measured. The question is, how do we most readily apprehend and express the relation between 1 and 4 I Ask a child, and he will answer, " the dif- ference is 3 " But when you ask him, " how many I's are there in 4 '" he will answer, " 4," using 1 as the standard. Thus, we begin to teach by using the standard 1 : that is, by dividing 4 by 1. Now, the relation between 3 and 12 is the same as that between 1 and 4 ; if then, we divide 4 by 1, we must also divide 12 by 3. Do wc, indeed, clearly apprehend the ratio of 3 to 12, until we have referred to 1 as a standard ] Is the mind satisfied until it has clearly perceived that the ratio of 3 to 12 is the same as that of 1 to 4 1 In the Rule of Three we always look for the result in the 4th term. Now, if we wish to find the ratio of 3 to 12, by referring to 1 as a stand ard, we have 3 : 12 : : 1 : ratio, which brings the result in the right place. But if we define ratio to be the antecedent divided by the consequent, we should have U : 12 : : ratio : I, wljitli would briiifi the ratio, or icqxivcd number, in the 3J place. RATIO AND PROPOKTIOIf. 216 What are the ratios of the proportions, 3 : 9 : 12 36? 2 : 10 : 12 : 60? 4 2 : 8 4? 9 : 1 : 90 10 ] 225. The 1st and 4th terms of a proportion are caJl^r^ the extremes t the 2d and 3d terms, the means. Thus, in th^ pm portion, 3 : 12 : : 6 : 24 Since (Art. 224), 3 and 24 are the extremes, and 12 and 6 the means: 12_24 3 "T' we shall have, by reducing to a common denominator, 1 2x6 24x3 ~3x6~ 6x3* But since the fractions are equal, and have the same deno- minators, their numerators must be equal, viz ; 12x6 = 24x3; that is, In any proportion^ the product of the extremes is equal to the product of the means. Thus, in the proportions, 1 : 6 : : 2 : 12; we have 1x12= 2x6; 4 : 12 : : 8 : 24 ; " " 4x24=12x8. 226. Since, in any proportion, the product of the extremes u equal to the product of the means, it follows that. In all cases, the numerical value of a quantity is the number of timep which that quantity contains an assumed standard, called its U7iit oj mcai ure. If we would find that numerical value, in its right place, we must aay, standard quantity : : 1 : numerical value : but if we take the other method, we have quantity stjtndard numerical value : 1. which brijiys the uuuieritfcil value in the wruntr place. 216 • RATIO ANP PRUPOKTION. 1st. If the product of the means be divided by one of ike extremes^ the quotient will be the other extreme. Thus, in the pioportion 3 : 12 : : 6 : 24, we have 3x24 = 12x6; then, if 72, the product of the means, be divided by one o* the extremes, 3, the quotient wi]] be the other extreme, 24: or, if the product be divided by 24, the quotient will be 3. 2d. If the product of the extremes be divided by either of the means ^ the quotient will be the other mean. Thus, if 3x24 =12x6 :rr 72 be divided by 12, the quotient will be 6 ; or if it be divided by 6, the quotient will be 12. EXAMPLES. 1. The first three terms of a proportion are 3, 9 and 12 : ■what is the fourth term 1 2. The first three terms of a proportion are 4, 16 and 15 : what is the 4th term '\ 3. The first, second, and fourth terms of a proportion are 6, 12 and 24 : what is the third term ? 4. The second, third, and fourth terms of a proportion are 9, 6 and 24 : what is the first term ? 5. The first, second and fourth terms are 9, 18 and 48 ; what is the third termi 227. Simple and Compound Ratio. The ratio of two single numbers is called a Simple Ratio^ and the proportion which arises from the equality of two such ratios, a Simple Proportion. 225. Which are thti extremes of a proportion ? Which the means ? What is the product of the extremes equal to 1 226. If the product of the means be divided by one of the extroinee, what will the quotient be 1 If the product of the means be divided by either extreme, what will the quotient be \ 227. What is a simple ratio ] What is the proportion called whicli comes from the equality of two simple ratios 1 What is a compouiul ratio \ AVLut is a cuuqtound j)r«>portion ? RAIIO AND PKOPOK'nON. 217 If the terms of one ratio be multiplied by the terms of an other, antecedent by antecedent and consequent by conse- quent, the ratio oi the products is called a Compound Ratio. Thus, if the two ratios 3 : 6 and 4 : 12 be multiplied together, we shall have the compound ratio 3x4 : 6x12, or 12 : 72; in which the ratio is equal to the product of the simple ratios. A proportion formed from the equality of two compound ratios, or from the equality of a compound ratio and a simple ratio, is called a Comjyound Proportion, 228. What part one number is of another. When the standard, or antecedent, is greater than the number which it measures, the ratio is a proper fraction, and is such a part of 1, as the number measured is of the standard. 1, What part of 12 is 3 '? that is, what part of the stand ard 12, is 3 ? T^=i ; or. 12 : 3 : : 1 : i; that is, the number measured is one-fourth of the standard. 2. What part of 9 is 2 ? 3. What part of 16 i&^l. 4. What part of 100 is 201 7. 3 is what part of 12 ? 8. 5 is what part of 20 '? 9. 8 is what part of 56 ? 5. What part of 300 is 200 ? j 10. 9 is what part of 8 ? 6. What part of 36 is 144 ? | 11. 12 is what part of 132 '? Note. — The standard is generally preceded by the word of, and in comparins: numbers, may be named second, as in examples 7, 8, 9, 10 and 11, but it must always be used as a divisor, and Bhould be placed first in the statement. 228 When the standard is greater than the consequent, hew may the ratio he compared ? WTiat part is 3 of 11 H of 1 ! Wliat part is 4of2' l2of:t' Vol;')' 8 21S SINGLE RULE OF 1 HREE. SINGLE RULE OF THREE. 229. The Single Rule of Three is an application of the principle of simple ratios. Three numbers are always giveii and a fourth required. The ratio between two of the given numbers is the same as that between the third and the required number. 1. If 3 yards of cloth cost $12, what will 6 yards cost at the same rate ? Note. — We shall denote the required term »f the proportion by the letter x. STATEMENT. 6 : : 12 OPERATION. 12 Ans. a; — $24. Analysis. — The condition, " at the same rate," requires that the quaidity 3 yards must have the same ratio to the quantity 6 yards, as $12, the cost of 3 yards, to x dol- lars, the cost of 12 yards. Since the product of the two extremes is equal to the product of the two means, (Art. 225), 3 X a; = 6 X 12 ; and if 3 x a; = 6 x 12, a; must be equal to this product divided by 8 : that is, The 4th term is equal to the product of the second and third terms divided by the first. 2. If 56 dollars will buy 14 yards of broadcloth, how many yards, at the same rate, can be bought for 84 dollars ? Analysis. — Fifty-six dollars, (being the cost of 14 yards of cloth), has the same ratio to $84, as 14 yards has to the number of yards which $84 will buy. N"oTB. — When the vertical line is used, the required term, (which is denoted by a;), is written on the left. STATEMENT. $ $ yd. yd, 56 : 84 : : 14 : a;. OPERATION. H $4 21 a; = 21 229. What is the Single Rule of Three? How many numbers are given ? How many required ? What ratio exists between two of the given numbers? SINGLE KULE OF THKEK. 219 230. Hence, we have the following Rule I. Write the number which is of the same kind with the answer for the third term^ the number named in connection with it for the first term, and the remaining number for the second term. II. Multiply the second and third terms together^ and divide the product by the first term : Or, Multiply the third term by the ratio of the frst and second. Notes. — 1. If the first and second terms have different units, they must be reduced to the same unit. 2. If the third term is a compound denominate number, it must be reduced to its smallest unit. 3. The preparation of the terms, and writing them in their pro- per places, is called the statement. EXAMPLES, 1. KI can walk 84 miles in 3 days, how far can 1 walk in 11 days? 2. If 4 hats cost %\2, what will be the cost of 66 hats at the same rate ? 3. If 40 yards of cloth cost $170, what will 325 yards cost at the same rate? 4. If 240 sheep produce 660 pounds of wool, how many pounds will be obtained from 1200 sheep 1 6. If 2 gallons of molasses cost 65 cents, what will 3 hogs- heads cost ? 6. If a man travels at the rate of 210 miles in 6 days, how far will he travel in a year, supposing him not to travel on Sundays ? 7. If 4 yards of cloth cost $13, what will be the cost of o pieces, each containing 25 yards? 8. If 48 yards of cloth cost $67,25, what will 144 y^rdg cost at the same rate ? 9. If 3 common steps, or paces, are equal to 2 yards, how many yards are there in 160 paces 1 10. If 750 men require 22500 rations of bread for a moiilh, how many rations will a garrLson of 1200 men require ? 235. Give the rule for the Ktatcmeut. Give the rule for findin/> the fourth tenn 220 SINGLE RULE OF THREE. 11. A cistern containing 200 gallons is filled by a pipe which discharges 3 gallons in 5 mniutes ; but the cistern has a leak which empties at the rate of" 1 gallon in 5 minuies. If the water begins to run in when the cistern is empty, how long will it run before filling the cistern 1 12. If 14^ yards of cloth cost $19^, how much will 19|- yards cost ? Note, — First make the statement ; then change the mixed numbers to im- proper fractions, after which arrange the terms, and cancel equal factors according to previous in- struction. yard of 144 X yd. 19^ STATEMENT. yd. $ 191 : : 19 0. 4 oth cost J 2 of a dollar, 53 = $26^ what will 13. If I of 1 2^ yards cost ? 14. If YS ^^ ^ ^^^^P ^^^^ <£273 2s. 6d., what will ^^ of her cost 1 15. If ly*, bushels of wheat cost $2|-, how much will 60 bushels cost 1 16. If 4^ yards of cloth cost $9,75, what will 131 yardb cost? 17. If a post 8 feet high cast a shadow 12 feet in length, what must be the height of a tree that casts a shadow 122 feet in length, at the same time of day ? 18. If 7cwt. Iqr. of sugar cost $64,96, what will be the cost uf Acwt. 2qr. 1 19. A merchant failing in trade, pays 65 cents ibr every dollar which he owes: he owes A $2750, and B $1975: how much does he pay each ? 20. If 6 sheep cost $15, and a lamb costs one-third as much as a sheep, what will 27 lambs cost ? 21. If 2tbs. of beef cost -J- of a dollar, what will oOlbs. cost ? 22. If 4^ gallons of molasses cost |2f , how much is it per quart ? 23. A. man receives | oi' his income, and finds it equal to $3724,10 : Jjuw iiiuch^is his whole incuine i BINGLE KULE OK TKiiKK. 221 24. If 4 barrels of flour cost $34|, how much can be bought for $175^? 25. If 2 gallons of molasses cost 65 cents, what will 3 hogsheads cost ? 26. What is the cost of 6 bushels of coal at the rate of £1 145. 6d, a chaldron '? 27. What quantity of corn can I buy for 90 guineas, at the rate of 6 shillings a bushel ? 28. A merchant failing in trade owes ^3500, and his efiects are sold for $2100 : how much does B. receive, to whom he owes $420 1 29. If 3 yards of broadcloth cost as much as 4 yards of cassimere, how much cassimere can be bought for 18 yards of broadcloth] 30. If 7 hats cost as much as 25 pair of gloves, .worth 84 cents a pair, how many hats can be purchased for $216] 31. How many barrels of apples can be bought for $1 14,33, if 7 ban-els cost $21,63] 32. If 27 pounds of butter will buy 45 pounds of sugar, how much butter will buy 36 pounds ol" sugar ] 33. If 42^ tons of coal cost $206,21, what will be the cost of 21 tons ] 34. If 40 gallons run into a cistern, holding 700 gallons, in an hour, and 15 run out, in what time will it be filled ] 35. A piece of land of a certain length and 12^ rods in width, contains 1^ acres, how much would there be in a piece of the same length 26| rods wide] 36. If 13 men can be boarded 1 week for $39,585, what will it cost to board 3 men and 6 women the same time, the women being boarded at half price ] 37. What will 75 bushels of wheat cost, if 4 ' bushels 3 pecks cost $10,687] 38. What will be the cost, in United States money, of 324 yards oqjs. of cloth, at 5ti. 46/. New York currency, for 2 yards? 39. At $1,12-| a square foot, what will it cost t(j pav<» a floor 18 fttet long and 12/^. Oin. wide] 222 CAUSE AND EFFECT. CAUSE AND EFFECT. 231. Whatever produces effects^ as men at work, animals eating, time, goods purchased or sold, money lent, and the like, may be regarded as causes. Causes are of two kinds, simple and compound. A simple cause has but a single element, as men at work, a portion of time, goods purchased or sold, and the like. A compound cause is made up of two or more simple ele- ments, such as men at work taken in connection tvith time, and the like. 232. The results of causes, as work done, provisions con- sumed, money paid, cost of goods, and the like, may be re- garded as effects. A simple effect is one which has but a single elttraent ; a compound effect is one which arises from the multiplication of two or more elements. 233. Causes which are of the same kind, that is, which can be reduced to the same unit, may be compared with each other ; and effects which are of the same kind may likewise be compared with each other. From the nature of causes and effects, we know that 1st Cause : 2d Cause : : 1st Effect : 2d Effect; and, 1st Effect ; 2d Effect : : 1st Cause : 2d Cause. 234. Simple causes and simple effects give rise to simple ratios. Compound causes or compound effects give rise to compound ratios. Note. — Professor H. N. Eobinson, author of a complete course of Diathemaiics, fir.>t made a practical application of the terms " Cause and Effect," in the development of proportion, as published in his arithmetic. By his permission, I have used the same terms, but have somewhat varied the method and rule. 231. Wlfat are causes? How many kinds of causes are there? What is a simple cause? What is a compound cause? 2o2. What are effects? What is a simple effect? What is a com- pound effect? 233 What causes are of the same kind? What causes maybe com- pared with each other? What do we infer from the nature of causes and effects ? 234. What gives rise to simple ratios? DOUBLE RULE OF THREE. 223 DOUBLE RULE OF THREE. 236. The Double Rule of Tliree is an application of the principles of compound proportion. It embraces all that class of" questions in which the causes are compound, or in which the efiects are compound ; and is divided into two parts : 1st. When the compound causes produce the same efiects; 2d. When the compound causes produce different effects. 237. When the cor/ipound causes produce the same effects. 1. If 6 men can dig a ditch in 40 days, what time will 30 men require to dig the same 1 Analysis. — The first cause is compounded of 6 men, and 40 days, the time required to do the work, and is equal to what 1 man would do in 6X40 = 240 days. The second cause is com- pounded of 30 men and the number of days necessary to do the same work, viz : But since the effects are the ^ same, viz : the work done, the causes must be equal ; hence, the products of the elements of the causes are equal. Therefore, in the solution of all like examples, Write the cause containing the unknown element on the left of the vertical line for a divisor^ and the other cause on the right for a dividend. Note— This class of questions has genera iy been arranged under the head of " Rule of Three Inverse." EXAMPLES. 1. A certain work can be done in 12 days, by working 4 hours a day : how many days would it require the same number of men to do the same work, if they worked 6 hours a day? 236. What is the double Rule of Three 1 What cla«s of oue.'stions does it embrace \ Into how many parts is it divided 1 ^^"hat are they 1 237. What is the rule when the effects are equal \ LiiUer what rule has this class of cases been arranged \ STATEMENT. men. men. \ ditch, ditch, 6 : 30 1 . . J . J days. days. \ ' ' 40 : X 240 : 30x« : : I : 1. $0 X 0^ ii2'l DOUBLE RULE OF THKEE. 2. A pasture of a certain extent supplies 30 horses for 18 days : how long will the same pasture supply 20 horses ? 3. If a certain quantity of food will subsist a family of 12 persons 48 days, how long will the same food subsist a family of 8 persons ? 4. If 30 barrels of flour will subsist 100 men for 40 days, how long will it subsist 25 men ? 5. If 90 bushels of oats will feed 40 horses for six days, how many horses would consume the same in 1 2 days 1 6. If a man perform a journey of 22^ days, when the days are 12 hours long, how many days will it take him to per- form the same journey when the days are 15 hours long? 7. If a person drinks 20 bottles of wine per month when it costs 2,s'. per bottle, how much must he drink without increas- ing the expense when it costs 2s. 6d. per bottle ] 8. If 9 men in 18 days will cut 150 acres of grass, how many men will cut the same in 27 days ? 9. If a garrison of 536 men have provisions for 326 days, how long will those provisions last ii' the garrison be increased to 1304 men? 10. A pasture of a certain extent having supplied a body of horse, consisting of 3000, wiih ibrage for J 8 days : how many days would the same pasture have supplied a body of 2000 horse ? 11. What length must be cut off from a board that is 9 inches wide, to make a square foot, that is, as much as is contained in 12 inches in length and 12 in breadth "? 12. If a certain sum of money will buy 40 bushels of oats at 45 cents a bushel, how many bushels of barley will the same money buy at 72 cents a bushel ? 13. If 30 barrels of flour will support 100 men for 40 days, how long would it subsist 400 men 1 14. The governor of a besieged place has provisions for 54 days, at the rate of 2lb. of bread per day, but is desirous of prolonging the siege to 80 days in expectation of succor : what muijt heans a divisor, EXAMPLES. 1. If I pay |24 for the transportation of 96 barrels of flour 200 miles, v/hat must I pay for the transportation of 480 bar- rels 75 miles ? 2. If 12 ounces of wool be sufficient to make Ij yards ol cloth 6 quarters wide, what number of pounds will be required to make 450 yards of flannel 4 quarters wide ? 3. V/hat will be the wages of 9 men for 11 days, if the wages of 6 men for 14 days be $84 '] 4. How long would 406 bushels of oats last 7 horses, if 154 bushels serve 14 horses 44 days? 6. If a man travel 217 miles in 7 days, travelling 6 hours a day, how ikr would he travel in 9 days if he travelled 1 1 hours a day ? 239. What is the rule for finding the unknown part DOUBLJi. RUhli OF THKEIi:. ' 227 6. If 27 men can mow 20 acres of grass in 5J days, work- ing 3| hours a day, how many acres can 10 men mow in 4^ days, by working 8^ hours a day '? 7. How long will it take 5 men to earn $11250, if 25 men can earn $6250 in 2 years ? 8. If 15 weavers, by working 10 hours a day for 10 days, can make 250 yards of cloth, how many must work 9 hours a day for 15 days to make 607 J yards ? 9. A regiment of 100 men drank 20 dollars' worth of wine at 30 cents a bottle : how many men, drinking at the same rate, will require 12 dollars' worth at 25 cents a bottle 1 10. If a Ibotman travel 341 miles in 7i^ days, travelling 12^ hours each day, in how many days, travelling 10 J hours a day, will he travel 155 miles'^ 11. If 25 persons consume 300 bushels of corn in 1 year, how much will 139 persons consume in 8 months, at the same rate 'i 12. How much hay will 32 horses eat in 120 days, if 9C horses eat 3 J tons in 7^ weeks 1 13. If $2,45 will pay for painting a surface 21 feet long and 13i feet wide, what length of surface that- is lOj feet wide, can be painted for $31,72 ? 14. How many pounds of thread will it require to make 60 yards of 3 quarters wide, if 7 pounds make 14 yards 6 quarters wide 1 15. If 500 copies of a book, containing 210 pages, require 12 reams of paper, how much paper will be required to print 1200 copies of a book of 280 pages ? 16. If a cistern 17^ feet long, 101 feet wide, and 13 feet deep, hold 546 barrels of water, how many barrels will a cistern 12 feet long, 10 feet wide, and 7 feet deep, contain? 17. A contractor agreed to build 24 miles of railroad in 8 months, and for this purpose employed 150 men. At the end of 5 months but 1 miles of the road were built : iiow many more men must be employed to finish the road in the time agreed upon ? 18. If 336 men, in 5 days of 10 hours each, can dig a trench of 5 degrees of hardness, 7 yards long 3 wide and 2 deep : what length of trench of 6 degrees of hardness, 5 yards wide and 3 yards deep, may be dug by 240 men in days of 12 hours each '] 228 PAKTNEK8HIP. PARTNERSHIP. 