UC-NRLF $B 27a fiMfl >- T^f LX^ L C. C IN MEMORIAM FLORIAN CAJORl / ' u I '^'^' bj- u ELEMENTARY ARITHMETIC; OR, SECOND B.Q°Q,K SERIES OF MATHEMATICS, BY Andrew "W. Baker, A.M., Ph.D., REVISED AND ENLARGED EDITION. NEW YORK: P. O'SHEA, PUBLISHER, 37 BARCLAY STREET. 1880. ■^32- Copyright, 1878, 1880, by P. O'SBEA. CAJORl J. CA.MPBKLI., PRINTER, 15 Vandewator St., N. Y, PREFACE rpHIS book contains the Elements of Arithmetic, or perhaps more properly the Elements of Mathematical Science. In its preparation I have endeavored to give it not only a mathematical basis, but also a scientific structure. This I have done, keeping steadily in mind that it is a book for the young, for whom the pathway of science should be made as easy and inviting as possible. We cannot insist too strongly on the advantages of the blackboard exercises in developing the principles of science, and rendering them more easily and more thoroughly under- stood. In a well-drilled class, whilst one student is at the board demonstrating a theorem or solving a problem, and the other members of the class are looking on with attention, the latter learn as much as the former. The teacher should be the guide of the class. He should not be satisfied with exemplification only, he should also endeavor to encourage and interest his pupils, carefully observing the progress made each day, each week, each month, and at the end of the year when all shall be summed up, such will be the delight of both teacher and class at the progress made, that they will begin to believe there is a Toyal road to learning. ^ii:^ CONTENTS PAGK Definitions 7 Mathematical Terms 8 Signs 9 Axioms 9 Notation and Numeration 10 Addition and Subtraction 12 Multiplication 21 Division 2T Kbvibw 35 Factoring 48 Least Common Multiple 50 Greatest Common Divisor 53 Cancellation 53 Fractions 56 Addition and Subtraction of Fractions 59 Multiplication of Fractions... 61 Division of Fractions 63 Complex Fractions. 66 Review 68 Decimal Fractions 79 Addition and Subtraction of Decimal Fractions 80 Multiplication of Decimal Fractions 80 FA«a Division of Decimal Frac- tions 83 Denominate Numbers 83 Simple Ratio 111 Compound Ratio 117 Percentage 119 Commission or Brokerage 119 Interest 123 Bank Discount 127 Time Discount 12i3 Exchange 129 Domestic Exchange 129 Foreign Exchange 131 Averaging Accounts 132 Bank Account 133 Alligation 134 Alligation Medial 134 Alligation Alternate 135 Involution and Evolution 139 Evolution 140 Series of Common Differ- ences 148 General Review 151 Answers 165 The Author cannot too highly recommend to the Teacher the use of the BlacJcboard described on the following page. Great facility in com- prehending the combinations and divisions of numbers will be acquired by this method. BLACKBOARD EXERCISE. 1 This page represents a blackboard with the num- 37 2 bers as high as 72 painted on its margins. 38 3 There is also a box containing slips which will cover 89 4 two, three, four, etc., as high as 12, and numbered 40 5 accordingly ; one of these the student will take in his 41 6 hand and apply it to the painted numbers to perform 42 7 addition or subtraction ; thus, begin at 1 and take a 43 8 slip marked 2, then 1 and 2 are 3, 3 and 2 are 5, 5 and 44 9 2 are 7, 7 and 2 are 9, etc., counting at least the left- 45 10 hand column ; then, to perform subtraction, begin at 46 11 the bottom of the 1st column; thus, 36 minus 2 equals 47 12 34, 34-2=32, 32-2=30, 30-2=28, etc., until the top 48 13 is reached ; then taking a slip marked 3, begin with 49 14 1 or 2, or first with 1 and then with 2, and return to 50 15 the top of the column as before, by subtraction ; let 51 16 this exercise be performed with all the slips, and as 52 17 the larger numbers are taken, continue the additions 53 18 to the bottom of the 2d column, and return as before. 54 19 For multiplication and division first make a chalk 55 20 mark after every two figures up to 24, and mul- 56 21 tiply ; thus, once 2 are 2, twice 2 are 4, 3 times 2 are 57 22 6, 4 times 2 are 8, etc. ; then the number of divisions 58 23 is 12 and each division has 2 numbers ; .*. 12 is con- 59 24 tained twice in 24, or 2 is contained 12 times, 2 is 60 25 contained once in 2, in 4 twice, in 6 three times, in 8 61 26 four times, in 10 five times, in 12 six times, etc. When 62 27 the student is familiar with multiplication and division 63 28 by 2, let the numbers be separated into 3's, then 4's, 64 29 etc., and let each be continued for 12 divisions ; when 65 30 all the divisions have been performed according to the 66 31 steps, beginning with 2 and ending with 12, a multi- 67 32 plication and division table will be made. 68 33 Rem. — In multiplication the product of any two f ac- 69 34 tors is the same by making either the multiplicand and 70 OK the other the multiplier ; so also in division, the divisor «,^ and* the quotient may be substituted, as the dividend 3^ is the product of the divisor and quoti**nt. *^ Rem. — The numbers, continued up W 144, should be painted on the sides of the board. ELEMEITARY AiifHIETld ■»»» Defij^itioj^s. 1. Arithmetic is the science of numbers. 3. A Unit is a single thing ; as, a book, one dollar, or simply one. 3. A Nwmhev is a unit or a collection of units ; as, one, ten, five books, twenty-five dollars. 4. The numbers used in Arithmetic are all formed by combinations of the ten Arabic characters, called Fig- ures; viz., 0, called zero or naught; 1, called one; 2, two; 3, three; 4, four; 5, five; 6, six; 7, seven; 8, eight ; 9, nine. 5. Expressing a number either in writing or figures is called dotation, and reading the expression is called Ifumeration. 6* When numbers are used without reference to any object, they are called Abstract Numbers ; as, five, twenty, etc. ; but when they are applied to things, they are called Concrete ; as, one book, ten men, four dol- lars, etc. 7. When concrete numbers express values of money, weights, measures, time, etc., they are called Denomi" b DEFINITIONS. nate Numbers ; as, dollars, pounds, shillings, pounds of weight, ounces, hours, minutes, etc. 8. When different denominations of either kind form burt One nUnibe'rv .it^ i^ called a Compound Number; as, ^4 3s. 6d.; ^ Ib/i oz. 3 pwt. and 2 gr. >'0v'^u.m'yejrs 6f>,.the ^me order and the same denom- ination are termed Like Num^bers; other numbers are termed Unlike Numbers. Rem. — Numbers expressing different species of the same genus are unlike, as horses and cows ; while the same numbers expressed in the term of the genus are alike, as animals. MATHEMATICAL TEEMS TJSED IN AEITH- METIC. 1, An affirmative sentence, or anything proposed for consideration, is a Proposition. 3. A self-evident proposition is called an Axiom. 3. A proposition made evident by a demonstration is called a Theorem. 4. When a proposition is used for developing a prin- ciple of Arithmetic, it is called a Problem* 5. Propositions given merely for solution, in order to impress the principles on the mind, are called JExam^ pies. 6. An obvious consequence of one or more proposi- tions is called a Corollary. H. An established custom, or an assumption without proof, is called a Postulate. Rem, 1 and 1 are 2, 3 and 1 are 3, 3 and 1 are 4, 5 and 2 are 7, 6 and 3 are 9, etc., is the postulate which forms the basis of Arith" metic. DEFINITIONS, AXIOMS, I. If equal numbers are added to equal numbers, the sums will be equal. 3. If equal numbers are subtracted from equal num- bers, the remainders will be equal. 3. If equals be multiplied by equals, the products will be equal. 4. If equals be divided by equals, the quotients will be equal. 5. If two numbers are each equal to the same number, they are equal to each other. 6. If the same number be added to and subtracted from another number, the latter number will not be changed. ?• If a number be both multiplied and divided by the same number, the former number will not be changed. 8. If two numbers be equally increased or diminished, the difference of the resulting numbers will be the same as the difference of the originals. 9. If two numbers are like parts of equal numbers, they are equal to each other. 10. The whole is greater than any of its parts. II. The whole is equal to the sum of all its parts. SIGN^S. 1. The sign +, called plus^ is the sign of addition, and indicates that the number on the right hand is to be added to the one on the left. 10 NOTATION AND NUMERATION 2. The sign — , called minuSj is the sign of sub- traction, and indicates that the number on the right is to be subtracted from that on the left. 3. The* sign x , called into, is the sign of multipli-^ cation, and indicates that the numbers between which it is placed are Victors of the same product, 4. The sign -^, divided hy^ the left-hand number ~ to be divided by the right hand. 5. The sign =, equal to^ indicates that the num- bers between which it is placed are equal. 6. 52, 53, the 2 and 3 placed to the right, a little above a number, indicates the power to which it is to be raised. 7. V , indicates the extraction of the square root ; and \/~, indicates the extraction of the citbe root. JYOTATIOJf AJfD J\[*UMERATIOJ^. 1st. A figure standing alone, as 1, 2, 3, holds the units place, or is of the 1st order, and is read, one, two, three, 2d. A number having two figures, as 14, 26, the right- hand figure holds the units place, and the left-hand figure that of tens, and they are read, fourteen, twenty-six. Cor. — The right-hand figure of a number is called units, or the 1st order ; the next figure to the left is called tens^ or the 2d order ; the third figure, hundreds, or the 3d order ; the fourth figure, thousands, or the 4th order ; and if a number be expressed with the nine figures in order, making 1 the right-hand figure, the figures will express their respective orders ; thus. NOTATION AND NUMERATION. 11 millions, thousands, units. m m nx •^ CM "^ «Ht 'O - fifty-four thousand three hundred and twenty-one. Ut>'±,0/vX ... six hundred and fifty-four thousand three hundred and twenty- I jDtlTjO/v'l. seven millions six hundred and fifty-four thousand three hundred and twenty-one, 07 £•«/ 091 J eighty-seven millions O 4 ,UU'±,0/Vi \ six hundred and fifty-four thousand three hundred and twenty-one. QR*? (\KA. ^91 [ °^°® hundred and eighty-seven millions VO i y\JO'±^0/iil. \ gix hundred and fifty-four thousand three hundred and twenty-one. Rem. — The column of I's is of the 1st order, the column of 2's is of the 2d order, the 3's the 3d order, the 4's the 4th order, etc. Cor. — The relation of any two consecutive orders is the same, for when in addition the sum of any column reaches 10, the left- hand figure belongs to the next column or order ; hence, a table may be formed, thus, 10 units = 1 ten. 10 tens = 1 hundred. 10 hundred = 1 thousand. 10 thousand = 1 ten-thousand. 10 ten- thousand = 1 hundred-thousand. 10 hundred-thousand = 1 million. etc. etc. jIdditioj^ Aj\rD Subtbactio:n'. Addition and Subtkaction Table. 1 2 3 4 5 6 7 8 9 '1 2 3 4 5 6 7 8 9 10 % 3 4 5 6 7 8 9 10 11 3 • 4 5 6 7 S 9 10 11 12 4 5 6 7 8 9 10 11 12 13 5 6 7 8 9 10 11 12 13 14 6 7 8 9 10 11 12 13 14 15 7 8 9 10 11 12 13 14 15 16 8 9 10 11 12 13 14 15 16 17 9 10 10 11 11 12 12 13 13 14 15 16 16 17 18 14 15 17 18 19 11 12 13 14 15 16 17 18 19 20 12 13 14 15 16 17 18 19 20 21 13 14 15 16 17 18 19 20 21 22 14 15 16 17 18 19 20 21 22 23 15 16 17 18 19 20 21 22 23 24 16 17 18 19 20 21 22 23 24 25 17 18 19 20 21 22 23 24 25 26 18 19 20 21 22 23 24 25 26 27 19 20 20 21 22 23 24 24 25 26 27 28 21 22 23 25 26 27 28 29 21 22 23 24 25 26 27 28 29 30 ADDITION AND SUBTRACTION 13 The square made by the ten Arabic characters forms an Addition and Subtraction Table. Beginning with the first line, thus. Zero and zero are zero ; zero and 1 are 1 ; zero and 2 are 2 ; zero and 3 are 3 ; zero and 4 are 4 ; zero and 5 are 5 ; zero and 6 are 6, etc. The second line, one and zero are one ; 1 and 1 are 2 : 1 and 2 are 3 ; 1 and 3 are 4; 1 and 4 are 5, etc. 2 and zero are 2 ; 2 and 1 are 3 ; 2 and 2 are 4 ; 2 and 3 are 5 ; 3 and 4 are 7, etc. 3 and are 3 ; 3 and 1 are 4; 3 and 2 are 5 ; 3 and 3 are 6 ; 3 and 4 are 7, etc. Continue this, taking the first figure of the 1st column and adding it to each successive figure in the first line ; the adding of zero is only nominal, as it makes no increase. It also becomes a subtraction table, the figures of the first column being the subtrahend, and those of the first line the remainders. Take zero from 1 and 1 remains ; from 2, 2 remain, etc. It may be thus expressed : 1— = 1; 2 — = 2; 3 — = 3; 4 — = 4; 5 — = 5; which is read, 1' minus zero equals 1, etc. Second line: take 1 from 2, 1 remains; 1 from 3, 2 remain ; or, 2-^1 = 1; 3 — 1 = 2; 4 — 1 = 3; 5 — 1 = 4, etc. Third line; 3-2 = 1; 4-2 = 2; 5-2=3; 6—2 = 4; 7 — 2 = 5, etc. In addition, we add two numbers at a time, never more, and in the first square we have the addition of every two units that can come together ; so also in subtraction. 14 ADDITION AND SUBTRACTION, In the second square, the units correspond with 'the first square^ and have an additional ten. In the third square, the units again are repeated, and another additional ten. As a column of tens, hundreds, and every higher or lower order is added and subtracted in the same way, the above table 'leveiops every principle of addition and subtraction. Add the column of units. One and 2 are 3 ; 3 and 3 are 6 ; 6 and 4 are 10 ; 10 and 5 are 15 ; 15 and 6 are 21 ; 21 and 7 are 28 ; or as is customary to begin at the bottom of the column, 7 and 6 are 13 ; 13 and 5 are 18; 18 and 4 are 22; 22 and 3 are 25; 25 and 2 are 27; 27 and 1 are 28. 9 + 7 = 16; 16 + 2 = 18; 18 + 4 = 22 ; 22 + 6 = 28; 28 + 5 = 33 ; 33 + 3 = 36. Eem. — Although many numbers may be added together, in - performing the operation only two at a time are added. Add the following numbers jointly and separately? thus, 35 24 43 52 = 67 = 221 The sum of the column of units is 21 ; that is, 1 unit and 2 tens ; the sum of the column of tens is 20; that is, 20 tens or 2 hundred; and the two sums united make 1 3 3 5 3 6 4 4 5 2 6 7 2 9 28 36 30 and 5 20 " 4 40 « 3 50 « . 2 21 60 " A 200 200 « '21 = = 221 ADDITION AND SUBTRACTION. 15 221 ; precisely the same as if the column of units is first added, and the units of the sum placed under the column of units, and the tens added with tiie column of tens ; and then the tens of the sum of the tens column placed under the column of tens, and the hundreds in place of hundreds. CoE. — As the relation of each successive order is the same, hence for every ten of any order, the 1, or left- hand figure, belongs to the next order ; and the process is the same in the addition of every column ; that is, one is carried to the next column for every ten in the addi- tion of each column. ADDITION-. STJBTEACTION". 3241 4365 643315 876432 987654 4356 5331 533684 543210 321334 6745 7546 478921 5364 8432 586432 876543 789654 19706 654331 331043 754331 864331 345678 869754 678643 678963 987654 654321 594721 987654 331987 367543 456789 654331 Add 7654 3897 11551 3465321 6354789 9830110 Subtract Minuend, 11551 Minuend, 11551 Subtrahend, 7654 Subtrahend, 3897 3897 7654 16 ADDITION AND SUBTRACTION, Minuend, 9820110 Minuend, 9820110 Subtrahend, 3465321 SuUraJiend, 6354789 6354789 3465321 Cor. 1. — The minuend is always equal to the sum of the subtrahend and remainder, and is therefore greatei than either. CoR. 2. — Arithmetic is based upon the postulate contained in 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ivhich is addition, etc. ; and the application of Axiom 6 (page 9) to this postulate proves the principle of subtraction; thus, 19 + 6 = 25, then 25—6 must be equal to 19. Rem. 1.— Wlien the figure of the subtrahend is larger than the one above it of the same order of the minuend, 1 of the next order of the minuend must be united to the figure of the minuend and then the subtraction be performed ; then in order to make up for this addition to the minuend, 1 must be added to the next order of the subtrahend, and then perform the subtraction ; this process i-s called carrying and requires all the attention of the student. Rem. 3. — I prefer few examples, but these may be often repeated, and if thought necessary the teacher can give others in which the columns are longer. Rem. 3.— Each order may be regarded as units, and the sum may reach one, two, or more hundred of its order. PRACTICAL EXAMPLES 1. A farmer has 17 sheep in one pasture, 41 in another, and 57 in a third; how many sheep has he ? 2. A fowler has 18 turkeys, 21 geese, 42 ducks, and 64 chickens ; how many fowls has he? 3. A man has 48 cattle in one field and 36 in another; ADDITION AND SUBTRACTION. 17 if he take 15 from the second field and put them in the first, how many will then be in each field ? 4. A granger owns 120 cattle, which are pastured in three fields ; in one field there are 45 cattle, in another 30 cattle ; how many cattle in the third field ? 5. Willie has 41 cents'in one pocket and 37 in another; if he buy a knife for 50 cents, how many cents will he have left ? 6. Willie had given him two books to read, one of 153 pages and the other of 226 pages ; he has read 240 pages, how many more has he to read ? 7. In one basket there are 51 eggs, in another 62, and in a third 42 eggs ; how many in the three baskets ? 8; One hen has 15 chickens, another 17, and a third has 14 ; how many chickens in all ? 9. Andrew learned 75 verses of poetry in one week, 94 in another, and 87 in a third ; how many verses did he learn in three weeks ? 10. John gave a beggar 53 cents, Willie gave him 72 cents, and Andrew gave him 65 cents ; how many cents did he receive ? 11. A man bought a horse for 95 dollars, a cow for 40 dollars, and a wagon for 65 dollars ; how much did he invest ? 12. A merchant commenced business with $9,875 ; the first year his net profits were $2,134, the second year $1,654, the third year $2,547, and the fourth year $2,620 ; what was then the amount of his funds ? 13. A merchant commenced with $12,650; the first year he gained $2,163, the second $1,875, the third $1,260, and the 4th year he lost $4,105 ; what funds had he left ? 18 ADDITION AND SUBTRACTION, 14 A man commenced with $4500 ; the first year he doubled his money, but at the beginning of the next year he lost $2500 ; he then doubled what remained, and lost $6000 ; then he doubled what remained and closed business with what amount of money ? 15. A man bought 3000 acres of land ; he then sold to one man 324 acres, to anather 236 acres, to a third 148 acres, to a fourth 465 acres, and to a fifth 634 acres ; how many acres were left ? 16. A merchant took with him $5000 to buy goods ; he purchased dry goods for $1864, groceries for $1256 ; hardware for $630, hats and boots for $362 ; how much had he left ? 