EXERCISES FOR THE SLATE. CASE I. 11. If 3 bushels of oats cost of a dollar, what will i bushel cost? Suggestion. 1 is 1 third of 3; therefore, 1 bushel will cost I third part as much as 3 bushels. I third of g is ß. Ans. į of a dollar. We divide the numerator of the fracOperation. tion g, which is the whole cost, by 3 the 9-3= . Ans, whole number of bushels, and place the quotient 2 over the given denominator. 12. If 4 yards of calico cost á of a dollar, what will 1 yard cost ? In this case we cannot divide Operation. the numerator of the dividend by 6 +4= bxz, or . Ans. 4 the given divisor, without a remainder. We therefore multiply the denominator by the 4, which is in effect dividing the fraction. (Art. 113.) Hence, 138. To divide a fraction by a whole number. Divide the numerator by the whole number, when it can be done without a remainder ; but when this cannot be done, multiply the denominator by the whole number. 13. Divide by 3. Second Method. First Method. 5--3=ş, or 4. Ans. 14. Divide 1} by 6. 16. Divide by 7. 15. Divide 19 by 8. QUEST.--138. How is a fraction divided by a whole number? 18. Divide 45 by 9. 20. Divide 121 by 25. 19. Divide 73 by 8. CASE II. 22. At $ of a dollar a pound, how many pounds of honey can be bought for & of a dollar ? Suggestion. Since of a dollar will buy 1 pound, $ of a dollar will buy as many pounds as & is contained times in f. Now $ is contained in 4, 3 times. Ans. 3 pounds. 23. At of a dollar a bushel, how much barley can be bought for of a dollar ? We first reduce the fracFirst Operation. tions to a common denominator; (Art. 125 ;) then divide the numerator of the dividend 16 2=13. Ans. by the numerator of the divi sor, as above. Obs. 1. After the fractions are reduced to a common denominator, it will be perceived that no use is made of the common denominator itself. In practice, therefore, it is simply necessary to multiply the numerator of the dividend by the denominator of the divisor, and the denominator of the dividend by the numerator of the divisor, in the same manner as two fractions are reduced to a common denominator; or, what is the same in effect, invert the divisor, and proceed as in multiplication of fractions. (Art. 135.) Note:-To invert a fraction is to put the numerator in the place of the denominator, and the denominator in the place of the numerator. Thus, in the example above, inverting the divisor ş, it becoines į; and f*=*, or 15, which is the same as before. Again, we may also illustrate the principle thus. Second Operation. Dividing the dividend & by 2, $=2= the quotient is . (Art. 113.) But it 3x5=1 is required to divide it by only of And =1. Ans. 2; consequently the f is 5 times too small for the true quotient. Therefore f multiplied by 5 will be the quotient required. Now X5=*, or lž, which is the same result as before. OBs. 2. By examination the learner will perceive that this process is precisely the same in effect as the preceding ; for in both cases the denominator of the dividend is multiplied by the numerator of the divisor, and the numerator of the dividend, by the denominator of the divisor. Hence, 139. To divide a fraction by a fraction. I. If the given fractions have a common denominator ; Divide the numerator of the dividend by the numerator of the divisor. II. When the fractions have not a common denominator; Invert the divisor, and proceed as in multiplication of fractions. (Art. 135.) Obs. 1. Compound fractions occuring in the divisor or dividend, must be reduced to simple ones, and mixed numbers to improper fractions 2. The method of dividing a fraction by a fraction depends upon the obvious principle, that if two fractions have a common denominator, the numerator of the dividend, divided by the numerator of the divisor, will give the true quotient. Multiplying the numerator of the dividend by the denominator of the divisor, and the denoininator of the dividend by the numerator of the divisor, is, in effect, reducing the two fractions to a common denominator. The object of inverting the divisor is simply for convenience in multiplying. 24. Divide of by 11. Solution. f of 4 =1, and 1}= Now == x), or 16. Ans. 25. Divide 75 by 21. 26. Divide 13} by š. 28. Divide by 46. Ans. 33. Quest.—139. How is one fraction divided by another when they have a common denominator? How when they have not common denomina. tors ? Obs. How proceed when the divisor or dividend are compound fractions or mixed numbers ? Upon what principle does the method of dividing a fraction by a fraction, depend? Why multiply the numerator of the dividend by the denominator of the divisor, &c. ? Why invert the divisor ? 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 foff by ¢ of io. Operation. After canceling all the factors com mon buth. to the numerators and de31 & 8 3 nominators, we divide the continued 14 product of the numbers remaining on 3 1 10,5 the right of the line by the continued 3 5=, or 13. Ans. product of those on the left, and the quotient is the answer. 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. 31. Divide 4} by 24. Ans. 2. 33. Divide 4f by 24. 35. Divide of 1 by . 37. Divide 16% by 2 39. Divide 256 by 6. 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 f 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 ? ; now inverting the divisor, we have 1 xj= 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 4. 42. Divide 35 by z. 44. Divide 165 by s. 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 fraction 41 is the same as a = ; for, the numerator 41=ġ, and 11' the denominator 1$=; but the numerator of a fraction is a dividend and the denominator a divisor. (Art. 109.) Now=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. 2} 46. Reduce to a simple fraction. 52 Operation. 2}=ž, 54= x . 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 ? |