240. Partnership is the joining together of two or more persons in trade, with an agreement to share the profits or losses. Partners are those who are united together in carrying on business. Capital, is the amount of money employed : Dividend is the gain or profit : Loss is the opposite of profit : 241. The Capital or Stock is the cause of the entire profit : Each man's capital is the cause of his profit : The entire profit or loss is the effect of the whole capital : Each man's profit or loss is the effect of his capital : hence, Whole Stock : Each man's Stock : : Whole profit or less : Each man's profit or loss. EXAMPLES. 1. A. and B buy certain goods amounting to 160 dollars, of which A pays 90 dollars and B, 70 ; they gain 32 dollars by the purchase : what is each one's share ? OPERATION. 160 : 90 : : 32 : A's share ; or, x n 18 ar=z|18. ^ 100 I U 1. 160 : 70 : : 32 : B's share ; or, ^ 70 ^^ a: = $14. !Si40. What is a partnership 1 What are partners 1 What is capita or slock ! What is dividend ! What is loss ^ 241. What is the cause of the protit ! What is the cause of each man's profit ? What is the ellcct of the whole capital ! What is the eflect of each man's capital ! \\ hat prooortion exis^ts between cause? and their eliects ' What is the rule .' (X)MP(>UND PARTNEKSiriP. 229 Hence, the following Rule. — An the whole stock in to each man's share, no is the whole guia or loss to each mail's share of the gain or loss. EXAMPLES. 1. A and B have a joint stock of $2100, of which A owns ^1800 and B $300 ; they gain in a year $1000 : what is each one's share of the profits 1 2. A, B and C fit out a ship for Liverpool. A contributes S3 200, B $5000, and C $4500 ; the profits of the voyage amount to $1905 : what is the portion of each ? 3. Mr. Wilson agrees to put in 5 dollars as olten as Mr. Jones puts in 7 ; after raising their capital in this way, they trade for 1 year and find their profits to be $3600 : what is the share of each ? 4. A, B and C make up a capital of $20,000 ; B and C each contribute twice as much as A ; but A is to receive one- third of the profits for extra services ; at the end of the year they have gained $4000 : what is each to receive i 5. A, B and C agree to build a railroad and contribute $18000 of capital, of which B pays 2 dollars and C, 3 dollars as often as A pa)s 1 dollar ; they lose $2400 by the opera- tion : what is the loss of each ? COMPOUND PARTNERSHIP. 242. When the causes ofprojit or loss are compound. "When the partners employ their capital for different periods of time, each cause of profit or loss is compound, being made up of the two elements of capital and tirne. The product of these elements, in each particular case, will be the cause of each man's gain or loss ; and their sum will be the cause of the entire gain or loss : hence, to find each share, Multiply/ each mans stock by the time he contimted it in trade ; then sag, as the sum of the products is to each pro duct , 80 is the whole gain or loss to each mam's sfiare of the gaiii or loss. 242. When is the cause of profit or loss compound 1 What are the elt'iiients of the compound cause ! What i? the rule hi this case ! 230 ooMPoum) pabtjsekshu* EXAMPLES. 1. A and B entered into partnersliip. A put in $840 foi 4 months, and B, $650 for 6 months ; ihey gained $363 ; what is each one's share ? OPERATION. A. 8840x4 = 3360 B. 650x6 = 3900 7260 : I nil : : 363 : j 3360 : : o^o . { $168 A's. $195 B's. 2. A puts in trade $550 for 7 months and B puts in $1625 for 8 months ; they make a profit of $337 : what is the share of each 1 3. A and B hire a pasture, for which they agree to pay $92,50. A pastures 12 horses for ^ weeks and B 11 horses for 7 weeks : what portion must each pay ? 4. Four traders form a company. A puts in $400 for 5 months ; B $600 for 7 months ; C ^960 for 8 months ; D $1200 for 9 months. In the course of trade they lost $750 : how much falls to the share of each ? 5. A, B and C contribute to a capital of $15000 in the following manner : every time A puts in 3 dollars B puts in $5 and C, $7. A's capital remains in trade 1 year ; B's Ij years ; and C's 2^ years ; at the end of the time there is a profit of $15000 : what is the share of each ? 6. A commenced business January 1st, with a capital of $3400. April 1st, he took B into pai Lnership, with a capital of $2600 ; at the expiration of the year they had gained $750 : what is each one's share of the gain ? 7. James Fuller, John Brown and William Dexter formed a partnership, under the firm of Fuller, Brown & Co., with a capital of $20000 ; of whioh Fuller furnished $6000, Brown $5000, and Dexter S9000. At the expiration of 4 months, Fuller furnished $2000 more ; at the expiration of 6 months, Brown furnished $2500 more ; and at the end of a year Dex- ter withdrew $2000. At the expiration of one year and a half, they found their profits amounted to $5400 : what was each partner's share ? PERCENTAGE. 281 PERCENTAGE. 243. Percentage is an allowance made by the hundred. The base of percentage, is the number on which the per- centage is reckoned. Per cent means by the hundred : thus, 1 per cent means 1 for every hundred ; 2 per cent, 2 for every hundred ; 3 per cent, 3 for every hundred, &c. The allowances, 1 per cent, 2 per cent, 3 per cent, &c., are called rates, and may be expressed decimally, as in the following TABLE. 1 per cent is .01 7 per cent is .07 3 per cent is .03 8 per cent is .08 4 per cent is .04 15 per cent is .15 5 per cent is .05 68 per cent is .68 6 per cent is .06 99 per cent is .99 ALSO, 100 per cent is 1. : for, \^ is equal to 1. 150 per cent is 1.50 : for, \^ is equal to 1.50 130 per cent is 1.30 : for, Ifa is equal to 1.30 200 per cent is 2. : for, \^ is equal to 2.00 ^ per cent is .005 : for, T^-r-2 is equal to .005 ^ per cent is .035 : for, 3^=.03 + .005 = .035 5f per cent is .0575 : for, 5^=:=. 05+ .075= 075 examples. Write, decimally, SJ per cent ; 9 per cent ; 6J per cent ; 65J per cent ; 205 per cent ; 327 per cent. 244. To find the percentage of any number. 1. What is the percentage of $320, the rate being 5 per cent? 243. What is per centage "? What is the base 1 What does per cent mean ] What do you understand by 3 per cent ] What is the rate, ot rate per cent ] 244. How du you find the percentage of any number I 232 PERCENTAGE. Analysis. — The . ate being 6 per cent, is ex- oferatiom. pressed decimally by .05. We are then to take 320 05 of the base (which is $320); this we do by ,05 multiplying $320 by .05. Hence, to tind the percentage of a number, $'16,00 Atis Multiply the number hy the rate expressed decimally^ and the product will be the jjercentage. EXAMPLES. 1. What is the percentage of $G57, the rate being 4-J per cent ? Note. — W^hen the rate cannot be reduced to an exact decimal, it is most cojivenient to multiply by the fraction, and then by that part of the rate which is expressed in exact decimals. OPERATION. 657 219=^ per cent. 26281=4 per cent. $28,47 = 4^ per cent. P'ind the percentage of the following numbers : 1. 2-^ per cent of 650 dollars. 2. 3 per cent of 650 yards. 3. 41 per cent of 875cwL 61 per cent of *?37,50. 5| per cent of 2704 miles. J per cent of 1000 oxen. 2| per cent of $376. 10. 66| per cent of 420 cows. 11. 105 per cent of 850 tons. 12. 116 percent of 875/^. 13. 241 per cent of $875,12. 14. 37i per cent of |!200. 15. 33i per cent of $687,24. 16. 87i per cent of $400. 17. 621 per cent of $600. 18. 308 per cent of $225,40. 4. 5. 6. 7. S, 2^*0 P^r cent of 860 sheep. 9. 5f per cent of $327,33. 19. A has $852 deposited in the bank, and wishes to draw out 5 per cent of it : how much must he draw for ? 20. A merchant has 1200 barrels of flour : he shipped 64 per cent of it and sold the remainder : how much did he Bell? 21. A merchant bought 1200 hogsheads of molasses. Oa getting it into his store, he found it short 3J per cent • how many hogsheads were wanting ? 22. What is the diflerence between 5^ per cent of $800 and 6^ per cent of $1050 V 23. Two men had each $240. One of them spends 14 per cent, and the other 18^ per cent : how many dollars more did one spend than the other ? 24. Aman has a capital of $12500: he puts 15 per cent of it in State Stocks : 33^ per cent in Railroad Stocks, and 25 per cent in bonds and mortgages : what per cent has he left, and what is its value ? 25 . A farmer raises 850 bushels of wheat : he agrees to sell 18 per cent of it at §1,25 a bushel ; 50 per cent of it at $1,50 a bushel, and the remainder at $1,75 a bushel : how much does he receive in all "? 245. To find the per cent which one number is of another. 1. What per cent of $16 is $4 ] Analysis. — The question is, what part of operation. $16 is $4, when expres.sed in hundredths: y^z=-i-zr:.25. The standard is $16 (Art. 228) : hence, the qj. 25 per cent. part is ^=^=.25 j therefore, the percent is 25 : hence, to find what per cent one number is of another, Divide by the sta^idard or base, and the quotient, reduced to decinuds, iv ill express the rate per cent. Note. — The standard or basse, is generally preceded by the word of. EXAMPLES. 1. What per cent of 20 dollars is 5 dollars'? 2. Forty dollars is what per cent of eighty dollars ? 3. What per cent of 200 dollars is 80 dollars ? 4. What per cent of 1250 dollars is 250 dollars? 5. What per cent of 650 dollars is 250 dollars 1 6. Ninety bushels of wheat is what per cent of IQQObush. ? 7. Nine yards of cloth is what per cent of 870 yards ? 8. Forty-eight head of cattle are what per cent of a drove of 1600? 9. A man has $550, and purchases goods to the amount of $82,75 : what per cent of his money does he expend ? 245. How do you find the per cent which one number is of another t 234 pekcentagp:. 10. A merchant goes to New York with $1500 ; he first tays out 20 per cent, after M'hich he expends $660 : \vhat per cent was his last purchase of the money that remained after his first 1 11. Out of a cask containing 300 gallons, 60 gallons are drawn : what per cent is this 1 12. If I pay $698,23 for 3 hogsheads of molasses and sell ihem for $837,996, how much do 1 gam per cent on the money laid outl 13. A man purchased a farm of 75 acres at $42,40 an acre. He afterwards sold the same farm for $3577,50 : what was his gain per cent on the purchase money 1 STOCK, COMMISSION AND BROKERAGE. 246. A Corporation is a collection of persons authorized by law to do business together. The law which defines theii rights and powers is called a Charter. Capital or Stock is the money paid in to carry on the business of the Corporation, and the individuals so contributing are called Stockholders. This capital is divided into equal parts called Shares, and the written evidences of ownership are called Certificates. 247. When the United States Government, or any of the States, borrows money, an acknowledgment is given to the lender, in the form of a bond, bearing a fixed interest. Such bonds are called United States Stock, or State Stock. The par value of stock is the number of dollars named in each share. The market value is what the stock brings per nhare when sold for cash. If the market value is above the par value, the stock is said to be at a premium, or above par ; but if the market value is below the par value, it is said to be at a discount, or below par. 246. What is a corporation 1 What is a charter 1 What is capital 01 stock 1 What are shares 1 217. What are United States Stocks^ What are State Stocks 1 What is the par value of a stock 1 What is the market value! If the market is above the par value, what is said of the stock 1 If it is below, what is said of the stock 1 What is the market value when above pari \Miat when below 1 COMMISSION AND BKOXERACJB. 235 Let 1 = par value of 1 dollar : I H- premium ^market value of 1 dollar, when above par : 1 — discount =:maket value of I dollar when below par. 248. Commission is an allowance made to an agent for buying or selling, and is generally reckoned at a certain rate per cent. The commission, for the purchase or sale of goods in the city of New York, varies from 2^ to 1 2^ per cent, and under some circumstances even higher rates are paid. Brokerage is an allowance made to an agent who buys or sells stocks, uncurrent money, or bills of exchange, and is generally reckoned at so much per cent on the par value of the stock. The brokerage, in the city of New York, is gene- rally one-fourth per cent on the par value of the stock. EXAMPLES. 1. What is the commission on $4396 at 6 per cent 1 OPERATION. Note. — We here find the commission, as $4396 in simple percentage, by multiplying by the de- 06 cimal which expresses the rate per cent. . . -,^ '„, ^ ^ Ans. $263,76. 2. A factor sells 60 bales of cotton at $425 per bale, and is to receive 2^ per cent commission : how much must he pay over to his principal ? 3. A drover agrees to purchase a drove of cattle and to sell them in New York city for 5 per cent on what he may re- ceive ; he expends in the purchase $4250, and sells them at an advance of 10 per cent : how much is his commission % 4. A commission merchant sells goods to the amount ol $8750, on which he is to be allowed 2 per cent, but in con- sideration of paying the money over before it is due, he is to receive 1 J per cent additional : how much must he pay ovei to his principal ? 5. A broken bank has a circulation of $98000 and pur- chases the bills at 85 per cent*: how much is made by the operation '\ 248. What is coinmissiou 1 What m brokerape! 236 TMCIMIONTAGE. 6. Merchant A sent to B, a broker, $3825 to be invested in stock ; B is to receive 2 per cent on the amount paid for the stock : what was the value of the stock purchased 1 Analysis. — Since the broker re- seives 2 per cent, it will require fl.02 to purchase 1 dollar's worth of stock; hence, there will be as 765 many dollars worth purchased as 714 $1.02 is contained times in $3825 that is, $3750 worth. OPERATION. 1.02)3825.00($3750Jn». 306 510 510 7. Mr.' Jones sends his broker $18560 to be invested in U. S. Stocks, which are 15 per cent above par ; the broker is to receive one per cent ; how many shares of $100 each can be purchased ? Analysis. — Since the premium is 15 per cent, and the brokerage 1 per cent, operation. each dollar of par value will cost $1 1.16)18560 plus the premium plus the brokerage^ " ^TTT.T^ nnntlpnt $1.16: hence, the amount purchased ^IbUUU quotient, will be as many dollars as $1.16 is or, IbO shares, contained times in $18560. 8. I have $5000,89 to be laid out in stocks, which are 15 per cent below par : allowing 2 per cent commission, how much can be purchased at the par value 1 Analysis. — Since the stock is at a dis- count of 15 per cent, the market value will operation. be 85 per cent; add 2 per cent, the broker- .87)5000,89 age. gives 87 per cent=.S7. The amount ^5747 Ans purchased will be as many dollars as .87 is ^ ' contained times in $5000,89. Hence, to find the amount at par value, Divide the amount to be expended by the market value of $1 plus the brokerage ]; and the quotient will be the amount in par value. 9. Messrs. Sherman & Co. rfeceive of Mr. Gilbert $28638,50 to be invested in bank stocks, which are 121 per cent above par, for which they are to receive one-fourth of one per cent commission : how many shares of $127 each can they buy 1 LOSS Oii GAIN. 237 10. The par value of Illinois Railroad stock is 100. It Eells in market at 72^: if I pay | per cent brokerage, how many shares can I buy for $5820 ? PROFIT AND LOSS. 249. Profit or loss is a process by which merchants dis- cover the amount gained or lost in the purchase and sale of goods. It also instructs them how much to increase or diminish the price of their goods, so as to make or lose so much per cent. EXAMPLES. 1. Bought a piece of cloth containing Ibyd. at $5,25 per yard, and sold it at $5,75 per yard : how much was gained in the trade % OPERATION. Analysis. — We first find the $5,75 price of 1 yard. profit on a single yard, and then $5,25 cost oi" 1 yard. multiply by the number of yards, 77- , r^ ■, ^ which is 75. ^'^^'^^■- P>-o^»t on 1 yard : then, $0,50x75:::= $37,50. 2. Bought a piece of calico containing 56 yards, at 27 cents a yard : what must it be sold ibr per yard to gain $2,24 ? OPERATION. 56 yards at 27 cents ==$15, 12 An.alysis. — First find the Profit - - - 2,24 cost, then add ilie profit and tx ^ n /• 7 ■, -> ^n ' divide the sum by the number It must sellfor - $17,36. of yards. 56)17,36 31 cents. 250. Knowing the per cent of gain or loss and tlie amount received, to find the cost. 1. 1 sold a parcel of goods for $195,50, on which I made 15 per cent : what did they cost me ? Analysis. — 1 dollar of the cost plus \6 per operation. cent, will be what that which cost Si sold lor, 1.15)195,50 viz., Si, 15 : hence, there will be as many ^Tto — /I dollars of cot«t, as $1.15 is contained times in % J . what the goods brought. '249. What is loss or gain 238 I'EKCEWTAGE. 2. [[ I sell a parcel of jroods for ^170, by which 1 losc 15 per cent, what did they cost I Analysis. — 1 dollar of the cost less 15 per operatiok. cent, will be what that which cost 1 dollar sold .85)170 for, viz., $0,85 : hence, there will be as many skQrul" A dollars of cost, as .85 is contained times in ^ . what the goods brought. Hence, to find the cost, Divide the aniou?tt received by 1 jplus the per cent when there is a gain, and by 1 minus the per cent when there is a loss, and the quotient will be the cost. EXAMPLES. 1. Bought a piece of cassimere containing 28 yards at IJ dollars a yard ; but finding it damaged, am f^'illing to sell it at a loss of 15 per cent: how much must be asked per yard ? 2. Bought a hogshead of brandy at $1,25 per gallon, and Bold it for |78 : was there a loss or gain ? 3. A merchant purchased 3275 bushels of wheat for which he paid |3517,10, but finding it damaged, is willing to lose 10 per cent : what must it sell for per bushel? 4. Bought a quantity of wine at $1,25 per gallon, but it proves to be bad and am obliged to sell it at 20 per cent less than I gave : how much must 1 sell it for per gallon ? 5. A farmer sells 125 bushels of corn for 75 cents per bushel ; the purchaser sells it at an advance of 20 per cent : how much did he receive for the corn ? 6. A merchant buys 1 tun of wine for which he pays $725, and wishes to sell it by the hogshead at an advance of 15 per cent : what must be charged per hogshead ? 7. A merchant buys 158 yards of calico for which he pays 20 cents per yard ; one-half is so damaged that he is obliged to sell it at a loss of 6 per cent : the remainder he sells at aii advance of 19 per cent : how much did he gain ? 8. If I buy coflee at 1 6 cents and sell it at 20 cents a pound, how much do I make per cent on the money paid '\ 250. Knowing the per cent of gain or loss and the amount received, £io\v do jou find the cost ' IN8UKAN0E. 289 9 A man bought a house and lot for '^1850,50, and sold it for $1517,41 : how much per cent did he lose. ? 10. A merchant bought 650 pounds oi' cheese at 10 cents per pound, and sold it at 12 cents per pound : how much did he gain on the whole, and how much per cent on the money laid out ? 11. Bought cloth at $1,25 per yard, which proving bad, I wish to sell it at a loss of 18 per cent: how much must 1 ask per yard ? 12. Bought 50 gallons of molasses at 75 cents a gallon, 10 gallons of which leaked out. At what price per gallon must the remainder be sold that I may clear 10 per cent on the cost ? 13. Bought 67 yards of cloth for $112, but 19 yards being spoiled, I am willing to lose 5 per cent : how much must 1 sell it for per yard ? 14. Bought 67 yards of cloth for $112, but a number of yards being spoiled, I sell the remainder at $2,21 6J per yard, and lose 5 per cent : how many yards were spoiled ? 15. Bought 2000 bushels of wheat at $1,75 a bushel, from which was manufactured 475 barrels of flour : what must the flour sell for per barrel to gain 25 per cent on the cost of the wheat? INSURANCE. 251. Insurance is an agreement, generally in writing, by which an individual or company bind themselves to exempt the owners of certain property, such as ships, goods, houses, &c., from loss or hazard. The Policy is the written agreement made by the parties. Premium is the amount paid by him who owns the property to those who insure it, as a compensation for their risk. The premium is generally so much per cent on the prop»ity in- sured. EXAMPLES. 1. What would be the premium for the insurance of a house valued at $8754 against loss by fire for one year, at J per cent? 251. What is insurance 1 What is the policy ? What is the pre- luiuui 1 How is it reckoned ? 240 PEllOENTAGE. 