17. A man owns three farms ; the first contains 275 acres, the second 483 acres, and the third 1230 acres; he sells the first farm, 236 acres of the second, and 584 acres of the third ; how many acres remain unsold ? how many of the second farm ? and how many of the third ? 18. A man bought a horse for $150, and a buggy and harness for $275; he sold them, gaining $42; what did he sell them for ? 19. John Jones bought a farm for $6875 ; he paid for repairs $2172, and then sold it for $10165 ; how much did he gain or lose by the transaction? 20. James Johnson bought one house for $6,540, another for $7,965, and a third for $12,384; he paid for repairs $3,165, and then sold the three houses for $31,236; how much did he gain or lose by the transaction ? 21. Invested 3245 dollars in property, and sold it so as to gain 534 dollars. What was it sold for ? ABmTlON AND SUB TB ACTION. 19 2^^. A merchant bought dry goods for 576 dollars, gro- ceries for $375, and hardware for $234 ; in selling he gains $156 ; what was the amount of sales ? Ans, $1,341. 23. A, B, and C engage in trade; A puts in $2,576, B $3,845, and $431 more than A and B ; what is the amount of their stock ? Ans. $13,273. 24. A man is 9 years older than his wife, and she is 22 years older than their son, who is 12 years old ; what is the age of each ? Ans. Wife's age, 34 y.; man's, 43 y. 25. A man bequeathed to his wife $5,650, to each of his three sons $3,775, and to each of his two daughters $3,550 ; what was the amount of his bequests ? Ans. 24,075. 26. A man owing $9,856, pays at one time $3,453, and at another $2,176; how much does he still owe? Arts. $4,227. 27. The sum of two numbers is 8,643, and one of the numbers is 5,756; what is the other number ? 28. The number of pupils registered is 1,253, and the number at school is 1,168 ; how many are absent? 29. The sum of two numbers is 6,432, and one of the numbers is 2,541 ; what is the difference of the numbers ? Ans, 1,350. 30. The sum of two numbers is 5,680, and their differ- ence is 596 ; what are the numbers ? Ans, 3,138 and 2,542. 31. The difference of two numbers is 265, and the smaller number is 576 ; what is the larger number? 32. The difference of two numbers is 175, and the larger number is 651 ; what is the sum of the two num- bers? Ans. 1,127. 20 AVDITION AND SUBTRACTIONS 33. Add 32,745, 276, 304,721, 5,640,' 216, 243, 976,- 874, 2,176, 81,275, 9,406, 20,045, 6,320,754, and 7,105,006. Arts, 14,859,377. 34. Fifteen gallons leaked out of a cask of 81 gallons ; how many gallons were left ? 35. A merchant engaged in trade with $5,481, and gained each year for five years $1,254 ; but the sixth year he lost $2,162 ; what was then the amount of his capital ? Ans. $9,589. 36. A father left an estate of $32,600 to his five chil- dren, the eldest to have $1,000 more than the next, and the next $1,000 more than his next younger brother, and so on to the youngest ; what was the share of each ? Ans, Eldest, $8,520, 7,520, 6,520, 5,520, 4,520. 37. A man purchased 1,200 acres of land, and sold it in parcels as follows : to one man, 364 acres; to another, 204 acres; to a third, 468 acres, and the balance (how many acres ?) to a fourth. Ans, 164 acres. 38. A man bought one farm for $6,748, and another for $4,482 ; he then sold both for $12,000 ; how much did he gain or lose ? Ans, $770 gain. 39. A man sold pork for $75; beef for $36; butter for $25, and cheese for $18 ; he then bought sugar for $12 ; coffee for $15 ; salt for $5, and dry goods for $31 ; how much cash had he left? Ans. $91. 40. Columbus discovered America in 1492, and Wash- ington achieved our Independence in 1783; what time elapsed between those two great events? 41. Washington died in 1799, at the. age of 67 years ; in what year was he born ? 42. A merchant purchased goods for $2,154, on which he paid $1,465 ; how much did he still owe ? MULTIPLICATIOJf. Multiplication and Division Table. 1 3 3 4| 5 6 7 8 9 10 11 13 3 4 6 8 1 10 1 12 14 16 18 20 22 24 3 4 5 6 9 12 15 1 18 21 24 27 30 33 36 8 12 16 1 30 1 24 28 32 36 40 44 48 10 15 20 1 25 1 30 35 40 45 50 55 60 6 12 18 24 30 36 42 48 54 60 66 72 7 14 21 28 1 35 42 49 56 63 70 77 84 8 16 24 32|40 48 56 64 72 80 88 96 9 18 27 36|45|54|63 72 81 90 99 108 lO 20 30 40 1 50 1 60 70 80 90 100 110 120 11 22 33 44|55 66 77 88 99 110 121 132 13 24 36 48|60 72 1 84 96 108 120 132 144 As a Multiplication Table, begin with the first line ; thus, Once 1 is 1 ; twice 1 are 2 ; three times 1 are 3, etc. Second line, Once 2 are 2; twice 2 are 4; 3 tiipoies 2 are 6 ; 4 times 2 are 8, etc. Third line, Once 3 are 3 ; twice 3 are 6 ; 3 times 3 are 9 ; 4 times 3 are 12, etc. Eecite each line similarly. Kem. 4 times 3 are 12, and 3 times 4 are 12 ; hence, alternating the factors does not change the product. 22 MULTIPLICATION, As a Division Table, begin with the first line ; thus, 1 is contained in 1, once; in 2, twice; in 3, 3 times; in 4, 4 times, etc. Second line, 2 into 2 = 1 ; 2 into 4 = •^; 2 into 6 = 3; 2 into 8 = 4, etc. Third line, 3 into 3 = 1 ; 3 into 6 = 2, etc. Rp'\i.— As a Multiplication Table, it may also be read by the column, \>Y which the factors are alternated, without changing the prod"\ct. Any number is multiplied by 10 by adding a zero to it. As J' Division Table, the first column has all the divisors, the first \mc all the quotients, and every number in each line is a dividend, wh>c>\ is always in the same line and the same column with the quotimt and divisor. Any number having a zero in the units placr is divided by 10 by removing the zero. THEOREM I. At^y number is jnultipUed by 10 by annexing a zere to it, f^ince the product of any number multiplied by 1 is equal to the number itself, the product of any number multiplied by 2 is double the number, etc, For, as 10 X 1 = 10, and 10 x 2 = 20, and 10 x 24 = 240, and as alternating the factors does not change the product, hence, 1 X 10 = 10, and 2 x 10 = 20, and 24 x 10 = 240. .-. Any number is multiplied by 10 by annexing a zero to it. CoK. — Any number is multiplied by 100 by annexing two zeros to it, and annexing three zeros multiplies it by 1000, etc. MULTIPLICATION, 23 THEOREM II. The product of any tivo factors will have as many figures, or one less, than both factors, 600 500 5 50 1 3 3 4 9 50 1 3 4 4 9 5 i 9 12 16 81 250 2500 25000 The products of the smaller figures of units will be but one figure until above 3, when there will be two figures, but never more, as 9 x 9 = 81, and every additional figure annexed to each or either factor, whether small or large, will make an increase of one figure and no more ; therefore the product of any two factors will have as many figures, or one less than both factors. CoE. 1. — The product of any two figures cannot be less than one figure, nor more than two. CoR. 2. — The product of units by units must be units, and when there are two figures, the left-hand figure will be tens. The product of tens by units must be tens, and when there are two figures, the left-hand figure will be hundreds ; and if any order be multiplied by units, the right-hand figure of the product will be the same order as the multiplicand, and if there be two figures in the product, the left-hand figure will belong to the next highej order. CoR. 3. — When the multiplier is tens, the product will be ten times as great as if the multiplier were units ; that is, each product will have one zero to the right of it, holding the units place, or the first figure of the product must be placed in the column of tens ; when the multi- 24 MULTIPLICATION. plier is hundreds, the right-hand figure must be placed in the column of hundreds; and, in general, whatever the order of the multiplier is, the right-hand figure must be in the column of that order. CoE. 4. — If there be one or more zeros in the multi- jilier, the product of the next figure will be put back one figure for every zero. Rem. — 111 the multiplication, each figure may be regarded as the unit of its order. PROBLEMS. 1. 10 X 10 = 100. 2. 11 X 11 = 121 = 11 X (10 + 1) = 11 X 1 = 11 11x10 — 110 121 3. 12 X 12 = 144 = 12 X (10 + 2) = 12 X 2 z= 24 12 X 10 = 120 144 4. Multiply 432 by 4 = (400 + 30 + 2) x 4. 2x4= 8 and 432 30 x 4 = 120 4 400 X 4 = 1600 1728 1728 5. Multiply 432 by 14 = 432 x (10 + 4). .-. 432 X 4 = 1728 or 432 432 X 10 = 4320 14 6048 1728 432 6048 Rem. — The problems should be carefully impressed on the mind before proceeding. MULTIPLICATION. 25 6. 432 432 X 4 = 1728 124 432 X 20 = 8640 1728 432 X 100 = 43200 864 53568 432 53568 Cor. 1. — When the multiplicand has several figures and the multiplier one that is only units, the first product of units by units will be units, or units and tens ; the units must be placed in the right-hand or units place ; if there be tens, it must be reserved and placed in or added to the column of tens ; in the next product of tens by units, the right-hand figure will be tens, and must be united with the tens reserved, and placed in the column of tens ; the left-hand figure, if there be one, must be treated as the previous one, reserved until the next product is obtained, and united with the right-hand figure; the process is the same in every successive order. Cor. 2. — When the multiplier also has several figures, the process of each successive multiplier is the same, except that the right-hand figure of each product must be placed in the order of its multiplier. (Cor. 3, Prob. 2, page 22.) Rem. — A multiplicand may be eitlier an abstract or a concrete number, but a multiplier cannot be concrete, as it cannot refer to tilings, but merely indicates how many times the multiplicand is to be taken ; but the product will be of the same name as the mul- tiplicand; for twice $5 are $10; 3 times 20 yards of cloth are 60 yards of cloth ; twice 4 are 8 ; 3 times 4 are 12, etc. In computation, it is best to regard all numbers as abstract. 26 MVLTTPLICA TION. (7.) (8.) (9.) 36435 26432 26433 334 104 3004 145700 105728 105728 72850 26432 79296 109275 2748928 79401728 11801700 (11.) (12.) (10.) 234 123 26432 123 234 50004 702 492 105728 468 369 132160 234 246 1321705728 28783 28782 Rem. — The product is not changed by altemathig the multipli- cand and multiplier. EXAMPLES. 1. Multiply 54326 by 346. 2. Multiply 23748 by 543. 3. Multiply 46874 by 697. 4. Multiply 36975 by 476. 5. Multiply 236874 by 2134. 6. Multiply 9876325 by 35a 7. Multiply 879654 by 2175. 8. Multiply 986432 by 8704. 9. Multiply 326875 by 3005. 10. Multiply 468753 by 2100. Examples may be added, or the same repeated, as the student will more readily comprehend by repetition than by different examples. Rem. 1. — In multiplication, two factors are given to find their product. Rem. 2. — In division, two numbers also are given to find the third ; the one called the dividend corresponds to the product in multiplication, the other given number is called the divisor, and the required number is called the quotient ; the two latter corre- spond to the factors in multiplication. Bivisioj^. PROBLEMS. When the product of two numbers is 4, and one of the numbers is 2, the other number is also 2 ; for 2x2=4^ and 4 divided by 2, or 4 divided into 2 equal parts, each part is 2, that is, the quotient is 2. 1. 9 ~ 3 = 3. 4. 16 -r- 4 = 4. 3. 12 -=- 2 = 6. 5. 15 -=- 3 = 5. 3. 12 -7- 3 = 4. 6. 15 -^ 5 = 3. CoR. 1. — The product of the divisor and quotient equals the dividend. CoR. 2. — The divisor and quotient may be alter- nated. 24 CoR. 3. — Division is the reverse of multi- 6 plication and addition, and is similar to sub- 13 traction ; for, it is separating a number in tr equal parts, which is the same as subtracting rr the same number from a larger one ; that is, g subtracting the divisor from the dividend and — then from the remainder, repeating this pro- cess until there is no remainder, or until the remainder is less than the divisor. 6 is sub- ^ tracted 4 times, hence it is contained four times. 24 ~ 6 = 4. 28 DIVISION. (1-) (2.) (3.) 10 ) 100 ( 10 11 ) 121 (11 12 ) 144 ( 12 10 11 12 11 24 11 24 (4.) (5.) 11 ) 121 ( 10 + 1 12 ) 144 ( 10 + 3 110 120 11 24 11 24 6, 48 -> 13 = 4. 12. 120 -^ 10 = 12. 7. 64 -J- 8 = 8. 13. 130 -^ 10 = 13. 8. 96 H- 12 = 8. 14. 140 -f- 10 = 14. 9, 12 X 4 = 48. 15. 10 X 13 = 120. 0. 8x8 = 64. 16. 10 X 13 = 130. .1. 12 X 8 = 96. 17. 10 X 14 = 140. Cor. 1. — Adding a zero to the right of a number mul- tiplies the number by 10 ; taking a* zero away from the right of a number divides the number by 10. Divide 60536 by 4; thus, or 4 ) 60536 15134 ) 60536 { 10000 40000 30536 ( 5000 30000 536 ( 100 400 136 ( 30 120 16 ( 4 16 15134 The divisor 4 is contained once in the unit of the highest order of the dividend, which is one ten- thousand ; into the remainder 5000 times, then 100, 30 and lastly 4. DIVISION, 29 Rem. 1. — The same result is obtained by short division, by putting the first figure of the quotient under the left-hand figure of the dividend (when it is contained in it), as it is of the same order. Rem. 8. — If the unit of the divisor is not contained in the first unit of the dividend, then the first figure of the quotient will be of the same order as the second figure of the dividend and should be placed under it. ivide 60536 by 14 ; thus, 14 ) 60536 ( 4324 and 214 ) 925336 ( 4324 56 856 45 693 43 643 33 513 28 428 56 856 56 856 4334 X 14 = 60536. 4334 X 214 = 925336. Cor. 1. — Since the product of any two factors will have as many figures or one less than both factors, so in division the number of figures of the divisor and quotient will either be equal to or one greater than that of the dividend. CoR. 2. — When the divisor is contained in the same number of figures of the dividend as is in the divisor, then the number of figures of the divisor and quotient will be one more than that of the dividend ; but when it requires an additional figure of the dividend to contain the divisor, then the number of figures of fche divisor and quotient will be equal to that of the dividend. 30 DIVISION. PROBLEMS. 1. Divide 9253360 by 2140; thus, (2.) 21410 ) 925336|0 ( 4324 26432 856 ' 104 105728 26432 26432)2748928(104 26432 105728 105728 3. Divide 987654321 by 12300. 123100 ) 9876543|21 ( 80297 ' 80297 984 12300 365 240891 246 160594 1194 80297 693 4324 642 2140 513 ^ 17296 428 4324 856 8648 856 9253360 1107 987653100 873 1221 861 987654321 1221, remainder. Rem. 1. — When the dividend is not the exact product of two integral numhers, there will be a remainder, and the dividend is equal to the product of the quotient and divisor plus the remainder. Rem. 2. — When there are the same number of zeros in divi- dend and divisor, beginning with the order of units they may be canceled ; and when there are zeros in the divisor only, they may be omitted, and also the same number of figures in the dividend, which after the division is performed must be brought down as a part or the whole of the remainder. DIVISION. 31 EXAMPLES. 1. Divide 235643 by 123. 2. Divide 345678 by 234. 3. Divide 234567 by 891. 4. Divide 1357916 by 248. 5. Divide 369875432 by 1768. 6. Divide 487698425 by 625. 7. Divide 987654321 by 1234. 8. Divide 876543219 by 2345. 9. Divide 678956732 by 1546. 10. Divide 34567890 by 2564. 11. Divide 786954321 by 176543. 12. Divide 678900432 by 1004000. The student must not proceed until he is famihar with division. EXAMPLES. 1. What cost 5 lbs. of sugar at 10 cts. per lb. ? 2. At 10 cts. per lb., how many lbs. can be bought for 50 cts. ? 3. What cost 10 lbs. of sugar at 10 cts. per lb. ? 4. At 10 cts. per lb., how many lbs. can be bought for 100 cts. ? 5. What cost 15 lbs. of sugar at 10 cts. per lb. ? 6. At 10 cts. per lb., how many lbs. can be bought for 150 cts. ? 7. What cost 20 lbs. of sugar at 10 cts. per lb. ? 8. At 10 cts. per lb., how many lbs. can be bought for 200 cts. ? 9. What cost 25 lbs. of sugar at 10 cts. per lb. ? 10. At 10 cts. per lb., how many lbs. can be bought for 250 cts. ? 32 DIVISION. 11. What cost 245 lbs. of beef at 8 cts. per lb. ? 12. At 8 cts. per lb., how many lbs. of beef can be bought for 1960 cts. ? 13. What cost 348 acres of land at $45 per acre ? 14. At $48 per acre, how many acres can be bought for $15660 ? 15. What cost 3245 acres of land at $64 per acre ? 16. At $64 per acre, how many acres can be bought for $207680 ? 17. What is the cost of 15 horses at $125 each ? 18. How many horses at $125 each can be bought for $1875 ? 19. What is the cost of 35 oxen at $75 each ? 20. How many oxen at $75 each can be bought for $2625 ? 21. What is the cost of 84 cows at $45 each ? 22. How many cows at $45 each can be bought for $3780 ? . 23. Multiply 54682 by 9 ; thus, 54682 X 10 = 646820 54682 X 1 = 54682 492138 24. Multiply 54682 by 99. 54682 X 100 = 5468200 54682 X 1 = 54682 5413518 25. Multiply 54682 by 999. 54682 X 1000 = 54682000 54682 X 1 = 54682 54627318 DIVISION, 33 26. Multiply 54682 by 25. Instead of 25, multiply T)y J-f^. 4 ) 5468200 1367050 27. Multiply 54682 by 50. Instead of 50, multiply byH^. 28. Multiply 54682 by 75. Multiply the result ot 26th by 3. 29. Multiply 54682 by 150. 54682 X 100 = 5468200 Adding |, 2734100 8202300 30. Divide 546825 by 25. Multiply by 4 and strike off two figures. 31. Divide 546850 by 50. 5468|50 2 10937.00 Rem. — As a number is multiplied by 100 by adding two zeros, so any number is divided by 100 by cancelling the two right-hand figures, which will then form the remainder. 32. Bought 24 horses at $84 each, 54 cows at $36 each, and 364 sheep at $4 each ; what did the horses cost ? the cows ? the sheep ? what did all cost ? 33. Sold all the stock of the last example at a profit of $324 ; what did I sell them for ? 34. Bought 3064 acres of land at $25 per acre, and sold it at a loss of $1245. What did the land cost ? and what was it sold for ? 35. Bought 3840 acres of land at $10 per acre ; divided it into twenty farms of an equal number of 34 DIVISION, acres each; twelve of the farms I sold at $15 per acre, 3 of the farms at ^12 per acre, and the five remaining farms at $4 per acre; did I gain or lose, and how much ? 36. A farm of 364 acres was bought for $9100 and sold at a gain of $1456; what was paid per acre ? and at what rate was it sold ? 37. Bought 184 acres of land for $11960, and sold it for $13064; what was the cost, per acre? aod at what rate was it sold ? 38. The cost of 12 horses and 15 cows was $2760 ; the horses cost $180 each; what was the average cost of each cow ? 39. In a square mile there are 640 acres ; how many farms of 160 acres each in a State that has 9,000 square miles ? how many in one of 46,000 sq. miles ? how many in one of 257,000 sq. miles ? 40. What is the value of the land in the first State at $25 per acre? in the second at $20 per acre ? and in the third at $5 per acre ? 