the same in effect as the preceding; for, in both cases dividend is multiplied by the numerator of the divisor the dividend, by the denominator of the divisor. 12. Divide of by 21. 13. Divide 8 by 34. 14. Divide 2 by . 15. Divide by 18. Ans. 16. Divide 17. Divide 231. The process of dividing fractions n tracted by canceling equal factors in the divis (Art. 146;) or, after the divisor is inverted, by which are common to the numerators and denomi 18. Divide of 4 of by of of ✈. Operation. 31 74 11 45 23 17 115 Ans. rule above. Hence, For convenience we arra tors, (which answer to di right of a perpendicular li nominators, (which answer the left; then canceling the and 7, which are common (Art. 151,) we multiply the tors in the numerators tog remaining in the denominat 232. To divide fractions by CANCELATION. Having inverted the divisor, cancel all the factor to the numerators and denominators, and the prod maining on the right of the line placed over the pr remaining on the left, will be the answer required. OBS. 1. Before arranging the terms of the divisor for cancela necessary to invert them, or suppose them to be inverted. 2. The reason of this contraction is evident from the princ numerator and denominator of a fraction are both divided by th the value of the fraction is not altered. (Arts. 148, 191.) 19. Divide 183 by 63. Answer QUEST. 232. How divide fractions by cancelation? How arrange th given fractions? Obs. What must be done to the divisor before arranging does it appear that this contraction will give the true answer? 20. Divide of & by of. 23. Divide of 73 by of §. 24. Divide of 4 of 3 by #. 25. Divide 2 of 7 by 5% of 42. 21. Divide of 14 by 63. 22. Divide 154 by of 3. 26. Divide 4 of 11 of 12 of 19 by 3 of 30 of 3 of 5. CASE III. 233. Dividing a whole number by a fraction. 27. How many pounds of tea, at of a dollar a pound, can be bought for 15 dollars? Reducing the Analysis. Since of a dollar will buy 1 pound, 15 dollars will buy as many pounds as is contained times in 15. dividend 15, to the form of a fraction, it becomes ; (Art. 197. Obs. 1;) then inverting the divisor and proceeding as before, we have 4x4, or 20. Ans. 20 pounds. Or, we may reason thus: is contained in 15, as many times as there are fourths in 15, viz: 60 times. But 3 fourths will be contained in 15, only a third as many times as 1 fourth, and 60÷3=20, the same result as before. Hence, 234. To divide a whole number by a fraction. Reduce the whole number to the form of a fraction, (Art. 197. Obs. 1,) and then proceed according to the rule for dividing a fraction by a fraction. (Art. 229.) 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. Or, reducing the dividend to a fraction having the same denominator, (Art. 197. Obs. 2,) we may divide one numerator by the other. (Art. 229. I.) 2. If the divisor is a unit or 1, the quotient is equal to the dividend; if the divisor is greater than a unit, the quotient is less than the dividend; and if the divisor is less than a unit, the quotient is greater than the dividend. 28. How much cloth, at 34 dollars per yard, can you buy for 28 dollars? QUEST.-234. How is a whole number divided by a fraction? Obs. How by a mixed number? 235. When the divisor is 31, 331, 3331, &c. Multiply the dividend by 3, divide the product by 10, 100, or 1000, as the case may be, and the result will be the true quotient. (Art. 131.) OBS. The reason of this contraction will be understood from the principle, that if the divisor and dividend are both multiplied by the same number, the quotient will not be altered. (Art. 146.) Thus 31X3=10; 33×3=100; 333X3=1000, 333X3 1000, &c. 35. At 3 dollars per yard, how many yards of cloth can be bought for 561 dollars? Operation. dolls. 561 We first multiply the dividend by 3, then divide the product by 10; for, multiplying the divisor 3 by 3, it becomes 10. (Art. 146.) 3 10)1683 Ans. 168 yds. 36. Divide 687 by 331. 37. Divide 453 by 331, Ans. 20. 38. Divide 2783 by 333. 236. When the divisor is 13, 163, 166%, &c. Multiply the dividend by 6, and divide the product by 10, 100, or 1000, as the case may be. OES. This contraction also depends upon the principle, that if the divisor and dividend are both multiplied by the same number, the quotient will not be altered. (Art. 146.) Thus, 13×6=10; 163×6=100; 1663×6=1000, &c. 39. What is the quotient of 725 divided by 16? Solution.-725×6=4350; and 4350÷100=43 40. Divide 367 by 14. 41. Divide 507 by 16. Ans. 42. Divide 849 by 16. 43. Divide 1124 by 166. 237. When the divisor is 11, 111, 1111, &c. Multiply the dividend by 9, and divide the product by 10, 100, or 1000, as the case may be. OBS. This contraction depends upon the same principle as the preceding Thus, 19-10; 1149=100; 1114X9-1000, &c. 44. Divide 587 by 11. Solution.-587X9=5283, and 5283÷100=52,83 Ans. 45. Divide 861 by 13. 46. Divide 4263 by 114. Ans. 7741. 47. Divide 6037 by 111. Note.--Other methods of contraction might be added, but they will naturally suggest themselves to the student, as he becomes familiar with the principles of fractions. 238. From the definition of complex fractions, and the manner of expressing them, it will be seen that they arise from di41 vision of fractions. (Art. 183.) Thus, the complex fraction is 11' the same as÷; for, the numerator, 41–2, and the denominator 14; but the numerator of a fraction is a dividend, and the denominator a divisor. (Art. 184.) Now, ÷÷4=16. which is a simple fraction. Hence, 239. To reduce a complex fraction to a simple one. Consider the denominator as a divisor, and proceed as in division of fractions. (Arts. 229, 232.) OBS. The reason of this rule is evident from the fact that the denominator of a fraction denotes a divisor, and the numerator, a dividend; (Art. 184;) nence the process required, is simply performing the division which is expressed by the given fraction. QUEST.-238. From what do complex fractions arise? 239. How reduce them to simple fractions? Solution.-444, and 74-22. (Art. 197.) Reduce the following complex fractions to simple ones: 240. To multiply complex fractions together. First reduce the complex fractions to simple ones; (Art. 239;) then arrange the terms, and cancel the common factors, as in multiplication of simple fructions. (Art. 219.) OBS. The terms of the complex fractions may be arranged for reducing them to simple ones, and for multiplication at the same time. Operation. 217 125 712 92 Place The numerator 31. (Art. 197.) the 7 on the right hand and 2 on the left of the perpendicular line. The denominator 2 =2, which must be inverted; (Art. 239;) i. c. place the 12 on the right and the 5 on the left of the line, 14, and 44, 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 answer is §. (Art. 219.) 955. Ans. QUEST.-240. How are complex fractions multiplied together? 241. How is one complex fraction divided by another? |