2. What Avauld be the premium tor insuring a ship anJ cargo, valued at ^37500, liom New York to Liverpool, at 3^ per cent ? 3. What would be the insurance on a ship valued at $47520 at ^ per cent ; also at ^ per cent? 4. What would be the insurance on a house valued at $14000 at 1^ per cent? 5. What is the insurance on a store and goods valued at $27000, at 2J percent? 6. What is the premium of insurance on $9870 at 14 per cent? 7. A merchant w^ishes to insure on a vessel and cargo at sea, valued at $28800 : what will be the premium at Ij per cent ? 8. A merchant owns three-fourths of a ship valued at $24000, and insures his interest at 2^ per cent : what does he pay for his policy ? 9. A merchant learns that his vessel and cargo, valued at S36000, have been injured to the amount of $12000 ; he eflects an insurance on the remainder at 5^ per cent ; what premium does he pay ? 10. My furniture, worth $3440, is insured at 2| per cent ; my house, worth $1000, at 1^ per cent ; and my barn, horses and carriages, worth $1500, at 3;J- percent: what is the whole amount of my insurance ? 11. A man bought a house, and paid the insurance at 2^ per cent, the whole of which amounted to $1845 : what was the value of the house and the amount of the insurance? 12. What would it cost to insure a store, worth $3240, at I per cent, and the stock, worth $7515,75, at ^ per cent ? 13. A merchant imported 250 pieces of broadcloth, each piece containing 36^ yards, at $3,25 cents a yard. He paid 4J per cent insurance on the selling price. $4,50 a yard. If the goods were destroyed by fire, and he got the amount of insurance, how much did he make ? 14. A vessel and cargo, worth $65000, are damaged to the amount of 20 per cent, and there is an insurance of 50 per cent on the loss : how much will the owner receive ? INTEREST. 241 INTEREST. 252 Interesi is an allowance made for the use of money that is borrowed. Phincipal is the money on which interest is paid. Amount is the sum of the Principal and Interest. For example : If I borrow 1 dollar of Mr. Wilson for 1 vear, and pay him 7 cents for the use of it ; then, 1 dollar is the pinncipal, . 7 cents is the interest, and ^1,07 the amount The RATE of interest is the number of cents paid for the use of 1 dollar for 1 year. Thus, in the above example, the rate is 7 per cent per annum. Note. — The term per cent meaus, by the hundred; and fer annum means by the year. As interest is always reckoned by the year, the term per annum is understood and omitted. CASE I. 253. To find the interest of any principal for one or more years. 1. What is the interest of $1960 for 4 years, at 7 per cent ? Analysis. — The rate of interest being 7 per cent, is expressed deci- operation. mally by .07 : hence each dollar, in $1960 1 year, will produce .07 of itself, and .07 rate. $1960 will produce .07 of $1960, ,o^ o n • ^ V i or $137,20. Therefore, $137,20 is the 1^7,20 int. for \yr. interest for 1 year, and this interest j^ ^^- ^^ y^ars. rauhiplied by 4, gives the interest for |548,80 4 years : hence, the following Rule. — Multiply the pi-incipal by the rate, exjyressed decimally, and the product by the number of years. 252. What is interest ■ What is principal ^ What is amount ? What is rate of interest 1 What does per annum mean 1 253. How do you find the interest of any principal for any number ot' years ^ Tiivp the analy:-is 10 242 SIMPLE INTJCliEBT. EXAMPLES. 1. What is the interest of $365,874 for one year, at 5j percent? OPERATION. Analysis.— We first find the in- ^^^^'^^ti terest at i per cent, and then the '}_zJ interest at 5 per cent; the sum is 1,82937 ^ per cent, the interest at 5i per cent. 18,29370 5 per cent. Ans. 12042307 5^ per cent 2. What is the interest of $650 for one year, at 6 per cent ? 3. What is the interest of $950 for 4 years, at 7 per cent ? 4. What is the amount of 13675 in 3 years, at 7 per cent ? 5. What is the amount of $459 in 5 years, at 8 per cent? 6. What is the amount of $375 in 2 years, at 7 per cent ? 7. What is the interest of $21 1,26 for 1 year, at 4^ per ct. ? 8. What is the interest of $1576,91 for 3 years, at 7 per ct. ? 9. What is the amount of $957,08 in 6 years, at 31 per ct. ? 10. What is the interest of $375,45 for 7 years, at 7 per ct. ? 11. What is the amount of $4049,87 in 2 years, at 5 per ct. 1 12. Whatisthe amount of $16 199,48 in 16 yrs.,at 5^perct.? Note. — When there are years and months, and the months are aliquot parts of a year, multiply the interest for 1 year by the yean i»nd months reduced to the fraction of a year. EXAMPLES. 1. What is the interest of $326,50, for 4 yeais and 2 months, at 7 per cent '? 2. What is the interest of $437,21, for 9 years and 3 months, at 3 per cent ? 3. What is the amount of $1119,48, after 2 years and 6 months, at 7 per cent 1 4. What is the amount of $179,25, after 3 years and 4 months, at 7 per cent ? 5. What is the amount of $1046,24, after 4 years and 3 uiuulhs, at 5J per cent? grMl'LlC INTKRKST. 243 CASE II. 254. To find the interest on a given principal for any rate »nd time. 1. What is the interest of $876,48 at 6 per cent, for 4 years 9 months and 14 days? Analysis. — The interest for 1 year is the product of the jirinci- pal rnujtiplied by the rate. If the interest for 1 year be divided by 12, the quotient will be the interest for 1 month : if the interest for 1 month be divided by 30. the quotient will be the interest for 1 day. The interest for 4 years is 4 times the interest for 1 year : the interest for 9 months, 9 times the interest for 1 month ; and the interest for 14 days, 14 times the interest for 1 day. OPERATION. $876,48 .06 12)52,5888r=int. for lyr. 52,5888 X 4=$210,3552 4yr. 30)4^3824 rz: int. for 1 mo. 4,3824 x 9 = $ 39,4416 9mo ,14608=:int. for Ida. ,14608 xl4r^ ^ 2,.0451 Uda Total interest, $251,84424- Hence, we have the following Rule. — I. Find the interest for 1 year: II. Divide this interest by 12, and the quotient will be the interest for 1 month: III. Divide the interest for 1 month by 30, and the quo- tient will be the interest for 1 day. IV. Multiply the interest for 1 year by the number of years., the interest for 1 month by the number of months., and the interest for 1 day by the number of days, and the sum f the products will be the required interest. Note. — In computing interest the month is reckoned at 30 days. 2. What is the interest of $132,26 for 1 year 4 months ..nd 10 days, at 6 per cent per annum? 3. W'hat is the interest of 825,50 for 1 year 9 months and 12 days, at per cenf? 2S4. How do ywu find the iiittrekt for uiiy tunc at any rule ' 244 8IMPI.E INTEKEfeT. 2d method. 255. There is another rule resulting from the last analysis, which is regarded as the best general method of computing interest. Rule. — I. Find Ike interesl for 1 year and divide it by 12 : the quotient will be the interest for 1 month. II. Multiply the interest for 1 month by the time expressed in months and parts of a month^ and the product will be the required interest. Note. — Since a month is reckoned at 30 days, any number of days is reduced to decimals of a month by dividing the days by 3. EXAMPLES. 1. What is the interest of $327,50 for 3 years 7 mouths and 13 days, at 7 per cent? OPERATION. 3y/rs. = 36wos. $327,50 Imos. .07 13 days- .^mos. 12)2 2.9250 zuint. fbr 1 year. Time=z43.4^-mos. 1.91044- =int. tor 1 month. Note.— Tiie method em- 43.4 1 —time in months. j)loyed, and the number of .6308 decimal places used, in com- 76416 puting interest, may affect 57312 the mills, and possibly, the nf,A-\a last figure in cents. It is best to use 4 places of decimals. $82.97504 Ans. 2. What is the interest of ^1728,00, at 7 per cent, for 2 years 6 months and 21 days'? 3. What is the interest of $288,30, at 7 per cent, for I year 8 months and 27 days 1 4. What is the interest of $576,60, at 6 per cent, for 10 months and 18 days ? 5. What is the interest of $854,42, at 6 per cent, fbr 3 months and 9 days ? 6. What is the interest of $1153,20, at 6 per cent, for I I months and 6 days ? 265. How do you find the interest for ycirb, njoiitlis and (hiyt* by the sccund lactliod ! SIMPLE i:^'TEKl!:ST. 2iL 7. "What is the interest of $2306,54, at 5 per cent, fot 7 mouths and 28 days'? 8. What is the interest of $4272,10, at 5 per cent, ibi 1 months and 28 days ? 9. What is the interest of $1620, at 4 per cent, for 5 years lid 24 days ? 10. What is the interest of $2430,72, at 4 per cent, for 10 years and 4 months ? 11. What is the interest of $3689,45, at 7 per cent, for 4 years and 7 months ? 12. What is the interest of $2945,96, at 7 per cent, for 7 years and 3 days 1 13. What is the interest, at 8 per cent, of $675,89, for b years 6 months and 6 days ? 14. What is the interest, at 8 per cent, on $12324, for 3 years and 4 months ? 15. What is the interest, at 9 per cent, on $15328,20, ibr 4 years and 7 months ? 16. What is the interest of $69450 for 1 year 2 montha and 12 days, at 9 per cent ? 17. What is the interest of $216,984 for 3 years 5 months and 15 days, at 10 per cent? 18. What is the interest of $648,54 for 7 years 6 months, at 4i per cent ? 19. What is the interest of $1297,10 for 8 years 5 months, at 5 J per cent ? 20. What is the interest of $864,768 for 9 months 25 day?, at 6 J per cent? 21. What is the interest of $2594,20 for 10 months and 9 days, at 7^ per cent i 22. What is the amount of $2376,84 for 3 years 9 months and 12 days, at 8^ per cent '? 23. What is the amount of $5148,40 for 7 years 11 months and 23 days, at 9^ per cent ? 24. What is the amount of $3565,20 lor 3 years 9 months, at 10-J [^»er cent ^ 246 SIMPLE INTEREST. 25. What is the amount of $125,75 for 1 year 9 months and 27 days, at 7 per cent 1 26. What is the amount of $256 for 10 months 15 days, at 7^ per cent ? 27. What is the interest on a note of $264,42, given Janu- ary 1st, 1852, and due Oct. 10th, 1855. at 4 per cent? 28. Gave a note of $793,26 April 6th, 1850, on interest at 7 per cent : what is due September 10th, 1852 1 29. What amount is due on a note of hand given June 7th, 1850, for $512,50, at 6 per cent, to be paid Jan. 1st, 1851 ? 30. What is the interest on $1250,75 for 90 days, at 10 per cent? 31. What is the amount of S71,09 from Feb. 8th, 1848, to Dec. 7th, 1852, at 6f per cent? 32. What will be due on a note of $213,27 on interest after 90 days, at 7 per cent, given May 19th, 1836, and pay- able October 16th, 1838? 33. What is the interest of $426,54, from Augu<3t 15th, 1837, to March 13th, 1840, at 7 per cent ? 34. What is the interest of $2132,70, from Nov. 17th, 1838, to Feb. 2d, 1839, at 7i per cent? 35. What is the interest of $38463, from April 27th, 1815, to Sept. 2d, 1824, at 8 per cent? 36. What is the interest of $14231,50, from June 29th, 1840, to April 30th, 1845, at 8} per cent? 37. What is the interest of $426,50, from Sept. 4th, 1843, to May 4, 1849, at 9 per cent? 38. What is the interest of $4320, from Dec. 1st, 1817, to Jan. 22d, 1833, at 9i per cent? 39. What is the amount of $397,16, from March 23, 1824, to March 31st, 1835, at 101 per cent ? 40. What is the amount of $328,12, from July 4th, 1809, to Feb. 15th, 1815, at 3 per cent ? 41. What is the amount of $164,60, from Sept. 27th, 1845, to March 24th, 1855, at 1^ per cent? 42. What is the amount of $1627,50, from July 4th, 1839. to August Ifct, 1855, at 8 per cent ? PAKTIAI. rATMENTS. 247 CASE III. 256. When the principal is in pounds shillings and pence. 1. What is the interest, at 7 per cent, of £27 I65. Qri., for 2 years ? OPERATION. Analysis.— The interest on pounds £27 155. 9^Z.=^27.7875 and decimals of a pound is found in _07 the same way as the interest on doJ- '^ lars and decimals of a dollar: after l.J4ol20 which the decimal part of the interest ^ may be reduced to shillings and £3.890250 pence : hence, £.89025 =z 1 75. ^d. Ans, £3 175. ^d. 1. Reduce the shillings and pence to the decimal of a pound and afinex the result to the pou7ids. II. Find the interest as though the sum were United States Money, after which reduce the decimal part to shil- ings and pence. 2. What is the interest of £67 195. 6c?., at 6 per cent, for 3 years 8 months 16 days ? 3. What is the interest of £127 155. 4(i., at 6 per cent, for 3 years and 3 months '] 4. What is the interest of £107 I65. lOc?., at 7 per cent, for 3 years 6 months and 6 days ? 5. What will £279 135. 80 for 2 years, the interest being added yearly ? OPERATION. $3750,000 principal for 1st year. 13750 X. 07= 262,500 interest for 1st year. 4012,50^0 principal for 2d " $40 2,50x.07= _280^875 interest for 2d ** 4293,375 amount at 2 years. 1st principal 3750,000 iraount of interest $543,375. ?, If the- interest be computed annually, what will be the compound interest on $100 for 3 years, at 6 per cent? 3. What will be the compound interest on $295,37, at 6 per cent, for 2 years, the interest being added annually ? 4. What will be the compound interest, at 5 per cent, of ^1875, for 4 years'? 5. What is the amount at compound interest of $250, for 2 years, at 8 per cent % 6. What is the compound interest of $939,64, for 3 years, at 7 per cent 1 7. What will $125,50 amount to in 10 years, at 4 per cent compound interest? 5i(iO. What of compound interetit 1 How do you compute it 1 252 OOMJ'OUNI) INTEKEST. NoT£. — The operation is rendered much shorter .and easier, by taking the amount of 1 dollar for any time and rate given in the following table, and multiplying it by the given principal; the product will be the required amount, from which subtract the given principal, and the result will be the compound interest * TABLE. Which shows the amount of $1 or £1. compound interest, from I year to 20, and at the rate of 3, 4, 5, 6, and 7 per cent. Years. 3 per cent. 4 per cent. ,5 per cent. 6 per cent. 7 percent. /ears. 1 1 1.03000 1.04000 1.05000 1.06000 1.07000 2 1.06090 1.18160 1.10250 1.12360 1.14490 2 3 1.09272 1.12486 1.15762 1.19101 1.22504 3 4 1.12550 1.16985 1 1 21550 1.26247 1.31079 4 5 1.15927 1.21065 j 1.27628 1.33822 1.40255 5 6 1.19405 1.20531 I 1.34009 1.41851 1.50073 6 7 1.22987 1.31593 1.40710 1.50363 1.60578 7 S 1.26677 1.36856 1.47745 1.59384 1.71818 8 9 1.30477 1.42331 1.55132 1.68947 1.83845 9 10 1.34391 1.48028 1.62889 1.79084 1.96715 10 11 1.38423 1.53945 1.71033 1.89829 2.10485 11 12 1.42576 1.60103 1.79585 2.01219 2.25219 12 13 1.46853 1.66507 1.88564 2.13292 2.40984 13 14 1.51258 1.73167 1.97993 ' 2.26090 2.57853 14 15 1.55796 1.80094 2.07892 2.39655 2.75903 15 16 1.60470 1.87298 2.18287 2.54035 2.95216 16 17 1.65284 1.94790 2.29201 2.69277 3.15881 17 18 1.70243 2.02581 2.40661 2.85433 3.37993 18 19 1.75350 2.10684 2.52695 3.02559 3.61652 19 20 1.80611 2.19112 2.65329 3.20713 3.86968 20 Note. — When there are months and days in the time, find the amount for the years, and on this amount cast the interest for the months and days : this, added to the last amount, will be the re- quired amount for the whole time. 8. What is the amount of $96,50 for 8 years and 6 raoiiths,, interest being compounded annually at 7 per cent ? 9. What is the compound interest of ^300 for 5 years £ months and 15 days, at 6 per cent ? 10. What is the compound interest of $1250 lor 3 years 3 months and 24 days, at 7 per cent 1 11. What will $56,50 amount to in 20 years and 4 ruonthjj, ut 5 per cent compound interest l * Tlio n;>ti't may dill'tr in tiic mills jilatc (nnii thai (»l»tui"cd l-v t'i» other rule DIBOOUNT. 253 DISCOUNT. 261. Discount is an allowance made lor the payment of money before it is due. The face of a note is the amount named in the note.* Note. — Days of grace are days allowed for the payment ol a nole after the expiration of the time named on its face. By mercantile usage a nole does not legally fall due until 3 days after the expiration of the time named on its face, unless the note specifies without grace. Days of grace, however, are generally confined to mercantile paper and to notes discounted at banks. 262. The present value of a note is such a sum as being put at interest until the note becomes due, would increase to an amount equal to the face of the note. The discount on a note is the difference between the face of the note and its present value. 1. I give my note to Mr. Wilson for $x07, payable in 1 year : what is the present value of the note, if the uiterest is 7 per cent ? what the discount ? ePERATlON. Analysis.— Since 1 dollar in 1 year' $107-4-1,07 = $100. at 7 per cent, will amount to $1,07, the proof. present value will be as many dollars Int. $100 lyr.=^$ 7 as $1,07 is contained times in the face Principal, 100 of the note : viz., $100: and the dis- . ^ STTTT^ . „.: n t. . tfi. , rv'n lI^,r^/._rT . V Amount, ^107 Discount, 7 count will be $107— $100^$7 : hence, Divide the face of the note by 1 dollar plus the interest oj 1 dollar for the yiven tirne^ and the quotient will be tlie pre- sent value: take this sum from the face of the note and tlu reTiiainder will be the discount. 261. What is discount 1 What is the face of a note'' What are ilays of grace 1 262. What is present value ? What is the discount 1 How do yuv fuiJ the present value of a note ? * Sec Appendix, page UJO. 254 DISCOUNT. EXAMPLES. 1. What is the present value of a note for $1828,75, due in 1 year, and bearing an interest of 41 per cent ? 2. A note of $1651,50 is due in 11 months, but the person to whom it is payable sells it with the discount off at 6 per cent : how much shall he receive ? Note. — When payments are to be made at different times, find the present value of the sums separately, and their sum will be the present value of the note. 3. What is the present value of a note for $10500, on which 1900 are to be paid in 6 months ; $2700 in one year ; $3900 in eighteen months / and the residue at the expiration of two years, the rate of interest being 6 per cent per annum ? 4. What is the discount of £4500, one-half payable in six months and the other half at the expiration of a year, at 7 per cent per annum 1 5. What is the present value of $5760, one-half payable in 3 months, one-third in 6 months, and the rest in 9 months, at 6 per cent per annum ? 6. Mr. A gives his note to B for $720, one-half payable m 4 months and the other half in 8 months ; what is the present value of said note, discount at 5 per cent per annum ? 7. What is the diiierence between the interest and discount of $750, due nine months hence, at 7 per cent? 8. What is the present value of $4000 payable in 9 months, discount 4J per cent per annum ? 9. Mr. Johnson has a note against Mr. Williams for $2146,50, dated August 17th, 1838, which becomes due Jan. 11th, 1839 : if the note is discounted at 6 percent, what ready money must be paid for it September 25th, 1838 1 10. C owes D $3456, to be paid October 27th, 1842; C wishes to pay on the 24th of August, 1838, to which D con- sents ; how much ought D to receive, interest at 6 per cent ? 11. What is the present value of a note of $4800, due 4 years hence, the interest being computed at 5 per cent per annum ? 12. A man having a horse for sale, offered it for $225 cash in hand, or $230 at 9 months ; the buyer chose the latter : did tlie seller lose or make by his offer, supposing money to be worth 7 per cent ? BANK DISUUUNT. 256 BANK DISCOUNT. 263. Bank Discount is the charge made by a bank for the payment of money on a note before it becomes due. By the custom of banks, this discount is the interest on the amount named in a note, calculated from the time the note is discounted to the time when it falls due ; in which time the three days of grace are always included. The interest is always paid in advance. Rule. — Add 3 da^JS to the time which the note has to run, and then calculate the interest for that time at the given rate, EXAMPLES. 1. What is the bank discount of a note for $350, payable months after date, at 7 per cent interest ? 2. What is the bank discount of a note of $1000 payable in 60 days, at 6 per cent interest ? 3. A merchant sold a cargo of cotton for $15720, for which he receives a note at 6 months : how much money will he receive at a bank for this note, discounting it at 6 per cent interest ? 4. What is the bank discount on a note of $556,27 paya- ble in 60 days, discounted at 6 per cent interest ? 5. A has a note against B for $3456, pa) able in three months ; he gets it discounted at 7 per cent interest . how much does he receive % &^ What is the bank discount on a note of $367.47, having 1 year, 1 month, and 13 days to run, as shown by the face of the note, discounted at 7 per cent 1 7. For value received, I promise to pay to John Jones, on the 20th of November next, six thousand five hundred and seventy-nine dollars and 15 cents. What will be the discount on this, if discounted on the 1st of August, at 6 per cent per annum 1 263. What is bank discount ? How is interest calculated by the custom of banks ^ How is the intereftt paid 1 How do you liuJ U»« mteicKtl '256 liANX DIBOOUNT. 