41. A man having an estate worth $15,000, increases it $2500 every year for twenty-five years, when he dies, leaving it as follows : To his wife $20,000, to his eldest son $8000, to his second son $7000, to his third son $7000, and the remainder to be divided equally among his five daughters ; what is the share of each daughter ? 42. A merchant commences business with a capital of $25000 ; at the end of the first year he finds that he has increased it $5000, the second year the increase is $4500, and the same at the end of the third year, when he trans- fers the whole business to his three sons. What is the capital of each son ? Arts, $13000. Review. NUMEEATION". Write in figures the following numbers : Twenty-five, One hundred and twenty-five, Two hun- dred and five, Three hundred and six. Four hundred and eight, One thousand, One thousand three hundred and forty-five. Five thousand eight hundred and four. Ten thousand five hundred and nine. Thirty-four thousand and twenty, Fifty-three thousand and five. Five hundred thousand, Six hundred thousand and four, One million, One million four hundred and twenty-five thousand six hundred and twelve. ADDITION ORAL QUESTIONS. John has 8 apples, WilHe has 7, and Andrew 5 ; how many apples have the three boys ? Bought coal for $8, wood for $6, and a barrel of flour for $5 ; what was the amount of the purchase ? How many are 6 + 4 + 5 2 + 5 + 8 8 + 7—5 5 + 6 + 8—5 8 + 7 + 4 5 + 9 + 2 9 + 8—7 4 + 5 + 9—8 3+8+7 1+3+6 7+9+8 3+9+7-9 4 + 6 + 9 9 + 6 + 4 6 + 5 + 4—6 2 + 7 + 8—4 How many are 3 + 7; 13 + 7; 23 + 7; 43 + 7; 53 + 7; 63 + 7; 73 + 7; 83 + 7; 93 + 7? Howmany are 3+8; 13 + 8; 23 + 8; 33 + 8; 43 + 8; 53 + 8; 63 + 8; 73 + 8; 83 + 8; 93 + 8? 36 ADDITION. How many are 4 + 7; 14 + 7; 24 + 7; 34+7; 44 + 7; 54 + 7; 64 + 7; 74+7; 84+7; 94 + 7? How many are 5 + 7; 15 + 7; 25 + 7; 35 + 7; 45 + 7; 55 + 7; 65 + 7; 75 + 7; 85 + 7'; 95 + 7? How many are 5 + 8; 15 + 8; 25 + 8; 35 + 8; 45 + 8; 55 + 8; 65 + 8; 75 + 8; 85 + 8; 95 + 8? How many are 5 + 9; 15 + 9; 25 + 9; 35 + 9; 45 + 9; 55 + 9; 65 + 9; 75 + 9; 85 + 9; 95 + 9? How many are 6 + 7; 16 + 7; 26 + 7; 36 + 7; 46 + 7; 56 + 7; 66 + 7; 76 + 7; 86 + 7; 96+7? How many are 6 + 8; 16 + 8; 26 + 8; 36 + 8; 46 + 8; 56 + 8; 66 + 8; 76 + 8; 86 + 8; 96 + 8? How many are 7 + 7; 17 + 7; 27 + 7; 37 + 7; 47 + 7; 57 + 7; 67 + 7; 77 + 7; 87 + 7; 97 + 7? How many are 7 + 8; 17 + 8: 27 + 8; 37 + 8; 47 + 8; 57 + 8; 67 + 8; 77 + 8; 87 + 8; 97 + 8? These questions should be frequently repeated or simi- lar ones asked. EXAMPLES. Add the foUowinj y numbers: 1 2.10 3.100 4.111 2 20 200 223- After adding the first 3 30 300 333 three examples, add their 4 40 400 444 sums, and observe the cor- 5 50 500 555 respondence with Ex. 4, 6 60 600 666 when the tens and hun- 7 70 700 777 dreds are added with the 8 80 800 888 columns of tens and hun- 9 90 900 999 dreds. SUBTRACTION. 6. 28 6. 53 7. 45 8. 56 9. 79 36 U 61 43 86 37 10. 84 73 11. 121 12. 312 13. 276 14. 751 15. 542 15. 374 343 256 543 324 324 236 621 461 631 263 261 573 17. What is the sum of 4,321, 604, and 36 ? 18. What is the sum of 264, 301, and 5 ? 19. What is the sum of 205, 25, and 6 ? 20. What is the sum of 4, 24, 324 ? 21. What is the sum of 3,276, 2,132, and 5,437 ? 22. What is the sum of 57,896, 98,765, and 78,901 ? Similar examples emi larger ones may be given for the slate or blackboard. SUBTRACTION. ORAL QUESTIONS. Take 7 from 9; 7 from 19; 7 from 29; 7 from 39; 7 from 49 ; 7 from 59 ; 7 from 69 ; 7 from 79 ; 7 from 89 ; 7 from 99 ; 7 from 109. Take 6 from 13 ; 6 from 23 ; 6 from 33; 6 from 43 ; 6 from 53 ; 6 from 63 ; 6 from 73 ; 6 from 83 ; 6 from 93 ; 6 from 103. Take 8 from 14; 8 from 24; 8 from 34; 8 from 44 ; 8 from 54; 8 from 64; 8 from 74; 8 from 84; 8 from 94; 8 from 104. Take 9 from 16 ; 9 from 26 ; 9 from 36 ; 9 from 46 ; 9 from 56; 9from 66; 9 from 76 j 9 from 86; 9 from 96; 9 from 106. 38 SUBTRACTION. Take 2 from 11 ; 2 from 21 ; 2 from 31 ; 2 from 41 ; 2 51; 2 from 61; 2 from 71; 2 from 81; 2 from 91; 2 from 101. Take 3 from 12 ; 3 from 22 ; 3 from 32 ; 3 from 42 ; 3 from 52 ; 3 from 62 ; 3 from 72 ; 3 from 82 ; 3 from 92 ; 3 from 102. Take 4 from 13 ; 4 from 23 ; 4 from 33 ; 4 from 43 ; 4 from 53 ; 4 from 63 ; 4 from 73 ; 4 from 83 ; 4 from 93 ; 4 from 103. Take 5 from 14 ; 5 from 24 ; 5 from 34 ; 5 from 44 ; 5 from 54 ; 5 from 64 ; 5 from 74 ; 5 from 84 ; 5 from 94 ; 5 from 104. Take 2 from 10 ; 2 from 20 ; 2 from 30 ; 2 from 40 ; 2 from 50 ; 2 from 60 ; 2 from 70 ; 2 from 80 ; 2 from 90 ; 2 from 100. These questions should be ofteu repeated. EXAMPLES. 1. From 10 take 1 ; 2, 3, 4, 5, 6, 7, 8, 9. 2. From 20 take. 1 . 2, 3, 4, 5, 6, 7, 8, 9. 3. From 30 take 1 ; 2, 3, 4, 5, 6, 7, 8, 9. 4. From 40 take 1 ; 2, 3, 4, 5, 6, 7, 8, 9. 5. From 50 take 1 2, 3, 4, 5, 6, 7, 8, 9. 6. From 60 take 1 , 2, 3, 4, 5, 6, 7, 8, 9. 7. From 70 take 1 . , 2, 3, 4, 5, 6, 7, 8, 9. 8. From 80 take 1 , 2, 3, 4, 5, 6, 7, 8, 9. 9. From 90 take 1 , 2, 3, 4, 5, 6, 7, 8, 9. 10. From 100 take 1 ; 2, 3, 4, 5, 6, 7, 8, 9, 99. 11. From 200 take 1 , 2, 3, 4, 5, 6, 7, 8, 9, 199. 12. From 300 take 1 ; 2, 3, 4, 6, 6, 7, 8, 9, 299. 13. From 400 take 1 ; 2, 3, 4, 5, 6, 7, 8, 9, 399. SUBTRACTION. 39 14. From 1,000 take 1 ; 2, 3, 4, 5, 6, 7, 8, 9, 999. 15. From 10,000 take 1 ; 2, 3, 4, 5, 6, 7, 8, 9, 9,999. 16. From 100,000 take 99,999. 17. From 2,000,000 take 1,999,999. 18. From 3,246,532 take 2,164,320. 19. From 800 take 400. 20. From 8,000 take 4,000. 21. From 545 take 116. 22. From 576 take 144. 23. From 624 take 304. 24. From 5,324 take 1,304. 25. From 6,725 take 2,432. 26. From 74,832 take 68,741. 27. From 83,754 take 48,623. 28. Columbus discovered America in the year 1492; how many years since ? 29. Washington was born in 1732, and lived 67 years; in what year did he die ? 30. I owed $1,000, and I paid one man $324, another $204, a third $120, and a fourth $67; how much do I still owe? Ans. $285. 31. A man dying leaves $5,000 to his widow and son, the widow to have $1,000 more than the son ; what is the portion of each ? Ans. Son, $2,000 ; widow, $3,000. 32. A father divides his estate of $12,000 as follows : To each of his two daughters, $2,000 ; to one son, $1,500 ; to another, $2,000 ; to a third son, $2,500, and the bal- ance to the widow ; what is her share ? Ans. $2,000. 4:0 ADDITION AND SUB1*RACTI0N. ADDITION AND SUBTKACTION. EXAMPLES, 1. A man has six farms; in the 1st 6,312 acres ; in the 2d 3,241 acres ; in the 3d 4,276 acres ; 4th, 272 acres ; 5th, 304 acres, and in the 6th, 63 acres ; how many acres in all ? Ans. 14,468 acres, s^ 2. A merchant sold goods for $6,032, gaining thereby $326. What did the goods cost ? Ans, $5,706. 3. A man owning 2,142 acres of land, gave to his eldest son 834 acres, to his second son 612 acres, and to the third 420 acres ; how many acres were left ? Ans. 276 acres. ?^4. A man left real estate worth $9,376, and personal property worth $2,142 ; he owes one man $1,236, another $875, and a third $450 ; what is the net value of his estate? Ans. 88,957. 5. A, B, and C commenced business ; A put in $3,275, B $4,150, and C $5,180; when they closed they found that they had lost $1,200; what had they at the begin- ning and at the close ? Alls. Commenced with $12,605 and ended $11,405. ^6. A horse and buggy together are worth $500; but the horse is worth $100 more than the buggy ; what is the value of each ? A7is. Horse, $300 ; buggy, $200. 7. The sum of two numbers is 5,439, and the one number is 215 greater than the other; what are the numbers ? 8. The minuend is 5,746, and the subtrahend 3,825 ; what is the difference ? y 9. The subtrahend is 3,825, and the difference 1,921 ; what is the minueud ? MULTIPLICATION, 41 10. The minuend is 5,746 and the difference 1,921; what is the subtrahend ? 11. When you know the minuend and difference, how do you find the subtrahend ? 12. How do you find the minuend, knowing the sub- trahend and difference ? MULTIPLICATIOK ORAL EXERCISES. I If an apple is worth 2 cents, what are 2 apples worth ? 3?4?5?6?7?8?9?10?11?12? If an apple is worth 3 cents, what are 2 apples worth ? 3? 4? 5? 6? 7? 8? 9? 10? 11? 12? If an apple is worth 4 cents, what are 2 apples worth ? 3 ? 4 ? 5 ? 6 ? 7 ? 8 ? 9 ? 10 ? 11 ? 12 ? If an apple is worth 5 cents, what are 2 apples worth ? 3 ? 4 ? 5 ? 6 ? 7 ? 8 ? 9 ? 10 ? 11 ? 12 ? If an apple is worth 6 cents, what are 2 apples worth ? 3? 4? 5? 6? 7? 8? 9? 10? 11? 12? If an apple is worth 7 cents, what are 2 apples worth ? 3 ? 4 ? 5 ? 6 ? 7 ? 8 ? 9 ? 10 ? 11 ? 12 ? If an apple is woiTth 8 cents, what are 2 apples worth ? ^3? 4? 5? 6? 7? 3? 9? 10? 11 ? 12? If an apple is worth 9 cents, what are 2 apples worth ? 3? 4.^ 5? 6? 7? 8 ? 9? 10? 11 ? 12? If an apple is worth 10 cents, what are 2 apples worth ? 3? 4? 5? 6? 7? 8? 9? 10? 11? 12 ? If an apple is worth 11 cents, what are 2 apples worth ? 3? 4? 5 ? 6? 7? 8? 9? 10? 11? 12? If an apple is worth 12 cents, what are 2 apples worth ? 3 ? 4 ? 5 ? 6 ? 7 ? 8 ? 9 ? 10 ? 11 ? 12 ? 48 MULTIPLICATION, EXAMPLES. 1. Multiply 13 by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. 2. Multiply 14 by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. 3. Multiply 15 by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. 4. Multiply 16 by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. 5. Multiply 17 by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. 6. Multiply 18 by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. 7. Multiply 256 by 4, 24, 124. 8. Multiply 2,145 by 4, 24, 124. 9. Multiply 3,636 by 5, 20, 100, 125. 10. Multiply 6,435 by 5, 20, 300, 1,000, and by 1,325„ Analyze. 11. Multiply 3,674 by 104, 206, 3,005. Ans, 382,096, 756,844, and 11,040,370. 12. Multiply 21,564 by 2,100, 2,010, 2,001. Ans. 45,284,400, 43,343,640, and 43,149,564. 13. Multiply 54,000 by 1,200, 14,000, 630. Ans. 64,800,000, 756,000,000, and 34,020,000. 14. Multiply 60,401 by 1,040, 1,004, 1,400. Ans. 62,817,040, 60,642,604, and 84,561,400. 15. Multiply four thousand and three by three hundred and one. • Ans, 1,204,903. 16. Multiply forty-six thousand one hundred and three by three thousand and two. A71S, 138,401,206. 17. Multiply four hundred and fifty-six thousand three hundred and fifty-four by three thousand five hundred and thirty-six. Ans. 1,613,667,744. 18. Multiply four millions three hundred and fifty-four thousand six hundred and twenty-five by thirty-six thou- sand four hundred and sixty-seven. Ans. 158,800,109,875. DIVISION. 43 19. Multiply sixty-four millions seven hundred and ninety-five thousand eight hundred and sixty-nine by six hundred and fifty-four thousand three hundred and seventy-eight. Ans. 42,400,991,164,482. 20. Multiply six hundred and four millions two hun- dred thousand four hundred and four by four hundred and five thousand two hundred. Ans. 244,822,003,700,800. 21. What is the product of 3,754 and 268. A71S, 1,006,072. 22. The factors of a number are 57 and 29 ; what is the number? Ans, 1,653. 23. Two of the three factors of the number 1,188 are 9 and 12 ; what is the third factor ? Ans. 11. DIVISION. ORAL EXERCISES. If one apple cost 2 cents, how many .can you buy for 4 cents? 6? 8? 10? 12? How often is 2 contained in 4 ? 6 ? 8 ? 10 ? 12 ? 14 ? 16 ? 18 ? 20 ? 22 ? 24 ? How often is 3 contained in 6 ? 9 ? 12 ? 15 ? 18 ? 21 ? 24? 27? 30? 33? 36? How often is 4 contained in 8 ? 12 ? 16 ? 20 ? 24 ? 28 ? 32? 36? 40? 44? 48? How often is 5 contained in 10? 15? 20 ? 25 ? 30? 35 ? 40 ? 45 ? 50 ? 55 ? 60 ? How often is 6 contained in 12 ? 18 ? 24 ? 30 ? 36 ? 42 ? 48 ? 54 ? 60 ? 66 ? 72 ? How often is 7 contained in 14 ? 21 ? 28? 35 ? 42 ? 49 ? 56 ? 63 ? 70 ? 77 ? 84 ? 44 MULTIPLICATION AND DIVISION. How often is 8 contained in 16 ? 24 ? 32 ? 40 ? 48 ? 56 ? 64 ? 72 ? 80 ? 88 ? 96 ? How often is 9 contained in 18 ? 27 ? 36 ? 45 ? 54 ? 63 ? 72 ? 81 ? 90 ? 99 ? 108 ? How often is 10 contained in 20 ? 30 ? 40? 50? 60? 70? 80? 90? 100? 110? 120? How often is 11 contained in 22? 33 ? 44? 55? 66? 77? 88? 99? 110? 121? 132? How often is 12 contained in 24? 36? 48? 60? 72? 84? 96? 108? 120? 132? 144? Divide 12 into 2 equal parts; into 3, 4, 6. Divide 24 into 2 equal parts ; into 3, 4, 6. Divide 36 into 2 equal parts; into 3, 4, 6. Divide 48 into 2 equal parts ; into 3, 4, 6-. If 10 yards of cloth cost $50, what is the cost of one yard? If 4 pecks make one bushel, how many bushels in 36 pecks ? If 48 bushels of wheat cost $48, what is the cost of one bushel ? If 24 bushels of wheat cost $48, what is the cost of one bushel ? Divide 48 apples equally among 4 boys ; 6 boys ; 8, 12. MULTIPLICATIOlSr AND DIVISION. EXAM PLE S. 1. At $5 each, how many sheep can be bought for $25? $50? $100? $200? $300? $400? $500? $600? $625? $650? $675? 2. At $6 each, how many for $24 ? $48? $96? $120? $132 ? $144 ? $240 ? $264 ? $300 ? $336 ? $348 ? MULTIPLICATION AND DIVISION, 45 3. Divide $1,200 into 5 equal parts; how many in each part ? 4. Divide 48,000 into 4 parts, or two equal pairs, so that each number of the first pair shall be one-half of one of the numbers of the second pair. Ans, Each of the first pair 8,000, and each of the sec- ond 16,000. 5. Divide 24,000 into three parts, so that the difference of any two consecutive parts shall be 1,000. Ans. 7,000, 8,000, and 9,000. 6. How long would it take a man to travel around the earth, whose circumference is about 25,000 miles, if he travel 50 miles a day ? 7. A man purchased the same number of horses and cows for $5,640, the horses at $85 each, and the cows at $35 each. How many of each kind ? Ans, 47 of each. 8. A drover bought sheep, hogs, and cows, of each an equal number, for $2,752, the sheep at $4 each, the hogs at $11, and the cows at $28 each; how many of each kind ? Ayis, 64 of each. 9. The divisor is 463 and the quotient 264; what is the dividend? Ans. 122,232. 10. The quotient is 327 the dividend 160,557 ; what is the divisor ? Aiis. 491. 11. A farmer sold 8 cows and 12 horses for $1,092 ; the horses were rated at $75 each ; what were the cows sold at? Ans. $24. 12. A man bought a farm of 225 acres for $10,125, and sold it for $11,250 ; what did he pay per acre, and how much per acre did he gain ? Ans. He paid $45 and gained $5 per acre. 46 MULTIPLICATION AND DIVISION. 13. A clerk has a salary of $80 per month ; he pays for board $25 per month and other expenses $15 per month; how much does he save per month ? how much per year ? Ans, $40 per month and $480 per year. 14. If I pay $225 for 75 sheep, and they cost $1 each for keeping, and I sell them at $5 each, how much is my profit ? Ans. $75. 15. If 8 cords of wood cost $32, what is the cost of 1 cord? of 2 cords? 3 ? 4? 5? 6? 7 ? 8? 9 ? 16. If 9 lbs. sugar cost 63 cents, what is the cost of 11 lbs.? 17. If 12 lbs. coffee cost 300 cents, what cost 5 lbs.? 18. If 8 lbs. wool cost 280 cents, what cost 15 lbs. ? Ans. 525 cents. 19. A farmer sold 235 barrels of flour at $6 per barrel ; he bought goods for $830, and for the balance took coal at $4 per ton ; how many tons did he get ? Ans, 145 tons. 20. The quotient is 164, the divisor 235, and the re- mainder 143; what is the dividend? Ans, 38,683. 21 . How many revolutions will a wheel 20 feet in cir- cumference make in running a mile (5,280 ft.) ? Ans, 264. 22. If two men can do a piece of work in 3 days, how long will it take one man to do it ? Ans. 6 days. 23. If 4 men can do a piece of work in 6 days, how long will it take 3 men to do it? Ans, 8 days. 24. If 15 acres of land cost $375, what is the cost of 1 acre ? 2 acres ? 3 acres ? 4 acres ? 5 acres ? 55 acres ? Ans, $25, $50, $75, $100, $125, $1,375. 25. If 5 tons of hay cost $60, what is the cost of 1 ton ? 2 tons? 3 tons? 45 tons? 75 tons? Ans, $12, $24, $36, $540, $900. MULTIPLICATION AND DIVISION. 47 26. If 36 acres of land cost $1,440, what will 57 acres cost? 84 acres? Arts, $2,280 and $3,360. 27. Sold 12 horses at $75 each and 15 cows at $24 each, and invested the proceeds in sheep at $5 each ; how many sheep did I get ? Ans. 252 sheep. 28. Sold 275 acres of land at $24 per acre, 340 acres at $32 per acre, and invested the proceeds in government land at $2 per acre ; how many acres did I get. Ans. 8,740 acres. 29. Of what number is 36 both divisor and quotient ? Ans. 1,296. 30. Bought 12 horses at $80 per head ; at what price must I sell each horse so as to gain $72. A^is, $86. 31. A party of 45 men has provisions sufficient for 40 days; how long would it last 1 man ? 2 men? 4 men? 5 men ? 20 men ? Ans. 1 man, 1,800 days; 2 men, 900 days; 4 men, 450 days; 5 men, 360 days; 20 men, 90 days. 32. A garrison of 900 men have provisions for 6 months; how many men would the same provisions last 9 months? Ans. 600 men. 33. If 14 men do a piece of work in 3 days, how many men would do it in 1 day ? how many in 2 days ? in 6 days ? Ans. 42 men in 1 day, 21 men in 2 days, 7 men in 6 days. 34. If 4 men mow a field of grass in 3 days, how long will it take 1 man ? 2 men ? 3 men ? 6 men ? Ans. 1 man, 12 days; 2 men, 6 days; 3 men, 4 days; 6 men, 2 Factoring, Any number that is the product of two or more num- bers is a Composite Ntmiber ; as, 12 is the product of 4 and 3, and 4 is the product of 2 and 2 ; hence, the factors of 12 are 2, 2, and 3 ; these factors cannot be reduced, they are therefore called Prime Factors^ as any number is called Prime which is not formed by other factors than itself and unity. There are sixteen Prime Numbers in the first fifty ; viz., 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47, and they are found thus : 5, 0, 7, %, 0, 10, n, 10, 17, n, 19, n, t$, n, n, n, 29, $0, 1, % 3, i, 11, n, 13, U, n, n. 23, u, 31, n, %t, u. 41, it, 43, u. $$, $0, 37, $$, $% 0, 0, ^6, 47, 4$, 40, $0. Every even number after 2 is composite, as it is divisi- ble by 2 ; strike these. Every third number after 3 is divisible by 3 ; strike these. Every fifth number after 5 ; every seventh number after 7 ; the ninth numbers are canceled by 3 ; every eleventh number after 11, etc. CoR. 1. — Every even number is divisible by 2. CoR. 2. — Every number whose last two figures express a number which is a multiple of 4, is divisible by 4 ; for if the number expressed by these two figures is subtracted from the whole number, the remainder will be a certain number of hundreds which is divisible by 4. CoR. 3. — Every number ending in 5 is divisible by 5. CoR. 4. — Every number ending in zero is divisible by 10, consequently by 2 and 5. FACTORING, 49 CoE. 5, — Every number is divisible by 3, when the sum of its figures taken as units is divisible by 3 ; for if from 1000 one be subtracted, the remainder is divisible by 9 ; if from 100 one be subtracted, the remainder is divisible by 9 ; so also of 10 ; hence if from 2000 two be subtract- ed, the remainder is divisible by 9 ; so also take 2 from 200, and 2 from 20, etc. ; therefore, in dividing any num- ber of thousands, hundreds, or tens, the remainder will always be the unit of thousands, hundreds, or tens, and if the sum of these as units and ialso of the units of the given number equals 9, or any number of nines, the whole number is divisible by 9, and consequently by 3. PROBLEMS. Resolve the following numbers to their prime fact( 1. 