Ans. s. 3 56 53 101 18079 18. Find the value of (41 x 3) + 13 x (3}:— 1%). 19. Find the value of 28 + (7 — 23) x ã x ( + 3). Note 3. — The word of between fractions is equivalent to the sign of multiplication; and such an expression is sometimes called a compound fraction. Find the values of the following indicated products : . Ans. 24. of į of of of of 4 of of of 10 In the following examples, cancellation may be employed by the aid of the Factor Table. 25. What is the value of 0 x 1313 x 3933? Ans. 18 26. What is the value of 3581 388 491? Ans. 67. 27. What is the value of 371 x x $594? Ans. 13031 28. What will 7 cords of wood cost at $3 per cord ? Ans. $25 29. What is the value of (3) X li x ()*? Ans. T1873 30. If 1 horse eat of a bushel of oats in a day, how many bushels will 10 horses eat in 6 days ? Ans. 252 31. What is the cube of 127 ? 32. At $9} per ton, what will be the cost of of á of a ton of Ans. $4. 33. At $, a bushel, what will be the cost of 13 bushels of corn? 34. When peaches are worth $7 per basket, what is of a basket worth? 35. A man owning 1 of 1563 acres of land, sold i of of his share; how many acres did he sell? Ans. 47. 36. What is the product of (3): X (1)’ X (%)’ (33)"? Ans. $72. 37. If a family consume 17 barrels of flour a month, how many, barrels will 6 families consume in 8 10 months ? 38. What is the product of 1507—(9 of 1211 + $ of 483)—75, multiplied by 3 x (1 of 1} 4) — 21 ? Ans. 342% hay? 39. A man at his death left his wife $12,500, which was į of of his estate; she at her death left of her share to her daughter; what part of the father's estate did the daughter receive? Ans. 25 40. A owned ã of a cotton factory, and sold of his share to B, who sold of what he bought to C, who sold of what he bought to D; what part of the whole factory did each then own? Ans. A, Á ; B, &; C, As; D, 41. What is the value of 24x}+s of 4;X() +(33)–(3})?? Ans. 36448 DIVISION. 5 FIRST OPERATION. SECOND OPERATION, 25 FIRST OPERATION. 15 · 194. 1. Divide ž} by 3. ANALYSIS In the first ope ration we divide the fraction by 3:3= 73 3 by dividing its numerator by 3, and in the second operation we divide the fraction by 3 by j} = 3 = }} multiplying its denominator by 3, (190, II or III). 2. Divide 15 by . ANALYSIS. To divide by 1, we must divide by 3 and multiply x = 5 x 7=35 by 7, (191, II or III). In the first operation, we first 15 : * = 105 -- 3 = divide 15 by 3, and then mul tiply the quotient by 7. In the second operation we first multiply 15 by 7, and then divide the product by 3. 3. Divide 4 by : ANALYSIS. To divide by · Ist step, * = 3 , we must divide by 3 and 20 step, 4 x 5 ! multiply by 5, (191, II or III). In the first operation SECOND OPERATION. 35 FIRST OPERATION. 4 45 S Ans. we first divide is by 3 by * * j = ! = multiplying the denominator by 3. We then multi SECOND OPERATION. THIRD OPERATION. ply the result, a's, by 5, by 4 $ multiplying the numerator Х 16 3 by 5, giving a=for the required quotient. By in specting this operation, we observe that the result, å s, is obtained by multiplying the denominator of the given dividend by the numerator of the divisor, and the numerator of the dividend by the denominator of the divisor. Hence, in the second operation, we invert the terms of the divisor, }, and then multiply the upper terms together for a numerator, and the lower terms together for a denominator, and obtain the same result as in the first operation. In the third operation, we shorten the process by cancellation. We have learned (107) that the reciprocal of a number is 1 divided by the number. If we divide 1 by }, we shall have 1 : = 1 x . Hence 195. The Reciprocal of a Fraction is the fraction inverted. From these principles and illustrations we derive the following general RULE. I. Reduce integers and mixed numbers to improper fractions. II. Multiply the dividend by the reciprocal of the divisor. NOTES. — 1. If the vertical line be used, the numerators of the dividend and the denominators of the divisor must be writton on the right of the vertical. 