8. A merchant bought 175 barrels of flour at $7,50 cents a barrel, and sells it immediately for $9,75 a barrel, for which he receives a good note, payable in 6 months. If he should get this note discounted at a bank, at 6 per cent, what will be his gain on the flour '? 264. To make a note due at a future iime, whose present value shall be a given amount. 1. For what sum must a note be drawn at 3 months, so that when discounted at a bank, at 6 per cent, the amount received shall be $500 1 Analysis. — If we find the interest oii 1 dollar for the given time, and then subtract that interef^t from 1 dollar, the remainder will be the present value of 1 dollar, due at the expiration of that time. Then, the number of times which the present value of the note contains the present value of 1 dollar, will be the num- ber of dollars for which the note must be drawn : hence, Divide the present value of the note by the present value of 1 dollar, reckoned- for the saw.e time and at the same rate of interest^ and the quotient will be the face of the note. OPERATION. Interest ol $1 for the time, 3mo. and 3\Ti : the remainder will be the amount to be raised on the properly. Having done this, divide the whole tax to be raised by the amount of taxable 'pmpeny, and the quotient will be tJte tax on ^1. Then multiply thie quotient by the inventory of each individual, and the product will be the tax on his property. EXAMPLES. 1. A certain town is to be taxed $4280 ; the property on which the tax is to be levied is valued at $1000000. Now there are 200 polls, each taxed $1,40. The property of A is valued at S2800, and he pays 4 polls B's at $2400, pays 4 polls. E's at $7242, pays 4 polls. C's at $2530, pays 2 " F's at $1651, pays 6 " D's at $2250, pays 6 " G's at $1600,80 pays 4 " What will be the tax on 1 dollar, and what will be A's tax, and also that of each on the list ? First, $1,40x200 = $280 amount of poll-tax. $4280— $280 = 4000 amount to be levied on proi.»erty. Then, $4000-h$1000000 = 4 mills on $1. Now, to find the tax of each, as As, for example, A's inventory - - - $2800 ,004 11,20 4 polls at $1,40 each 5,60 A's whole tax - - - $16,80 In the same manner the tax of each person in the town- ship may be found. Having found the per cent, or the amount to be raised on each dollar, form a table showing the amount which certain sums would produce at the same rate, per cent. Thus, after having found, as in the last example, that 4 mills are to be raised on every dollar, we can, by multiplying in succession by the numbers 1, 2, 3, 4, 5, 6, 7, 8, &c., form the following 267. WTiat is the first thing to be done in assessing a tax 1 If there IS a poll-tax, how do you find the amount] How then do you find the per cent of tax to be levied on a dollar 1 How do you then find the ajjjount to be levied on cat-h individual t 202 ASSESSING TAXES. TABLE. $ $ $ $ $ $ 1 gives 0.004 20 gives 0.080 300 gives 1.200 2 ' ' 0.008 30 ' 0.120 400 " 1.600 • 3 ' * 0.012 40 ' 0.160 500 " 2.000 4 ' ♦ 0.016 50 ' 0.200 600 " 2.400 5 ' ' 0.020 60 « 0.240 700 •' 2.800 6 ' ' 0.024 70 ' 0.280 800 " 3.200 7 ' ' 0.028 80 ' 0.320 900 " 3.600 8 ' ' 0.032 90 ' 0.360 1000 " 4 000 9 ' ' 0.036 100 ' 0.400 2000 " 8.000 10 * ' 0.040 200 ' 0.800 3000 " 12.000 This table shows the amount to be raised on each sum in the columns under $'s. EXAMPLES. 1. Find the amount of B's tax from this table. B's tax on 12000 - - is - $8,000 B's tax on 400 - - is - $1,600 B's tax on 4 polls, at $1,40 - $5,600 B's total tax - - is - $15,200 2. Find the amount of C's tax from the table. C's tax on $2000 C's tax on 5a0 C's tax on 30 C's tax on 2 polls C's total tax - - $8,000 - $2,000 - $0,120 - $2,800 s - $12,920 paid In a similar manner, we might find the taxes to be by D, E, &c. 3. If the people of a town vote to tax themselves $1500, to build a public hall, and the property of the town is valued at $300,000, what is D's tax, whose property is valued at $2450? 4. In a school district a school is supported by a tax on the property of the district valued at $121340. A teacher is employed lor 5 months at $40 a month, and contingent ex- penses are $42,68 ; what will be a farmer's tax whose property is valued at $3125 ? CX)TNS AND GUKEENOY. 268 COINS AND CURRENCY. 268. Coins are pieces of metal, of gold, silver, or copper, of fixed values, and impressed with a public stamp prescribed by the country where they are made. These are called specie, and are declared to be a legal tender in payment of debts. The Constitution of the United States provides, that gold and silver only shall be a legal tender. 269. Currency is what passes for money. In our country there are three kinds. 1st. The coins of the country : 26?. Foreign coins, having a fixed value estabhshed by law : 3d. Bank notes, redeemable in specie. Note. — Tfie foreign coins most in use in this country are the English shilling, valued at 22 cents 2 mills ; the English sove- reign, valued at $4,84 ; the French franc, valued at 18 cents 6 mills ; and the five-franc piece, valued at $0,93. Although the currency of the United States is in dollars, cents and mills, yet in some of the States accounts are still kept in pounds, shillings and pence. In all the States the shilling is reckoned at 12 pence, the pound at 20 shillings, and the dollar at 100 cents. The following table shows the number of shillings in a dol- lar, the value of £1 in dollars, and the value of $1. in the fraction of a pound : In English currency, In N. E., Va., Ky., ) Tenn., J 4s. 6d. 16s. - £l = $4,84, and $l = £^l^. . £1 = $3^, andll^XyV In N. Y., Ohio, N. Carolina, In N. J., Pa., Del. ) Md., 8s. 7s. 6d. - £1-$2|, - £1=$2|, and $1=£ |. and $! = £ f. In S. Carolina & Ga. In Canada & Nova ) Scotia, J 4s. 8d. 5s. -£l = $4f, -£1=S4, and$l=£ ^. and8l=£ i 268. What are coins- legal tender ? What are they called 1 What is iiiatie a 204 KI'ID'.C'J ION OF CUKICKNOIES. REDUCTIOxN OF CURRENCIES. 270. Reduction of Currencies is changing their denoniiiia tions without changing their values. There are two cases of the Reduction of Currencies : 1st. To change a currency in pounds shillings and pence, to United States currency. 2d. To change United States currency to pounds, shillings and pence. 271. To reduce pounds, shillings and pence to United States currency. 1. What is the value of £3 126". 6c/., New England cur- rency, in United States money ? OPERATION. Analysis.— Since £l=$3i, the £3 125. 6rf. = £3.625 number of dollars in £3 125. 6d.= (JqHs in £1 = 3i £3.625, will be equal to £3.625 " To^l_ taken 3i times : that is, to $12,08 : l.^U»-h hence, 10.875 Ans. $12.0S3-f- Multiply the amount reduced to pounds and the decimals of a pound by the number of dollars in a jjound^ and the product will be the answer. 272. To reduce United States money to pounds, shillings and pence. 1. What is the value of |375,87, in pounds, shillings and pence, New York currency ] Analysis. — Since $1 =£f; the number of pounds in $375.87 will be operation. equal to this number taken f times : $375.87 xf = £150.348 thatis, equal to £150.348 =£150 65. =£150 6s. \l^d. ll^d. : hence, 269. What is currency ■? How many kiuds are there 1 "WTiat foreijm coins are most used in this country ! What are the denominations of United States currency ] What denominations are sometimes used in the States ? 270. What is reduction of currencies ? How many kinds of reduc- tion are there 1 What are they ! 271. What is the rule for reducing from pounds, slulliug.s and pence to United States uiouey ] EXCHANGE. 205 Multiply the amount by that fraction of a pound which denotes the value of$'[, and the product will be the anawei in pounds and decimals of a pound. EXAMPLES. 1. What is the value of £127 l&s. 6c?., New England currency, in United States moiiey 1 2. What is the value of ^2663.75 in pounds, shillings and pence, Pennsylvania currency ? 3. What is the value of X'4&9 Ss. 6rf., Georgia currency, in United States money ? 4. What is the value of $973,28 in pounds, shillings and pence, North Carolina currency? 5. W^hat is the value in United States money of £637 18«. 8cZ., Canada currency] 6. Reduce ^102,85 to English money ; to Canada cur- rency ; to New England currency ; to New York cunency ; to Pennsylvania currency ; to South Carolina currency. 7. Reduce £51 136-. 0\d. English money; £62 lO*-. Can- ada currency; £75 New England currency; £100 New York currency ; £193 15.v. Pennsylvania currency; and £58 66. l\d. Georgia currency, to United States money. EXCHANGE. 273. Exchange denotes the payment of a sum of money by a person residing in one place to a person residing in an- other. The payment is usually made by means of a bill of exchange. A Bill of Exchange is an order liom one person to anothei directing the payment to a third person named therein of a certain sum of money : 1. He who writes the open letter of request is called tne drawer or maker of the bill. 2. The person to whom it is directed is called the drawee. 272 What is the rule for reducing from United States money to pounds, shillings and pence ? 273. What does exchange denote \ How is the payment generally made ] What is a bill of exchange ] Who is the drawer 1 Who the drawee 1 Who the buver or remitter ^ 260 PORMGN BILLS. 3. The person to whom the money is ordered to be paid is called the 'payee ; and 4. Any person who purchases a bill of exchange i& called the buyer or remitter. 274. A bill of exchange is called an inland bill, when the drawer and drawee both reside in the same country ; and when they reside in different countries, it is called a foreign bill. Exchange is said to be at par, when an amount at the place from which it is remitted will pay an equal amount at the place to which it is remitted. Exchange is said to be at a premium, or above par, when the sum to be remitted will pay less at the place to which it is remitted ; and at a dis- count^ or below par, when it will pay more. EXAMPLES. 1. A merchant at Chicago wishes to pay a bill in New York amounting to $3675, and finds that exchange is 1^ per cent premium : what must he pay for his bill ? 2. A merchant in Philadelphia wishes to remit to Charles- ton $8756,50, and finds exchange to be 1 per cent below par ; what must he pay for the bill ? 3. A merchant in Mobile wishes to pay in New York $6584, and exchange is 2^ per cent premium : how much must he pay for such a bill ? 4. A merchant in Boston wishes to pay in New Orleans $4653,75 ; exchange between Boston and New Orleans is 1-^ per -cent below par : what must he pay for a bill 1 5. A merchant in New York has $3690 which he wishes to remit to Cincinnati ; the exchange is Ij per cent belo"w par : what will be the amount of his bill 1 FOREIGN BILLS. 275. A Foreign Bill of Exchange is one in which the drawer and drawee live in different countries. Note. — In all Bills of Exchange on England, the £ sterling is the unit or base, and is still reckoned at its former value of $4| =$4,4444+, instead of its present value $4,84. 274. When is a bill of exchange said to be inland 1 When foreign 1 When is exchange said to be at pari When at a premium '' When at a discount ^ FOREIGN HILLS. 267 Hence, £1=^4.4444-1- Add 9 per cent, -3999 Gives the present value of £l $4.8443. Hence, the true 'par value of Exchange on England is 9 per cent on the nominal base. 1. A merchant in New York wishes to remit to England a bill of Exchange for £125 155. 6<^ : how much must he pay for this bill when exchange is at 9^ per cent premium ? - £125 155. 6^/. =£125.775 Add 91 per cent - - - - 11.948 6 + gives amount in £'s, at $4| = ^. £137.7236 + Note. — The pounds and decimals of a pound are reduced to dollars by multiplying by 40 and dividing by 9 — giving, in this case, $612,105. Rule. — 1. Reduce the amount of the bill to pounds and decimals of a pound, and then add the preTniumof exc]ia7ige. II. Multiply the result by 40 and divide the product by 9 : the quotient will be the answer in United States Money. 2. A merchant shipped 100 bales of cotton to Liverpool, each weighing 450 pounds. They were sold at l\d. per pound, and the freight and charges amounted to £187 IO5. He sold his bill of exchange at 9|- per cent premium : how much should he receive in United States Money ? 3. There were shipped from Norfolk, Va., to Liverpool, S6hhd. ol' tobacco, each weighing 450 pounds. It was sold at Liverpool for 12^ri. per pound, and the expenses of freight and commissions were £92 I5. 8d.' If exchange in New York is at a premium of 9^ per cent, what should the ownei receive for the bill of exchange, in United States Money 1 276. The unit or base of the French Currency is the French franc, of the value of 18 cents 6 mills. The franc is divided into tenths, called decimes, corresponding to our dimes, and into centimes corresponding to mills. Thus, 5.12 is read, 5 francs and 12 centimes. 275. What is a foreign bill of exchange 1 In bills on England, what is the unit or base 1 What is the exchange value of the £ sterling '' How much is the true value above the commercial value of the £ ster- ling 1 How do you find the value of a bill in English currency in Uiiited States money 1 26S UUTfES. All bills of exchange on France are drawn in francs. Ex- change is quoted in New York at so many francs and centimes to the dollar. 1. What will be the value of a bill of exchange for 4536 francs, at 5.25 francs to the dollar ? Analysis.— Since 1 dollar will buy operation 5.25 francs, the bill will cost as many ^ 25)4536(^864 Arts. dollars as 5.25 is contained times in the ^ ^ amount of the bill : hence, Divide t}ce amount of the bill by the value of%l i7t francs : the quotient is the amount to be paid in dollars. 2. What will be the amount to be paid, United States money, for a bill of exchange on Paris, of 6530 francs, — exchange being 5.14 francs per dollar? 3. What will be the amount to be paid in United States money for a bill of exchange on Paris of 10262 francs, ex- change being 5.09 francs per dollar ? 4. Wliat will be the value in United States money of a bill for 87595 francs, at 5.16 francs per dollar 1 DUTIES. 277. Persons who bring goods or merchandise into the United States, from foreign countries, are required to land them at particular places or Ports, called Ports of Entry, and to pay a certain amount on their value, called a Duty. This duty is imposed by the General Government, and must be the same on the same articles of merchandise, in every part of the United States. Besides the duties on merchandise, vessels employed in commerce are required, by law, to pay certain sums for the privilege of entering the ports. These sums are large or small, in proportion to the size or tonnage of the vessels. The moneys arising from duties and tonnage, are called revenues. 276. What is the unit or base of the French currency 1 W'hat is its value 1 How is it divided 1 In what currency are French bills of ex- cliange drawn "? 277. What is a port entry '' W^hat is a duty 1 By whom are duties imposed ^ What charges are vessels required to pay 1 What are the moneys arising from duties and tonnage called ? DUTIKS. 269 278. Tlie revenues of the country are under the general direction of the Secretary of the Treasury, and to secure their faithful collection, the government has appointed various officers at each port of entry or place where goods may be landed. 279. The office established by the government at any port of entry is called a Custom House, and the officers attached to it are called Custom House Officers. 280.' All duties levied by law^ on goods imported into the United States, are collected at the various custom houses, and are of two kinds, Specific and Ad valorem. A specific duty is a certain sum on a particular kind of goods named ; as so much per square yard on cotton or wool- len cloths, so much per ton weight on iron, or so much per gallon on molasses. An ad valorem duty is such a per cent on the actual cost of the goods in the country from which they are imported. Thus, an ad valorem duty of 15 per cent on English cloths, is a duty of 15 per cent on the cost of cloths imported from Eng- and. 281. The laws of Congress provide, that the cargoes of all vessels freighted with foreign goods or merchandise shall be weighed or gauged by the custom house officers at the port to which they are consigned. As duties are only to be paid on the articles, and not on the boxes, casks and bags which con- tain them, certain deductions are made from the weights and measures, called Allowances. Gross Weight is the whole weight of the goods, together with that of the hogshead, barrel, box, bag, &c., which con- tains them. 278. Under whose direction are the revenues of the country ? 279. What is a custom house \ What are the officers attached to it called 1 280. Where are the duties collected 1 How many kinds are there, and what are they called 1 What is a specific duty ^ An ad valorem duty 1 281. What do the laws of Congress direct in relation to foreign goods I \\'hy are deductions made from their weight \ What are these deductions called ! What is gross weight \ W' hat is draft ? What is the greatest draft allowed 1 What is tare ! What are the diflereiit kinds of tare 1 - What allowances are made on liquors 1 270 DUTIES. Draft is an allowance from the gross weight on account o! waste, where there is not actual tare. On \\2lh, it is Mh. From 112 to 224 " 2, 224 to 336 " 3, 336 to 1120 " 4, " 1120 to 2016 " 7, Above 2016 any weight " 9; fonseqiiently, 9Z6. is the greatest draft allowed. Tare is an allowance made for the weight of the boxes, barrels, or bags containing the commodity, and i& of three kinds : U/, Legal tare, or such as is established by law ; 2<:/, Customary tare, or such as is established by the custom among merchants ; and 3c/, Actual tare, or such as is found by re- moving the goods and actually weighing the boxes or casks in which they are contained. On liquors in casks, customary lure is sometimes allowed on the supposition that the cask is not full, or what is called its actual wants ; and then an allowance of 5 per cent for leakage. A tare of 10 per cent is allowed on porter, ale and beer, in bottles, on account of breakage, and 5 per cent on all other liquors in bottles. At the custom house, bottles of the com- mon size are estimated to contain 2|- gallons the dozen. Note. — For tables of Tare and Duty, see Ogden on the Tariff ol" 1842. EXAMPLES. 1. What will be the duty on 125 cartons of ribbons, each containing 48 pieces, and each piece w^eighing Zoz. net, and paying a duty of |2,50 per pound ? 2. W%at will be the duty on 225 bags of coffee, each weigh- ing gross 160/6., invoiced at 6 cents per pound ; 2 per cent being the legal rate of tare, and 20 per cent the duty ? 3. What duty must be paid on 275 dozen bottles of claret, estimated to contain 2| gallons per dozen, 5 per cent being allowed for breakage, and the duty being 35 cents per gallon '] 4. A merchant imports 175 cases of indigo, each case weighing 196/6.S'. gross ; 15 per cent is the customary rate of tare, and the duty 5 cents per pound : what duty must he pay on the whole ? ALLIGATION MEDIAL. 271 ALLIGATION MEDIAL. 28.2. Alligation Medial is the process of finding- the price of a mixture when the quantity of each simple and it? price are known. 1. A merchant mixes Sib. of tea, worth 75 cents a pound with 161b. worth ^1,02 a pound : what is the price of the mixture per pound '^ Analysis. — The quantity, 8/6. of operation. tea, at 75 cents a pound, costs $6; 6Ib. at '75cts.z=$ 6,00 and 16/6. at Sl.02 costs $16,32: 16/^. at $1,02 = $16,32 hence, the mixture. = 24/6., costs ^r — oas^o ^o $32,32 ; and the price of 1/6. of the ^* "^^^'^^ mixture is found by dividing this $0,93 cost by 24 : hence, to find the price of the mixture, I. Find the cost of the entire mixture : II. Divide the entire cost of the mixture by the sum of the simples, and the quotient toill be the jynce of the mixture. examples. 1. A farmer mixes 30 bushels of wheat worth 55. per bushel, with 72 bushels of rye at 35. per bushel, and with 60 bushels of barley worth 2s. per bushel : what should be the price of a bushel of the mixture ? 2. A wine merchant mixes 15 gallons of wine at $1 per gallon with 25 gallons of brandy worth 75 cents per gallon : what should be the price of a gallon of the compound ? 