4 = 2,2. 13. 22 = 2, 11. 2. 6 = 2,3. 14. 24 = 2, 2, 2, 3. 3. 8 = 2, 2, 2. 15. 26 = 2, 13. 4. 9 = 3,3. 16. 27 = 3, 3, 3. 5. 10 = 2,5. 17. 28 = 2, 2, 7. 6. 13 — 2, 2, 3. 18. 30 = 2, 3, 5. 7. 14 = 2,7. 19. 32 = 2, 2, 2, 2, 3. 8. 15 = 3,5. 20. 33 = 3, 11. 9. 16 ^ 2, 2, 2, 2. 21. 34 = 2, 17. 10. 18 z± 2, 3, 3. 22. 35 = 5, 7. 11. ao = 2, 2, 5. 23. 36 = 2, 2, 3, 3. 12. 21 = 3,7. 24. 38 = 2; 19. 50 1 FACTORING • EXAMPLES. Resolve the following numbers into their prime f actors 1. 60. 10. 86. 19. 106. 28. 135. 2. 64. 11. 88. 20. 108. 29. 128. 3. 65. 13. 90. 21. 110. 30. 130. 4. 70. 13. 95. 22. 112. 31. 132. 5. 73. 14. 96. 23. 114. 32. 136. 6. 75. 15. 98. 24. 116. 33. 140. 7. 78. 16. 100. 25. iia 34. 325. 8. 80. 17. 103. 26. 120. 35. 500. 9. 84. 18. 104. 27. 124. 36. 635. LEAST COMMON MULTIPLE. Def. — One number is called a Wiiltijjle of another when it is" exactly divisible by that other number. When a number is resolved into its prime factors, the original number is a multiple of all the prime factors, and of all the quotients arising from these factors divided into the original number; when the same factor occurs several times, the different products of this factor multi- plied by itself must also be divided into the original number. PROBLEMS. Find the numbers of which the following are multiples: 1. 24. Ans. t, ty 2, 3, 4, 6, 8, 12. 2. 25. Ans. 5, 5. 3. 30. Ans. 2, 3, 5, 6, 10, 15. 4. 32. Ans. t, %,. % % 2, 4, 8, 16. 5. 36. ' Ans. t, 2, $, 3, 4, 6, 9, 12, 18. FACTORING* 51 6. 40. Ans. t, t, 2, 5, 4, 8, 10, 20. 7. 48. Ans. % t, % % 3, 4, 6, 8, 12, 16, 24. 8. 54. Ans. 2, 3, $, $, 6, 9, 18, 27. 9. 56. Ans. $, t, 2, 7, 4, 8, 14, 28. Rem. — First find the prime factors, then the quotients arising by dividing each prime factor and the different products of these factors into the original number. PROBLEMS. 1. Find the least common multiple of 8 and 12. 8 and 12 It is evident that any number 2, 2, 2. 2, 2, 3. which contains all the prime 2x2x2x3 = 24. factors of each number is a common multiple of the given numbers ; and the least common multiple must contain these factors and no others. CoR. — Any factor must enter the L. C. M. as often as it does any given number. 2. Find the least common multiple of 6 and 15. 6 and 15 As 3 is common to both num- 2, 3. 3, 5. bers, it must be taken but once. The L. 0. M. is 2 X 3 x 5=30. 3. Find the least common multiple of 6 and 12. 6 and 12 As 12 is a multiple of 6, it is 2, 3. 2, 2, 3. evident that it contains all the prime factors of 6 ; hence, when- ever one of the given numbers is a multiple of another given number, th^t other number need not be considered, and may be canceled. 2 cancel 5 ; and as 2 is now a common factor of two of the quotients, divide it into them, reserving as before ; and as 15 is now a mul- tiple of 3, cancel 3 ; the product of all the divisors and of the remaining quotients and original numbers, if there be any, will be the least common multiple. 3. Find the least common multiple of 4, 6, 8, and 24. Since 24 is a multiple of each of the other numbers, it is a common multiple of all the numbers. 4. Find the least common multiple of 6, 8, and 10. Ans. 120. 5. Find the least common multiple of 10, 12, and 14. Ans. 420. 6. Find the least common multiple of 9, 12, 18, 27, and 36. Ans. 108. 7. Find the least common multiple of 7, 14, 18, 21, and 28. Ans. 252. 8. Find the least common multiple of 5, 15, 20, 30, and 60, Ans. 60. FACTORING, 53 GREATEST COMMON DIVISOE. The Greatest Common Divisor of two or more numbers is the largest number that will exactly divide all the numbers. PROBLEM. Find the greatest common divisor of 4 and 10. Ans. 2, 2, 2. 2, 5. The greatest common divisor of any number of terms, must be the factor, or the product of the factors, common to the terms. EXAMPLES. 1. Find the G. C. D. of 6 and 15. Ans. 3. 2, 3. 3, 5. 2. Find the G. C. D. of 12 and 18. Ans. 3 x 3=:a 2, 2, 3. 2, 3, 3. 3. Find the G. C. D. of 43 and 70. Short method, 3 43 70 7 31 35 3 5 ^Ms, 3x7=14. CANCELLATION. THEOREM. The dividend contains all and exactly the same factors as the divisor and quotient. Any composite number is the product of all its prime factors, and may be resolved into them. The product of any two integral numbers is a composite number and must contain all the factors of both numbers ; and as- a 54 FACTORING. dividend is the product of its divisor and quotient, it must contain the same factors as its divisor and quotient. Cor. 1. — The same is true if one or both divisor and quotient be fractional ; for when reduced to a common denominator, their numerators • may be regarded as integral. CoR. 2. — Every factor of the divisor will cancel the same factor in the dividend. Cor. 3. — The factors which are not canceled by those of the divisor will be the factors of the quotient CoR. 4. — Canceling a factor in the dividend divides the quotient by the same factor. CoR. 5. — Canceling a factor in the divisor multiplies the quotient by the same factor. PROBLEMS. 1. Divide 648 by 36. 648 ^ t, WW^^ ^- = 18, Ans. 2. Divide 625 by 125. g| ^ M^!l^ = 5, Ans. 125 0, ^, 0. 3. Divide 500 by 100. 100 4. A man bought 30 yards of cloth at $5 a yard ; he then exchanged it for other cloth at $3 a yard. How many yards of the latter did he get ? FACTORING, 55 — - — == 50 yards. The 3 of the divisor is canceled into 30 of the dividend. 5. Sold 48 cattle at $60 each, and invested the proceeds in sheep at $5 each ; how many sheep were purchased. ^^i? = 576 sheep. 6. A farmer sold 150 bushels of wheat at 125 cents per bushel, and invested the proceeds in oats at 25 cents per bushel ; how much oats did he get ? Xt$ X 150 U = 750 bu. oats. 7. Sold 160 acres of land at $50 per acre, and bought another tract for the amount of sales, at $40 per acre ; how much land was bought ? $50 X 1:00 ■:i-^ = 200 acres. 8. Divide 18 x 15 x 16 x 24 x 32 by 9, 5, 8, 12, and 16. = 48. 2x3x2x2x8 9. Divide 100 x 102 x 96 x 45 by 102 x 100 x 16 x 9. 3:00 X nt X 00 X ^0 _ „ 10^ X 100 X i0 X ~ FUACTIOJfS. Def. 1. — If a unit or any other number is divided into equal parts, one or more of these parts is a fraction of the whole, and all the parts constitute the whole. If a unit is divided into two equal parts, each part is called one-half, and is written \y and the two halves constitute the whole ; thus, f = 1. If a unit is divided into three equal parts, each part is one-third {\) ; two of the parts, f ; and the three thirds constitute the whole; thus, -1 = 1. If 5 is divided into two equal parts, each part is (five-halves (f); two of the parts, -^ = 5, etc.; if 6 is divided into three equal parts, each part is f == 2 ; and two-thirds of 6 is 4, etc. Fourths, fifths, sixths, etc., are similarly constructed. 2. When a unit is divided into equal parts, any num- ber of the parts less than the whole, expressed fraction- ally, is called a Proper Fraction ; as, |, |, |, -^, etc. The quotient is the same in division when the dividend is less than the divisor ; as, -^, |f , etc. ; but when the divisor is less than the dividend, the quotient is called an Improper Fraction; as, f, -^/, % etc. 3. When the division indicated by an Improper Frac- tion is performed, and the divisor is not contained an exact number of times in the dividend, the quotient is partly integral and partly fractional, and is termed a FRACTIONS. 57 Mixed Number ; thus, f = li ; -^^ — 8| ; and ■V- = lof . CoE. — The denominator expresses the number of parts into which a unit or any other number is divided, and the numerator expresses the number of parts of a unit taken, or the number divided. ExEMPLiFiCATioiq^. — If each half is divided into two equal parts, the whole number of parts is four, and the one-half has made two of those parts ; hence, ^ = ^ ; if each half is divided into three equal parts, the whole number of parts is six, and ^ = -1; ,\^=z^=zf = ^=: ■^jy, etc., and ^ = ^z=^ = -^=z^, etc. ; hence, if both terms of a fraction are multiplied by the same number, the value of the fraction is not changed, and by this principle fractions are reduced to a Common Denom- inator. \ PROBLEMS./ 1. Eeduce |- and ^ to a common denominator. 1-3=3^ and iilzl 2. Eeduce ^, ^, and J to a common denominator. ixQ=T29 3x4=T¥^ ^^^ Tx3=T^' 3. Eeduce f and f to a common denominator. |-?=|f, and 4-s^|6. Cor. 1. — The least common multiple of all the denom- inators is the least common denominator. CoR. 2. — Fractions are reduced to a common denom- inator thus : Multiply both terms of each fraction by the quotient obtained by dividing its denominator into the least common denominator ; or, when all the denom- 58 FRACTIONS, inators are prime to each other, multiply both terms of each fraction by all the other denominators. Cor. 3. — An integral number is reduced to a fraction by multiplying it by the denominator of the fraction. EXAMPLES. 1. Eeduce |, f , and ^ to a common denominator. ^ns. ^, II, and ff. 2. Eeduce \, f, and f to a common denominator. ^ns, -^j If, and ^. 3. Reduce 4? h ^^^ | to a common denominator. ^ns. m^ m, and ||f. 4. Eeduce |, -|> I? and ^^ to a common denominator. ^ns. m^, \m, ^Wj> and ^WV- 5. Eeduce \y ^, ^-J^, and -^q to a common denom- inator. Ans. T^o, ^oP^, -^fo, and y^. Exemplification. — Since ^ = 1 — ^=:^ = ^, etc., and 1 = 1 = 1 = ^ = 36^, etc., and fl:|=^, |l:i=i ftt=i, etc., and fl|=i, 111=^, -Ati=i, etc. ; hence, if both terms of a fraction are divided by the same number, the value of the fraction is not changed. CoR. — When both terms of a fraction have a common factor, it may be canceled, and the fraction is thereby reduced to its lowest terms. Rem. — The common factor may be either a prime or a composite number, and it is the greatest common divisor of the terms. EXAM PLES. 1. Eeduce -^ to its lowest terms. Am. Atl'^. 2. Eeduce W to its lowest terms. Ans, -J. 3. Eeduce y||-j to its lowest terms.- Ans, ^h:. FRACTIONS. 59 ADDITION AND SUBTRACTION OF FRACTIONS. EXAMPLES. 1. Add and subtract | and ^. 4 is L. C. D. 2. Add and subtract | and ^, 6 is L. C. D. Arts. Sum, -J; difference, -J-. 3. Add and subtract | and |. 20 is L. C. D. Ans. Sum, fj; difference,^. 4. Add and subtract f and f . 30 is L. C. D, Ans, Sum, |^; difference, -gig^. 5. Add and subtract f and 4- 42 is L. C. D. Ans. Sum, If; difference,:^. Cor. — When the denominators have no common fac- tor, then their product is the L. C. D., and each numer- ator is multiplied by all the denominators except its own 6. Add I and |. L. C. D., 12. 2x4__^ 3x3__^ 3 X 4 "" 12* 4 X 3 "" 12* 8 + 9 = 17. Sum z=:\i=:l^. 7. Add f , f , and f Sum = -W = ^^f . As no two numbers hav6 a common factor, the L. C. D. is the product of all the denominators ; and then as each denominator is multiplied by the other two denomina- tors, so each numerator must be multiplied by the product of all the denominators except its own. 8. Addf, I, and 3^. 4 )5 8 12 5 2 3 4x5x2x3 = 120. 1^ = 24. if i^ =15. J^ = 10. .•. 5, 8, and 12 are the multipliers of the fractions. 60 FRACTIONS. 9. Add and subtract -^ and -^^. 25 ) 275 and 350 11 14 25x11x14 = 3850, L. CD. Sum = -J-m, and difference = -^-^. 10. Add and subtract 3| and 2|^. 3f = 3,^ Difference =. 1-^. Sum = 6^. ^^ .3 11 X 4 44 ^'' ^* = yx4 = 12 21 ___^ x3 _ 27 "^ ~ 4 X 3 "~ 12 Sum = fl- = Sli. Difference = ^ = 1^^. Rem. — Mixed numbers may be united by either of the above jnethods ; the latter is generally preferred. 11. Cast up the following account: cents. •Bought 3 pairs hose, at 37|- cts. = 112^ 5 pocket handkerchiefs, at 62^ cts. =r 312 J- 3 pocket knives, at 31;^ cts. =r 93|- 7 lbs. sugar, at 12|^ cts. = 87| 9 inkstands, at 6 J cts. = 56} 5 qrs. paper, at 15| cts. — 76^;^ 738f Paid on account, 549;^ Balance due, 189^ cts. The numerators of the 4ths show the value, but those of the halves must be doubled, as 4 is the common denominator. FRACTIONS. 61 MULTIPLICATION OF PEACTIONS. THEOREM. The -product of two proper fractions is less than either fraction. For, if a number is multiplied by one, the product is the same as the number multiplied. If the multiplier is greater than one, the product is greater than the num- ber ; and if the multiplier is less than one, the product is less than the number. In the multiplication of two proper fractions, each factor is less than one ; hence the product is less than either fraction. PROBLEMS. 1. Multiply I by 1. Ans. | x 1 = f . 2. Multiply I by 2. Ans. f x 2 = |. 3. Multiply f by J. Ans. f x i = f It is evident that f multiplied by 1 =: f , that f x 2 is ^, and f X 1^ or 1^ time f is -J ; and, as alternating the factors does not change the product, therefore, 1 x f =1 , 2 X I = t, and i X I = i-. 4. Multiply I by f Jw5. | x | = |. CoR. 1. — In multiplying by a fraction the numerator is a multiplier and the denominator a divisor. CoR. 2. — In the multiplication of fractions, the pro- duct of all the numerators is the numerator of the product ; and the product of all the denominators is the denominator of the product. 62 FRACTIONS. Cor. 3. — The product of two improper fractions is greater than either fraction. EX A M PLE s. 1. Multiply i I, I, f , I, 4, 1, 1, ji^, ^, ^. ixtx|x|x|x|xix|x^xi|x» = ^. By analysis, i of f = i, i of f = i, i of f ==: i, ^ of f = i, i of f =:. I, I of I = i, i of f = i, i of ^^ = 3V> 2. Multiply I, If, and |; thus, 2 I X ^ X I = ii, product. 3 3. Multiply -^ and f . A X i = U, product- 4. Multiply 35| by 9. f X 9 = ^ = 6| 35 X 9 = 315_ 321f = Product. Axiom 7. — If any number be both multiplied and divided by the same number, the value of the original number is not changed. CoR. 1. — If the multiplier is greater than the divisor, the product is greater than the number multiplied, but if the multiplier is less than the divisor, then the pro- duct is less than the multiplicand. CoR. 2. — Multiplying the numerator or dividing the denominator by any number, multiplies the fraction by the same number. FRACTIONS* 63 5. Multiply 37 by 8f 37 18i 296 Product = 314J 6. Multiply 37i by llf. 37| X llf = H^ X ^ =r iffi. = 440^^ = Product. Rem. — When only one of the factors is a mixed number, they may be solved as the 4tli and 5th examples ; but when both fac- tors are mixed numbers, it is better to reduce them to improper fractions as in 6th example. 7. Multiply 14| by 4. 13. Multiply 3147 by 35f. 8. Multiply 17i by 6. 14. Multiply 4156| by 2124|. 9. Multiply 18i by 9. Ih. Multiply l'^l^\ by 13f 10. Multiply 24| by 8. 16. Multiply 15f by 9f . 11. Multiply 1456 by 15f 17. Multiply 29^ by 13^. 12. Multiply 2375 by 27-J. 18. Multiply 104^ by 20f. DIVISION OF FEACTIONS. PROBLEMS. 1. Divide 10 by 5. Ans. 10 -^ 5 = Y = 2. 2. Divide 6 by 3. Ans, 1 = 2. 3. Divide 3 by 3. Ans. 1 = 1. 4. Divide 1 by 2. Ans. \. 5. Divide 1 by 3. Ans. \. 6. Divide 2 by 3. Ans. f. 7- Divide 3 by 4. A71S. f. 8. Divide 5 by 4. Ans. f = IJ. 9. Divide | by 2, or divide ^ into two equal parts. Ans. 1^3 = f 64 FRACTIONS. 10. Divide f by 2, or divide | into two equal parts. Ans. f-^2 _ 1^ 11. Divide \ by 2, or divide ^ into two equal parts. Ans. ix3 = f 12. Divide f by \, or how often is, -J- contained in f. ^^s. Evidently twice. 13. Divide ^ by J, or how often is J contained in 4=1. Ans. Evidently twice. 14. Divide ^ by -J-, or how often is -J- contained in ^. Ans. ^ = 3 and J = f ; f -V- 1 = I = H. 15. Divide \ by J, or how often is ^ contained in \. Ans. i = -^ and i = ^^; ^ -r- -^V = f CoR. 1. — The numerator of a fraction is a dividend, the denominator a divisor, and the fraction itself the quotient. CoE. 2. — To divide a fraction by a fraction, reduce both to a common denominator and then divide the numerator of the dividend by the numerator of the divisor. CoR. 3.— Dividing the numerator or multiplying the denominator by any number, divides the fraction by the same number. THEOREM. To divide any numher hy a fraction, invert the fractional divisor and malce it a multiplier. 1st. When the dividend is integral. If the dividend is multiplied by the denominator of the divisor, the product will be the numerator of the quotient, and the numerator of the divisor will be the denominator. FRACTIONS. 65 2d. When the dividend is also a fraction. If the numerator of the dividend is multiplied by the denominator of the divisor, the product will be the numerator of the quotient, and if the numerator of the divisor is multiplied by the denominator of the dividend, the product will be the denominator of the quotient. PROBLEMS. 1. Divide 15 by |. 15x4=1-6/, and ^-r-f =-^ z=20. 15 X I = 20. 2. Divide ^- by f . ' -V-xt=fi and fx|=:^^, and ^z=^- = ^. Any number is divided by a fraction by inverting the fractional divisor and making it a multiplier. EXAMPLES 1. Divide 36 by f . 9. Divide 83| by ^. 2. Divide 12 by f. 10. Divide 15 by ^V 3. Divide 16 by ^. 11. Divide ^- by ^V 4 Divide f by f . 12. Divide || by f . 5. Divide | by f . 13. Divide |f by -^. 6. Divide | by f 14. Divide 208| by 27|. 7. Divide 112i by 21f. 15. Divide 109^ by 29f 8. Divide 14| by 9f Eem. — Mixed numbers should be reduced to improper fractions. 16. Divide 5436f by 3. Ans. 1812f 66 FRACTIONS. 17. Divide 6,478| by 9. Ans. 719||. 9 ) 64781 719, remainder ^=:^ and 3/-^9=ff. 18. Divide 250f by 12|. Ans. 19|4. 19. Divide 450 by |. ^^5. 600. 20. Divide! by 20. Ans. ^. 21. Divide i X f X f X 1 by I, ^0. -H. H- -^^5. f. 22. Divide f by J. ^/i5. 3. 23. Divide | by \. Ans. 5. 24. Divide 1 by i, i, i, \. Ans. 2, 3, 4, 5. 25. Divide 1 by f, |, |. ^7^5. |, !, |. 26. Divide 2 by f , f, f. ^t^s. |, |, J^-. CoK. 1. Any fraction having unity for its numerator is contained in 1 as many times'as there are units in its de- nominator ; thus, ^ is contained in 1 twice, ^ three times, &c. If the fraction has 2 for its numerator, it is con- tained one-half as many times ; if the numerator is 3, one- third as many times, &c. CoK. 2. If the dividend is increased, the quotient is in- creased just as many times ; if the divisor is increased, the quotient is diminished just as many times ; the converse is also true in both cases. CoE. 3. To divide any number, either integral or frac- tional, by a fraction, invert the divisor and multiply the dividend by it. COMPLEX FEACTIONS. When one or both terms of a fraction are either frac- tions or mixed numbers, it is called a Complex Fraction; thus, 34 i 3 37i 100^ , IPX- -/' f ' 9* zrsT' -TKK^ etc., are complex fractions. Off 62J 100 ^ FB ACTIONS, 67 Eem. — ^When we consider that the numerator of a fraction represents a dividend, and the denominator a divisor, a complex fraction is readily reduced to a simple fraction. 