2. Since a compound fraction is an indicated product of several fractions, its reciprocal may be obtained by inverting each factor of the compound fraction. EXAMPLES FOR PRACTICE. 1. Divide jy by 4. 33 X =, Ans. 2. Divide 10 by 5, and 13 8 by 80. 3. Divide 10 by 2. Ans. 35. 4. Divide 28 by 4, and 3 by 12. 5. Divide 56 by 15. Ans. 36. 6. Divide by 7. Divide by , i byn, and 38 by 5%. 8. Divide 17 by 15 Ans. 13. 9. Divide 14 by 15. Ans. 14. 5 14 OPERATION. = 5 18 10. Divide şof by of ANALYSIS. The dividend, } reduced to a simple fraction, 1 X = is }; the divisor, reduced { x = = 1; Ans. in like manner, is jg; and 3 Or,' 2 ß divided by is is 1}, the quotient required. Or, we X X X = 1} may apply the general rule directly by inverting both factors of the divisor. Note 3. The second method of solution given above has two advantages. 1st, It gives the answer by a single operation; 2d, It affords greater facility for cancellation. 11. Divide of ý by i of 's. Ans. 1. : 12. Divide iz of x by of Ans. 181 13. Divide 2' x 7} by 33 x 3,3% 14. Divide 11 by x 51 x 7. 15. Divide 31 x 19 by } x 73 x 15. Ans. 25. 16. Divide in x 1; by x x V x ff x 31. Ans. 333 17. Divide 38 by 8675. Ans. 1, 18. Divide 1289 by i' x 88 x 64. 19. Divide į x ý xix | by x x x x 5. 20. What is the value of ? 43 5 5 Ans. : 13 4554 OPERATION. 22 51 1 x = = 1; 43 ANALYSIS. The fractional form indicates division, the numerator being the dividend and the denominator the divisor, (168, II); hence, we reduce the mixed numbers to improper fractions, and then treat the denominator, , as a divisor, and obtain the result, 11, by the general rule for division of fractions. 5! Note 4.-Expressions like are sometimes called complex fractions. 43 5. In the reduction of complex fractions to simple fractions, if either the numerator or denominator consists of one or more parts connected by + or -, the operations indicated by these signs must first be performed, and afterward the division. and 21. What is the value of 4 ? Ans. 4. Ans. 2. Ans. 73. 22. What is the value of į x 14, TB X 5.1 7 + 3 23. What is the value of ? 1, 24. Reduce go 을 to its simplest form, } + 25. Reduce to its simplest form. 83 545 Ans. 25 98 Ans. $45. 26. Reduce to its simplest form. 61 – 27. If 7 pounds of coffee cost $1, how much will 1 pound cost ? 28. If a boy earn $; a day, how many days will it take him to earn $61 ? Ans. 173 29. If of an acre of land sell for $30, what will an acre sell for at the same rate ? Ans. $671 30. At of of a dollar a pint, how much wine can be bought for $10 ? Ans. 2 pints. 31. If % of a barrel of flour be worth $27, how much is 1 barrel worth? Ans. $75. 32. Bought of 4cords of wood, for sof of $30; what was 1 cord worth at the same rate ? 33. If 235, acres of land cost-$1725%, how much will 1251 acres cost? Ans. $918441 34. Of what number is 264 the' [ part ? Ans. 313 - 35. The product of two numbers is 27, and one of them is 23; what is the other ? B 36. By what number must you multiply 161to produce 1481? 37. What number is that which, if multiplied by of 6 of 2, will produce z? Ans. 113 38. Divide 720 — 5 * 28 — 74) by 401 + % = ) ()'. 39. What is the value of (34 (3)° + { of })• - (174— + 1 = (3)" x 5)? of (3) X 3 of 53 40. Divide by Ans. 31543 1 of (91) X (3) ( 143 |