140. After inverting the divisor, the operation of dividing fractions may often be abbreviated, by canceling the factors common both to the numerators and denominators, as in multiplication of fractions. (Art. 136.) 30. Divide of by of Operation. 31 28 3 After canceling all the factors common both. to the numerators and denominators, we divide the continued product of the numbers remaining on the right of the line by the continued | 5=5, or 13. Ans. product of those on the left, and the quotient is the answer. 1 4 3 10,5 OBS. 1. The numerators (which answer to dividends) should always be arranged on the right of a perpendicular line, and the denominators (which answer to divisors) on the left. 2. Before arranging the terms of the divisor for cancelation, it is necessary to invert them, or suppose them to be inverted. 40. A merchant sent 12 barrels of flour to supply some destitute people, allowing of a barrel to each family. How many families shared in his bounty? Solution. If of a barrel supplied 1 family, 12 barrels will supply as many families as is contained times in 12. Reducing the dividend 12 to the form of a fraction, it becomes 12; now inverting the divisor, we have 2×}= 36, or 18. Ans. 18 families. QUEST.-140. How may the operation of dividing fractions be contracted? Obs. How are the terms of the given fractions arranged for cancelation? What must be done to the divisor before arranging its terms? Or thus; is contained in 12, as many times as there are thirds in 12; and 12 x3=36 Now 2 thirds are contained in 12 only half as many times as 1 third; and 36 2=18. Ans. Hence, 141. To divide a whole number by a fraction. Reduce the whole number to the form of a fraction, (Art. 122. Obs. 1,) and then proceed according to the rule for dividing a fraction by a fraction. (Art. 139.) Or, multiply the whole number by the denominator, and divide the product by the numerator. OBS. 1. When the divisor is a mixed number, it must be reduced to an improper fraction, then proceed as above. 41. Divide 25 by . 43. Ans. 33. 42. Divide 35 by 2. 44. Divide 165 by 7. 45. Divide 237 by 11. 142. From the definition of complex fractions and the manner of expressing them, it will be seen that they arise from division of fractions. Thus, the complex fraction is the same as÷4; for, the numerator 4, and the denominator 14; but the numerator of a fraction is a dividend and the denominator a divisor. (Art. 109.) Now 36, which is a simple fraction. Hence, 5 143. To reduce a complex fraction to a simple one. Consider the denominator as a divisor, and proceed as in division of fractions. QUEST.-141. How is a whole number divided by a fraction? Obs. How by a mixed number? 142. From what do complex fractions arise? 143. How reduce them to simple fractions? 144. To multiply complex fractions together. First reduce the complex fractions to simple ones; (Art. 143;) then arrange the terms, and cancel the common factors as in multiplication of simple fractions. (Art. 136.) Oss. 1. The terms of the complex fractions may be arranged for reducing them to simple ones and for multiplication at the same time. 2. To divide one complex fraction by another, reduce them to simple fractions, then proceed as in Art. 139. Operation. 37 94 29 74,2 3|8=23. Ans. 14 The numerator 23. (Art. 122.) Place the 7 on the right hand and 3 on the left of the perpendicular line. The denominator 242, which must be inverted; (Art. 143;) i. e. place the 4 on the right and the 9 on the left of the line. 42, and 13 = , both of which must be arranged in the same manner as the terms of the multiplicand Now, canceling the common factors, we divide the product of those remaining on the right of the line by the product of those on the left, and the quotient is 23. (Art. 136. Ex. 43.) QUEST.-144. How are complex fractions multiplied together? Obs. How is one complex fraction divided by another? 1. At dollar per bushel, how many bushels of pears can be bought for 5 dollars? 2. At of a penny apiece, how many apples can be bought for 18 pence 3. At of a dollar a pound, how many pounds of tea will 7 dollars buy? 4. How many bushels of pears, at 14 dollar a bushel, can be purchased for 15 dollars? 5. How many gallons of molasses, at 2 dimes per gallon, will 10 dimes buy? 6. How many yards of satinet, at 1 of a dollar per yard, can be purchased for 20 dollars? 7. At 4 dollars per yard, how many yards of cloth can be obtained for 25 dollars? 8. At 64 cents a mile, how far can you ride for 62 cents? 9. At 12 cents a pound, how many pounds of flax will 67 cents buy? 10. At 164 cents per pound, how many pounds of figs can you buy for 87 cents? 11. How many cords of wood, at 6 will it take to pay a debt of 671⁄2 dollars? 12. How many barrels of beer, at 11 rel, can be obtained for 95 dollars? dollars per cord, dollars per bar 13. A man bought 15 barrels of beef for 124§ dollars how much did he give per barrel? : 14. A man bought 131⁄2 pounds of sugar for 94 cents: how much did his sugar cost him a pound? 15 A lady bought 15 yards of silk for 145 shillings: how much did she pay per yard? 16. Bought 15 baskets of peaches for 244 dollars : how much was the cost per basket? 17. Bought 30 yards of broadcloth for 181 dollars: what was the price per yard? 18. Paid 375 dollars for 125 pounds of indigo: what was the cost per pound? 19. How many tons of hay, at 16 dollars per ton, can be bought for 196 dollars? 20. How many sacks of wool, at 17 can be purchased for 1500 dollars? 21. How many bales of cotton, at 15 can be bought for 2500 dollars? dollars per sack, dollars per bale, 23. Divide 163 by 25. 25. Divide 12563 by 681}. 27. Divide 1052 by 8228. 29. Divide of 16 by of. 31. Divide of by 21. by 2 of 31. 33. Divide of by off. 22. Divide 145,7 by 16. 8 10 of 15 of 30 by 19. SECTION VII. COMPOUND NUMBERS. ART. 146. Numbers which express things of the same kind or denomination, are called simple numbers. Numbers which express things of different kinds or denominations, are called compound numbers. Thus 6 oranges, 5 dollars, 7 roses, &c. are simple numbers; 5 pounds and 2 ounces, 7 feet and 3 inches, &c. are compound numbers. OBS. Compound Numbers, by some late authors, are called Denominate Numbers. QUEST.-146. What are simple numbers? What are compound numbers ? |