3- 1. Eeduce -^ to a simple fraction. Rem. — As the denominator is a divisor, it must be inverted. 2. Eeduce J to a simple fraction. I I = i X f := f . 3 3. Eeduce ^ to a common fraction. I |=^^f = ¥ = ^i. 3. Eeduce -^ to a common fraction. 62|- 5. Multiply -^ X ^^ = i|i X ^ = ^px^ X 6. Divide|by| = |x| = 4x|x|x| = i. Jxix|xi = f. In changing the division to multiplication, the whole divisor must be inverted ; that is, f becomes the numer- ator and f becomes the denominator, with the sign of multiplication ; then again the two denominators must be inverted ; then they are all in the form of multiplication. Review. EEDUCTION. Problem. — Eeduce f, |, and 4 to a common denomi- nator. Since all the denominators are prime to each other, the common denominator is their product; thus, 3x5x7= 105, the denominator; And 2x5x7=J?0; 3x3x7=63; 5x3x5z=75, the numerators; Or |x|x^=tVV; |x|x^=t^^; 4x|x|=t^. Cor. Each numerator is multiplied by all the denomi- nators but its own. EXAMPLES. 1. Eeduce |, f , and f to a common denominator. Ans. 6 is the L. C. D. ; |, -f-, f. 2. Eeduce ^, f, and f to a common denominator. ^/^5. L. C.D.,60; |i,f|.if. 3. Eeduce |, f, and 4 to a common denominator. Ans. \;f%y trf^ TTo"' 4. Eeduce f , 3^, and -^ to a common denominator. Ans. TJWTy TTsJy 1 2 8 7 • 5. Eeduce y\, \\, ^, and ^, to a common denomina- tor. ^7?.^. tVV t¥o. t¥o. tVo. 6. Eeduce f, f , ^, ^, and ^ to a common denominator. ^«*'- n, u, u, a, u- 7. Eeduce y^y, ^fy to a common denominator. FRACTIONS. 69 ADDITION. PROBLEMS 1. What is the sum of ^ and ^ ? Ans. f + f =|-. 2. What is the sum of \ and J ? Ans. ^. 3. What is the sum of ^ and ^ ? ^;^5. -J^. 4. What is the sum of \ and ^ ? Ans. \f. CoR. The sum of any two fractions whose numerators are unity, will have for its numerator the sum of its de- nominators, and for its denominator their product. 5. What is the sum of i, J, and ^ ? Ans. |f . 6. What is the sum of ^ \, and ^ ? Ans. J|4. CoR. The sum of any three fractions whose numerators are unity will have for its numerator the sum of the pro- ducts of the denominators, taken two at a time, and for its denominator the product of all the denominators. EXAMPLES. 1. Add f , I, 3^, and ^. Ans. 2f| _ _ 2. Add f, I, I -^, and H- ^^s. 4^^, 3. Add H, if. Ans.liU' 4. Add if, H- ^^^- hh 5. Add f , -j^, ^V, and l-J-. -4w5. ly^. 6. Add I, f , 3^, A. Ans. 2^, 7. Add I, ^, i^, ^V -i^^. IH- 8. Add i, ^, i, i, i, T^^, ^= ^/i5. If =li 9. Add 2^, 3|, 4^, and 5|. ^^5. 16^, 10. Add 12f , 15f, and 25f ^7^s. 54|f. 11. Add i, I, f, and f. Ans. 2^, 12. Add 3^, ^, ii, and if. ^7i5. 1^ 13. Add \, I, and ^. ^/^5. tf|, 14 Add ^ and ^V- ^^5- i^^ 70 FRACTIONS. SUBTEACTIOK PROBLEMS. 1. From f take ^. Ans. ^. 4. From f take f . Ans, ^. 2. From | take f. Ans, ^. 5. From ^ take f . Ans. -^^» 3. From f take |. Ajis. gV ^' Fi'om -^ take f. ^^5. -^^. Cor. Two fractions whose numerators differ by unity, and each denominator is greater by unity than its numer- ator, the fraction having the larger terms is the greater, and their difference has unity for its numerator, and the product of the two denominators for its denominator. EXAMPLES. 1. From f take }. Ans. J. 2. From 1^ take }. Ans. |. 3. From 5| take f Ans. m. 4. From 5i take 2|. Ans. 2J. 5. From 51 take 3f. Ans. 2^V 6. From 5f take 4|-. Ans. 1^. 7. From ^ take 4f . J[^5. 2^. 8. From l^ take 5|. ^7^5. 7^. 9. From 15f take 12|. Ans. 3t^. 10. From 37f take 21f . ^7^5. 16t^. 11. From 45| take 34^. Ans. llyV- 12. From 271 take 15|. ^^5. llf^. 13. From a piece of cloth containing 16f yards, a mer- chant sold 7f yds.; how much was left ? Ans. 8|^ yd. 14. A merchant has 2 pieces of muslin, the one has 41f yards, and the other 43| yd.; he sells 61|-yd.; how much has he left ? Ans. 231^. 15. A man earned $36|, and spent $15| ; how much had he left? Ans. $21^. FB ACTIONS, 71 16. James had $45|, and lost $16|; how much had he left? Ans. $28i-f. 17. Subtract 9^^ from lO^^. Ans. f|}. 18. Subtract 15-| from 25|. Ans. 10-^. 19. Subtract 36f?- from 451^. Ans, 9^^^. 20. Subtract 56| from 97|. Ans. 40^^. 21. Subtract 21f from 35-|. ^^5. l^^. 22. Subtract 105t»j- from 205t^. ^/^5. 99^^. ORAL EXERCISES. Whatisjof f ? ioff? Jof f? iof f ? ioff? | of f?ioff? JofA? One-third of 61s i of what? i? i? i? ^P -J? i? -^i^? One-third of 9 is i of what? i? i? i? 4^? i? i? ^? A? A? Two-thirdsof 9 is^of what? i? i? i? ^? i? i? ^? 1^? 1^^? One-third of 12 is i of what? i? i? i? |? i? i? iV? 1^? 1^? Two-thirdsof 12isiofwhat? J? |? |? ^? ^? j? What part of 5 is 1 ? 2 ? 3 ? 4? Ans. |, |, |, |. What part of b\ is ^ ? H ? 2| ? 3^ ? ^^5. _^, ^3^, ^, ^. What part of o| is i? IJ ? 2^? 3^? Ans. -g^, ^, -f^, f^. What part of 5i is i ? f ? 4? 2^? What part of 100 is 6i ? 13^ ? 25 ? 37^ ? Am. -^y \, \, |. 72 FRACTIONS. What part of 24 hours is 5 h.? 6 h.? 8^ h.? 9^ h.? Ans. ^, I, H, 1^. What part of 1 is ^? ^ ? J? ^ ? Ans. i i, h \. What part of | is ^ ? i ? i ? i? Ans. |, i, i, ^. PROBLEMS. 1. Multiply 1 by 2. 2. xMultiply 1 by 1. 3. Multiply 1 by f 4. Multiply 1 by J. 5. Multiply 1 by |. 6. Multiply ^ by f. 7. Divide 2 by 2. 8. Divide 2 by 1. 9. Divide 2 by i. 10. Divide 2 by J. 11. Divide 2 by |. ' 12. Divide ^ by |. 13. If i cord of wood. cost $2, what will 1 cord cost? Ans. 2xf=$4. 14. If i cord of wood cost $1, what will 1 cord cost ? Ans. lxf=$4. 15. If f cord of wood cost $3, what will 1 cord cost ? Ans. 3x|=S4. 16. Iff ton of coal cost $5, what will 1 ton cost ? Ans. 5x|=$6. 17. If ^ ton of coal cost 17, what wiU 1 ton cost ? Ans. 7xf=$8. 18. If ^ cord of wood cost $2, what will 3 cords cost ? Ans. 2x-|x3=$12. 19. If i cord of wood cost $1, what will 5 cords cost ? A71S. 4xf xf=$20. Ans. 1x2= =3. Ans. 1x1= = 1. Ans. lxi= =i- Ans. lxi= =h Ans. lx|= =f- Ans. ixf= =|. Ans. 2-^2= =1. Ans. 3^1= =3. Ans. 3xf= =4. Ans. 2xt= =8. Ans. 3 X 1 =^= 2|. Ans |-x i=i= =1- f FRACTIONS. 73 20, If f cord of wood cost $3, what will 6 cords cost ? Ans. 3x^x6=124. 21. If 3 cords wood cost $15, what will 8 cords cost ? Ans. 15-r-3x8=$40. CoK. 1. When a given number of articles, each of the same value, cost a certain sum of money, in order to find the cost of one article, divide the certain sum by the given number, whether integral or fractional. CoE. 2. Multiply the cost of one article by any re- quired number of articles, and the product will be the cost of that number. EXAMPLES, 1. If 4^ of a yard of cloth cost 40 cents, what will a yard cost ? what will | of a yard cost ? f ? t^j ? ii ? Ans. 280 cents, 186|- cts., 210 cts., 100 cts., 110 cts. 2. At $9 a barrel, what will f of a barrel of flour cost ? Ans. $6. 3. At $8 a ton, what will f of a ton of coal cost ? Ans. $6. 4. If f of a yard of linen cost 18 cts., what will 1 yard cost? Ans. 45 cents. 5. If f of a yard of linen cost 21 cts., what will 1 yard cost ? Ans. 49 cents. 6. If ^ of a yard of linen cost 36 cts., what will \ yard cost ? Ans. 10 cents. 7. If f of a yard of linen cost 15 cts., what will 1 yard cost? • Ans. 20 cents. 8. If f of a yard of linen cost 24 cts., what will f yard cost ? Ans. dO cents. 9. If f of a yard of cloth cost $f , what will 1 yard cost ? 2 yd.? 3 yd.? 5 yd.? Ans. %1\, $2^, $3|-, $5f. 4 FRACTIONS. 10. If I of a ton of hay cost $7, what will 1 ton cost? gftons? 3|? 7i? 9J? ^/^5. $8, $22, $29, $60, $74. il. What is the value of ix|x|xt-r-fx4x|xf. Ans. ^. 12. What is the number whose factors are \^ and -^ ? Ans. f. 13. The divisor is f and the quotient f ; what is the dividend? Ans, f. 14. In what time will a man make the round of the earth — 24,894 miles— by traveling 33^ miles a day ? Ans. 746f^days. 15. In one mile there are 1,760 yards ; how many yards in 5| miles? 7^ miles? 9| miles ? 12f miles ? A71S. 10,120 yds., 12,540 yds., 16,896 yds., 22,586| yds. 16. In one square mile there are 640 acres; how many acres in 3| sq. m.? 5| sq. m.? 9| sq. m.? 12^ sq. m.? A71S. 2,304 A., 3,680 A., 6,000 A., 7,786f A. 4^17. If f of a ton of coal cost $9, what will 15J tons cost? Ans. $184. /C 18. If f of a ton of coal cost $4, what will 31| tons cost? Ans. $441. 19. Divide f^ by |, and multiply the quotient by 24|. Ans. 14^f. 20. Divide ^ by f|, and then multiply by ^. Ans. \. 21. Eeduce | to a simple fraction. Ans. f|. 22. Eeduce |— -| to a simple fraction. Ans. ffj-. 23. Eeduce | — | to a simple fraction. Ans. 2^. T 6 FRACTIONS. 4 4 24. Eeduce | x f to a simple fraction. Ans. |. 25. Eeduce Tq -7--- to a simple fraction. Ans. f-i. PRACTICAL EXAMPLES. 1. A man owns -j^ of a gold mine, and sells f of his share for $50,000. What is the whole mine worth at that rate? Ans, $400,000. 2. The distance from Baltimore to Philadelphia is 97 miles; A starts from Baltimore to Philadelphia at the same time that B starts from Philadelphia to Baltimore ; A travels 6^ miles per hour, and B 7^ miles. In what time will they meet, and how far will each have traveled ? 3. What number multiplied by | will give a product of 3^? Ans. ^. 4. What number divided by f will give a quotient of ^ ? Ans. f. 5. If a man travel 237^ miles in 21f hours, how far does he travel each hour ? Ans. ll iVg\ . 6. If a traveler perform a journey in 17f hours, by traveling 6f miles in an hour, what is the length of the journey? Ans. 11 If miles. 7. What number multiplied by ^ of f of f of ^ will give a product of 3^ ? Ans. If. 8. What cost 15| yards of cloth at $2| per yard ? Ans. $41^. 9. A man owning f of a farm sold f of his share ; what part did he sell, and what part had he left ? Ans. He sold ^ and had left ■^. 76 FRACTIONS, 10. James is 12 years old, and | of his age is 9 years less than that of his brother; how old is his brother? A71S. 18 years. 11. If 8^ yards of muslin at 9 cents a yard is worth f of a gallon of syrup, what is a gallon of syrup worth ? Ans. 90 cents. 12. If f of a yard of linen at 70 cts. a yard costs as much as ^ of a yard of muslin, what will a yard of muslin cost ? Ans. 25 cents. 13. If a man build f of a rod of wall in a day, how many rods will he build in 11 J days ? Ans. 6|- rods. 14. A and B bought a barrel of flour; A paid $3 J, and B $4| ; what part should each get ? Ans, A If, and B ||. 15. In an orchard ^ of the trees bear apples, J peaches, \ plums, and 2^ are pear trees; how many trees in the orchard? Ans. 120 trees. |<16. One-fourth of a pole is in the mud, \ in the water, and 22 feet above the water; what is the length of the pole ? Ans, 40 feet. 17. What is the difference between f of ^ and i of ^? Ans. -gV. 18. A owns 1^ of a gold mine, and B ^ of the same; A sells f of his share and B ^ of his to C ; what is now the part of each ? Ans. A has \y B -J-, and C ^. 19. A, B, and C purchased 160 acres of land; A is to have i of it, which is | as much as B's, and the balance ; how much is C's part ? Ans. 60 acres. "^20., My horse is worth $150, which is | the value of the Duggy; what is the buggy worth ? Ans. $225. FRACTIONS. '^>i^ 77 21. A can do a piece of work in 4 days and B in 5 days ; how much will A and B do in one day ? How long will it take both to do it ? Ans. They will do ^ in 1 day, and in 2| days the whole. 22. A can do a piece of work in 12 days, and B in 15 days ; how long will it take both to do it ? ^ Ans, 6f days. 23/Si cistern has two pipes for supply and one for dis- charge ; by the first it would be filled in 6 hours, and by the second in 8 hours, and the third would empty it in 4 hours ; in what time will it be filled if all run together ? Ans, In 24 hours. 24. How does multiplying the numerator of a fraction affect its value ? 25. How does diyiding the numerator of a fraction affect its value ? 26. How does multiplying the denominator of a frac- tion affect its value ? 27. How does dividing the denominator of a fraction affect its value ? 28. What effect has multiplying both terms of a frac- tion by the same number ? 29. What effect has dividing both terms of a fraction by the same number ? 30. To what term in division does the numerator cor- respond ? the denominator ? the fraction itself ? 31. A can do a piece of work in 7 days; B in 9 days ; how much will each do in 1 day ? how much will both do in a day, and how long will it take both to do it ? Ans. A will do | and B -| in a day ; both will do ^| in a day, and it will take both ^\\ days. f\A 78 FRACTIONS, 32. A man who had purchased a house, found that after he had paid ^ and \ of the cost he still owed $500 ; what was the cost of the house ? Ans. $3,000. 33. A, B, and C purchase a farm ; A pays ^, B ^, and the balance, which was $4,700 ; how much did A and B each pay ? Ans. A, $3,000, B, $3,300. 34. A and B together have $170, and f of A's money is equal to f of B's ; how much has each ? A'sx f =:B'sx f A'sx-i^=:B'sXtV 8 A's=9 B^s; hence A has $9 as often as B has $8. Ans. A, $90; B, $80. rS5. The sum of two numbers is 56, and | of the first is equal to f of the second ; what are the numbers ? Ans, First, 32; second, 24. 36. The product of two numbers is 33^, and one of the numbers is 5^; what is the other number ? Ans, 6^^. 37. The factors are 25f and 37|; what is the product ? Ans, 969^^. 38. At $5| a ton, how many tons of coal can be bought for $123| ? Ans, 21ff tons. 39. The weight of 3 loads of hay were: 1st, 2 T. 5 cwt. 30 lbs.; the 2d, 3 T. 2 cwt. 25 lbs. and the 3d, 2 T. 8 cwt. 60 lbs.; what did the three loads weigh ? Ans, 7 T. 16 cwt. 15 lbs. or 7.8075, or 7Hf tons. 40. The dividend is 205f and the quotient 25^ ; what is the divisor ? Ans, S-^. 41. My horse is worth 1^ times as much as my car- riage ; both are worth $250 ; what is each worth ? Ans, Horse, $150; carriage, $100. 42. If If acres of land cost $87^, what will 12| acres .cost ? Al71s. $630. 'Decimal Fbactioj^s. Fractions whose denominators are 10, 100, 1000, etc., are rendered decimals of the same name by a little change in form ; thus, a decimal point is placed on the left of the decimals, or on the right of the units, and the same relation exists between the successive orders, as in abstract numbers, but the orders themselves are reversed. T^ = .1, ii)W = -001, tU = .01, ttW = .0001, and are read alike; thus, one tenth, one thousandth, one hundredth, one ten-thousandth. Also, -^ = .3, read three tenths ; yfg- = .07, seven hundredths ; ■^ = .36, thirty-six hundredths ; ^^ = .456, four hundred and fifty-six thousandths. Hence, to enumerate a decimal fraction, read it as you would an integral number, adding to this the name of the denominator, when a common fraction, which will be expressed by 1 with as many zeros attached to it as there are numbers of decimal figures. 80 DECIMAL FRACTIONS. ADDITION AND SUBTKAOTION. EXAM PLES 1. Add .1, .01, .001, .0001, and .00001 ; thus : (1.) (2.) (3.) .1 Add .0234 Add 5.634 .01 .213 21.321 .001 .3146 .654 .0001 M .012 .00001 .6 5.364 .11111 (*•) 1.4710 32.985 (5.) From 4.36315 FVom .326159 Take 1.83754 Take .234573 Rem. 2.53461 Rem. .091586 Cor. — As the relation of the orders are the same, and the decimals rise in value in the same direction, whilst in name they take the opposite direction ; hence, addi- tion and subtraction of decimals are performed as in Integral Numbers. MULTIPLICATION. THEOREM. In the v^ultiplicatio^^ of decimals, the product will have as many places of decimals as hath factoids. t\ X -r^ = tU ••• .1 X .1 = .01, and ^^xi^ = T(5W ••• -01 X .1 = .001. DECIMAL FRACTIONS. 81 1st ool. 2d OOL. 1x1 = 1 and 1 X .1 = .1 .1x1= .1 and .1 X .1 = .01 .01 X 1 = .01 and .01 X .1 = .001 or, The first column of products is the same as the first column of multiplicands, as 1 is the multiplier. The multiplier in the second case is one-tenth, consequently the products of the second column must be one-tenth of the first. Therefore the product of two decimal factors will have as many decimal places as both factors. 1x1 =1, units. .1 X .1 = .01, hundredths. .01 X .01 = .0001, ten thousandths. 1x1 =1, units. \ 10 X 10 z= 100, hundreds. 100 X 100 = 10000, ten thousands. Rem. — Observe the correspondence in name, when the contrary orders are multiplied. PROBLEMS. 1. Multiply 2. Multiply 3.156 .534 .215 .136 15780 3204 3156 1602 . 6312 534 .678540 .072624 Rem. — Each product must have six decimals, hence ir. the second example a zero must be prefixed. 82 DECIMAL FRACTIONS. (3.) (4.) .01 .00001 .01 .00001 .0001 .000000001 DIVISION. Corollaries to Theorem, Page 54. Cor. 1. — As the product of the divisor and quotient is equal to the dividend, therefore the dividend has as many decimal figures as both divisor and quotient. CoR. 2. — If the divisor has decimal figures and the dividend has none, or less than the divisor, as many must be added to the dividend as to make the number equal to that of the divisor, and then the quotient will be inte- gral. If more decimals are added to the dividend, the quotient wiU contain as many. PROBLEMS. 1. Divide 21.4263 l?y 3.12. 3.12 ) 21.42|63 ( 6.86+ As the divisor has two 18 72 places of decimals, the 2 706 quotient will be integral 2 496 for two places of decimals 2103 in the dividend ; after that 1872 the quotient will be deci- 231, remainder. mal. 2. Eeduce the fraction J to a decimal. 4 ) 1.00 3 .25 5 ) 3.0 .6 CoR. — Any common fraction may be reduced to a deci- mal by performing the division indicated by the terms. DeJSTOMIJ^ATE JSf'UMBERS. All arithmetical numbers may be considered Denom^ inate, even abstract numbers, as every figure in each successive order, beginning at the right and going to the left, is ten times the value of the same figure in the pre- vious order, and may be arranged in a table ; thus, 10 units = 1 ten. 10 tens = 1 hundred. 10 hundred = 1 thousand. 10 thousand = 1 ten-thousand. In the United States currency, the orders have the same relation ; thus, 10 mills (m.) =z 1 cent {ct.). 10 cents = 1 dime. 10 dimes = 1 dollar ($). 10 dollars = 1 eagle. Dimes and eagles are coins, but are not regarded in subtraction like orders must be -placed under each other, and in every other way the same methods are followed. PROBLEMS. $25,365 1. What is the sum of twenty-five 12.184 dollars, thirty-six cents and five mills; 9.100 twelve dollars, eighteen cents and four 30.005 mills; nine dollars and ten cents; thirty 15.030 dollars and five mills; fifteen dollars $91,684 ^^^ three cents. Arts., Ninety-one dollars, sixty-eight cents and four mills. 2. Add the following sums of money : Five dollars, thirty cents and four mills. $5,304 Three dollars and two mills .... 3.002 Two dollars and three cents .... 2.030 Seven dollars and three mills . . . . 7.003 Twelve dollars and one cent .... 12.010 Nine dollars 9.000 «3&349 DENOMINATE NUMBERS. 