30. Divide of by 1 of 3. Operation. 31 8 For convenience we arrange the numerators, (which answer to dividends,) on the right of a perpendicular line, and the denominators, (which answer to divisors,) on the left; then canceling the factors 3 and 2, which are common to both sides, (Art. 91. a,) we multiply the remaining factors in the numerators together, and those remaining in the denominators, as in the rule above. Hence, 85-4. Ans. 140. To divide fractions by CANCELATION. Having inverted the divisor, cancel all the factors common both to the numerators and denominators, and proceed as in multiplication of fractions. (Art. 136.) OBS. Before arranging the terms of the divisor for cancelation, it is always necessary to invert them, or suppose them to be inverted. 31. Divide 4 by 24. Ans. 2. 33. Divide 43 by of 33. 35. Divide of 3 by 4. 37. Divide by 3 of 7. 39. Divide 25+ by of 26. 4 CASE III. 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 12×3=36, or 18. Ans. 18 families. QUEST.-140. How divide fractions by cancelation? How arrange the terms of the given fractions? Obs. What must be done to the divisor before arranging its terms? Or, we may reason thus: is contained in 12, as many times as there are thirds in 12, viz: 36 times. 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 120 by 33. 42. Divide 35 by . 44. Divide 165 by 7. Ans. 33. 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 frac tion is the same as÷; for, the numerator 412, 11 and the denominator 14=; but the numerator of a fraction is a dividend, and the denominator a divisor. (Art. 109.) Now 2-16, which is a simple fraction. Hence, 143. To reduce a complex fraction to a simple one. Consider the denominator as a divisor, and proceed as in division of fractions. (Art. 139.) 24 46. Reduce to a simple fraction. 531 Now ÷23X1, or 28. Ans. QUEST.-141. How is a whole number divided by a fraction? Obs. How by a mixed number? 142. From what do complex fractions 143. How reduce them to simple fractions? arise? 6 5월 7급 6급 125 254 144. 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.) OBS. 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 sim ple fractions, then proceed as in Art. 139. Operation. 3 21 14 38 2. Ans. (Art. The numerator 21 122.) Place the 7 on the right hand and 3 on the left of the perpendicular line. The denominator 24-2, which must be inverted; (Art. 143;) i. e. place the 4 on the right and the 9 on the left of the line. 4, and 13-4, 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 2. (Art. 136.) 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? of a penny apiece, how many apples can be 2. At 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 21 gallon, will 10 dimes buy? dimes per 6. How many yards of satinet, at 13 of a dollar per yard, can be purchased for 20 dollars? 7. At 45 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 621 cents? 9. At 12 cents a pound, how many pounds of flax will 673 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 dollars per cord, will it take to pay a debt of 67 dollars? 12. How many barrels of beer, at 113 dollars per barrel, can be obtained for 954 dollars? 13. A man bought 15 barrels of beef for 1245 dollars, how much did he give per barrel? 14. A man bought 134 pounds of sugar for 944 cents: how much did his sugar cost him a pound? 15. A lady bought 15 yards of silk for 145 shil· lings: how much did she pay per yard? 16. Bought 15 baskets of peaches for 24+ dollars how much was the cost per basket? 1 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 can be bought for 1963 dollars? 20. How many sacks of wool, at 17 can be purchased for 1500 dollars? dollars per ton, dollars per sack, 21. How many bales of cotton, at 157 dollars per bale, can be bought for 2500 dollars? 22. Divide 145 by 16. 23. Divide 1635 by 25. SECTION VII. COMPOUND NUMBERS. ART. 146. Numbers which express things of the same kind or denomination, as 3 pears, 7 roses, 15 horses, are called simple numbers. Numbers which express things of different kinds or denominations, as the divisions of money, weight, and measure, are called compound numbers. Thus 6 shillings, 7 pence; 5 pounds, 2 ounces; 7 feet, 3 inches, &c., are compound nnmbers. OBS. Compound Numbers, by some late authors, are called Denominate Numbers. QUEST. 146. What are simple numbers? numbers? What are compound |