3. A grocer mixes 40 gallons of whisky worth 31 cents per gallon with 3 gallons of M'ater which costs nothing : what should be the price of a gallon of the mixture? 4. A goldsmith melts together 2/6. of gold of 22 carats fine, 602. of 20 carats fine, and 602. of 16 carats fine : what is the fineness of the mixture ? 5. On a certain day the mercury in the thermometer was observed to average the following heights : from 6 in the morning to 9, 64° ; from 9 to 12, 74° ; from 12 to 3, 81° ; and from 3 to 6, 70° : what was the mean temperature of the day ? '^82. What is Alligation Medial 1 What is the rule for determhiing the price of the mixture 1 272 ALIJGATION ALTICRNATE. A. B. C. D. E. i ^ 2 1 3 1 2 2 J. 4 1 1 ALLIGATION ALTERNATE. 283. Alligation Alternate is the process of firuliiio- wha; proportions must be taken of each of several simples, whos€ prices are known, to form a compound of a ^riven price. Ti is the opposite of Alligation Medial, and may be proved by it. 284. To find the 'proportional parts : 1. A farmer would mix oats at 35. a bushel, rye at 6v., and wheat at 9.s'. a bushel, so that the mixture shall be worth 5 shillings a bushel : what proportion must be taken of each Bort? OPERATION oats, 3-1 rye, GJ wheat, 9 Analysis. — On e/ery bushel put into the mixture, whose price is less than the mean price, there will be a gain; on every bushel whose price is greater than the mean price, there will be a /o.v.s ; and since there is to be neither gain nor loss by the mixture, the gains and losses must balance each other. A bushel of oats, when put into the mixture, will bring 5 shil- lings, giving a gain of 2 shillings ; and to gain 1 shilling, we must take half as much, or -^ a bushel, which we write in column A. On 1 bushel of wheat there will be a loss of 4 shillings ; and to make a loss of 1 shilling, we must take |- of a bushel, which we also write in column A : -J and ^ are called proportional numbers. Again : comparing the oats and rye. there is a gain of 2 shil- lings on every bushel of oats, and a loss of 1 shilling on every bushel of rye: to gain 1 shilling on the oats, w^e take ^ a bushel, and to lose 1 shilling on the rye, we take 1 bushel : these num- bers are written in column B. Two simples, thus compared, are called a couplet: in one. the price of unity is less than the mean piice, and in the other it is greater. If, every time we take ^ a bushel of oats we take ^ of a bushel of wheat, the gain and loss will balance; and if every time we take -^ a bushel of oats we take 1 bushel of rye. the gain and loss 283. What is Alligation Alternate ? 284. How do you find the proportional numbers 1 ALLIGATION ALTERNATE. 273 Will balance : hence, if the proportional numbers of a couplet be multiplied by any number^ the gain and loss denoted by the products, will balance. When the proportional numbers, in any column, are fractional (as in columns A and B), multiply them by the least common divi.sor of their denominators, and write the products in new- columns C and D. Then, add the numbers in columns C and D, standing opposite each simple, and if their sums have a common factor, reject it : the last result will be the proportional numbers, Rule. — I. Write the prices or qualities of the simples in a column, beghming with the lowest, and the mean price or quality at the left, II. Opposite the first simple write the part which must he taken to gain 1 of the mean price, and opposite the other simple of the couplet, write the part which juust he taken to lose 1 of the mea.7i price, and do the same for each simple. III. When the proportional numbers are fractional, reduce them to integral numbers, and then add those which stand oppo- site the same sifnple : if the sums have a common factor, reject it: the result will denote the proportional parts. 2. A merchant would mix wines worth IGs., ISs., and 22s. per gallon, in such a way, that the mixture may be w^rth 20s. per gallon : what are the proportional parts 1 OPERATION. A. 20 J 18-, (22J PROOF. 1 gallon, at 16 shillings, = 16s. 1 gallon, at 18 shillings, = 18s. 3 gallon, at 22 shillings, = 66s. 5)100(20s., mean price. Note. — The answers to the last, and to "all similar questions, will be infinite in number, for two reasons : Ist. If the proportional numbers in column E be multiplied by any number, integral or fractional, the products will denote pro- tiortional parts of the simples. 2d. If the proportional numbers of ayiy touyUt be muUiplied by A. B. C. D. E. i 1 1 i 1 1 i i 2 1 3 274: ALLIGATION ALTERNATE. any, number, the gain and loss in that couplet will still balancCj arid ,the proportional numbers in the final result will be changed. 3. What proportions of tea, at 24 cents, 30 cents, 33 cents and 36 cents a pound, must be mixed together so that the mixture shall be worth 32 cents a pound ? 4. What proportions of coffee at IQcts., 20cts. and 28cf,s. per pound, must be mixed together so that the compound shall be worth 24:Ct.s. per pound ? 5. A goldsmith has gold of 16, of 18, of 23, and of 24 carats fine . what part must be taken of each so that the mixture shall be 21 carats fine ? 6. What poition of brandy, at 14s. per gallon, of old Ma deira, at 24s. per gallon, of new Madeira, at 21s. per gallon, and of brandy, at 10s. per gallon, must be mixed together so that the mixture shall be worth 1 8s. per gallon 1 285. When the quantity of one sitnple is given : 1. How much wheat, at 9s. a bushel, must be mixed with 20 bushels of oats w^orth 3 shillings a bushel, that the mix- ture may be worth 5 shillings a bushel ? Analysis. — Find the proportional numbers : they are 2 and 1 ; hence, Ihe ratio of the oats to the wheat is ^ : therefore, there must be 10 bushels of wheat. Rule. — I. Find the 2^foportional numbers^ and write the given simple opposite its proportional number. II. Muliiply the given simiole by the ratio which its proper' tional number bears to each of the others, and the products will denote the quantities to be taken of each, EXAMPLES. 1. How much wine, at 5s., at 5s. 6d., and 6s. per gallon must be mixed with 4 gallons, at 4s. per gallon, so that the mixture shall be worth 5s. 4d. per gallon ? 2. A farmer would mix 14 bushels of wheat, at $1,20 per bushel, with rye at 12cts., barley at 486'^s., and oats at 36rfs. : how much must be taken of each sort to make the mixture worth 64 cents per bushel 1 3. There is a mixture made of wheat at 4s. per bushel, rye at 3s., barley at 2s., \Vith 12 bushels of oats at \Bd. pei bushel : how much is taken of each sort when the mixture is worth 3 5. 6^/. '\ ALLIGATION ALTER 4. A distiller would mix iOgal. of per gallon, \vith English at 7^. and spiril what quantity must be taken of each sort may be afforded at 85. per gallon? 286. When the quantity of the mixture is given. I. A merchant would make up a cask of wine containing 60 gallons, with wine worth 16^., I8s. and 226'. a gallon, in such a way that the mixture may be worth 20s. a gallon • how much must he take of each sort 1 Analysis. — This is the same as example 2, except that the quantity of the mixture is given. If the quantity of the mixture be divided by 5, the sum of the proportional parts, the quotient 10 vsi 11 show how many times each proportional part must be taken to make up 50 gallons: hence, there are 10 gallons of the first, 10 of the second, and 30 of the third : hence. Rule. — I. Find the jM-oporttonal parts. II. Divide the quantity of the mixture by the sum of the proportional parts, and the quotient vnll denote how many titnes each part is to he taken. Multiply this quotient by the parts separately, and each product vnll denote the quan- tity of the corresponding simple. EXAMPLES. 1. A grocer has four sorts of sugar, worth 12cZ., lOo?., 6c?. and 4c?. per pound ; he would make a mixture of 144 pounds worth 8c/. per pound : what quantity must be taken of each sort '? 2. A grocer having four sorts of tea, worth 5s., 6s., 8s and 96-. per pound, wishes a mixture of 87 pounds worth 7s. per pound : how much must he take of each sort ? 3. A' silversmith has four sorts of gold, viz., of 24 carats fine, of 22 carats fine, of 20 carats fine, and of 15 carats fine ; he would make a mixture of A2oz. of 17 carats fine; hov.' much must be taken of each sort % Proof. — All the examples of AUigation Medial may be found by Alligation Alternate. 285. How do you find the quantity of each simple when the quantity of one simple is known 1 286. How do you find the quantity of eac-li simple when the quaniitj tif each mixture i? known ' 276 INVOLUTION. INVOLUTION. 287. A POWER is the product of equal factors. The equal factor is called the root of the power. The first power is the equal factor itself, or the root : The second power is the product of the root by itself : The third power is the product when the root is taken 3 times as a factor : The fourth power, when it is taken 4 times : The fifth power, when it is taken 5 times, &c. 288. The number denoting how many times the root is taken as a factor, is called the exponent of the power. It is written a little at the right and over the root : thus, if the equal factor or root is 4. 4= 4 the 1st power of 4. 42 = 4x4= 16 the 2d power of 4. 4^ = 4x4x4= 64 the 3d power of 4. 4* ==4x4x4x4= 256 the 4th power of 4. 4^ = 4x4x4x4x4=1024 the othpowerof4. Involution is the process of finding the powers of , lumber 8. Notes. — 1. There are three things connected with every power • 1st, The root ; 2d3 The exponent ] and Sd, The power or result of the multiplication. 2. In finding a power, the root is always the 1st power : hence, the number of multiplications is 1 less than the exponent : Rule. — Multiply the number by itself as many times less 1 as there are units in the exponent^ and the last jn'oduct will be the power. EXAMPLES. Find the powers of the following numbers : 10. 5th power of 16. 11. 6th power of 20. 12. 2d power of 225. 1. Square of 1. 2. Square of ^. 3. Cube of f 4. Square ol" j. 5. Square of 9. 6. Cube of 12. 7. 3d power ol 125, 8. 3d power of 16. y. 4 th power of 9. 13. Square of 2167. 14. Cube of 321. 15. 4th power of 215. 16. 5th power of 906. 17. 6th power of 9. IS. ;^ 3x6- 32 + 3x6 30 to be taken 6 units' times, giving 6^ + 3X6: then taking it 3 tens' times, we have 3 X6-t-32, and the sum is 32+2(3x6) + 62: ^^'^^'"' 32 + 2(3x6) + 6'-^ The square of a number is equal to the square of the tens, plus twice the product of the tens by the units, plus the square of the units. The same may be shown by the figure : Let the hne AB re- present the 3 tens or 30, and BC the six units. Let AD be a square on AC, and AE a square on the ten's line AB. Then ED will be a square on the unit line 6, and the rectangle EF will be the product of HE, which is equal to the ten's line, by IE, which is equal to the unit line. Also, the rectangle BK will be the product of EB, which is a 30 B t; equal to the ten's line, by the unit line B C. But the whole square on AC is made up of the square AE, the two rectangles FE and EC, and the square ED. 1. Let it now be required to extract the square root of 1296. Analysis. — Since the number contains more than two places ot figures, its root will contain tens and units. But as the square of one ten is one hundred, it follows that the square of the tens of the required root must be found in the two figures on the left ol 96. Hence, we point off the number into periods of two figures each. 290. "What is the square root of a number? What are perfect squares \ How many are there between 1 and 100 1 291. Into what parts may a number be dcnonijmiied 1 Whou so dc- Lomposod. wliut is its S(|u;ue rqiiiil to ! 30 6 6 6 180 36 K 30 E 900 + 180 + 180+36 = 1296. 30 30 S 30 6 900 180 EXTUACTION OF THE SQUARE liOOT. 279 We next find the greatest square contained in operation. 12, which is 3 tens or 30. We then square 3 |2 96(36 tens which gives 9 hundred, and then place 9 un- g der the hundreds' place, and subtract ; this takes away the square of the tens, and leaves 396, 66)396 which is twice the product of the tens by the units 396 •plus the square of the units. If now, we double the divisor and then divide this remainder, exclusi^De of the right hand figure, (since that figure cannot enter into the product of the tens by the units) by it, the quotient will be the units figure of the root. If we annex this figure to the augmented divisor, and then multiply the whole divisor thus in- creased by it, the product will be twice the tens by the units plus the square of the units ; and hence, we have found both figures of ihe root. This process may also be illustrated by the figure. Subtracting the square of the tens is taking away the square AE and leaves the two rectangles FE and BK, together with the square ED on the unit line. The two rectangles FE and BK representing the product of units by tens, can be expressed by no figures less than tens. If, then, we divide the figures 39, at the left of 6, by twice the tens, that is, by twice AB or BE, the quotient will be BC or EK, the unit of the root Then, placing BC or 6, in the root, and also annexing it to the divisor doubled, and then multiplying the whole divisor 66 by 6, we obtain the two rectangles FE and CE. together with the square ED. 292. Hence, for the extraction of the square root, we have the following E.ULE. — I. Separate the given number into periods of two figures each, by setting a dot over the place of units, a se- cond over the place of hundreds, and so on for each alternate figure at the left. II. Note the greatest square contained in the period on ihe left, and place its root on the right after ihe manner of a quotient tn division. Subtract the square of this root from the first period, md to the remainder bring down the secmid period for a dividend. 292. What is the first step in extracting the square root of numbers 1 What is the secoivJ ( What is the third I What the fourth ? What Ihe fifth 1 (Jive the entire rule 280 EXTiiAcnoN OF Till; sc^uark root. III. Double the root thus found for a trial divisor and •place it on the left of the dividend. Find how many times the trial divisor is contained in the dividend, exclu- sive of the right-hand figure, and place the quotient in tit^ root and also a?inex it to the divisor. IV. Midtij)ly the divisor thus increased, by the last figure of the root ; subtract the product from, the dividend, and to the remainder bring down the next period for a new divi^ dend. V. Double the whole root thus found, for a new trial di- visor, and continue the operation as before, until all the periods are brought down. EXAMPLES. 1. What is the square root of 263169 % Analysis^. — We first, place a dot over the operation. 9, making the riglit-hand period 69. We 26 Si 69(513 then put a dot over the 1 and also over the «< 6, making three periods. The greatest periect square in 26 is 25, 101)1^V the root of whicli is 5. Placing 5 in the lOl root, sublracting its square from 26, and 1023 ")30(')I) ' bringing down the next period 31, we have SO (9 131 for a dividend, and by doubling the root v^-e have 10 for a trial divisor. Now. 10 is contained in 13, 1 time. Place 1 both in the root and in the divisor : ii.cn multi- ply 101 by 1 ; subtract the product and bring down the lext period. We must now double the whole root 51 for a new trial divisor ; or we may take the first divisor after havfng doubled the last figure 1 ; then dividing, we obtain 3, the third figure of the root. Notes. — 1. The left-hand period may contain but one figure ; each of the others will contain two. 2. If any trial divisor is greater than its dividend, the corres- ponding quotient figure will be a cipher. 3. If the product of the divisor by any figure of the root exceeds the corresponding dividend, the quotient figure is too large and muft be diminished. . 4. There will be as many figures in the root as there are periods in the given number. 5. If the given number is not a perfect square there will be a remainder after all the periods are brought down. In this case, periods of ciphers may be annexed, forming new periods, each of V'liich will give one decimal place iuthe root. EXl'BAOTION OF TLHO SQTTAKK liOOT. 281 2 Wliat ig the square root of 36729? operation. Ill this example there are two periods of decimals, which give two places ef decimals in the root. 3 67 29(191 64-f-, 1 29)267 261 381)629 381 3826)24800 22956 38324)184400 153296 31104 Rem 293. To extract the square root of a fractiim 1. What is the square root of .5 1 Note. — We first annex one cipher to make even decimal places. We then ex- tract the root of the first period : to the remainder we annex two ciphers, forming a new period, and so on. OPERATION. .60(.707-|- 49 140)100 000 1407)10000 9849 151 Renu 2. What is the square root of J ? Note. — The square root of a fraction is equal to the square root of the numerator divided by the square root of the denomi- nator. 3. What is the square root of J ? Note. — When the terms are not per- fect squares, reduce the common fraction to a decimal fraction, and then extract the square root of the decimal. OPERATION '4 v/: v/4_2 OPERATION. :.75; v/|=V?75=..854d-|- 29'J. How do you extract the w^uare root of a deciuial fraction 1 uf ;. cumumui fraction ! 10 IJow 282 SQUAKE KOOT. Rule. — 1. If the fraction is a decimal, j^oint off tft^ periods from the decimal point to the right, annexing ci- phers if necessary, so that each period shall co7itain two places, and then extract the root as in integral numbers. II. If the fraction is a common fraction, and its terms perfect squares, extract the square root of the numerator and denominator separately ; if they are not perfect squares, re- duce the fraction to a decimal, and then extract the square root of the result, EXAMPLES. What are the square roots of the following numbers? 1. of 3"? 2. of lU 3. of 10691 4. of 2268741 ? 5. of 7596796 1 6. of 36372961 ? 7. of 22071204? 8. of 3271.4207 ? 9. of 4795.25731 ? 10. of 4.372594 ? 11. of .0025? 12. of .00032754? 13. of .00103041 ? 14. of 4.426816? 15. of 8|? 16. of 9 J ? 17. 18. 19. 20. of /A? ofi|f1 APPLICATIONS IN SQUARE ROOT. 294. A triangle is a plain figure which has three sides and three angles. If a straight line meets another straight line, making the adjacent angles equal, each is called a right angle ; and the lines are said to be perpendicular to each other. 295. A right angled triangle is one which has one right angle. In the right angled triangle ABC, the side AC opposite the right angle B is called the hypothenuse ; the side AB the base; and the side BC tlie perpendicular AI'FLICATIONS. 2S? 296. In a right angled triangle the square described in the hypothenuse is equal to the sum of the squares described in the other two sides. Thus, if ACB be aright angled triangle, right an- gled at C, then will the large square, D, described in the hypothenuse AB, be equal to the sum of the squares F and E described on the sides AC and CB. This is called the carpen- ter's theorem. By count- ing the small squares in the large square D, you will find their number equal to that contained in the small squares F and E. In this triangle the hypothenuse AB:^5, AC =4, and CB = 3. Any numbers having the same ratio, as 5, 4 and 3, such as 10, 8 and 6 ; 20, 16 and 12, &c., will represent the sides of a right angled triangle. I. Wishing to know the distance from A to the top of a tower, I measured the height of the tower and found it to be 40 feet ; also the distance from A to B and found it 30 feet; what was the distance from A to C ^ = 30 ; AB2 = 302=: 900 BC=40 ; BC2=402 = 1600 AC^^AB^+BC^zzz 2500 AC=:V2500 = 50 feet. iB D 297. Hence, when the base and perpendicular are known •lid the hypothenuse is required. 294 295 Wliat is a triangle 1 What is a right angle 1 What is a right angled triangle 1 Which side is the hypotlie- 296. In a right aiigled triangle wliat i? the square on the hypothe- aU€e equul tui 284 t;QUARE ROOT. Square the base and square the perpendicular, add the t- suits and then extract the .square root of their sum. 2. What is the length of a rafter that will reach from th eaves to the ridge pole of a house, when the height of the roof is 15 feet and the width of the building 40 feet ? 298. i'o find one side when we know the hypothenuse and the other side. 3. The length of a ladder which will reach from the mid- dle of a street 80 feet wide to the eaves of a house, is 50 feet : what is the height of the house ? Analysis. — Since the square of the length of the ladder is equal to the sum of the squares of half the street and the height of the house, the square of the length of the ladder diminished by the square of half the street will be equal to the square of the height of the house : hence, Square the hypothenuse and the known side, and take the difference ; the squure root of the difference will be the other side, EXAMPLES. 