86 Add (3.) $97,548 68.754 97.633 198.564 (4.) Add $386,946 5372.875 64759.654 876943.687 (5.) Add 387,643 milla 548,753 659,864 3,317,634 $463,498 . $947463.163 4,713,893 mills. Rem. 1. — The sum of the Ikst example may be numerated thus : Four millions seven hundred and thirteen thousand, eight hun- dred and ninety-three mills ; or, thus : Four thousand seven hundred and thirteen dollars, eighty-nine cents and three mills. Hem. 2. — Mills are numerated the same as abstract numbers. SUBTEACTIOH. $287,304 1. From two hundred and eighty- 194.293 seven dollars, thirty cents and four $93,011 mills, take one hundred and ninety- four dollars, twenty-nine cents and thvc^ mills. Eejnainder, Ninety-three dollars, one cent an^i 9ne milL (2.) (3.) (4.) $475648.364 $9,486,397,213 $21795.375 387654.875 6,397,423.875 10963.625 $87993.489 $3,088,973,338 $10831.750 (5.) (6.) (7.) (8.) (9.) 100000 100 100 100.00 100.00 99999 99 1 1.50 2.50 1 1 99 98.50 97.50 Rem.— As in addition and subtraction, so also in multiplication, the process is the same as that of abstract integers and decimals ; he«»ie there is no need of further exemplification. 86 DENOMINATE NUMBERS. English money is reckoned in pounds, shillings, pence, and farthings; sometimes also in guineas; thus, TABLE. 4 farthings {far,) = 1 penny (d.). 12 pence = 1 shilling (s.). 20 shillings = 1 pound (£). 21 shillings = 1 guinea. PROBLEMS. Reduce £1 to shillings, pence, and farthings. £1 20 £1 = 20 shillings, 20 = shillings. ^1 = ^^0 pence. 12 £1 = 960 farthings. 240 = pence. 4 960 = farthings. As there are twenty shillings in one pound, there will always be twenty times as many shillings as pounds; and as there are twelve pence in every shilling, there will be twelve times as many pence as shillings; and four times as many farthings as pence. CoE. — A higher denomination is reduced to a lower one by multiplication. Reduce 960 farthings to pence, shillings and pounds; thus, 4 ) 960 farthings. 12 )^40 pence. 20 ) 20 shilHngs. 1 pound. As four farthings make one penny, there will be one- DENOMINATE NUMBERS, 87 fourth as many pence as farthings, one-twelfth as many shillings as pence, and one-twentieth as many pounds as shillings. CoR. — A lower denomination is reduced to a higher one by division. Eeduce 1095 farthings to pence, shillings, and pounds. 4 ) 1095 farthings. 12 ) 273 ... 3 far. 20) 22 . . . 9d. £1 2s. 9d. 3 far. The first remainder is farthings, the second pence, and the third shillings. Eeduce £1 2s. 9d. 3 far. to farthings. 20 22 shillings. 12 273 pence. 4 1095 farthings. ■ In reducing a higher denomination to a lower one, begin by multiplying by the number of the next lower denomination that makes one of the higher, and if it be a compound number, add to the product the number of the lower denomination, and continue this process until you reach the lowest denomination. In reducing a lower to a higher denomination, divide by the number of the lowest denomination that makes one of the next higher, and if there be a remainder, it will be of the lowest denomination, etc. 88 DENOMINATE NUMBERS, Cor. — In the computation of compound numbers, in- stead of carrying a unit to a higher order for every ten, as in abstract numbers, a unit is carried to a higher denomination as often as the sum : -caches the number that it takes of the lower denomination to make one of the next higher denomination ; thus, as 4 farthings make '' 1 penny, as often as the sum of the farthings reaches four, one must be carried to the pence ; and as 12 pence make 1 shilling, in computing pence as many must be carried to shillings as the number of times 12 is contained in the number of pence ; 1 from shillings to pounds for every 20. In division, the order ia reversed, as then we begin with the highest denomination and descend. 1 Add 24 5 11 1 The sum of the first column is 9 farthings, which is 2 times 4 and 1 ; the 1 is farthings, and must be placed under the farthings ; the 2 is carried to the next denom- ination and added with the pence, the sum of which is 35; that is, 2 times 12 and 11, that is, 2 shilhngs and 11 pence ; the 2 is added with the shillings, making the sum 45, which is £2 5s. ; the shillings are placed under the shillings and the 2 carried to the pounds, the sum of which is 24. EXAMPLES. £ ». d. fa/r. 3 8 1 3 5 9 6 3 6 11 9 1 8 15 11 3 Z^ d DENOl^ TIN ATE AUJiBEBS, f ^ ^ £ S. d. far. 54 6 5 1 28 7 6 3 89 2. From £25 18s. lOd. 2 far. As you cannot subtract 3 farthings from 1 farthing, you must borrow 1 penny, which is 4 farthings ; this 4 and the 1 make 5 ; then 3 from 5, 2 remains ; the 1 penny borrowed must be carried to the 6, which makes 7, which cannot be subtracted from 5 ; 1 shilling, that is, 12 pence, must be borrowed and added to the 5, which makes 17 ; 7 from 17, 10 remains ; 1 shilling to carry to 7 makes 8, which cannot be taken from 6 ; 1 pound, that is, 20 shil- lings, must be borrowed and added to the 6, making 26, from which subtract 8 and 18 remains; and £1 to carry to 28, making 29, which is subtracted from 54 and 25 remains. Rem. — When the subtrahend is less than the minuend, the dif- ference can be taken directly. & 8. d. far. 3. Multiply 4 6 5 3 by 5 £31 12s, 4 3 4 6 5 3 _5 ^ 5 5 21 20)32 25 )15 £1 12s. • 3 12)28 3d. 3 2s. 4d. Cob. — Multiply each denominate number, and diyide the product by the number of this denomination that it 90 DENOMINATE NUMBERS, takes to make one of the higher, and carry the number of times it is contained to the higher denomination, and place the remainder under its kind. 4. Multiply £48 12s. 7d. 2 far. by 6. 5. Divide 4 ) £5 6s. 3d. 1 fa r, by 4. £1 6s. 6d. 3J^far. 4 is contained in 5, once and £1 over; this £1 is 20 shillings, which added to the 6 shillings make 26 shillings, into which 4 is contained 6 times and 2 shillings over; this 2 shillings is 24 pence, which added to the 3 pence, makes 27 pence, in which 4 is contained 6 times and 3 pence over, which is 12 farthings, and 1 more make 13. in which 4 is contained 3J times. 6. Divide £754 15s. 9d. 3 far. by 27. 27 ) £754 15s. 9d. 3 far. ( £27 54 214 189 25 20 515 ( 19s. Add the 15s. 27_ 245 243 2 12 33 ( Id. Add the 9d. 27 • 6 _4 27(1 far. Add the 3 far. 27 Quotient = £27 19s. Id. 1 far. DENOMINATE NUMBERS. 91 7. Multiply £5 4s. 6d. 1 far. by 35. 35 £183 18s. 3d. 3 far. 35 35 35 4)35 5 175 4 140 6 310 8d. 3 far. 7 £183 18 30)158(7 8 13)318 140 18s. 3d. 18 Rem. — Observe these solutions carefully j for if they are under- Btood, there is no further difficulty in denominate numbers ; the principle is the same in all, the tables alone differ. EXAM PLES. 1. In 2 dollars, how many cents ? How many mills ? $2x 100= 200 cents. 2 X 1000 = 2000 mills. 2. In 5 dollars, how many cents ? How many mills ? 3. In 7 dollars, how many cents? How many mills? 4. In 5 dollars 15 cents^ how many cents? How many mills ? $5 = 500 cents. 15 515 cents = 5150 mills. 5. In 6 dollars 15 cents and 3 mills, how many mills ? 6. In 500 cents; how many dollars ? fg^ = $5, Ans. 7. In 625 cents, how many dollars and cents ? Ans. 16.25. 92 DENOMINATE NUMBERS. 8. In 5325 mills, how many dollars, cents, and mills ? Ans. $5,325. 9. In 63257 miUs, how many dollars, cents, and mills? 10. In 75325 cents, how many dollars and cents ? 11. If 1 bushel of wheat cost $1,125, what will 8 bushels cost ? 12. If 1 bushel of wheat cost $1.05, what will 10 bushels cost ? 13. If 1 bushel of wheat cost $1.05, what will 100 bushels cost ? 14. If 8 bushels of wheat cost $9, what cost 1 bushel? 15. If 8 bushels of wheat cost $9, what cost 35 bushels ? 16. If 10 bushels of wheat cost $10.50, what cost 53 bushels ? 17. Bought dry goods for $243.37; groceries for $146,294; hardware for $71.96; notions for $21,512. What was the amount of the bill ? Sold the same at a profit of $157,192. What did I sell the whole for ? 18. If 5 lbs. sugar cost 50 cents, what will 6 lbs. cost ? 7 lbs.? 8? 9? 10? 11? 12? 19. If 6 lbs. cost 72 cts., what wiU 7 lbs. cost ? 8 lbs. ? 9? 10? 11? 12? 20. In 15 farthings, how many pence ? Ans. 3| pen 21. In 18 farthings, how many pence? How mg-ny pence in 21 far. ? 23? 25? 27 ? 29 ? 31 ? 33? 34? 35? 22. How many shillings in 25 pence ? in 28 ? 35 ? 38 ? 45? 51? 56? 65? 23. How many pounds in 35 shillings? in 40? 50? 60? 65? 70? 75? 80? 85? 90? 95? 100? 105? 110? 120? DENOMINATE NUMBERS. 93 ^ 24. How many farthings in £9 13s. 9d. 3 far. ? \ 25. How many pounds, shillings, pence, and farthings in 37864321 farthings? ft 26. Multiply £4 8s. 9d. 3 far. by 9. 27. Divide £25 9s. 4d. 1 far. by 13 ? AVOIRDUPOIS, OR COMMERCIAL WEIGHT, is used in commercial transactions, when goods are bought or sold in quantity, and for all metals except gold and silver. * TAJiLE. 16 drams {dr.) = 1 ounce {oz.) 16 ounces * = 1 pound {lb). 25 pounds =: 1 quarter {qr). 4 quarters ~ 1 hundredweight (cwt). 20 cwt. --:= 1 ton (T.). EXEMPLIFICATIOIT. 1 T. 20 — 16 ) 512000 dr. 20 cwt. ■- . 16 ) 32000 oz. — 25 ) 2000 lbs. 80 qrs. ^- 25 ^ )^_ ^^s- 2000 lbs. ^^ )1^_ c^^ 16 1 T. 32000 oz. 16 512000 dr. 94 DENOMINATE NUMBERS. 16 T. cwt. Reduce 3 4 ST. 2 8 OS. dr. 6 10 20 64 4 16)1653354 258 16 ) 103334 ... 10 dr. 25 25)6458 ... 6oz. 6458 16 103334 4)258 . . . 8 1b. 20 ) 64 . . . 2 qr. 3 4 2 8 6 10 T. cwt. qr. lb. oz. dr. 1653354 drams. Kediice 1653354 drams to the original denominations. TEOY WEIGHT is used for gold, silver, and jewels ; also in philosophical experiments. TABLE. 24 grains {gr) = 1 pennyweight {pwt.y 20 pennyweights = 1 ounce. 12 ounces = 1 pound. lib. 12 24 ) 5760 gr. 12 oz. 20 ) 240 pwt 20 12 ) 12 oz. 240 pwt. 1 lb. 24 5760 qr. DENOMINATE NUMBERS. ^5 Eeduce 5 lb. 6 oz. 10 pwt. 16 gr. 5 1b. 6 oz. 10 pwt. 12 66 Add the 6 oz. 20 1330 Add the 10 pwt. 24 31936 Add the 16 gr. Eeduce 31936 grs. to the original denominations. 24 ) 31936 gr. 20 ) 1330 . . . 16 gr. 12)^ . . . 10 pwt. 5 lb. 6 oz. 10 pwt. 16 gr. DIAMOND WEIGHT. Used for diamonds and other precious stones. TABLE, 16 parts = 1 grain = .8 grain Troy. 4 grains = 1 carat =. 3.2 grains Troy. APOTHECAEIES' WEIGHT is used by druggists in putting up prescriptions ; the pound, ounce, and grain are the same as in Troy Weight. TABLE. 20 grains = 1 scruple (3). 3 scruples = 1 dram ( 3 ). 8 drams = 1 ounce ( § ). 12 ounces = 1 pound. 96 DENOMINATE NUMBERS. 1 lb. 13 12 ! 8 96 3 3 288 3 20 20 ) 5760 gr. 3 ) 288 3 8)96 3 12)12 I 1 lb. 5760 gr. APOTHECAEIES FLUID WEIGHT is used for liquids in medical prescriptions. TABLE. 60 minims (m) = 1 fluid dram (f 3 ). 8 fluid drams = 1 fluid ounce (f | ). 16 fluid ounces = 1 pint (0.). 8 pints = 1 gallon {Cong). For ordinary use, 1 teacup = 2 wine glasses = 8 table, spoons = 32 tea-spoons = 4 f | . COMPARISON- OF WEIGHTS. 1 lb. Avoirdupois = 7000 gr. Troy. 1 lb. Troy = 5760 gr. Troy. LIISTEAE MEASUEE is used for lengths and distances. TABLE. 12 inches {in.) = 1 foot {ft). 3 feet = 1 yard {yd.). 5^ yds., or 16^ ft. =1 rod {rd.). 40 rods = 1 furlong {fur.). 8 furlongs = 1 mile {m.). 3 miles = 1 league {lea.). DENOMINATE NUMBERS, 97 MAEINEE^S MEASUKE. 6 feet = 1 fathom. 120 fathoms = 1 cable length. 880 fath., or 7-J^ cable lengths = 1 mile. Rem. — 1 nautical league = 3 equatorial miles = 3.45771 statute miles. 60 equatorial miles = 69.1542 statute miles = 1 equatorial degree (°). 360° = the circumference of a circle. 360 equatorial degrees = the circumference of the earth. CLOTH MEASUKE. i' i •.* 1»'K A CT.I C A L QUE ST IONS. 1. Reduce £3 9s. lid. 3 far. to farthings, 2. Reduce £12 15s. 8d. to pence. 3. Eeduce £7 Os. 2d. to pence. 4. Keduce 2354 farthings to the higher denominations. 5. Reduce 543 pence to the higher denominations. 6. Reduce 731 shillings to the higher denominations. 7. Eeduce 3 T. 6 cwt. 2 qr. 12 lb. 6 oz. and 9 dr. to drams. 8. Reduce 672432 drams to the higher denominations. 9. Reduce 5 lb. 8 oz. 9 pwt. 15 gi\ to grains. 10. Reduce 64324 grains Troy to the higher denomi- nations. 11. Reduce 2 lb. 6 ? 4 3 23 10 gr. to. grains. 12. Reduce 6742 gr., Apothecaries weight, to the higher denominations. 13. Reduce 3 lea. 2 mi* 5 fur. 24 rods 2 yd. 1 ft. 6 in, to inches. 14. Reduce 802456 inches to the higher denomina- tions. 15. Reduce 4 yards 3 qrs. 2 na. and 2 inches to inches. 16. Reduce 5 ells Flemish to inches. 17. Reduce 4 ells English to inches. 18. Reduce 3 ells French to inches. 19. Reduce 4 sq. rods 8 sq.yd. 105 sq. ft. and 112 sq. in. to square inches. 20. Reduce 3 cu. yd. 12 cu. ft. and 1236 cu. in. to cubic inches. 2L A pile of wood is 16 ft. long, 4 ft. high, 16x4x4 and the length of the wood is 4 feet. How 8x4x4 many cords of wood ? DENOMINATE NUMBERS. 103 22. Eeduce 25 gal. 3 qt. 1 pt. and 3 gills to gills. 23. Kediice 9 bu. 3 pk. 4 qt. 1 pt. to pints. 24. How many years, months, and days, from April 15tli, 1842, to June 20tli, 1850 ? yr, mo. 1850 6 1842 4 da. 20 15 8 2 5 25. How many years, months, and days, from October 25th, 1845, to August 18th, 1850? yr. mo. 1850 8 1845 10 da. 18 25 4 9 23 Rem.— In this question, instead of using the eighth and tenth months, some authors prefer calling them 7 months and 9 months ; but if there is an inaccuracy in the months, there is also in the years ; hence if we read the above thus : The one thousand eight hundred and fiftieth year, the 8th month and 25th day, there is no inaccuracy. In computations of time we always take 30 days as a month. 26. The difference in time of two places is 2 hr. 2 min. and 2 sec. ; what is the difference in longitude ? hr. min. sec. 2 2 2 15 30^ 30' 30" Ans. Thirty degrees,. 30 minutes, and 30 seconds. 104 DENOMINATE NUMBERS, 27. If the difference of longitude of two places is 16^^ the difference of time will be one hour; the eastern place will have the latest time. If the difference in longitude is 16° 24' 30", what is the difference in time? 15 ) 16° 24' 30" ( 1 hr. 5 min. 38 sec. 15 10 cts. = $yV 1 12^ cts. = li. eo 16f cts. = ^. - 20 cts. = .- ^^ ^ 25 cts. = ! — 33icts. = !„ ^ 37icts. = $|. > -^ 50 cts. = ^, 570(38 62icts. = 1 45__ 66f cts. = $|. 120 75 cts. = \ 120 87icts. = \ 28. Multiply 576 by 100 = 57600. 29. Multiply 576 by 25 = i == 14400. Take i of the above. 30. Divide 576 by 100 = 5.76. 31. Divide 576 by 25 = 23.04. Multiply by 4. 32. Multiply 576 by 50 = ^ (57600) = 28800. 33. Divide 576 by 50 = 5.76 x 2 = 11.52. 34. Multiply 576 by .12^ = ^ = $72. 35. Multiply 576 by .16f = \ = $96. 36. Multiply 576 by .33| = 1 = 8192. 37. Multiply 576 by .62^ = 576 x 5-^8 = 360. 38. Multiply 576 by .87^ = 576 x 7-T-8 — 504. 39. .What cost 342 yds. muslin at 8 )_342 12J cts. per yard ? "~^42|= $42f. DENOMINATE NUMBERS. 105 40. Wliat cost 342 yds. linen at 37^ cts. per yard ? Multiply by 3 = $128^. 41. What cost 342 yds. linen at 62^ cts. per yard? Multiply by 5 = $213f . 42. What cost 342 yds. linen at 87|^ cts. per yard? Multiply by 7 = $299i. 43. What cost 548 yds. muslin at 16f cts. 6 )_548 per yard ? ~~$9l-J- 44. What cost 345 yds. muslin at 20 cts. 5 ) 345 per yard ? $69' 45. What cost 469 yds. linen at 33| cts. 3 ) 469 per yard ? "IT56-J 46. What cost 469 yds. linen at 66| cts. per yard ? Multiply by 2 == $312|. 47. What cost 500 yds. linen at 25 cts. per yard? 500 -^ 4 = $125. 48. What cost 500 yds. linen at 75 cts. per yard? Multiply by 3 = $375. 49. What cost 500 yds. linen at 50 cts. per yard? 500 -^ 2 = $250. 50. Bought 648 yards muslin at 12^ cts. a yard, and sold it at 16f cts. per yard. What was the profit ? ■J- of 648 = $108 I of 648 = 81 $27/Profit. 51. Bought 500 yds. clolli at 20 cts., and sold it at 25 cts. ; what profit ? i of 500 == $125 i of 500 = _12? $25, Profit. 106 DENOMINATE NUMBERS, 52. Bought 480 yds. cloth at 66-| cts., and sold, it at 87^ cts. ; what was the profit ? \ of 480 = $60; | = $420 I of 480 = 160; | = ^ $100, Profit. 53. Bought 480 yds. at 37|- cts., and sold it for 50 cts. per yard ; what was the profit ? i of 480 = $240 ^ of 480 = 60 ; | = _180 $60, Profit. 54. Bought 600 yds. cloth at $1 per yard, and sold it for $1.25 per yard; what was the profit ? 600 X li = $750 600 X 1 = 600 $150, Profit. 55. How many yards of cloth, at 12^. cts. per yard, can be bought for $240 ? 8 yds. can be bought for every dollar. 240 X 8 = 1920 yds. 56. How many yards at 16| cts. ? 20 cts. ? 25 cts. ? 37i cts. ? 50 cts. ? 62| cts. ? 75 cts. ? 87^ cts. ? 37i^ I; ^^0x1 = 640yds.; for 62^ cts. =. 240 x f . 57. Sold 500 barrels flour at $6.62^ per barrel, and invested the proceeds in different kinds of dry goods, averaging 87|^ cts. per yard. What were the proceeds of the flour, and how many yards of goods did I get ? DENOMINATE NUMBERS. 107 58. How many years, months, and days from January 21st, 1874, to AprU 30th, 1876 ? Ans. 2 y. 3 m. 9 days. 59. From June 18th,- 1865, to April 15th, 1869 ? Ans. 3y. 9 m. 27 d. 60. Prom Dec. 30th, 1872, to Jan'y 5th, 1875 ? Ans. 2y. m. 5 d. 61. The difference in longitude of two places is 15°; what is the difference in time ? Ans. 1 hour. 6^. The difference in longitude of N"ew York and Cin- cinnati is 10° 35'; what is the difference in time? Ans. 42 min. 20 sec. 63. The difference in longitude of New York and St Louis is 16° 14'; what is the difference in time ? Ans. 1 h. 4 min. 56 sec. 64. The difference in longitude of Philadelphia and Cincinnati is 9° 20' ; what is the difference in time ? Ans, 37 min. 20 sec. 65. The difference in the time of London and Wash- ington is 5 h. 8 m. 4 sec. ; what is the difference in lon- gitude ? Ans. 77° 1'. 66. When it is noon at New York, what is the time 15° east of New York ? 15° west ? Ans. 1 P.M. and 11 A.M. 67. When it is noon at Cincinnati, what is the time at New York ? A7is. 12 h. 42 min. 20 sec. P.M. 68. When it is noon at New York, what is the time at Cincinnati ? Ans, 11 h. 17 min. 40 sec. A.M. 69. When it is noon at New York, what is the time at St. Louis ? Ans, 1 h. 4 min. 56 sec. P.M. 70. When it is noon at St. Louis, what is the time at New York ? Ans. 10 h. 55 min. 4 sec. A.M. 108 DENOMINATE NUMBERS. 71. When it is noon at Philadelpliia, what is the time at Cincinnati? Ayis. 11 h. 22 min. 40 sec. A.M. 72. When it is noon at Cincinnati, what is the time at Philadelphia ? Ans. 12 h. 37 min. 20 sec. P.M. 73. When it is noon at Washington, what is the time at London ? Ans, 5 h. 8 min. 4 sec. P.M. 74. Reduce 4 shil. 10 pence and 2 far. to the decimal of a <£. Ans. £.24375. Cor. Reduce the compound number to the lowest given denomination, which will be the numerator of a common fraction, the denominator of which must be the number of the lowest denomination that makes a unit of the one to which it is to be reduced ; then perform the division indicated by the fraction, adding decimal zeros to the numerator. Rem. In general, I prefer the common fraction to the decimal. 75. Express £5 12s. 4d. 3 far. in pounds and decimals. Ans. £5.61979f 76. Reduce the qrs. and lbs. to the decimal of cwi; 6 cwt. 2 qrs. 10 lbs. Ans. 6.6 cwt. 77. Reduce 57 lbs. to the decimal of a cwt. Ans. .57 cwt. Rem. Pounds are as readily reduced to the decimal of cwt. as pounxis are hundredths. 