140. Hence, to divide fractions by Cancellation. Invert the divisor, then cancel the factors common to the numerators and denominators, and proceed as in multiplication of fractions. (Art. 136.) OBS. 1. Before arranging the terms of the divisor for cancellation, it is always necessary to invert them, or suppose them to be inverted. 2. The reason of this contraction is manifest from the fact, that it simply divides the numerator and denominator of the quotient by the same numbers, and therefore does not alter its value. (Art. 116.) of 4. 21. Divide 41 by 21. 28. Divide by 3 of 25. Ans. 2. 23. Divide 41 by } of 33. 25. Divide of? by 3. 27. Divide by 3 of 7. 29. Divide 25 by ¦ of 26. CASE III.-Dividing a whole number by a fraction. 30. 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? First operation. 12x=30 36-18 Ans. Analysis. 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; then inverting the divisor, we proceed as in dividing a fraction by a fraction. Or, we may reason thus: is contained Second operation. 12 3 2)36 in 12, as many times as there are thirds in 12, and 12×3=36. Now, 2 thirds are contained in 12, only half as many times as 1 third; and 36÷2=18. (Art. 85.) That is, we multiply the whole number by the denominator, and divide the product by the numerator. Hence, Ans. 18 fam. QUEST.-140. How divide fractions by cancellation? Obs. What must be done to the divisor before arranging its terms? How does it appear that this Contraction gives the true answer ? T.P. 141. To divide a whole number by a fraction. Reduce the whole number to the form of a fraction, and proceed according to the rule for dividing a fraction by a fraction. Or, multiply the whole number by the denominator, and divide the product by the numerator. OBS. When the divisor is a mixed number, it must be reduced to an improper fraction, then proceed as above. 31. Divide 120 by 33. 32. Divide 35 by 3. 141.a. The preceding cases following general Ans. 33. 33. Divide 47 by 5. 35. Divide 237 by 11. 37. Divide 643 by 2417. may be summed up in the RULE FOR DIVISION OF FRACTIONS. Reduce compound fractions to simple ones, whole and mixed numbers to improper fractions; then invert the divisor, and proceed as in multiplication of fractions. (Art. 187.a.) EXAMPLES FOR PRACTICE. 1. At dollar per bushel, how many bushels of pears can be bought for 65 dollars? 2. At of a penny apiece, how many apples can be bought for 78 pence? 3. At of a dollar a pound, how many pounds of tea will 87 dollars buy? 4. How many bushels of wheat, at 1 dollar a bushel, can be purchased for 215 dollars? 5. How many gallons of molasses, at 21 dimes per gallon, will 310 dimes buy? 6. How many yards of satinet, at 13 of a dollar per yard, can be purchased for 120 dollars? 7. At 4 dollars per yard, how many yards of cloth can be obtained for 25 dollars? 8. At 6 cents a mile, how far can you ride for 62 cents? QUEST.-141. How is a whole number divided by a fraction? Obs. How by a mixed number? 141.a. What is the general rule for division of fractions ? · 9. At 12 cents a pound, how many pounds of flax will 67} cents buy? 10. At 16 cents per pound, how many pounds of figs can you buy for 871 cents? 11. How many cords of wood, at 61 dollars per cord, will it take to pay a debt of 671⁄2 dollars? 12. How many barrels of beer, at 113 dollars per barrel, can be obtained for 95 dollars? 13. A man bought 15 barrels of beef for 1245 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 153 yards of silk for 145,5 shillings: how much did she pay per yard? 16. Bought 15 baskets of peaches for 24 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 1961 dollars? 20. How many sacks of wool, at 171 dollars per sack, can be purchased for 1500 dollars? 21. How many bales of cotton, at 157 dollars per bale, can be bought for 2500 dollars? COMPLEX FRACTIONS. 142. From the definition of complex fractions, and the manner of expressing them, it will be seen that they arise from division of fractions. (Art. 108.) 1. Reduce to a simple fraction. Suggestion.-Since the numerator of a fraction answers to the dividend, and the denominator to the divisor, the operation is Operation. the same as dividing by. We therefore +}×}, or }. invert the denominator, and then multi ply the numerators together, and the denominators, as in dividing a fraction by a fraction. The result 8, is the answer required. Hence, 143. To reduce a complex fraction to a simple one. Consider the denominator as a divisor, and proceed as in di vision of fractions. (Art. 139.) QUEST. 142. From what do complex fractions arise? 143. How reduce them to simple fractions? 114. To add, subtract, multiply, or divide complex fractions. Reduce them to simple ones, then proceed according to the rules of Simple fractions. Note.-If the following examples are found too difficult for beginners, they can be omitted ull review. QUEST.-144. How do you add, subtract, multiply, or divide complex frac tions? |