1. If an acre of land be laid out in a square form, what will be the length of each side in rods ? 2. What will be the length of the side of a square, in rods, that shall contain 100 acres? 3. A general has an army of 7225 men : how many must be put in each line in order to place them in a square form 1 4. Two persons start from the same point ; one travel? due east 50 miles, the other due south 84 miles : how far are they apart ? 5. What is the length, in rods, of one side of a square that shall contain 12 acres ? 6. A company of speculators bought a tract of land foi ^6724, each agreeing to pay as many dollars as there were partners : how many partners were there ? 297. How do you find the hypothenuse when you know the base and perpendicular ! 298. If you know the hyputh^iuise and cue side, how il'i vou finiJ '.c othr r sic]«> ■ CVBK ROOT. 285 7. A farmer wishes to set out an orchard of 3844 trees, so that the number of rows shall be equal to the number of trees in each row : what will be the number of trees ? 8. How many rods of fence will enclose a square field of 10 acres ? 9. If a line 150 feet long will reach from the top of a steeple 120 feet high, to the opposite side of the street, what is the width of the street ? 1^0. What is the length of a brace whose ends are each 3^ feet from the angle made by the post and beam ? CUBE ROOT. 299. The Cube Root of a number is one of three equal factors of the number. To extract the cube root of a number is to find a factor which multiphed into itself twice, will produce the given number. Thus, 2 is the cube root of 8 ; for, 2x2x2 = 8: and 3 is the cube root of 27 ; for 3 X 3 X 3 = 27. 1, 2, 3, 4, 5, 6, 7, 8, 9. I 8 27 64 125 216 343 512 729 The numbers in the first line are the cube roots of the corresponding numbers of the second. The numbers of the second line are called perfect cubes. By examining the num- bers of the two lines we see, 1st. That the cube of units cannot give a higher order than hundreds. 2d. That since the cube of one ten (10) is 1000 and the cube of 9 tens (90), 81000, the cube of tens will not give a lower de7ioini7iation tlian thousands^ nor a higher denomi- nation thun hundreds of thausayids. Hence, if a number contains more than three figures, its cube root will contain more than one : if it contains more than six, its root will contain more than two, and so on ; every additional three figures giving one additional figure in the root, and the figures which remain at the left hand, although less than three, will also give a figure in the root. This law explains the reason for pointing off into periods ol tliite liguit> each 280 CUBE ROOT. 300. Let us now see how the cube of any number, as 1 6, is formed. Sixteen is composed of 1 ten and 6 units, and may ^e written 10-f 6. To find the cube of 16 or of 10 -fO, we must multiply the number by itself twice. To do this we place the number thus 16= 10-f 6 10+ 6 60+~36 product by the units product by the tens Square of 16 Multiply again by 1 6 product by the units product by the tens Cube of 16 100+ 60 100 + 120 + 10 + 36 6 - 600+ 720 + 216 1000+1200+ 360 1000 + 1800+1080 + 216 1. By examining the parts of this number it is seen that the first part 1000 is the cube of the tens ; that is, 10x10x10 = 1000. 2. The second part 1800 is three times the square of the tens multiplied by the units ; that is, 3 X (10)2x6 = 3x100x6 = 1800. 3. The third part 1080 is three times the square of the units multiplied by the tens ; that is, 3x62x10 = 3x36x10=1080 4. The fourth part is the cube of the units ; that is, 62 = 6x6x6 = 216. 1. What is the cube root of the numbei* 40961 OPERATION. 4 096(^6 1 Analysis. — Since the number contains more than three figures, we know that the root will eon- tain at least units and tens. Separating the three right- hand figures from the 4, we know that the cube of the tens will be found in the 4 ; and 1 is the greatest cube in 4 12x3 = 3)3 (9-8-7-6 163 = 4 096. 299. What is the cube root of a number 1 How many perfect cubes are there between 1 and 1000 ! UOO. Of how many parts is the cube of a numbrr c(iiii|M)«{>il I Wlial an- ihfv '' OUBK KOUl. 287 ' Hence, we place the root 1 ou the right, and this is the tens of the required root. We then cube 1 and subtract the result from 4, and to the remainder we brine down the first figure of the next period. We have seen that the second part of the cube of 16, viz 1800^ is three times the square of the tens multiplied by the units : and hence, it can have no significant figure of a less denomination tha» hundreds. It must, therefore, make up a part of the 30 hundreds above. But this 30 hundreds also contains all the hundreds which come from the 3d and 4th parts of the cube of 16. If it were not so, the 30 hundreds, divided by three times the square of the tens, would give the unit figure exactly. Forming a divisor of three times the square of the tens, we find the quotient to be ten; but this we know to be too large. Placing 9 in the root and cubing 19. we find the result to be 6859. Then trying 8 we find the cube of 18 still too large; but when we take 6 we find the exact number. Hence, the cube root of 4096 is 16. 301. Hence, to find the cube root of a number, Rule. — 1. Separate the given number into periods of thret figures eadi^ by placing a dot over the place of units, a second over the place of thousands, and so on over each third figun to the left; the left hand period will often contain less thai three places of figures. II. JVote the greatest perfect cube in the first period^ and set its root on the right, after the manner of a quotient in di- vision. Subtract the cube of this number from the first j^^iod^ and to the remainder bring dovjn the first figure of the 7iext period for a dividend. III. Take three times the square of the root just found fo^ a trial divisor, and see how often it is contained in the divi dend^ and place the quotient for a second figure of the root TJien cube the fig\ires of the root thus found, and if theif cube be greater than the first two periods of the given num- ber, diminish the last figure, but if it be less, subtract it from the first ttco periods, a7id to the remainder bring doicn the first figure of the next period for a new dividend. IV. Take three times the square of the whole root for a second trial divisor, and find a third figure of the root. Cube the whole root thus fou/nd and subtract the result from the first three periods of the given number ivhen it is less than that 7iumher, but if it is greater, diniihish the figu^i of the root ; jnoceed in a similar way for all the periods. 288 *^ ^ICUJiE KOOT. EXAMPLES. 1. What is the cube root of 99252b47 1 99 252 847(463 4^ = 64 42x3 = 48)352 dividend. First two periods - - - 99 252 (46)^ = 46 X 46 X 46= ^ 336 3 X (46)2 = 6348 ) 19168 2d dividend. The first three periods - 99 252 847 (463)3 =99 252 848 Find the cube roots of the following numbers : 1. Of 389017? 2. Of 5735339 ? 3. Of 32461759? 4. Of 84604519? 5. Of 259694072 ] 6. Of 48228544 ? 302. To extract the cube root of a decimal fraction, Annex ciphers to the decimal^ if necessary, so that it shall consist of 3, 6, 9, ^c, places. Then put the first point over the place of thousandths, the second over the place of millionths, and so on over every third place to the right ; after tvhich extract the root as in whole numbers. Notes. — 1. There will be as many decimal places in the root as there are periods in the given number. 2. The same rule applies when the given number is compoaed of a whole number and a decimal. 3. If in extracting the root of a number there is a remainder after all the periods have been brought down, periods of ciphers may be annexed by considering them as decimals. EXAMPLES. Find the cube roots of the following numbers ; 1. Of .157464? 2. Of .870983875? 3. Of 12.977879 ? 4. Of .7510894291 5. Of .353393243 1 6. Of 3.408862625? 301. What is the rule for extracting the cube root? 302. How do you extract the cube root of a decimal fraction 1 How many decimal places will there be in the root \ Will the same rule apply when there is a whole number and a decimaH If in extracting the root of any f; umber you fmd a decimal, how do you proceed I :bi=? APPLICATIomS. z "v>0 %^ 303. To extract the cube root of a o^mon fraction. J. B.educe compound fractiofis to simjdeov^es, mixed numr hers to improper fractions, and then reduce the fraction to its lowest terms. II. Extract the cube root of the numerator and denomi- lator separately, if they have exact roots ; but if either of hem has not an exact root, reduce the fraction to a decimal, and extract the root as %n the last case, EXAMPLES. Find the cube roots oi" the following fractions : 1 Of ^^^- 2 I 4 Of A 9 2. Of 31^1^? 5. Of f ? 3. OfAVo^ I 6. Off] APPLICATIONS. 1. What must be the length, depth, and breadth of a box, when these dimensions are all equal and the box contains 4913 cubic feet 1 2. The solidity of a cubical block is 21952 cubic yards : what is the length of each side % What is the area of the surface ? 3. A cellar is 25 feet long 20 feet wide, and 8^ feet deep : what will be the dimensions of another cellar of equal capactiy in the form of a cube 1 4. What will be the length of one side of a cubical granary that shall contain 2500 bushels of grain % 5. How many small cubes of 2 inches on a side can be sawed out of a cube 2 feet on a side, if nothing is lost in sawing ? 6. What will be the side of a cube that shall be equal to the contents of a stick of timber containing 1728 cubic feet ? 7. A stick of timber is 54 feet long and 2 feet square ; what would be its dimensions if it had the form of a cube ] Notes. — 1. Bodies are said to be similar when their like part* are propoitional. 2. It is touud that the contents of similar bodies are to each other as the cubes of their like dimensions. 3()3. How do you extract the cube root of a vul{iar fraction \ 11) 21)0 AKITUMETICAL PR0GKE6SI0N. 3. Ail bodies named in the examples are supposed to be simi lar. 8. If a sphere of 4 feet in diameter contains 33.5104 cubic feot, what will be the contents of a sphere 8 feet in diameter 1 43 : 83 : : 33.5104 : Ans. 9. If the contents of a sphere 14 inches in diameter is 1436.7584 cubic inches, what will be the diameter of a sphere which contains 11494-0672 cubic inches? 10. If a ball weighing 32 pounds is 6 inches in diameter, what will be the diameter of a ball weighing 964 pounds ? 11. If a haystack, 24 feet in height, contains 8 tons of hay, what will be the height of a similar stack that shall contain but 1 ton? ARITHMETICAL PROGRESSION. 304. An Arithmetical Progression is a series of numbers in which each is derived from the preceding one by the addition or subtraction of the same number. The number added or subtracted is called the common dif- ference. 305. If the common difference is added, the series is called an increasing series. Thus, if we begin with 2, and add the common difference, 3, we have 2, 5, 8, 11, 14, 17, 20, 23, &c., which is an increasing series. If we begin with 23, and subtract the common difference 3, we have 23, 20, 17, 14, 11, 8, 5, &c., which is a decreasing series. 304. What is an arithmetical progression T What is the number added or subtracted called 1 305. When the common diiference is added, what is the series called 1 What is it called when the common difference is subtracted 1 What ' are the several numbers called 1 What are the first and last called \ What are the intermediate uney called ! AlilTHMETIOAL PliOGKILSSION. 291 The several numbers are called the terms of the progres- sion or series : the first and last are called the extremes, and the intermediate terms are called means. 306. In every arithmetical progression there are five parts : 1st, the first term ; 2d, the last term ; 3d, the common difference ; 4th, the number of terms ; 5th, the sum of all the terms. If any three of these parts are known or given, the remam- ing ones can be determined. CASE I. 307. Knoiving the first term, the common difference, and the number of terms^ to find the last term. 1. The first term is 3, the common difference 2, and the number of terms 19 : what is the last term? Analysis. — By considering the manner in which the increasing progression is formed, we see that the 2d term is obtained by adding the common difference to the 1st term; the 3d, by operation. adding the common difference to the 2d ; the 18 No. less 1 4th, by addmg the common difference to the 2 Com. dif. 3d, and so on ; the number of additions being 1 o^ less than the number of terms found. q i ^ + But instead of making the additions, we may ^ erm. multiply the common difference by the number 39 last term, of additions, that is, by 1 less than the number of terms, and add tlie fii-st term to the pro- duct : hence, Rule. — Multiply the common difference hy 1 less than the number of terms ; if the progression is increasing, add tlie 'product to the first term and the sum will be the last term ; if it is decreasina^ subtract the product frorri t le first term and tJie difference will be the last term. 306 How many parts are there in every arithmetical progression ' What are they 1 How many parts must be given before the remaining ones can be found 1 292 AlilTUMETlCAL PKOGliKSSION. EXAMPLES. 1. A man bought 50 yards of cloth, for which he was to pay 6 cents for the 1st yard, 9 cents for the 2d, 12 cents for the 3d, and so on increasing by the common difference 3 : how much did he pay for the last yard 1 2. A man puts out $100 at simple interest, at 7 per cent : at the end of the. 1st year it will have increased to $107, at the end of the 2d year to $114, and so on, increasing $7 each year : what will be the amount at the end of 16 years 1 3. What is "the 40th term of an arithmetical progression of which the first term is 1, and the common difference 1 i 4. What is the 30th term of a descending progression of which the first term is 60, and the common difierence 2 ? 5. A person had 35 children and grandchildren, and it so happened that the difierence of their ages was 18 months, and the age of the eldest was 60 years : how old was the youngest ] CASE II. 308. Knowing the two extremes and the number of terms^ to Jind the common difference. 1. The extremes of an arithmetical progression are 8 and 104, and the number of terms 25 : what is the common dii- ference ? 4^ Analysis. — Since the common difference multiplied by 1 less than the number of operation. ^erms gives a product equal to the differ- 1.04 eiice of the extremes, if we divide the dif- g ference of the extremes by 1 less than the —: oTVdr'TT number of terms, the quotient will be the '^^— 1 =i/i4) Jo(4. common difference : hence, Rule. — Subtract the less extreme from, the grealer and divide the remainder by 1 Less than the number of tertns ; the quotient will be the common difference. 307. When you know the first term, the cuniiiioti difierence, and the atimber of terms, how do you find the last, term ! 308. When you know the extremes and the number of tenus, how do you find the common difference \ AJirmMi. rioAL pikkjrkssion. 29'J EXA?.irLES. 1. A man has 8 sons, the youngest is 4 years old and the eldest 32 : their ages increase in arithmetical progression : what is the common diflerence of their ages ? 2. A man is to travel from New York to a certain place in 12 days ; to go 3 miles the first day, increasing every day by the same number of miles ; the last day's journey is 58 miles: required the daily increase. 3. A man hired a workman for a month of 26 working days, and agreed to pay him 50 cents for the first day, with a uniform darily increase; on the last day he paid $1,50: what was the daily increase ? CASE III. 309. To find the sum of the terms of an arithmetical progression. 1. What is the sum of the series whose first term is 3, common diflerence 2, and last term 19 ? Given series - 3+ 5-f- 7+ 9-1-114-134-14+17 + 19= 99 Sarae; order ) . of terms in-[ 19+17+15+13+11+ 9+ V+ 5+ 3= 99 inverted. J Sum of both. 22 22 22 22 22 22 22 22 22 = 198 Analysis. — The two series are the sarae ; hence, their sum is equal to twice the given series. But tlieir sum is equal to the sum of the two extremes 3 and 19 taken as many times as there are terms : and the given series is equal to half this sum, or to the sum of the extremes multiplied by half the number of terms. Rule. — Add the extremes together and multiply their sutn by Jialf the number of terms ; the 'product will be the sum of tlie series. EXAMPLES. 1. The extremes are 2 and 100, and the number of terms 22 : what is the sum of the series 1 OPERATION. 2 1st term, 100 last terra. Analysis. — We first add together the two extremes and then multiply by half 102 sum of extremes. the number of terms. 1 ] half the number of terms. 1122 sum of series. 1}0U. How do you find the sum of the tcrmbl 2t5 I GLOMLTKICAL J'KOOKESSION. 2. How many strokes does the hammer of a clock strike in 12 hours? 3. The first term of a series is 2, the common diflerence 4, and the number of terms 9 : what is the last term and sum of the series ? 4. James, a smart chap, having learned arithmetical pro- gression, told his father that he would chop a load of wood of 1 5 logs, at 2 cents the first log, with a regular increase of I cent for each additional log : how much did James receive for chopping the wood ? 5. An invalid wishes to gain strength by regular and in- creasing exercise ; his physician assures him that he can walk 1 mile the first day, and increase the distance half a mile for each of the 24 following days : how far will he walk? 6. If 100 eggs are placed in a right line, exactly one yard from each other, and the first one yard from a basket : what distance will a man travel who gathers them up singly, and places them in the basket ] GEOMETRICAL PROGRESSION. 310. A Geometrical Progression is a series of terms, each of which is derived from the preceding one, by multi- plying it by a constant number. The constant multipher is called the ratio of the progression. 311. If the ratio h greater than 1, each term is greater than the preceding one, and the series is said to be in- creasing. 310. What is a geometrical progression! What is the constant multipher called 1 311. If the ratio is greater than 1, how do the terms compare with each other 1 What is the series then called 1 If the ratio is less than 1, how do they compare? WTiat is the series then called ? What are the several numbers called 1 What are the first and last called ? What are the intermediate ones called 1 312. How many parts are there in every geometrical progression 1 What are they 1 How n.'any must be known before the others can be fuund ** 1,. 2, 4, 8, 16, 32, &c.- 52, 16, 8, 4, 2, 1, &c.- GEOMEITJICAL PROGRESSION. 20*i If the ratio is less than 1, each term is less than the preceding one, and the series is said to be decreasing ; thus, -ratio 2 — increasing series : -ratio ^^-decreasing series. The several numbers are called terms of the progression. The first and last are called the extremes, and the intermedi- ate terms are called means. 312. In every Geometrical, as well as in every Arithmeti- '^al Progression, there are five parts : 1st, the first term ; 2d^ the last term ; 3d, the common ratio ; 4th, the number of terms ; 5th. the sum of all the terms. If any three of these parts are known, or given, the re- training ones can be determinied. CASE I. 313. Having given the first term, the raXio, and the nwinber of terms, to find tlie last term. 1. The first term is 3 and the ratio 2 : what is the 6th term 1 Analysis, — The se- opkration. oond term is formed by 2 X 2 X 2 X 2 X 2 = 2^ =r 32 multiplying the first 3 ^^^ ^^ term by the ratio ; the third term by multiply- Ans. 96 ing the second term by the ratio, and so on ; the number of multiplicators being I /«5i ihan the number of terms : thus, 3 = 3 1st term, 3x2 = 6 2d term, 3x2x2 = 3x22=12 3d term, 15x2x2x2 = 3x23 — 24 4th term, &c. 2D(J GEOMETRICAL FKOGKESSION. Therefore, the last term is equal to the f/st ternCmulti- plied by the ratio raised to a power 1 less than the numbet of tenns. Rule. — Raise the ratio to a power whose ex-ponent is 1 less than the number of terms, and then multiply this powei by the first term. EXAMPLES. 1. The first term of a decreasing progression is 192 ; the ratio J, and the number of terms 7 : what is the last term '? Note. — The 6th power of the ratio, (i,) is operation. ^^ and this multiplied by the first term 192, (i)'^=:Jj gives the last term 3. 