78. Reduce £12 9s. 4d. 3 far. to farthings. Ans. 11,971 far. 79. Reduce £27 14s. 8d. 1 far. to farthings. Ans. 26,625 far. 80. Reduce £8 12s. 9d. to pence. ' Ans. 2,073d. 81. Reduce £9 8s. to shillings. Ans. 188s. DENOMINATE NUMBERS. 109 82. Eeduce 24,862 far. to the higher denominations. Ans. £25 17s, lid. 2 far. 83. Reduce 3,684d. to the higher denominations. Ans. £15 7s. 84. Add £3 10s. 8d. 2 far., £12 7s. 8d. 3 far., and £7 4s. 2d. Ans. £23 2s. 7d. 1 far. 85. From £12 8s. 4d. 1 far. take £7 9s. 5d. 3 far. Ans. £4 18s. lOd. 2 far. 86. Multiply £3 5s. 7d. 3 far. by 5. A71S. £16 8s. 2d. 3 far. 87. Multiply £8 6s. 8d. 2 far. by 12. Ans. £100 Os. 6d. 88. Divide £24 8s. 9d. by 12. Ans. £2 Os. 8d. 3 far. 89. Divide £46 8s. 6d. by 9. Ans. £5 3s. 2d. 90. Eeduce 12s. 8d. 2 far. to the fraction of a £. Ans. f|. 91. Eeduce 7s. 3d. to the decimal of a £. Ans. ff^. 92. Eeduce 2 qrs. 15 lbs. to the fraction and to the decimal of a cwt. Ans. ^ and .65. 93. Eeduce 12 oz. avoirdupois to the fraction and to the decimal of a cwt. Ans. -^^-^ and .0075. 94. Eeduce 8 oz. Troy to the fraction and to the deci- mal of a lb. Ans. | and .666f. 95. 12 cents is what part of one dollar ? Ans. $^or$.12. 96. 12 lbs. is what part of a cwt.? Ans. -^ cwt. or .12 cwt. 97. 3 qrs. and 7 lbs. are what part of a cwt. ? Ans. fj- cwt. or .82 cwt. 98. If chestnuts are worth 11.60 a bushel, what is a quart worth ? Ans. h cents. 110 DENOMINATE NUMBERS. 99. How many pint bottles can be filled from 31| gaL cider? Arts. 252 bottles. 100. What is the value of a pile of wood 32 ft. long, 12 feet high, and 4 feet wide, at $5 a cord ? Ans, $60. 101. If a man earn $9.25 a week, how much will he earn in 5 weeks ? Arts, $46.25. 102. A man earned $95.37^, and spent $34.87^ ; how much had he left ? Ans, $60.50. 103. Bought 37f yards of cloth at $5.25 per yard ; what was the cost ? Ans. $198.18|-. 104. What is the cost of 257| acres of land at $37.45 per acre? Ans. $9,652. 73f. 105. What is the cost of 542:^ bushels of wheat at $1.12^ per bushel? Ans. $610. 03|. 106. What is the product of ^ and .625 ? ■^ns. ^, or .0925. 107. What part of 5 gals. 3 qts. 1 pt. is 3 gal. 2 qt. 1 pt.? Ans, 1^. Def. The reciprocal of a number is formed by invert- ing the number thus: 2, 3, 4 may be written f , |, \', their reciprocals are |, |, ^; so also of any fraction — f, f, I, the reciprocals |, -f, |. 108. What is the value of any number multiplied by its reciprocal? Ans, 1. 109. How much alloy must be mixed with 2 lb. 8 oz. pure gold to make it 18 carats fine ? Pure gold is 24 carats fine. Ans. 10 oz. 13 pwt. 8 gr. 110. If 4 equal angles are made at the center of a circle, how many degrees in each ? Ans, 90 degrees. These angles are termed right angles. 37- Ratio. Two fractions can be formed with any two integral numbers, the one a proper fraction and the other an improper fraction ; thus, ^ and \ can be formed with 7 and 9. When the proper fraction is a multiplier of any number, the product is less than the number multiplied; therefore, this fraction is termed a JDifninishing Ratio, But when the improper fraction is a multiplier, the product is greater than the multiplicand; hence, the Improper fraction is termed an Increasing Ratio. PROBLEMS. 1. If 5 lbs. sugar cost 50 cents, what will 9 lbs. cost ? It is evident that 9 lbs. will cost more than 5, and just as much more as is indicated by the increasing ratio formed by the two like terms, 5 lbs. and 9 lbs. 10 If 5 lbs. cost 50 cts., 9 lbs. will cost ^0 cts. x f = 90 cts. ; this may be further demonstrated thus, 5 lbs. = 50 cts. 1 lb. = 10 cts. 9 lbs. = 90 cts. In a problem of ratios, the one ratio is given, and one of the terms of the other ratio, to get the second term ; thus, in the above : 112 RATIO. Given, 5 lbs. sugar and 50 cents. Kequired, 9 ibs. sugar and ? The ratio of the money will be the same as of the sugar. As the required sugar is more than the given, the ratio must be increasing ; that is, |. .*. 50 x -^ = 90, the ratio of the required money to the given, \^=.\, the same as of the sugar. EXAMPLES. 1. If 5 bushels of wheat cost $6.25, what will 8 bushels cost? Given 5 bu. and $6.25. 'Eequired, 8 bu. and %^.U X I = $10.00. 8 __25)1000 __ 5)40 __ 8 5 "" 625 ~" 25 ~ 5* The ratios of the wheat and of the money is the same. Rem. — Ratios can only be formed by two like terms. 2. If 5 bushels of oats cost $1.50, what will 21 bushels cost? ^ z=i fll = ^; the ratio is the same. Given 5 bu. and $1.50. Eequired, 21 bu. and $1.^0 X -^ = $6.30. Rem. — ^Write the given terms in a line and the like term of the required immediately under that of the given. One term of the required is wanting, and the given like term may be called the term of demand, and should be placed first and multiplied by the ratio, having for its numerator the required term of the ratio, and for its denominator the given term. RATIO, 113 As a general thing, an increase in the required term of the ratio will take more of the unknown to accomplish it ; an increased amount of goods will cost a greater sum ef money ; an enlarged piece of work, an additional sum of money ; and the greater the work, the longer time to perform it, etc. In examples of this kind, the ratios are direct, and the required term of the ratio holds the place of the numerator and the given term that of the denomi- nator, and the product of the ratio and the odd given term is the term required. 3. If a man travel 40 miles in 8 hours, how many miles will he travel at that rate in 18 hours ? Given 40 miles and 8 hours. Eequired, ? miles and 18 hours. 6 ^0 miles X J^ == 90 miles. 4. If 15 bushels of wheat yield 3 barrels of flour, how many bushels will yield 10 barrels of flour? Given 15 bu. and 3 barrels of flour. Kequired, ? and 10 barrels of flour. SOLUTION. 6 -10 X -^ = 50 bushels. 5. If a man travel 30 miles in 2 days, how long will it take him to travel 240 miles ? Given 30 miles and 2 days. Eequired, 240 miles and ? 6. If a staff 4 feet long cast a shadow 3 feet, what is the height of a steeple which casts a shadow 90 feet? STAFF. SHADOW. Given 4 ft. 3 ft. Eequired, ? 90 ft. 114 RATIO, 7. If the interest of $100 for one year is $5, what would be the interest of $500 for the same time ? PRmCIPAL. INTEREI Given $100 and $5. Required, "$500 and V 8. If f of a barrel of flour cost $4, what will 4| barrels cost ? Given f barrel and $4. Eequired 4f barrel and ? 7 Rem. 4| is a multiplier, and f is a divisor ; the | must be inverted. 9. If 4| bushels of wheat cost $5.40, what will 8| bu. cost? 9ibu.? 23i? 31f? 47^? 39^? 58f? 97^? 106|? 10. If 8 bushels of wheat cost $10, what will be the cost of ^ bu. ? lOi bu. ? 15| bu. ? 37^ bu. ? 95f bu. ? 125f bu. ? 150| bu. ? 279f bu. ? ,^ 11. If 5| acres of land cost $280, what is the cost of 6i acres? 7| acres? 12^ acres? 13^ acres ? 17|?18|? 19|? 20|? 37|? 49|? 12. If 2\ acres of land cost $110, what will | of an acre cost? |acre? J acre? \ acre? | acre? -f acre? f acre ? 1-^ acres ? 1 1 acres ? 13. If -^7^ of a yard of cloth cost y^^^ of a dollar, what will "I of a yard cost ? Given -^^ yd. and %^. Eequired, f yd. and ? tV X f X ^. BATIO. 115 14. If I yard of cloth cost $2, what will 3 ells Fl. cost? 2 x| = 15. If f yard of cloth cost $2.25, what will 5 ells English cost ? 2.25 X ^. What will 5 ells French cost ? In the preceding examples, the ratios were all direct; as in those cases any increase in the required term of the ratio demanded a similar increase of the unknown ; but . there are cases which require the ratio to be inverted, such as, the more men employed, the less time will be required to perform a piece of work; the more hours employed in the day, the less days ; the wider the ma- terial, the less yards it will take to make a garment, etc. These cases of inverse ratio are readily detected by asking this question : " Will an increase of the required term of the ratio demand an increase in the unknown term ?" If it does, the ratio is direct; but if an increase in the required term of the ratio demand a diminution of the unknown term, the ratio must be inverted ; thus, PROBLEM. If 4 men can do a piece of work in 10 days, how long will it take 8 men to do the same work ? Given 4 men in 10 days. Required, 8 men in ? 5 ^0 days X I =: 5 days. 116 RATIO. Cor.— It is evident that 8 men will do it in less time ; that is, in one-half the time that it will take 4, which ratio is expressed by the diminishing ratio of 4 and 8, that is, I = 1^, in which the given term is the numerator of the ratio and the required term the denominator. EXAMPLES. 1. If 5 men can dig a ditch in 20 days, how many men will dig it in 25 days ? Given 5 men and 20 days. Eequired, ? and 25 days. 4 $ men x |f = 4 men. Ik Hem. — An increase in the d&js will require less men. 2. If 6 horses eat a certain quantity of hay in 30 weeks, how many horses will consume the same quantity of hay in 9 weeks ? Given 6 horses and 30 weeks. Eequired, ? and 9 weeks. 2 10 horses x ^ = 20 horses. 3. K a man perform a journey in 12 days, when the days are 9 hours long, how many days of 12 hours will it take him ? Given 12 days and 9 hours. Eequired, ? and 12 hours. RATIO. 117 4. How many yards of lining | yd. wide will it take to line 3 yards of cloth f yd. wide ? Given f yard wide and 3 yards long. Eequired, f yard wide and ? $ X I = 6 yds. COMPOUND KATIO. When there are two or more ratios, it is termed Compound Ratio ; thus, If 3 men in 12 days build 40 rods of wall, how many rods will 9 men build in 24 days ? Given 3 men, 12 days, 40 rods. 9 men, 24 days, ? 40 X I X fl =: 240 rods. 4 Rem. — Each ratio is direct. If 12 men dig a ditch 20 rods long in 18 days by work- ing 8 hours a day, how many men will dig a ditch 40 rods long in 24 days, working 6 hours a day ? Given 12 men, 20 rods, 18 days, 8 hours. ? 40 rods, 24 days, 6 hours. 2 ^ 12 men x||xi|x| = 24 men. Exemplification.— The longer the trench, the more men it will take, and the ratio is direct ; but the greater the number of days and the more hours of each day, the less men would be re- quired ; hence these two ratios are inverse. Cor. — Each ratio must be dealt with as in the pre- ceding article. 118 RATIO, EXAMPLES. 1. If 3 men in 8 days of 9 hours each build a wall 20 ft. long, 2 ft. thick, and 4 ft. high, in how many days of 8 hours each will 12 men build a wall 100 ft. long, 3 ft. wide, and 6 ft. high ? Given 3 men, 8 days, 9 hrs., 20 ft. 1., 2 ft. w., 4 ft. h. 12 men, ? 8 hrs., 100 ft. 1., 3 ft. w., 6 ft. h. $ days X A X I X -W X I X } = ^^^^^/^ = W = ^^^ days. Bem. — The men and hours are inverse, the other ratios direct. > 2. If 6 men mow 12 acres of grass in 2 days of 10 hrs. each, how many hours a day must 8 men work to mow 40 acres in 4 days ? Ans, 12|^ hours a day. I- 3. If 6 horses eat 36 bushels of oats in 18 days, how many bushels will be sufficient for 12 horses 24 days ? A71S. 96 bushels. 4. If $100 in 12 months gain $6, how long will it take $500 to gain $15 ? Given Eequired, $100 $500 prin., prin.. 12 mo., ? $6 int. $15 int. l^mo, .xi x4 = = 6 months. 5. If 6 men manufacture 300 pairs of shoes in 30 days, how many men will make 900 pairs of shoes in 60 days ? Ans, 9 men. Percentage. Pev Cent, means per hundred, and is generally ex- pressed fractionally; thus, 5 per cent., 6 per cent., marked b% and 6^, is expressed y^, yg^, etc., or .05, .06 ; thus, yfo of 100 = 100 X y|^ = 5, and -^ of 100 is 6. EXAMPLES. 1. What is b% of 200 ? ^00 x ,1^ = 10, Ans. What is 5^ of 300 ? Arts. 15. What is b% of 400 ? Ans. 20. 2. What is 5^ of 245 ? 2.45 X yf^ = 12.25, Ans. The 100 is canceled in the 245 by pointing off two places of decimals. 3 3. What is %% of 300 ? $00 x li^ = 18, Ans. What is 6^ of 400 ? Ans. 24. What is %% of 500 ? Ans. 30. 4. What is 6^ of 368 ? 3.68 X yf^ = 22.08. COMMISSIOlSr, OR BEOKEKAGE. The business of a commission merchant or broker is to make purchases and sales, on which he receives a per- centage. PROBLEM I. A purchase of $100 worth of goods, at 1% commission, will cost $101 ; that is, \^ of the amount of the purchase. 120 PERCENTAGE, PROBLEM II. In a sale of goods for $100, at 1% commission, the owner will realize $99 ; that is, -^ of the amount of sale. 'PROBLEM III. When stocks, bonds, drafts, or currency, are purchased at a discount of 2%, the cost of $100 worth will be $98; that is, -^ of the face of the bond, etc. ; but when they are purchased at a premium of 2^, the cost of $100 worth is $102 ; that is, \^ of the face. PROBLEM IV. In the exchange of currency, when there is a premium on the funds on hand, as that of English money to be exchanged into United States, the premium in favor of England is about 9^ ; it is computed as follows : Eng. £ X ^^ X i^ = $ U. S., and $ TJ. S. X A X iU = ^ Eng. that is, England gets $109 for every $100 of her money, and the United States must pay $109 of her money for $100 English money. Rem. — This is according to the old exchange value. Now, however, the exchange value of £1 is fixed at $4.86. EXAMPLES. 1. A broker sold goods to the amount of $6000, at 2% commission, and invested the balance of the proceeds, after deducting 2% on the amount of purchase ; what was the owner's portion of the sale, and what amount of goods were purchased ? PERCENT A GE. 121 $6000 Xt^ = $5880 — owner's portion of the sale. $5880 x|if = $5764.7011 ^ amt. of goods purchased. .-. 6000x4\xMI = 6000xif 51 = $576441-, amt. of goods purchased. Rem. — Observe the difference in the ratios of the sale and purchase. ^. What is the cost of a bond for $5000 at h% discount, stocks whose face indicate $2000 at \.% premium, $1000 currency at 2^ discount, and $3000 gold at 8^ premium ? $5000 X ^^ = $4750 2000 X itt = 2080 1000 X ^ = 980 3000 X HI = 3240 $11050 Cor. — When brokerage is paid in the exchange of money, the percentage is on the amt. purchased, which, if the rate is 2%, is |f| of the funds on hand. 3. If a broker makes sales to the amount of $500, on which he receives S%, what is his commission ? 5 2. What is the cost of a draft for $1000 at a premium $1000 X « X ^^^ =^ $1005. 3. What is the face of a draft at i% premium, costing 11005 ? 122 PERCENTAGE, 4. A broker makes sales for $4325, at 2^ ; what is the brokerage, and what does the owner reahze ? 43.25 X xt^ = $86.50, commission. 43.25 X ^^ = $4238.50, owner reahzed. 5. A merchant sells to a broker $3275 uncurrent funds at 5% discount ; what does he realize ? 163.75 X 19 $$M-$x ^\\ = $3111.25. 6. An architect charges 1^% for plans and specifica- tions, and 2^% for superintending a building, the cost of .which is $10000 ; what is the architect's fees ? Ans, $400. 7. A broker has 2% commission, and 3% for guarantee- ing payment ; what does he receive on sales amounting to $42325? A71S. $2116.25. 8. I sent my broker $4000 to purchase goods; what amount of goods did he purchase after deducting com- missions at 2% on the amount of goods? $4000 x fgf. For every $102 he gets $100 worth of goods. 9. Bought a draft on New York, the face of it $500 premium, i% ; what is the whole cost and the premium ? «» - S X ^ = *^-^ 500.00 Premium, $1.25 10. Sold goods to the amount of $4444, and invested the proceeds, after retaining my commissions, which were 2% on the sales, and 1% on the investment; what was the amt. of investment ? • $MM X ^Ax^ = $4312. PERCENTAGE. 123 INTEREST. Interest is an allowance for the use of money. It is reckoned by percentage; thus, b%, 6%, etc., meaning for a year, when not otherwise expressed; for any other time it is as the ratio of the time ; thus, the interest of $100 at 6% is $6 for a year, for two years $12, and for six months $3. PROBLEMS. 1. Find the interest of $150, at 6%, for 1 year, $150 X tU =:= $d. For 8 months. $1.50 X tI^ X A = 1.50 X ^ = $6.00. For 6 months. $1.50 X ^ X A = $4.50. For 14 months. 7 $1.50 X tI^ X M = 1-50 X xfe = 110.50. Cor. — At 6%, the rate per cent, for any number of months is ^ the number of months; thus, for 8 montb« it is 4^, for 6 months it is 3%, and for 14 months 7%. 2. Find the interest of $150, at 6%, for 129 days ? $150 X T^ X ilt = 1150 X i^ = $3,225. 60 CoR. — The interest of a sum of money for any number of days is equal to the product of the sum of money and 124 PERCENTAGE. the number of days divided by 6000; or, if the number of dollars be multiplied by the number of days and this product divided by 6, the quotient is the interest in mills ; point off three decimals and it is reduced to dol- lars, cents, and mills. If the rate of interest is 7^, add \ ; if 8^, add -J- ; if 9^ *^add ^ ; if 5^, deduct -J-; if 4^, deduct ^ ; if 3^, take ^. The rate for 200 months is 100^ ; that is, the interest is equal to the principal. 200 months of $100 is $100. 20 months of $100 is $10. 2 months of $100 is $1. 30 days, or 1 month $0.50. 3 days of $100 is $.05. 1 day of $100 is $.01f . 2 days of $100 is $.03-^. EXAM PLES. 1. Find the interest of $625, at 6^, for 8 months. Rate for 8 mo. is y^. $6.25 X T*^ = $25.00. For 8 mo. and 20 days, 8| mo., rate 4t\%. $625 X ^^ = ml X ^ = $27.08^. If months and days are computed separately. $625 X tU = $25.0* $625 X ^h = _^3 $27. 08^ PER CENT A GE, 125 2. What is the interest of $650, at 6^, for 1 year 6 months and 24 days ? At 7^ ? At %% ? Eeduce to days 1 year = 360 Eeduce to days | year = 180 24 564 $650 X 'KB.U^ — $61.10 = int. at 6^. 1 000 iofl8|| = H^% = 9, .4^. $650 X loJo — $61.10 = int. at e%. At 6^ .= : $61.10 At 6^ = $61.10 Addi = : 10.18i Addi = 20.361 Mt% = : $71.28^ At 8^ = $81.46| 3. What is the interest of $575, at 6%, for 2 years 4 months and 18 days ? 2 years = 720 days. 4 months 120 days. 18 days. 858 days. 143 $575 X ^ = 575 X ^-^ = $82,225. 1000 -V- 4. A note due April 1st, 1872, for $2000, bearing Interest at 6%. On the back of the note were the follow- ing credits: May 1st, 1873, $230; June 1st, 1874, $223.50 ; July 1st, 1875, $217. What was due July 1st, 1877, when the note was taken up ? Ans. $1904. First credit $230. Second credit $223.50. Third credit $217. v j 126 PER CENTA G^. 