Rule. 1. Reduce the fractions to the same unit, and then subtract as in Case I.: Or, II. Find the value of each fraction in units of lower de nominations, and then subtract as in denominate numbers. Examples. 1. From of a pound troy, take 2. From of a ton, take of 3. From 4. From quart. of an ounce. of a pound. of of a hogshead of wine, take of of a of a league, take of a mile. 5. What is the difference between 13s. and of 71⁄2d? 28 6. What is the difference between 2 of a degree and of of a degree. 7. From 18 of a square mile, take 367 acres. 8. From of a ton, take 9. From 13 lb. troy, take 10. From 2 cords, take 11. From of a yard, take of 12 cwt. of an ounce. of a cord foot. of an inch. 12. From of of a pound, take ecaries' weight. of of a dram, apoth 13. From a piece of ground containing 25 acres take 1 A. 1 P. and 9 square yards. 14. A pound avoirdupois is equal to 14 oz. 11 pwt. 16 gr troy what is the difference, in troy weight, between the ounce avoirdupois and the ounce troy? 169. How do you subtract when there are mixed numbers? 170. What is the rule when the fractions have different units. MULTIPLICATION. 171 MULTIPLICATION OF FRACTIONS is the operation of taking one number as many times as there are units in another, when one or both are fractional. 1. If 1 pound of tea cost of a dollar, what will of a pound cost. ANALYSIS.-The cost will be equal to the price of 1 lb. taken as many times as there are units in the multiplier (Art. 84). OPERATION. 5X3 $ × = ;; = $38. 8X7 = One-seventh of a pound of tea will cost one-seventh as much as 1 lb. Since 1 lb. cost $, of 1 lb. will cost of $85% (Art. 142). But 3 sevenths of 1 lb. will cost three times as much as (Art. 139). Hence, to multiply one frac ; that is, $5 × 3 tion by another : = 56 Rule. Cancel all factors common to the numerator and denominator; then multiply the numerators together for a new numerator, and the denominators together for a new denom inator. 172. Principles of the operation. 1. When the multiplier is less than 1, we do not take the whole of the multiplicand, but only such a part of it as the multiplier is of 1. 2. When the multiplier is a proper fraction, multiplication does not increase the multiplicand, as in the multiplication of whole numbers. The product is the same part of the multiplicand as the multiplier is of 1. 3. When either of the factors is a whole number, write 1 under t for a denominator. 4. When either of the factors is a mixed number, reduce it to an improper fraction. 171. What is multiplication of fractions? What is the rule ?-172 What is the first principle of the operation? What is the second? 4. Multiply 9. Multiply 67 by 912 10 Multiply 842 by 73. 11. Multiply 360 by 123. 15. Multiply by 8. 16. Multiply 15 by . 12. Multiply 460 by 112. 33. Multiply T, 33, and 48 together. 6 34. Multiply 4, 2, 3, and 38 together. 9 28 40. What will 7 yards of cloth cost, at a yard? 41. What will 123 bushels of apples cost, at $ a bushel? 42. If one bushel of wheat costs $13, what will of a bushel cost? 43. If one horse eats of a ton of hay in one month, how much will 18 horses eat in the same time? 44. If a man earns $ in one day, how much can he earn in 24 days? 45. What will 3 yards of cloth cost, at of a dollar a yard? 46. At $16 a ton, what will cf a ton of hay cost? 47. If one pound of tea costs $11, what will 6 pounds cost? 48. What will 3 boxes of raisins cost, at $2 a box? 49. At 75 cents a bushel, what will of a bushel of corn cost? 50. If a lot of land is worth $75, what will of it be worth? 51. What will 17 yards of cambric cost, at 2 shillings a yard? 52. Bought 15 barrels of sugar, at $20 a barrel: what did the whole cost? 53. If one bushel of corn is worth of a dollar, what is of a bushel worth? 54. If I own of a farm, and sell of my share, what part of the whole farm do I sell? 55. I bought a book for of a dollar, and a knife for the cost of the book: how much did I pay for the knife? 56. At of of a dollar a pound, what will 3 pound of tea cost? of 15 of 57. If hay is worth $9 a ton, what is of 3 tons worth? 58. If a man can dig a cellar in 22 would it take him to dig of it? days, how many days 59. If a railroad train runs 1 mile ir of an hour, how long will it be in running 1063 miles? 60 A owned of a farm 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 did D have? 61. A owned to B, who sold had each ? of 200 acres of land, and sold of his share of what he bought to C: how many acres DIVISION. 173. DIVISION OF FRACTIONS is the operation of finding how many times one number is contained in another, when one or both are fractional. 1. What is the quotient of divided by ? ANALYSIS.-How many times is OPERATION. contained in? If be divided by } ÷ 4 = 7 × 1. 14, the quotient will be Since the true divisor is the divisor used is 5 times too large; hence, the partial quotient , is 5 times too small. Multiplying this by 5, we have the true This result is produced by inverting the terms of quotient, = the divisor and multiplying. Rule.-Invert the terms of the divisor, cancel, and proceed as in multiplication. 174. Directions for the operation. 1. If either the dividend or divisor is a whole number, make it fractional, by writing 1 under it for a denominator. 2. Cancel all common factors. 3 If the dividend and divisor have a common denominator, they wil cancel, and the quotient of the numerators will be the answer. 173. What is division of fractions? What is the rule ?-174. What is the first direction for performing the operation? What the second? What the third? What the fourth? What the fifth? |