192 X ^~ 3. 2. A man purchased 1 2 pears ; he wa to pay 1 farthing for the 1st, 2 farthnigs for the 2d, 4 for ihe 3d, and so on, doubhng each time : what did he pay for the last ? 3. The first term of a decreasing progression is 1024, the ratio ^ : what is the 9th term 1 4. The first term of an increasing progression is 4, and the common difierence 3 : what is the 10th term ] 0. A gentleman dying left nine sons, and bequeathed his estate in the IblloM'ing manner : to his executors $50 ; his youngest son to have twice as much as the executors, and each son to have double the amount of the son next younger : what was the eldest son's portion \ 6. A man bought 12 yards of cloth, giving 3 cents ibr the 1st yard, 6 for the 2d, 12 for the 3d, &o. : what did he pay for the last yard V CASE n. 314. Knovnng the two extremes and the ratio, to find the sum of the terms. 1. What is the sum ol the terms, in the progression, 1, 4, 16, 64? 313. Knowing the first term, the r^tio, and the number of terms, how do you find the last term "? 314. Knowing the two extremes and the ratio, how do you fmd the sum of tlie terms ? GEOMLTKICAL PKoGKESSlO^. 297 Analysis. — If we multiply the terms of the progression by the ratio 4. we have a second pro- gression,. 4, 16, 64, 256, which operation. is 4 times as great as the first. 4+16 + 64 + 256= 4 times. If from this we subtract the 1+4 + 16 + 64 = once, first, the remainder, 256 — 1, Oof — 1 =Tf will be 3 times as great as ' ' the first; and if the remain- 256—1 255 der b« divided by 3, tlie quo- ~§ = — :=85 sum. tieut will be the sum of the terms of the first progression. But 256 is the product of the last term of the given progression multiplied by the ratio, 1 is the first term, and the divisor 3 is 1 less ihan the ratio : hence. Rule. — Multiply the last term by the ratio ; take the dif- ference between the jn-oduct and the first term and divide the remainder by the difference between 1 and the ratio. Note.-— When the progression is increasing^ the first term is subtracted from the product of the last term by the ratio, and the divisor is found by subtracting 1 from the ratio. When the pro- gression is decreasing^ the product of the last term by the ratio is EXAMPLES. 1 . The first term of a progression is 2, the ratio 3, and the last term 4374 : what is the sum of the terms ? 2. The first term of a progression is 128, the ratio i, and the last term 2 : what is the sum of the terms 1 ?>. The first term \& 3, the ratio 2, and the last term 192 : what is the sum of the series '? 4. A gentleman gave his daughter in marriage on New Year's day, and gave her husband \s. towards her portion, and was to double it on the first day of every month during the year : what was her portion % 5. A man bought 10 bushels of wheat on the condition that he should pay 1 cent for the 1st bushel, 3 for the 2d, 9 for the 3d, and so on to the last : what did he pay for the last bushel, and for the 1 bushels 1 6. A man has 6 children : to the 1st he gives $150, to the 2d $300, to the 3d $600, and so on, to each twice as much as the last : how much did the eldest receive, and what was Die amount received by them all ? 21)8 PKOMISCUOUS QUESTIONS. PROMISCUOUS EXAMPLES. 1. A merchant bought 13 packages of goods, lor which he paid $326 : what will 39 packages cost at the same rate? 2. How many bushels of oats at 62-J cents a bushel will pay for 4250 feet of lumber at $7,50 per thousand? 3. Bought Ihhd. of sugar which weighed as follows : the 1st '^cwi. \qr. 18lb., the 2d 6cwt. 10/6. : what did it cost at 7 cents per pound? 4. How many hours between the 4th of Sept., 1854, at 3 P.M., and the 20th day of April, 1855, at 10 A.M. ? 5. If I of a gallon of wine cost -f of a dollar, what will f of a hogshead cost ? 6. What number is thai which being multiplied by | will pro- duce ^? 7. A tailor had a piece of cloth containing 24j yards, from which he cut 6f yards : how much was there left ? 8. From | of ^ take J of ^^ ^12 ^5 9. What is the difference between 3| + 7f and 4 + 2^ ? 10. There was a company of soldiers, of whom \ were on guard, ^ preparing dinner, and the remainder, 85 men, were drilling : how many were there in the company ? 11. The sum of two numbers is 425, and their difference' 1.625 : what are the numbers ? 12. The sum of two numbers is ^. and their difference i|^: what are the numbers ? 13. The product of two numbers is 2.26, and one of the numbers is .25 : what is the other ? 14. If the divisor of a certain number be 6.66f, and the quo- tient I, what will be the dividend ? 15. A person dying, divided his property between his widow and his four sons; to his widow he gave $1780, and to each of his sons $1250 ; he had been 25^ years in. business, and had cleared on an average $126 dollars a year: how much had he when he began business ? 16. A besieged garrison consisting of 360 men was provisioned for 6 months, but hearing of no relief at the end of five months, dismissed so many of the garrison, that the remaining provision lasted 5 months : how many men were sent away ? 17. Two persons, A and B are indebted to C ; A owes $2173, which is the least debt, and the difference of the debts is Vo71 : what is the amount of their indebtedness ? 18. What number added to the 43d part of 4429, will make the 8uni 240 ? 20 PliOMlSCUUUS QUEBTIONS. 209 19. How many planks 15 feet long, and 15 inches wide, will floor a barn 60^ feet long, and 33^ feet wide? 20. A person owned -| of a mine, and sold | of his interest for $1710 : what was the value of the en lire mine? '21. A room 30 feet long, and 18 feet wide, is to be covered with painted cloth f of a yard wide : how many yards will cover it ? 22. A, B and C trade together and gain $120, which is to be shared according to each one's stock; A put in $140, B $300, and C $160 : what is each man's share? 23. A can do a piece of work in 12 days, and B can do the same work in 18 days : how long wil] it take both, if they work togethor ? 24. If a barrel of flour will iast one family 7^ months, a .second family 9 months, and a third 11^ months, how long will it last the three families together ? 25. Suppose I have ^| of a ship worth $1200 ; what part have I left after selling ^ of ^ of my share, and what is it worth ? 26. What number is that which being multiplied by | of | of l^, tho product will be 1 ? 27. Divide $420 between three persons, so that the second shall have ^ as much as the first, and the third ^ as much as the other two ? 28. What is the difference between twice five and fifty, and twice fifty-five ? 29. What number is that which being multiplied by three- thousandths, the product will be 2637 ? 30. What is the difference between half a dozen dozens and six dozen dozens ? 31. The slow or parade step is 70 paces per minute, at 28 inches each pace : how fast is that per hour ? 32. A lady being asked her age. and not wishing to give a direct answer, said, *• I have 9 children, and three years elapsed between the birth of each of them ; the eldest was born w/ien I was 1 9 years old. and the youngest is now exactly 19 :" what was her age ? 33. A wall of 700 yards in length was to be buxlt in 29 days : 12 men were employed on it for 11 days, and only completed 220 yards : how^ many men must be added to complete the wall in the required time ? 34. Divide $10429,50 between three persons, so that as often as one gets $4, the second will get $6, and the third $7. 35. A gentleman whose annual income is £1500, spends 20 guineas a week : does he save, or run in debt, and how much ? 36. A farmer exchanged 70 bushels of rye, at $0,92 per bushel, for 40 bushels of wheat, at $1,37-^ a bushel, and received the balance in oats, at $0,40 per bushel : how many bushels of oats did he receive ? 37. In a certain orchard -^ of the trees bear apples, ^ of them Lear peaches, ^ of them plums, 120 of them cherries, and 80 of Ihem pears : how many trees are there in the orcliard ? 000 PKOMlSClIOLife QUESTIONS. 38. A person being asked the time, said, the time past noon IS equal to \ of the time past midnight : what was the hour? 39. If 20 men can perform a piece of work in 12 days, how many men will accomplish thrice as much in one-fifth of the time ? 40. How many stones 2 feet long. 1 foot wide, and 6 inches thick, will build a wall 12 yards long, 2 yards high, and 4 feet thick ? 41. Four persons traded together on a capital of $6000. of which A put in ^, B put in ^. C put in J, and D the rest ; at the end of 4 years they had gained $4728 : what was each one's share of the gain ? 42. A cistern containing 60 gallons of water hah three unequal cocks for discharging it ; the largest will empty it in one hour, the second in two hours, and the third in three hours : in what time will the cistern be emptied if they run together ? 43. A man bought f of the capital of a cotton factory at par; he retained |- of his purchase, and sold the balance for $5000, which was 15 per cent advance on the cost ; what was the whole capital of the factory? 44. Bought a cow for $30 cash, and sold her for $35 at a credit of 8 months : reckoning the interest at 6 per cent, how much did 1 gain ? 45. If, when I sell cloth for 85. 9c?. per yard, I gain 12 per cent, what will be the gain when it is sold for IO5. 6d. per yard ? 46. How much stock at par value can be purchased for $8500, at 8-J per cent premium, \ per cent being paid to the broker ? 47. Twelve workmen, working 12 hours a day, have made, in 12 days, 12 pieces of cloth, each piece 75 yards long : how many pieces of the same stuff would have been made, each piece 25 yards long, if there had been 7 more workmen ? 48. A person was born on the 1st day of Oct., 1801, at 6 o'clock in the morning: what w^as his age on the 21st of Sept., 1854, at half-past 4 in the afternoon? 49. A, can do a piece of work alone in 10 days, and B in 13 days : in what time can they do it if they work together ? 50. A man went to sea at 17 years of age ; 8 years after he had a son born, who lived 46 years, and died before his father ; after which the father lived twice twenty years and died : what was the age of the' father ? 51. How many bricks, 8 inches long and 4 inches wide, will pave a yard that is 100 feet by 50 feet ? 52. If a house is 50 feet wide, and the post which supports the ridge pole is 12 feet high, what will be the length of the rafters? 53. A man had 12 sons, the youngest was 3 years old and the eldest 58, and their ages increased in Arithmetical progression what wai; tlic comnioii diflerence of their ages ? PKOlIlBCUOUtJ tiUlLSTlUNS. iiUl 54. If a quantity of provisions serves 1500 men 12 weeks, tit tlie rate of 20 ounces a day for each man, how many men will the same provisions maintain for 20 weeks, at the rate of 8 ounces a day for each man? 55. A man bought 10 bushels of wheat, on the condition that he should pay 1 cent for the 1st bushel, 3 for the 2d, 9 for the 3d, and so on to the last : what did he pay for the last bushel, and for the 10 bushels? 56. There is a mixture made of wheat at 45, per bushel, rye at 3s., barley at 25,, with 12 bushels of oats at 18f/. per bushel : how much must be taken of each sort to make the mixture worth 35. 6d. per bushel : 57. What length must be cut ofT a board 8^ inches broad to contain a square foot ? 58. What is the difference between the interest of $2500 for 4 years 9 mo. at 6 per cent, and half that sura, for twice the time, at half the same rate per cent ? 59. A person lent a certain sum at 4 per cent per annum ; had this remained at interest 3 years, he would have received for prin- cipal and interest $9676,80 : what was the principal ? 60. If 1 pound of tea be equal in value to 50 oranges, and 70 oranges be worth 84 lemons, what is the value of a pound of tea, when a lemon is worth 2 cents ? til. A person bought 160 oranges at 2 for a penny, and 180 more at 3 lor a penny ; after which he sold them out at the rate of 5 for 2 pence : did he make or lose, and how much ? 62. A snail in getting up a pole 20 feet high, was observed to climb up 8 feet every day, but to descend 4 feet every night : in what time did he reach the top of the pole ? 63. A ship has a leak by which it would fill and sink in 15 hours, but by means of a pump it could be emptied, if full, in 16 hours. Now, if the pump is -worked from the time the leak begins, how long before the ship will sink ? 64. A and B can perform a certain piece of work in 6 days. B and C in 7 days, and A and C in 14 days : in what time would each do it alone? 65. Divide $500 among 4 persons, so that when A has -J dollar, B shall have f C \, and D |, 66. A man purchased a building lot containing 3600 square feet, at the cost of $1.50 per foot, on which he built a store ar an expense of $3000. He paid yearly $180.66 for repairs and taxes: what annual rent must he receive to obtain 10 per cent, on the cc«t ? 67. A's note of $7851,04 was .dated Sept, 5th, 1837, on which were endorsed the following payments, viz.: Nov. 13th, 1839, $410,98; May lOtli, 1840, $152 : what wus due March Itt., 1841 ihe inteic:t hciu^^. (> per cent ? :30'2 I'KOMISCUOUS QUEBTlOiS'S. 68. A house is 40 feet from tlie ground to the eaves, and it is required to find the length of a ladder which will reach the eaves, supposing the foot of the ladder cannot be placed nearer to the house than 30 feet ? 69. Sound travels about 1142 feet in a second; now, if the flash of a cannon be seen at the moment it is fired, and the report heard 45 seconds after, what distance would the observer be from the gun ? 70. A person dying, worth $5460, left a wife and 2 children, a son and daughter, absent in a tbreign country. He directed that if his son returned, the mother should have one-third of the estate, and the son the remainder ; but if the daughter returned, she should have one-third, and the mother the remainder, Now it so happened that they both returned : how must the estate be divided to fulfill the father's intentions ? 71. Two persons depart from the same place, one travels 32, and the other 36 miles a day : if they travel in the same direction, how far will they be apart at the end of 19 days, and how far if they travel in contrary directions? 72. In what time will $2377,50 amount to $2852,42, at 4 per cent, per annum ? 73. What is the height of a wall, which is 14^ yards in length, and y'^ of a yard in thicknes.^, and which has cost $406, it having been paid for at the rate of $10 per cubic yard ? 74. What will be the duty on 22o bags of coffee, each weighing gross 1 60 lbs., invoiced at 6 cents per lb. : 2 per cent, being the legal rate of tare, and 20 per cent, the duty ? 75. Three persons purcliase a piece of property for $9202 ; the first gave a certain sum ; the second three times as much ; and the third one and a half times as much as the other two : what did each pay ? 76. A reservoir of water has two cocks to supply it. The first would fill it in 40 minutes, and the second in 50. It has likewise a discharging cock, by which it may be emptied when full in 25 minutes. Now, if all the cocks are opened at once, and the water runs uniformly as we have supposed, how long before the cisterr will be filled ? 77. A traveller leaves New Haven at 8 o'clock on Monday morning, and walks towards Albany at the rate of 3 miles an hour : another traveller sets out from Albany at 4 o'clock on the same evening, and walks towards New Haven at the rate of 4 miles an hour; now, supposing the distance to be 130 miles, where on the road will they meet ? MENSUKATION. 30:J MENSURATION. 315. A triangle is a portion of a plane bounded by three straight lines. BC is called the base ; and AD, perpendicular to BC, the altitude. 316. To find the area of a triangle Tke area or contents of a triangle is equal to half the product of its base by its altitude {Bk. IV. Prop. VI).* EXAMPLES. 1. The base, BC, of a triangle is 40 yards, and the perpendicu lar, AD, 20 yards : what is the area? 2. In a triangular field the base is 40 chains, and the perpendi- cular 15 chains : how much does it contain ? (Art. 1 10.) 3. There is a triangular field, of which the base is 35 rods and the perpendicular 26 rods : what are its contents ? 317. A square is a figure having four equal sides, ojid all its angles riuht angles. 318. A rectangle is a four-sided figure like a square, in which the sides are perpendicular to each other, but the adjacent sides are not equal. 319. A parallelogram is a four-sided figure which has its opposite sides equal and parallel, but it» angles not right angles. The line DE, perpendi- cular to the base, is called the altitude. 320. To find the area of a square, rectangle, or parallelogram, Multiply the base by the perpendicular height, and the product will be the area. {Book IV. Prop. V). EXAMPLES. 1. What is the area of a square field of which the sides are each 33.08 chains ? 2. What is the area of a square piece of land of which the sides are 27 chains ? 3. What is the area of a square piece of land of which the sides are 25 rods each ? D h E All the reference*: are to Davies' LefjenJre. 304 MENSURATION. 4. What are the contents of a rectangular field, the length oi which is 40 rods and the breadth 20 rods ? 5. What are the contents of a field 40 rods square ? 6. What are the contents of a rectangular field 15 chains long and 5 chains broad ? 7. What are the contents of a field 27 chains long and 9 rods broad ? 8. The base of a parallelogram is 271 yards, and the perpendi cular height 360 feet : what is the area ? 321. A trapezoid is a four-sided figure -0_ ABCD, having two of its sides, AB, DC, parallel. The perpendicular CE is called / the altitude. A E B 322. To find the area of a trapezoid. Multiply the sum of the two parallel sides by the altitude, and divide the product by 2, the quotient will be the area. (Bk. IV. Prop. VII). EXAMPLES. 1 . Required the area of the trapezoid ABCD, having given AB=321.51/i^., DC-=214.24/^, and CE=171.16/^ 2. What is the area of a trapezoid, the parallel sides of which are 12.41 and 8.22 chains, and the perpendicular distance between them 5.15 chains ? 3. Required the area of a trapezoid whose parallel sides are 25 feet 6 inches, and 18 feet 9 inches, and the perpendicular distance between them 10 feet and 5 inches. 4. Required the area of a trapezoid whose parallel sides are 20.5 and 12.25, and the perpendicular distance between them 10.75 yards. 5. What is the area of a trapezoid whose parallel sides are 7.50 chains, and 12.2^chains, and the perpendicular height 15.40 chains ? 6. What are the contents when the parallel sides are 20 and 32 chains, and the perpendicular distance between them 26 chains ? 323. A circle is a portion of a plane bounded by a curved line, called the circum- ference. Every point of the circumference is equally distant from a certain point within called the centre : thus, C is the centre, and any line, as ACB, passing through the centre, is called a diameter. If the diameter of a circle is 1, the circumference will bt 3.1416. Hence, if we know the diameter, we may find the circum- ference by multiplying by3A4\6 ; or, if we know the circvmfrrnice^ we may find the diavuUr by diiidin^^ by 3. 1416. MENSURATION. ^05 EXAMPLES. 1. The diameter ot a circle is 4, what is the circumference? 2. The diameter of a circle is 93, what is the circumference? 3. The diameter of a circle is 20, what is the circumference? 4. What is the diameter of a circle whose circumference is 78.54 ? 5. What is the diameter of a circle whose circumference is n652.1944? 6. What is the diameter of a circle whose circumference is 6850 ? 324. To find the area or contents of a circle, Multiply the square of the diameter by the decimal .7854 iBk. V. Prop. XII. Cor. 2). EXAMPLES. 1. What is the area of a circle whose diameter is 6 ? 2. What is the area of a circle whose diameter is 10? 3. What is the area of a circle whose diameter is 7 ? 4. How many square yards in a circle whose diameter is 3j feet ? 325. A sphere is a figure terminated ^rffj^jMi by a curved surface, all the parts of which are equally distant from a certain point within called the centre. The line AB passing through its centre C is called the diameter of the sphere, and AC its radius. 326. To find the surface of a sphere, Multiply'the square of the diameter by 3 1416 [Bk. VIII. Prop. X. Cor). EXAMPLES. 1 . What is the surface of a sphere whose diameter is 12 ? 2. What is the surface of a sphere whose diameter is 7 ? 3. Required the number of square inches in the surface of a sphere whose diameter is 2 feet or 24 inches. 