5. A note dated 1st April, 1870, for $200, bearing interest from date at Q%, has the following endorsements on the back of it. May 1st, 1871, paid 812. May 1st, 1872, paid 112. May 1st, 1873, paid $12. What was due on the note April 1st, 1875, when it was paid ? Int. of $200 for 1 year is $12 5 years. $60 Add Principal, 200 $260 Deduct sum of payments, 36 April 1st, 1875, balance, $224 paid. As the sum of the payments never equaled the interest, no computation need be made until the end, when the sum of the payments must be deducted. Pkob. 1. — In the case of Partial Payments, when the payment is greater than the interest until the time of the payment, the interest is computed until the date of the payment and added to the principal, and from this sum the payment is deducted, and the balance is regarded as the principal of the note. Peob. 2. — When the payment is less than the interest, no computation is made ; but whenever the sum of the payments is greater than the interest until that time, then the interest is to be computed to the date and added to the principal, and from this sum the sum of all the PER CENT A GE. 127 payments is deducted, and the balance regarded as the principal of the note. These computations are to be repeated until the note is paid, BANK DISCOUNT. Sanh Discount is reckoned the same as interest on the face of the note and for the time the note is given, plus three days, which are called days of grace, and the note need not be paid until the last day of grace, three days after the time specified in the note. It is called Discount, because the borrower does not receive the sum specified in the note, but the difference of this sum and the interest. Notes in Banks are usually given for a short time, viz., for 30 days, 60 days, or 90 days; and the interest is com- puted for 33 days, 63 days, or 93 days. 1. The bank discount on a note for $100 at 30 days. $100 X ^ X ^% = ii = $.55. 60 Borrower gets $100 — $.55 = $99.45. 2. Borrowergets$100— $1.05 = $98.95. $100 for 60 da. W00 X tIis X ^«A = li = $1.05. 60 3. Borrower gets $100— 1.55 rr $98.45. $100 for 90 days. $100 x^x i^ = ^ = $1.55. 60 4. Borrower gets $324— $5.02 =: $318.98. $324 for 90 days.^ '^ 16 2 31 UU X ^11^ = $5,022. ^ 128 PERCENTAGE, TRUE DISCOUISTT is the abatement made on a note not yet due, or the difference between the face of the note and a sum of money which, placed at interest,' will amount to the face of the note by the time the note is due ; thus, the present value of a note for $100, due in one year without interest, when money is worth 6^, is such a sum as will amount to $100 in one year at Q% interest. CoR.— As $100 now will be worth $106 at the end of the year, so the present worth of money due in one year is \l^ of the face of the note due in one year. CoR.— The ratio to obtain the present worth from the face of the note, has for its numerator 100, and the denominator is 100 increased by the interest for the time and rate. EXAMPLES. 1. What is the present worth of a note for $324, du© in one year, rate of money Q% ? $324 X iif = $305.66 + , present worth. $18.34, discount. 2. Due in 2 years 1 month. 100 Bate = 9 months = 100 104i 200 225 " 8 " 9 200 209 PER CENT A GE, 129 EXCHANGE. Def. 1. — Exchange is the system by which pay- ments are made at a distant place by means of Bills of Exchange or Drafts. 2. The person making or signing the Bill or Draft, is called the Maker or Drawer ; the one to whom it is addressed is the Drawee^ and the person to whom it is ordered to be paid is the Payee* 3. The person in possession of the draft is the Molder, and if he endorse it, he becomes responsible for the payment, unless otherwise specified. 4. Exchange between the different cities of one's own country is Domestic^ and that with a foreign country is Foreign Exchange. DOMESTIC EXCHANGE. PROBLEM. To find the cost of a draft on Philadelphia or New York when at a premium, and also when at a discount ; thus, if the premium is 1%, the draft will cost ^^ of its face ; if at a discount of 1%, it will cost: -^ of its face. Rem. — The face of the draft must be equal to the sum of money tliat we wish to remit. 130 PER CENT A GE. EXAM PLES. 1. What is the cost of a draft on Philadelphia for $500, at \% premium ? 500 X ^ = 500 X IM = ^^¥^ = 502i. 2. I owe $3000 in New York, and the premium on a New York draft is \%. What must I pay for the draft ? $$000 x^ = tM = lSx^=: $3007.50. 3. What is the cost of a draft on New Orleans for $500, at 1^ discount? 500 X II = 497i. Rem. — The computation may be made as interest ; thus, first example, $500 at 1 % is $5, and \ of $5 is $3J = premium ; second example, $3000 at 1^ is $30, at J% it is $7.50 premium ; in the third example, $500 at J% is $2.50, the discount. 5. What is the cost of a draft, at 60 days, on New York for $500, premium \%y interest off at 6% ? Discount = %%% = U%. Premium = i% = l%%. Discount above premium = ^%. X ^^ = $2.75, discount. 4 $500 2.75 $497.25, cost of draft. t i^'- PERCENTAGE. 131 FOREIGN EXCHANGE. EXAMPLES. 1. What is the cost in New York of a draft on London for £500, exchange at $4.80 to the £ ? £500 X 4.80 = 12400. 2. What amount of debt in London can be paid with $4000 in New York, rate of exchange as above ? ^ = -^ = £833 6s. 8d. 3. What will a draft on London for £480 15s. 6d. cost in New York, rate as above? 15s. 6d. 12 ■rfo = th £480|^ X 4.80 = $2307.72. 4. What will $500 in New York pay in Pans, exchange on London as above, and £1 = 25.2 Francs. X ^^- = 125 X 21 = 2625 Francs. 5. What sum in New York will pay 2625 Francs in Paris ? 6. What will $2540 in New York pay in Paris ? as above. 7. What sum in New York will pay 52500 Francs in Paris, exchange at 25.2 Francs to the £ and £1 = $4.80 p Rem. — As the rates of exchange are not constant, it is not proper to insert a table. AvERAGij^G Accou:n'ts. When sales are made at different times and on different terms, to find a mean time when the whole may be paid without loss to either party ; thus, A merchant sells goods as follows : January 1st for $100 on 1 month. January 18th for $200 on 1 month. February 1st for $300 on 2 months. * February 12th for $250 on 3 months. No payment has yet been made. In how many days should the whole be paid, in order that there be no lossi to either party ? Begin when the first account is due, February 1st. $100 X = 200 X 17 zzz: 3400 300 X 59 z=: 17700 ^-'250 X 100 = 25000 850 850 ) 46100 ( 4250 3600 3400 54 days. 300 = 1^ = ^. Due 54 days after February Isfc, that is, 27th March. AVERAGING ACCOUNTS, 133 BANK ACCOUNT. * In bank, accounts are closed at the end of the year, and the balance, which is always in favor of the depositor, is brought to the credit side of the account, as over-draw- ing is not permitted. To this balance each deposit is added at the date on which it is made, and from the credit side is subtracted each sum drawn by check at its date. Each balance, or sum, is multiplied by the number of days from its date until the next transaction, and lastly by the time between the last transaction and the end of the year. The sum of all these products will be the number df days that one dollar is at interest. Dr. John Johnson in acct. with Mechanics' Bank. Cr. 1877. Jan. 6. " 15. " 18. " 29. 1876. To check. $200 Dec. 31. « u 100 Jan. 10. (( ft 200 Jan. 25. it tt 350 By bal. old acct. By cash $300. By cash $750. DA. $600 6 400 4 700 5 600 3 400 7 1150 4 800 2 3600 1600 3500 1800 2800 4600 1600 6 ) 19,500 Interest, 3.25 The last sum or difference on the credit side is the Principal. $800 Interest = 3.25 Total balance = $803.25 The account is only rendered for one month. JLlligatio:n'. Alligation is the mixing of different qualities of grain, groceries, liquors, etc., either to obtain an average price of the mixture, which process is termed Alliga" Hon Medial^ or to make a mixture of several kinds that shall have a certain value, and this is called J Hi- gation Alternate. ALLIGATION MEDIAL. EXAMPLES. 1. Mix together 10 lbs. of tea at 40 cents a IK, i2 lbs. at 50 cts. a lb., 5 lbs. at 60 cts. per lb., and 3 1\3S. A 8 X 12 X 15 X 18 X 21 X 24 X 30 , . . 86. Eeduce ^^7; — ^^ — jtz — ts to a simple 20 X 36 X 42 X 48 ^ fraction. Ans, 270. 87. The product of two numbers is ^\, and one of the numbers is f ; what is the other number ? Ans, f. 88. The minuend is y\, and the remainder is f ; what is the subtrahend ? Ans, -^. 89. f is how many times f ? 1^ ? A ? A ? A? ^9 ? A? A? 90. I is how many times ,2^? ^? ^ ? ^ ? ^? ^ ? A? A? 91. How many bottles of 1 qt. 1 pt. each can be filled from a hogshead (63 gals.) of wine ? Ans. 168 bottles. 92. How many tiles 6 in. by 4 will cover a house 50 ft. long, and rafters on each side 24 ft. long ? Ans. 14,400 shingles. 93. How many square yards of carpeting will cover a room 48 ft. long and 30 feet wide ? Ans, 160 yds. PRACTICAL EXAMPLES. 161 94. How many cubic yards of excavation in a mass of earth 90 ft. long, 30 ft. wide, and 24 ft. deep. Arts. 2,400 cu. yd. 95. What is the cost of 1 T. 5 cwt. 65 lbs. beef at 8 cts. per lb.? 2,565 1b. Ans. $205.20. 96. How many cords of wood in a pile 28 ft. long, 8 ft. high, and 4 ft. wide ? Ans, 7 cords. 97. How many acres in a rectangular tract of land 480 rods long and 60 rods wide ?. A7is. 180 acres. 98. At 25 cents a gallon, how many gallons of beer can be bought for $57.25 ? Ans. 229 gallons. 99. Keduce 1,276 farthings to the higher denomina- tion. £1 6s. 7d. 100. The discount of a note at 90 days, 6^, was $31 ; Hhat was the face of the note given ? ^3^=:$31, and 31 ^^f^= the whole. Ans. $2,000. 101. A note with interest for 4 years 3 months, at 6%, was paid; the amount was $1,255 ; what was the face of the note ? f ^ 25|^ 51 . . , , . , ^251 ,^^^ l00'^200'^P^^^^P mterest=^-=1255; 126 6X||^^ ^/15. $1,000. 102. A farm of 375 A. 2 R. and 24 P. was sold at $51.50 per acre ; what did the sale amount to ? 3751^ A. Ans. $19,345.97f 103. How many rotations will a wheel 12 ft. in circum- ference make in a mile ? Ans. 440 rotations. 104. How many barrels, each containing 2 bu. 3 pk., can be filled with 275 bu. of apples ? Ans, 100 bar. 105. What is the difference between 25 lb. 8 oz. 9 dr. and 12 lb. 9 oz. 10 dr. of lead ? Ans. 12 lb. 14 oz. 15 dr. 162 PRACTICAL EXA3IPLES. 106. What is the sum of 12 bu. 3 pk. 5 qts. wheat and 25 bu. 2 pk. 4 qt. Ans. 38 bu. 2 pk. 1 qt. 107. Find the two equal factors 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484. Eem. One of two equal factors is the square root of the number, and one of three equal factors is the cube root. 108. Find one of the three equal factors of 8, 27, 64, 125, 216, 1,000. 109. How many seconds in 12 years of 365 days 5 h. 48 m. 48 sec. Ans, 378,683,136 sec. 110. How many suits of clothing can be made from 169 yds., each suit having 6 yds. 2 qrs.? Ans. 26 suits. 111. How many minutes in 31 days ? Ans. 44,640 min. 112. A man's income in 2 years was $2,333^, and the second year was 33^^ more than the first. What was it each year? Ans. 1st, $1,000; 2d, $l,333f 113. Muslin bought at 6 cts. and sold at 7^ cts.; what is the gain per cent.? Ans. 25^. 114. K receives $126 interest on a bond, rate 1% ; what is the face of the bond ? l^^l^^. Ans, $1,800. 115. On a note of $800, A received $300, which was the interest for five years ; what was the rate ? *^A=$60 for 1 year; rate WV=Jq^- ^^5- Hfc- 116. On a note of $300, at 6%, he received $76.50 in- terest; what was the time? -^^. ^^5. 4 y. 3 m. 117. In w^hat time will $600 produce $120 interest at 6% ? ^. Ans, 3 y. 4 m. 118. In what time will $500 at 6% amount to $650 ? J^. Ans. 5 years. PRACTICAL EXAMPLES, 163 119. For what sum must a note be drawn for 60 days, at 6^, to net $3,000 ? «3 o o o >'^^, Ans. $3,031.83. 120. A man owes $300, due in 4 months, $500 in 6 m., $1,000 in 7 m. 24 days, and $1,200 in 5 m. In what time may it all be paid without loss or gain ? Ans. 6 months. 121. Sold goods as follows: April 10th, $300 on 3 m. May 20th, $480 on 4 m. June 30th, $600 on 2 m. At what time may the whole be paid without gain or loss. Ans. August 26th. 122. A lot is 32 rods in length, and contains one acre ; how wide is it? Ans, 5 rods. 123. What is the interest of $486, at 6%, from April 16th, 1873, to July 1st, 1876 ? A71S. $93.55|. 124. What is the interest of $1,200, at 6%, from April 1st, 1874, t6 Jan'y 21st, 1877 ? What is the amount ? Ans. $202 Int., $1,402 Amt. 125. What is the interest of $3,600, at 6%, for 2 years 3 m. and 10 days ? 2 y. = 720 days 3m.= 90 days 10 days ^ "820 days .-. •'a«^^x,^=r$492. Sem. As a general rule it is simpler and shorter to re- duce the time to days. 126. What is the interest of $6,000, at 6%, for 3 years 5 m. 25 days? Ans. $1,255. Observe that the interest of $6,000 is $1 a day, reckon- ing 360 days to the year. 164 PRACTICAL EXAMPLES. 137. What is the interest of $4,000 fpr 636 days ? 4xt;36. Ans, $424. 128. What is the interest of $1,000 for 1,248 days ? ix 1,248. Ans. $208. 129. What is the interest of $2,000 for 624 days ? Ans. $208. 130. What is the interest of $60 for 3,000 days ? Ans, $30. 131. What is the interest of $742 for 3 y. 7 m. 24 d.? Ans, $162,498. 132. What is the cost of 12 gal. 3 qt. 1 pt. wine at $3.00 per gal.? 12|. Ans. $38|. 133. What cost 2 bu. 3 pk. 4 qt. apples at 80 cts. per bushel ? 2| bu. Ans, $2.30. 134. The product is 8f, and one of the factors is 3f ; what is the other factor ? " Ans, 2^. 135. The dividend is 12|, and the quotient 5f ; what is the divisor ? Ans. 2 J. 136. The product of three factors is 4,125, and one factor is 15 ; what is the product of the other two ? Ans. $275. 137. Two men engage a piece of work ; the one can do it in 5 days, and the other in 7^ days ; how long will it take both together ? Ans. 3 days. 138. A tradesman hired a journeyman at 24 pence for- every day he worked, and he was to forfeit 6d. for every idle day. At the end of 30 days his account balanced ; how many days did he work ? Ans. 6 days. ANSWERS NOT GIVEN IN THE PEECEDING PAGES, AbBITIOK AJ^B SUBTBACTlOJf. Page 16. 1. 115 sheep. 2. 145 fowls. 3. 63 in first and 21 in second. Page 17. 4. 45 cattle in third field. 9. 256 verses. 5. 28 cents left. 10. 190 cents. 6. 139 pages to read. 11. 200 dollars. 7. 155 eggs. 12. 18830 dollars. 8. 46 chickens. 13. 13843 dollars left, i Page 18. 14. 14000 dollars. 15. 1193 acres left. 16. 888 dollars left. 17. 893 acres unsold, 247 acres of second farm, and 64C acres of third farm. 166 DIVISION. 18. Sold for 467 doUars. 20. Gained $1182. 19. Gained 1118 dollars. 21. $3779. MULTIPLICATIOJV. Page 26. 1. 18796796. 4. 17600100. 7. 1913247450. 2. 12895164. 5. 505489116. 8. 8585904128. 3. 32671178. 6. 3515971700. 9. 982259375. 10. 984381300. Divi sioj^. I*affe 31. 1 Quo. 1915, rena L. 98. 7. Quo 800368, rem. 209. 2. u 1477, « 60. 8. « 373792, « 979. 3. iC 263, « 234 9. " 439169, « 1458 4. cc 5475, " 116. 10. « 13482, « 42. 5. 6C 209205, « 992. 11. " 4457, « 102170, 6. U 780317, « 300. 12. " 676, "196432. 1. 50 cts. 4. 10 lbs. 7. 200 cts. 3. 5 lbs. 5. 150 cents. 8. 20 lbs. 3. 100 cents. 6. 10. 15 lbs. 25 lbs. 9. 250 cts. Page 32. 11. 1960 cents. 13. $15660. 15. 1207680. 13. 245 lbs. 14. 348 acres. 16. 3245 acres. FACTORING. 167 17. $1875. 19. $2625. 21. $3780. 18. 15 horses. 20. 35 oxen. 22. 84 cows. Page 33. 32. Horses cost $2016, cows $1944, sheep $1456, all cost $5416. 33. $5740. 34. Cost $76600, sold for $75355. 35. Gained $6912. Page 34. 36. Paid $25 per acre, and sold for $29 per acre. 37. Paid $65 per acre, sold for $71. 38. Each cow cost $40. 39. Isf Ans., 36000 farms; 2d Ans., 184000 farms; Srd Ans., 1028000 farms. 40. 1st State, $144,000,000 ; 2d State, $588,800,000 ; 3rd State, $822,400,000. 41. Each daughter's share $7100. Factoeij^g Page 50. 1. 60 = 3, %, 3, 5. 9. 84 = 3, 3, 3, 7. 3. 64 = 3, 3, 2, 3, 3, 3. 10. 86 = 3, 43. 3. 65 = 5, 13; 11. 88 = 3, 3, 3, 11. 4. 70 = 2, 5, 7. 13. 90 = 3, 3, 3, 5. 5. 73 = 3, 3, 3, 3, 3. 13. 95. = 5, 19. 6. 75 = 3, 5, 5. 14. 96 = 3, 3, 3, 3, 3, 3. 7. 78 = 2, 3, 13. 15. 98 = 3, 7, 7. 8. 80 = 2, 2, 2, 2, 5. 16. 100 = 2, 2, 5, 5. 58 FB ACTIONS. 17. 102 = 2, 3, 17. 27. 124 = 2, 2, 31. 18. 104 = 2, 2, 2, 13. 28. 125 = 5, 5, 5. 19. 106 = 2, 53. 29. 128 = 2,2,2,2,2,2,3. 20. 108 = 2, 2, 3, 3, 3. 30. 130 = 2, 5, 13. 21. 110 = 2, 5, 11. 31. 132 = 2, 2, 3, 11. 22. 112 = 2, 2, 2, 2, 7. 32. 136 = 2, 2, 2, 17. 23. 114 = 2, 3, 19. 33. 140 = 2, 2, 5, 7. 24. 116 = 2, 2, 29. 34. 225 = 5, 5, 9. 25. 118 = 2, 59. 35. 500 = 2, 2, 5, 5, 5. 26. 120 = 2, 2, 2, 3, 5. 36. 625 = 5, 5, 5, 5. Multiplication of Fbactioj^s. Page 63. 7. i^ X 4 = 1^ = 58|. 8. ^ X 6 = ^ = 103f 9. ^ X 9 = ^ = 164}. 2 10. -^ X $ = 198. 11. 22204. 12. 64916|. 13. 3147 X J^^ = ^^ V ^ ^ = 112505I-. 532 14. i6|^ X ^H^ = 16627 X 532 = 8845564. 15. -H*^ X ^ = ^H^ = 1711H- 16. isj, X -V- = ^- = 143f|. 17. ij^ X W = ^WP = 380fff. 18. ^ X H^ = -^W^ = 2139ff DENOMINATE NUMBERS. 169 DirisiOM OF Fbactioj^s. Page 65. L /^ X I = 45. 4. I X I = A- 2. *^ X I = 18. 5. i X I = |. 3. «0 X J = 38. 6. f X f = f :« l\. 7. ^ X ^ = W = 5A- 8. 4x4 = 1 = 1*. 5 9. H^ X A = -W = 5A. 10. 15 X Jf = ^ = 32^. 11. ^ X ^ = W- = 85V. 12. H X J = W = 2M- 13. H X V- = «i = ItIs- 14. ^F X 3^ = Vs^ = 7^. 15. ^ X ^ = fiH = SHU- Page 75. . 1. f of T^ = ^; .-. i = 150,000, and the whole is eight times I = $400,000. 2, They will meet in 7^\ hours, A will have traveled 45|^ miles, and B 61-^ miles. Dej^omij^ate J^umbers. Page 90. 4. ^291 155. 9d. 170 DENOMINATE NUMBERS, Page 91. 2. Ans., 500 cents ; 5000 mills. 3. 700 cents ; 7000 mills. 4. 515 cents ; 5150 mills. 5. 6153 mills. Page 92. 9. 63 dollars 25 cents 7 mills; $63,257. 10. $753.25. 11. $9.00. 12. $10.50. 13. $105. 14. $1,125. 15. $39,375. 16. $55.65. 17. Cost $483.136 ; sold for $640,328. Page 93. 24. 9303 far. 26. £39 19s. 3d. 3 far. 25. £39442 Ifar. 27. £1 19s. 2d. ^tss. Page 102. 1. 3359 far. 7. 1705577 dr. 2. 3068d. 8. 1 T. 6 cwt. 1 qr 1 lb, 11 oz, 3. 1682d. 9. 32871 qr. 4. £2 9s. 2 far. 10. 11 lb. 2 oz. 4 qr. 5. £2 5s. 3d. 11. 14690 gr. 6. £36 lis. 12. lib. 21 13 2 gr. 13. 741402 in. 14. 4 Leagues 5 fur. 12 rods 4 yd. 1 ft. 4 in. 15. 177i inches. 18. 162 inches. 16. 135 inches. 19. 172048 sq. in. 17. 180 inches. 20. 161940 cu. in. RATIO. 171 Page 103. 22. 831giUs. 23. 633 pts. JPage 106. 56. 1440 yds. ; 1200 yds. ; 960 yds. ; 640 yds. ; 480 yds.; 384 yds.; 320 yds.; 274f yds. 57. Proceeds 83312.50 and 3785f yds. Ratio. JPage 113. 5. 16 days. 6. 120 feet. Page 114. 7. $25. 9. $10.50; $11.40; $28.50; $38; $57; $47; $70.50 $117; $128.10. 10. $11.87^; $12.81i; $19.50; $46.50; $119.50 $157; $188.50; $349.25. 11. $260; $310; $500; $525; $690; $735; $780 $825 ; $1510 ; $1995. 12. $37.50; $25; $12.50; $10; $20; $30; $40 $60; $70. 13. $lf|. 14. $9. 15. $18.75 and $22.50. Page 116. 3. Arts., 9 172 EVOLUTION. PEBCEJfTAGE. Page 131. 5. $500. 6. 13335 Francs. 7. $10,000 EVOLUTIOJr. Page 147. 10. 2700cti.ft. 13. 9.4248 cu. ft. 14 21.2058 cu. ft. THIS BOOK IS DUE ON THK LAST DATE STAMPED BELOW AN INITIAL FINE OP 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO 50 CENTS ON THE FOURTH DAY AND TO $1.00 ON THE SEVENTH DAY OVERDUE. FEB 19 1942 F I" YB 17349 ■■^ ^ ^ J 911326 THE UNIVERSITY OF CALIFORNIA LIBRARY \ BAKE R' S nVt^^HOVED AND SIMPLIFIED SRIES OF MiT":.^^!FICS. By ANDREW H. BA' mw^^'^^ ".USt Pibor. L '•le .us m.. a the pn ^.c^.'l ysis, thor- igbi,^>^ brief and ex; 'licit, iie books of tb- . eries <-| I PRIMARY ARltjJMT 11. ELEMENTARY III. COMPLETE A^ ITL- r ^^QMETRY .GEBRA i WIC a* 1