327. To find the contents of a sphere, Multiply the surface by the diameter and divide the product by % the quotient will be the contents. {Bk. VIII. Frop. XIV. Sch. 3). EXAMPLES. 1 . What are the contents of a sphere whose diameter is 1 2 ? 2. What are the contents of a sphere whose diameter is 4 ? 3. What are the contents of a sphere whose diameter is 14in. ? 4. What are the contents of a .sphere whose diauieler is ijft. 306 MENSUltATlON. 328. A prism is a figure whose ends are equal plane figures and whose faces are parallelograms. The sum of the sides which bound the base is called the perimeter of the base, and the sum of the parallelograms which bound the solid is called the convex surface. \^ i 329. To find the convex surface of a right prism, Multiply the perimeter of the base by the perpendicular height, and the product will be the convex surface {Bk. Vll. Pro]). 1). EXAMPLES. 1. What is the convex surface of a prism whose base is bounded by five equal sides, each of which is 35 feet, the altitude being 26 feet? 2 What is the convex surface when there are eight equal sides, each 15 feet in length, and the altitude is 12 feet? 330. To find the solid contents of a prism. Multiply the area of the base by the altitude., and the product will be the contents (Bk. VII. Prop. XIV). EXAMPLES. 1. What are the contents of a square prism, each side of the square which forms the base being 15, and the altitude of the prism 20 feet? 2. What are the contents of a cube each side of vfhich is 24 inches ? 3. How many cubic feet in a block of marble, of which the length is 3 feet 2 inches, breadth 2 feet 8 inches, and height or thickness 2 feet 6 inches ? 4. How many gallons of water will a cistern contain whose di- mensions are the same as in the last example ? 5. Required the contents of a triangular prism whose height ifc lO feet, and area of the base 350 ? 331 • A cylinder is a figure with circular ende. The line EF is called the axis or alti- tude, and the circular surface the convex sur- face of the cylinder. MENSURATION. ao7 332. To find the convex surface, Multiply the circumference of the base by the altitude^ and the pro- duct will be the convex surface. [Bk. VIII. Prop. I). EXAMPLES. 1. What its the convex surface of a cylinder, the diameter of whose ba.se is 20 and the altitude 50 ? 2 What is the convex surface of a cylinder, whose altitude is 14 te^t jind the circuniference of its base 8 feet 4 inches ? 3. What is the convex surface of a cylinder, the diameter of whose base is 30 inches and altitude 5 feet ? 333. To find the contents of a cylinder, Multiply the area of the base by the altitude : the product will be the contents. (Bk. VIII. Prop. II). EXAMPLES. 1 . Required the contents of a cylinder of which the altitude is 12 feet and the diameter of the base 15 feet? 2. What are the contents of a cylinder, the diameter of whose base is 20 and the altitude 29 ? 3. What are the contents of a cylinder, the diameter of whose base is 1 2 and the altitude 30 ? 4. What are the contents of a cylinder, the diameter of whoso base is 1 6 and altitude 9 ? 5. What are the contents of a cylinder, the diameter of whose base is 50 and altitude 15 ? 334. A pyramid is a figure formed by several triangular planes united at the same point S, and terminating in the different sides of a plain figure as ABODE. The altitude of the pyramid is the line SO. drawn \)erpendicular to the base 335. To find the contents of a p>Tamid, Multiply the area of the base by the 'iltitude. and divide the pf dud by 3 {Bk VII Proy XVI f) 308 MENSURATION. EXAMPLES. 1. Required the contents of a pyramid, of which the aren of the base is 95 and the altitude 15. 2. What are the contents of a pyramid, the area of who&e ba.se is 260 and the altitude 24 ? 3. What are the contents of a pyramid, the area of whose base is 207 and altitude 18? 4. What are the contents of a pyramid, the area of whose bas* is 403 and altitude 30 ? 5. W^hat are the contents of a pyramid, the area of whose base is 270 and altitude 16? ' 6. A pyramid has a rectangular base, the sides of which are 25 and 12: the altitude of the pyramid is 36: what are its con- tents ? 7. A pyramid with a square base, of which each side is 30, has an altitude of 20 : what are its contents ? 336. A cone is a figure with a circular base, and tapering to a point called the vertex. The point C is the vertex, and the line CD is called the axis or altitude. 337. To find the contents of a cone, Multiply the area of the base by the altitude^ and divide the pro- duct by 3. {Bk. VI 11. Prop. V). EXAMPLES. 1, Required the contents of a cone, the diameter of whose base is 5 and the altitude 10. 2 What are the contents of a cone, the diameter of whose base is 18 and the altitude 27 ? 3. What are the contents of a cone, the diameter of whose base is 20 and the altitude 30 ? 4. What are the contents of a cone, whose altitude is 27 feet, and the diameter of the base 10 feet? 5. What are the contents of a cone whose altitude is 12 feet, and tlie 414,75 3480-$4,50 EX. ANS. 15 I S2,33J 16 i$11000-o.500 17 ^23,16 KX. I ANS. 19 I ^o47/j2 20 ^DIG 21 ^27,685 II 22 | $290,82 jj 23 \ $90277,70 99. 99. ioo. 100. 101. 101. 101. 101. 101. 2 ' 30183/a/-. 3 I 84226>y. 4 I 39l679yar. 5 £84 £1 125. 3i^/. £25 14.5. \d. \2\)in. \9-Zin. ISyd. '62yd. 2 I \2yd. 3'3l6767fL 4 i 3597m. Ifur. 2Srd. 53796602/^5. 6 1 8201 miles. 7|240700858m. 3 I 4 I ( 109^ l2fL 8ft. 4ft. 24fur. 4.Sfur. 64far. 21^ mi. 3h/d. 2ft. Sin. Ifur. \rd. 8 1 Ana. 16/ja. 32//«. 96wa. \28yuu 2Wqr. 32qr. 2Sqr. 3 \3qr. Aqr. 5qr. 9qr. \Oqr. 102. 102. 102. 3\980?ia. 4 623?^.a. 5 204yc/. 3qr. 2na. 6 7 28 El. Fl. \qr. 95E. E. Aqr. 103. 103. 1 ( 288in. \32in. \8^Mn. 1152m. 2 40P. 120P. 640ii. 12807t. 3 160P. 320P. 9,jd. 104. 1 104. 2 104. 3 104. 4 104. 104: 3 4P. 16P. 20P. SOch. \60ch. 240ch. 16P. 64P. 96P. J 20sq. ck. 60sq. ch. (IOO57. c/t. I20sq.ch 3157P. 4 762300. 5 260sq. ft. 16.9^. in. 6 93.4. 2R. 12P. 7 35ilf. 563A IR. 19P. 8$12584,25. 9!t;15,25. 105, 105. 105. 105. 105. 105. 106. 106. 106. j 1728 0/^. m. 3i56Cu.in. I 5184 Cw. in. ] 27 C. ft. 540. ft. \08CfL ll(J2C.ft. 24 C.fl. 40 C.ff: 48 C. ft. 256 S. ft. 64 S. ft. 32 S. ft. 5|2C. ijd. 3C. yd. 6I3T. 7|2T. 4T. 8\8C. 120. 16C. 9\48C.ft. 592704. 200 c. ft. 3200 S. ft. 5 cords. 2 cord ft. 21870 cord.s-4C.ft. (88 tons. 24 S. ft. 1 1228 S. in. 107. 107. 107. 107, lljie^z. 6;?/. 12;;^. 18;;/. 20/^^. 2j8r;/. ICqt. 24qt. IGpt. 40pt. \3<2har. Sbar. I2bar. \4\i2qt. 20qt. SOqi. I26qt. 252qL I2002pt. I tuns 2hhd. 25 tuns Igcd. 136,64. 3i!0 A^iSWEKE P. m. 108. 108. ANS. n27<. i6qt. 36qt, [3672pt ANS. 12734 pt 129 hhd. ISgaL 109. 1 \l^t. 4:QqtAAqt.\ 3 (24^^. S2pk. ASpL (46m. 867/. I06z^. 109. 2 3pk. \pk. Spk. 109. - 4 726w. .10862^. 1446m. 110. 3 23808^^. 1 32^r. 5 3 tons. 110. 4 SAApk. 2 64o2. 110. 5 212bu. 3 \cwt.2qr. 110. 6 Uli. 2%u. 3pk. 6/t. 4 I2^r. 111. 3 2790366 drams. 7 er. 2cwt. 4Z6. 13o2. \4dr. 111. 4*^03136^^2. 8 299812802. 111. , \ 6T.Scwt.3qr.2Ub.\3oz. ^ \ Udr. 9 212 7. Wcwt. \qr. lib. Ill 10$118,99o-$10. 111. 6 2ST. icwt. \qr. 21/6. 11 $431,68-^160. 112. l,48^r. 72«'/-. 96^r. 4 1/6. lo^. 10^>w;^. lO^r. 112. 2 22Jwt. 3pwt. 5 25/6. 9oz. Opwt. 20 gr. 112. 3 202. Soz. 6 578618^r. 112. . ( 4802;. 1440Z 108o2. 7 ^ 1 84o;^. 8 36/6. 7o2. l^pwL 112. 38901^/-. 112. 5 2lb. 3lb. Sib. 9 K96^/. 112. 3 148340^r. 10 J $657. 113. 1 4:0 gr. 60 gr. 80 s^r. U lOgr, 3 40! 8 5. 113. S 129 219 159. 114. 3 8011 9. 7 .739 1 8 -r. 114. 4 91133. ^ i j 12fe 9! 73 ° 1 29 18^r. 114. 5 27 fe 9 § 6 3 19. 114. 6 94fe 11 ! 13. 115. 115. 115. 115. 1 2 3 4 2405ec. 3606-ec. 72hrA20hr.72hr. 4:2da. 66da. 3 4 5 379467l085ec. 47/ r. Ida. 24yr.. Ida. 26m. 5Ssec. 116. 116. ^ 3 9y//-. I4da. nhr. 1 7 '^ I lOm. A.Osec. \ 1 6600hr. ISOsec. 240sec. 300scc, ANSWEliS. P. EX. 116. 2 116. 3 116. 4 116. 1 116. 2 ANS. 360 w. 240m. 300m. 120° 180° 210° 240° 4° 12° 3s. 5s. 6s. 10765' 2592000" 117. 117. 117. 117 117 57953Ar. 10800' 1296000 Cw. in, £714 3 T. Icwt. 20lb. 6 7 8 9 10 EX. 3" 4 5 6 7 T Ic. 5s. 39468005ec. 921625.sec. 2° 23' 9" 9fe 8 5 13 2 9 I9^r. lib. Qoz. ^pwt. \9gr. 340\61gr. S\g. Is. 201 E. E. 3qr. 118. 11 118. 12 118. 118. 13 118. 14 118. 15 118. 16 118. 17 118. 18 118. 19 118. 20 118. 21 118. 22 118. 23 3320 half pints. 6539276dr. I 6 A. IR. 24P. 1 pound. 67953hr. 7s. 15° 24' 40" 12 cords. 1244 160 Cm. in, \20962)k. 311yd. 2qr. 48916gi. 41 8002432sq, in. 15359fa?: 4rd, 84mi. 3fur. 3yd. 2ft. 5A 3R. 35P. 2ft. 5in. 197111025ikr. 26880 times. .S93024 827 4 fniles. 40 yards. 57no.3wk. 5da. 16Ar 1008 bottles 110592 38 casks. 3iyd, 119. 119. 36 37 17097yy^ times. 10132992005ec. 38 39 248 miles $39,879 120. 120. 120. 120. 120. 120. 2 3 4 5 6 7 £1377 45. l\d. £1616 75. 6f^. 621Z^>. 8oz. Ipwt. \9gr. 9621b. 6oz. lOpwt. 2gr. 104}fe 3 5 33 2 9 4^r. 35! 7 3 2 9 llgr. 8 9 10 20 3 1 9 10^/. j 19cwt. 2qr. 18lb, j l5oz. lldr, J 340 T. 5cwt. 2qr. \ 20lb. 2oz. 121. 121. 121. 121. 121. 121. 121. 11 12 13 14 15 \6cwt. 2dr. j 432i. 2mi. 4fur. \ 39rd. 4yd. j \fur. 34rd. l^yd. \ \ft. 4in. 424E.Fl. Oqr. 3na. 42yd. 3qr. hm. 16 17 18 19 j 184E.E. 4qr. 2?ia, 263sq.yd. 5sq. ft 1 1 657. in. 21 M. 211 A. \R. ■ OP. 24I.S. yd. 159 A. 2R. 5P. Ml 3-22 ANSWERS. 121. m. 122. 122. 122. 122. 122. 122. 122. 122. 122. 123^ [23. 123. 123, 123. 123. 123. 123.i 123.1 EX 20 23 24 25 126 27 ANS. 176C. yd. \SC.ft. 614 C. iM. EX, ~2V 22 ANS. 90 C. ]06C.ft. 151 C. 3C.//. ij21hkd. Igat. \qt. \pt. j dOlmiOj). Ihhd. SSgcd. I Sqt. j 9467i. 22bu. 3])/c. Iqt. 2596-;i. \2bu. Opk. Oqt. Ipt. \12yr. 2mo. \wk. Ada. Akr. 28 29 30 1 6{)wk. 4 da. Ih? Aim. 346e6'. Icir. 9s. 28° 33' 59" Icir. 10s. 27° 2' 3" 291/Z*. 6o2. 15/^. 22-r. 244/A. 5o2. 4;^^. 'dgr. b2T. Hjctat. Oqr. l()lb. loz. 7 dr. 41 T. OcwL Sqr. 17 lb. Ouz. 5dr. 336.4. 111. 31 P. 2 [OSq.fi. 1 36 Sq. in, nOT.llC.ft.7UC.in. 1 (JSbu. Opk. 2qt. 9 10 11 12 13 14 45 A 3R. 3 IP. 3^Sq.ft. 22Sq.in. 15Sbu. Opk. Aqt. 2T. 5cwL 2c/r. 2llb. 86yd. 3lb. loz. 1 Ipivi. 17 gt 322mi.6fur. Urd. lOOA. IR. 13P. 124, 125^ 125. 125. 125. 125 125 126. 126. 126. 126. 126, 126. 1 26. 126. 126. 126. 3 I 174/6. loz. Ipivt. 3gr. 4 I Sib. lOoz. 14pwt.4gr ) 5 P. 7cwt. Iqr. 23/6. lloz. 7cwl. 2qr.20lb. lloz. bdr. 7 8 124 P. Ohhd. 69 gal lAyr. A6wk. Ada. 20h. 58m. 5Asec. '•1 1 1 2 3 ^yr. Amo. 2da. 2\yr. 9mo. 5da. 4 17 yr. Imo. 3da. 6 7 1 2 3 4 5 12yr. 3mo. 26da. 22hr. 30yr. Imo. 29da. 12/ir. J 27mo. 3ivk. Oda. \ 20kr. 20m. SAyr llmo.0wk.5da. £2 17 s. lib. lloz. Idpwt. Agr. 61fe 10 ! 53 19 j 7T. IScwt Iqr. Alb. \ Onz. 2dr. 7 8 9 10 11 12 13 11 2m. Afur. 21 rd. 7yr. 9mo. Ida. 362yr. 9mo. lAda. 68/6. 10o2. 3pwt. I5gr. { IT. 17 cwt. 3qr, \ 7/6. 14o^. 2dr. )84fe 9| 4 3 1 1 B lAgr. oyd. 2qr. Ina. \in. A cords 50 cubic feet. AJco 127. 18 ro6du. 2j)k. (5.|l 28 15' 3 U 1 S lOmi. Afur. 23rd. ■J2i ANSWEliS. P. EX. ANS. EX. 35 ANS. 133. 31 7s. 3d. \far. IT. Icwt. \qr. 19lb. 133. 32 J 24 r'ms 5 qrs, 1 12 sheets. 36 j £600 9s. 8d. ]£1050 16s. 11^. 133. 133. 33 £03 14s. Bd. 37 . 53iff^« . 133. 34 Ihhd. \9gaL \pL 38 J 294-IMria. 135. 1 1° 9'— time 4m. 36sec. 7 4/ .. 66m. 4sec. 135. 2 I2hr.4m 36sec. P.M. 8 4/ .. 66m. Asec. P.M., 135. 3 llhr.ddrn. 2Asec. A.M. 9 2h . 20m. 4sec. P.M. 135. 4 Um. 32sec. jio e/i . Om.. 8sec. A.M. 135. 5 llhr. I6m.28sec. A.M. 11 ] Ih. 6m. isec. A.M. 135. 6 10 hr. 59771. 56sec.A.M. — 137. 1 3x3 9 7x3x3 137. 2 3x5 10 19x2x2 137. 3 3x2x2x2 11 3 137. 4 2x2x2x2 12 3 and 7 137. 5 3x3x2 13 3 and 7 137. 6 2x2x2x2x2 14 2 and 7 137. 7 3x2x2x2x2 15 2 and 3 and 7 137. 8 7x2x2x2 16 2, 3, 5, 7 138.11 2 1 9 II 3 1 6 II 4 1 5 II 5 1 6 II 6 1 5 || 7 1 14 II 8 1 42 139.11 1 1 24 li 2 1 4 !| 3 1 45 II 4 1 630 || 5 j 267 |j 6 | 396 140.1 7 1 12 118 i 8 II 9 4 II 10 1 3 — — — — — 141. 141. 3 4 840 147 5 6 840 196 7 8 78 84 9 10 1008 156 11 223839 142.1 3 63 ||4 1 126 ||5 1 27 II 6 1 12 — — — — 144, 144, 144, 144. H 1 11 55 q i I « 10 i 27 12 16 13 14 15 16 7056 8 80 pounds 17 18 348'Ljds. 46yds. 145. 145. 145. 145. 145. 19 20 21 22 23 j 2662^. I 40\ pounds 9lbu. 300 pounds 600yds. 24 25 26 27 28 36 gal 40bu. 61 gal. I6yds. 6j\f.r'ns. 29 30 31 58 bcxf4 4s. 3 diee't. 151. 151. 24 35 5- ,14 6 4 17.'! '^> ^:^iRl^ ANSI r ro : ' "^ P. Iex. ANS. EX. T^sfe^^y ** EX. Jf^yjf 152. 1 2 f and 3 3 4 ^5i- j^^^S^^^ 162. m rWTw- 1 2 i 3 4 14 4 15' 15* 2 T60' JL6_ 50. 117' T 1 7 80 40 6 4 F3 0' 5 3 0' 5 3 0- 1 4* A- 6T 7 !%■ 1 loT* 16 78 7S- |Pi^ II 9 I 21 28 3.4 42 63 || Q I 150 225 300 37 5. 45.0 0' 504* 154.11 3 i if' 3V' T6' I- li 4 1 f«'^ ^' H' if' M . A. A 155. 1 i- 6 J- 11 *• 16 J- 155. 2 i- 7 i- 12 i 2 12#. 155. 3 3 8 a- 13 iH. 3 2iyds. 155, 4 i- 9 h- 14 tV 4 5JA«. 155, 5 h 10 h 15 J- — 156. 5 6 apples. 3 -V^ ro6/5 156. 6 2-,-'^, 24, nui isAxxif 4 ^F- 156. 7 21 9^. 5 2 3 6 -TTf-' 156. 8 2AV7. 45 t\% 9iMI|. 6 2 0|3^^^,5^ 156. 9 31504ilf i, 1345, 7947fm. 7 \'^-yrs. 156. 1 ^. 8 5_5_0_0p. 156. 2 i'fS^t/ds. 9 ifl^.^l^l, 157. 1 1 08 6 $2^. 3 2 5* 157. 2 W- 7 3 8 4^/^/5. 4 112. 157. 157. 3 182 8 1 68 1 67A 5 6 15 4 H^. 16 ''^' A- iA^^. 157. 5 27360 2 5 114 • 1 158.11 3 ! ,¥6- II 4 1 ^- II 5 1 Y-= =n- 169. 1 6 3 7 15 105' 105' 105- 7 IM. •e'Vo". -^-M*^- 13 if.A.A- 159. 2 Hh iH: m- 8 2 6 2 15 5 50 ' 50' 5 • 14 iie.il- 159. 3 HI a't. m- 9 1080 218 900 144 ' 144' 144- 15 H. VV- li- 159. 4 ff. A- 10 m. ¥Ai. m- 16 282 16 3 6 ' 3 6' 159. 5 ¥j'.il. ¥/■ 11 1 A. A- A' 3*6- 159. 6 7WT> TOT- 112 A. A. A- — 326 ijSSWjEKS. P. EX. ANS. EX. ANS. EX. ANS. 160. 2 H ' a* T5- 6 T«A.f*§.|g? 10 !i-§. If.fi 160. 3 H^. V. V- 7 w. W. %"• in- 160. 4 F60' Ffi'O' sTo- 8 A.A./^.lf U jil.lt.M, 160. 5 iVo.T'A.f^g- 9 ». ¥/• — > ■■*. I4- 161. li6f 4 Itt- 7 4- 10 mh 13 iiVb^' 161. 2|2f 5 lA- 8 m- 11 2-^. 14 H' t Ai Q 1 A2 6 lii 9 lOf I 10 1 25 )■ '■ '- : '126- " ! 162. 16 25B- 20|1426|a. 24 96||. Uf H-M 162. 17 S2|. 21!A- 25 64,^. - 162. 18 20|J. 22|39j|. 2 H- ia- - - I62.i|l9|2lTVo- 231 17^. 3 «• tW i -J.li- - - 163. 2 *• 5 m-\\ 3 #r- 6 Vj 9 31 163. 3 f- 1 41- M I'^iV- 7 /s- ~ _ 163. 4 ?^- 1 2 n- ;'5 IsV- 8 2M - . 164. 11 68^- 1 13 2*^. 15 6M 17172^. 164. 12 82Jff. 1 4 ^m 16 8H ■ lalAArlrA^rljill, 165. 1 3A. 4142^ h 7 S^cords 1( ) $8^. 13 $13f 165. 2 6f. 5J123 8 $10 1 I 14w. 14 6\har. 165. 3 T 35 1 6!l6ift. 9 13^6^*. V 2 27ci5. 15 %\^. 166. 21 3 75 5 $51. 7 $180 9 $161 166. 2 38^. 4 26f. 6 $43f. 8 36^m. 10 $440 167. 1 T^- 4 f- 7 $^. 2 A- 5 675 167. 2 A- 5 $f 8 $i|. 3 H- 6 204. 167. 3 A- 6 f 1 A- 4 8i- - — 168. 8 636^. 11 201 14 $25 17 546|c-^5. 20 ^yds. 168. 9 114 12 51J. 15 2 Z6> 10? 11/7 217 u £24 185. S^d-^ 5 £331 Is. 6d. 249.1! 2 1 $860,4194 || 3 | $167,983-f- 250.1 l|$i )50||2|7;?er ct.\\: 3!5?/r.i|4|$225||5|7 ,331— 7?/;-. ^nw. 251. 251. 2 3 $19,101 $36,50 + 4 5 $404,0625 $291,60 6 7 $211,456 $165,775 252. i 8|$171,597l||9i$118,528||10|$315,2438|i 11 | $15-2,408 253 254. 254. 254. 254. 254. 255. 255. 256. 257. 258. m 260. 260. $1750 present value. || 2 | $1565,402-|- pres. val. 8 %ob'6'd ,A01 -[- pres. val. $9677,50 + pres. val £223 6s.Qd. discount. 5 $5620,176+ j^-z-ei.mZ. $702,485 $1,94 difference. $2109,236 + $2763,694 + $4000 $6,473+ loss. il|$6,329l 2$10,50 $15240,54 $5,8408 $3393,504 $29.0096 7 $122,81+ 8 I $341.709+11 1 I $344,66+ i| 2 | $5734,32 + 3| $695,64 II 4 | $118,85+ || 5 | $1740,60 ||6 | $376,46 + 2 ll2mQ.!|3|8mQ. 22^da.\\^^mo.\\6\\T/\\da.\\^^mo . \7\61j\da. 9lh day of March. -— 9 10 60j-9^y/a. or Aug. ^Ist. Qfuo. (jda. 49jWda. or Jan. 25th 262.|i 3 I $12,25 || 4 | $6,25 334 ANSWERS. 265. 265. 265.! 265.! 265.1 $426,416 £1073 185. lid. $2033.4894- £389 6s. 2\d. ^2551,733 ANS.' ( £21 5s. -£25 14.S-. 3r/. £30 175. l6/.-(--£41 25. ^d.-^ ( £38 ll5. 4irZ.-£23 V^s. U\d i $250-1250425042004250 \ $516,66^4250. 266. 266. 1 2 $3720,937 $8668,935 3 4 $6748,60 $4583,94 + 5 $3643,875 267.11 2 1 $5944,791 || 3 | $9226,061+ 268.|i 2 1 $1270,428 || 3 | $2016,11 |j 4 | $16975,775 270.111 1-^28 12,50||2|$423,36 j| 3 | $251,45+ || 4 | $1457,75 271.11 1 1 35.||2| 846-^5. + ||3| 288 + 6-^5.||4|20|m'm/5.||5|73'' 274. 4 1/6.-1/6.- 3/6. 274. 5 3 0/ 16. 2 0/ 18. 3 of 23. 5 of 24 274. 6 Zgal. at 105.-3 at 145.-4 at 2\s. 4 at 245. 274. 1 4 «•«/. at 55.-8 at 55. 6r/.-8 a^ 65. 274. 2 ;46^^. TF. 28bu. RA^bu.B. 2Sbu. 0. 274. 3 96bu. W. \2bu. R. \2bu. B. \2bu. 0. 275. 4 40^a/. F. bOgal. E. 20gal. sjririts. 275. 1 10 of \st. 10 of 2d. 30 of 3d. 275. 1 36/6. at Ad. 36 at 6d. 36 a^ \0d. 36 aj5 12d 275. 2 2 If of each. 275. ! 3 4 each of the \st. three and 30 0/" 15 carats fine. 276. 1 12=1 9 . 9* = 6561 276. 2 13 1 10 16^=1048576 276. 3 11 206 = 64000000 276. ¥ -5TT- 12 2252 = 50625 276. 4 5 —TZ' 13 21672 = 4695884 276. 5 92 = 81 14 3213 = 33076161 276. 6 123=1728 15 215* = 213675062.5 276. 7 1253=1953125 16 = 610437195439771 276. 8 163=4096 17 96 = 531441 276. 18 360492=1299530401 282. 1 1.73205 + 66031 11 .05 16 3.12249 282. 23.31662 + 7J4698 12 .01809 17 0.71554 282.1 3 32.695 + 8157 19 + 13 .0321 18 0.64599 + 2^^2.1 1 1506.23 + 9'69.247 + 14 2.104 19!f 282.! 5 2756.22 + iO|2.oyi + 15 2.91547 + 20:f ANSWERS. 335 p. EX. 2 2 ANS. EX. 4 ANS. EX. 6 ANS. 284. 284.1 25ft. 1 2.6 4 9r. 290.1 tt 1 268.0832 1 9 1 2ft. iin. 10 \2/L\\ 11 1 V2ft. 292.1 1 1 $1,53 II 2 1 ^212 II 3 1 40 II 4 j 2 || 5 | 9yr. 293.1 1 1 47jr. II 2 1 67m. II 3 1 Acents. 294.||2i7 8/zwe5|!3|34-162||4i$l,3 5||5|175wz.||6|5mi.l300?/rZ. 296. |2!£2 2.9. 8d.\ 1 31 4 II 4 I 78732 || 5 | $25600 || 6 | $61,44 297. 1 6560 3 381 5 $196>83-$295,24 297. 2 254 4 £204 155 6 $48C0-$9450 298. 1 ^978 6 3 Ill J 213,3125 211,6875 ' 15 $3567 298. 2 516?*. 7 1 7 Jy(/5. 16 288 298. 3 S80,71 8 it- 12 nandii. 17 $4717 29 S. 4 5467/ir. 9 42V !l3 9.04 18 137 291 $26,25 10 120 we/i. '14 4i. 1 — 299. 19 10,^^7_ planks. 29 879000 299. 20 $3800 30:792 ^99. 21 80yds. ,, ( Imi. Ofur. 33r(L 31 |l5iA' 299. 22 \ ^'.9 128. ^'.s $60. 1 C's $32. 299. 32 62 years 299. 23 71 days. 33 4 299. 24 37no. ( $2450 l5i. 2^9. '''•'i \^\¥t $986,86-^ warth 34 \ $3681 2^/. 299. / $4294,50 3d, 299. 26|:i 35 £408 saves 299. ^« 3 l5^.$16C.2J.$120. "^'n 3(/. $140 II 28 1 50. . 36\23^bu. 299. 37J2400 300. 38 3 o'clock 42 32^^^^. 48 j 62yr. Umo. \20da, IOU7 BOO. 39 300 7nen 43 $34782,608 300. 40 864 44 $3,653 49 Hh 300. r.4'5 $2364 45 34 J per ct. 50 ]\\yr. 300. 41 B's $1182 46 $7816,0914- 51 22500 biicks 300. 1 C"..■ *¥;'i196.83 last term. \ $29o,24 tvhole am't ( 946m. wheat. 1 2 rye. \ 12 barley. 12 oats. 1 6l|zVi. 356,25 $8640 $1,20 lost 4 pence. 65 ANS. 4 f/dZ?/5 240 hours A21-B 8}-C42days. ^'5=$l94,801f J5'.«=$129,87yiy. C"s-^$97,40ff D'5^$77,92if $1020,66 $8925,544 + 302. 302. 302. 302. 302. 302. 50ft. 9mi. 6fur. 34:rd.-\- ( daughter $780. so?i\ (§3120. m> $1560. 76mi.-l292mi. 4//r. 11 wo. 22.8c?a. 73 74 75 76 77 4yds. ^423,36 ($920,20 l5i. $2760,60 \2d. 552 L20 3d. 3hr. 20m. 69^mi.f^7n N. Haven. 303. 303. 4006^. yd. 30A. 2 A. 3R. 15P. 109A lii. 28P. 12A. 3R. 3A. 3.R. 24P. 25P. 304.1 304 304. 304. oA. \0A. 7 A. 2R. 6A. OR. 12P 8 32520s-^. yd. 1|45849.485. 2i5A lil.9.95P. '^. 5^7"6'^' ;|230/^ 176.031256-5^. yi/. 15A OR. 33.2P. 67 A 2R. UP. 305, 305, 305.1 305. 395. 305j 306.1 306. 12.5664 292.1688 62.8320 25. 3709 2180.41 + 1 28.2744 3 2 78.5400 1 3 38.4846 2 4 1.069 + 3 1 452.3904 4 2 153.93841 - 809.5616 904.7804 33.5104 1436.7584 113.0976cm. y?; \\4.560 Sq.ft. 2 1440 " " I 4500. 2|l3824cw./^. 2\lcu.ft.\ 5\3500cu. ft. I r.72 ] 3 307. 307. 307. 308. 308. 308. 309. 309, 3141.6 \l0.666Sq.ft.-j- 6654.86 Sq. in. 212058 9110.64 3392.928 |18G9.5616 29452.5 475 2080 4030 1440 6000 65.45 1242 6 3600 2 2290.2264 6 706.86 3141.6 706.86 19G.52^aZ. 185.0688^aZ. \3i\sral 182.8Ugal. 6hr. 3m. 48|H UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW Books not returned on time are subject to a fine of 50c per volume after the third day overdue, increasing to $1.00 per volume after the sixth day. Books not in demand may I WHUtM expiration of 1 an period mm JCIF N) DEC 5 OCT i 4ftH20\96e65 RECD lId m 6 '66 -7 PM before 50m-7.'16 VB 17362 UNIVERSITY OF CAUFORNIA LIBRARY i ^ S. BARN':S & CO., 51 & 53 JOHX JSTF ; I, lnKW YORK. 1. MONTEITH AND McNALT Y'L Monteith's First Lessons in Geotrmiiliy. ( M MonteitU's Manual of Gc.)j:ra[)liy. f 2. DA VIES' SEEIES '? Davies' Primary Arithnunic. ! !• Davies' Intelloctnal Arilhm.-tic. Duvies' First Lessons in Aritlimetic. 3TIIES '^F GEOGRAPHY. ■'ly's Complete Sclinol Geography Lii Miiiiis ;uul enfrraviiigs. ARITHMETIC ^ » os' Ne»v Scliool ArithinPtic. i-ies' University Aritlimetic. ivies" GniiMinar of Aritlimetie. 3. ENGLISH GRAMMAR, COM'^Ol^ITION, READING, ETC. Clark's First Lessons in Grammar. Clarlv's Analysis of Enirlish Language. Clark's New'Kiiglisli Grammar. WelclTs Eiiglisii Sontunce. Brookfield's Firs; Book in Composi^'" • Martin's Orrlio-pist. Parkers liiiemrical Header. Iligli Scliool Lit(M-atnre. Sh.'r\^oo(rs Seif-Cnltnre in Elocution. .Par;..;r'3 Word Builder. Parker and Watson's series o:' K. i-Ur.^ 4. 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