Suggestion.-Since in 1 unit there are 3 thirds, in 13, there are 13 times as many. We therefore reduce the 13 to thirds, by multiplying it by 3, because 3 thirds make a whole one; and adding the 2 thirds, we have 41 thirds. Hence, 91. To reduce a mixed number to an improper fraction. Multiply the whole number by the denominator of the fraction, and to the product add the given numerator. The sum placed over the given denominator, will form the improper fraction required. OBS. 1. A whole number may be expressed in the form of a fraction without altering its value, by making 1 the denominator. 2. A whole number may also be reduced to a fraction of any denominator, by multiplying the given number by the proposed denominator; the product will be the numerator of the fraction required. Thus, 25 may be expressed by 25, 100, or 400, &c. 26. Reduce 7 to an improper fraction. Ans. 1. 27. Reduce 83. 29. Reduce 25%. 31. Reduce 43,7. 33. Reduce 108%.. 35. Reduce 63 to 4ths. 28. Reduce 144. 34. Reduce 21011. 36. Reduce 225 to 11ths. CASE IV. Reducing compound fractions to simple ones. 37. Reduce of to a simple fraction. ៖ Suggestion. Multiplying the two numerators together, and the two denominators, we have; and re 12 duced to its lowest terms is equal to , which is the answer required. Hence, 12. Obs. QUEST.-91. How reduce a mixed number to an improper fraction? How express a whole number in the form of a fraction? How reduce a whole number to a fraction of a given denominator? 92. To reduce compound fractions to simple ones. Multiply all the numerators together for a new numerator, and all the denominators for a new denominator. OBS. If the example contains whole or mixed numbers, they must be reduced to improper fractions, before multiplying. 38. Reduce of of 21 of 4 to an improper fraction. Solution. The expression 2 of 4 of ; and ×××1=1, or 21. Ans. 39. Reduce of 4 of 7. 41. Reduce 4 of 12 of. 43. Reduce 1 45. Reduce 5 46. Reduce of 2 of 12. 40. Reduce of of 1 of 5. 42. Reduce § of 3 of 3 of 31. 44. Reduce 7 of 41 of 45. of 2 of 3 of 17. of 3 of 21 of 73. 8 0 Contraction by Cancellation. 47. Reduce of of 3 of to a simple fraction. Suggestion. Since the product 1 Operation. 2.3 5 5 of of of 3 7 28 of the numerators is to be divided by the product of the denominators, we cancel the factors 2 and 3, which are common to both. This divides the terms of the new fraction by the same number, and therefore does not alter its value. Then multiplying the remaining factors together, we have, the answer required. Hence, 93. To reduce compound fractions to simple ones by CANCELLATION. Cancel all the factors common to the numerators and denominators; then multiply the remaining factors together as before. 48. Reduce of of to a simple fraction. Ans. 4. 49. Reduce of 2 of 4. 51. Reduce of of 1⁄2 of 11. 53. Reduce of 15 of 3 of 5. 54. Reduce 3 of 3 55. Reduce of 13 of 33. 56. Reduce of 51 60. Reduce 21 of 2 of 7 of 110. Note. For method of reducing Complex Fractions to Simple ones, see Art. 143. CASE V.-Reducing fractions to a common denominator. 94. Two or more fractions have a common denominator, when they have the same denominator. 61. Reduce 2, 3, and 3 to a common denominator. Suggestion.-After each numera 1 2 Operation. 1 × 3 × 5 15 -= 2 x X 5 30 2 2 × 2 × 5 20 tor, we write all the denominators of the given fractions, except its own, with the sign x between them, and under each line place all the denominators in like manner. Then performing the multiplications indicated, the products are equal to the given fractions, and are the answer required. Hence, 95. To reduce fractions to a common denominator. 3 5 Multiply each numerator into all the denominators except its own, for a new numerator, and all the denominators together for a common denominator. OBS.-Compound fractions must be reduced to simple ones, and mixed numbers to improper fractions, before attempting to reduce them to a common denominator. 62. Reduce of †, 21, and 2 to a com. denominator. Suggestion of 3, 212, and 2. Reducing ,, and to a common denominator, they become 54, and 4. Ans. QUEST. 95. How reduce fractions to a common denominator? 8 247 Reduce the following to a common denominator. 63. Reduce 3, 3, 2 64. Reduce,, and 13. 66. Reduce, 21, and 37. 68. Reduce 1, 5, and 83 70. Reduce, 41, and 3. 250 CASE VI.-Reducing fractions to their least common de nominator. 71. Reduce 2, 3, and to the least com. denominator. Suggestion.We first find the Operation. 2)4"6"8 2)2" 3"4 1"3"2 2 × 2 × 3 × 2=24 least common multiple of all the given denominators, which is 24; and this is the least common denominator required. The next step is to reduce the given fractions to twenty-fourths without altering their value. The denominator of the first fraction, is contained in 24, 6 times; and multiplying both terms of the fraction by 6, it becomes 1. The denominator 6 is contained in 24, 4 times; and multiplying the fraction by 4, it becomes. The denominator 8 is contained in 24, 3 times; and multiplying the fraction by 3, it becomes. Now!,, and are å are equal to the given fractions 3, 2, and 3. Hence, 96. To reduce fractions to their least common denominator. I. Find the least common multiple of all the denominators of the given fractions, and it will be the least common denominator. II. Divide the least common denominator by the denominator of each of the given fractions, and multiply the numerator by the quotient; the products will be the numerators required. QUEST. 96.-How reduce fractions to the least common denominator. OBS.-If the example contains compound fractions, whole or mixed numbers, they must first be reduced to simple fractions, then all must be reduced to their lowest terms; otherwise the least common multiple of their denominators, may not be the least common denominator. 72. Reduce 3, 3, to the least common denominator. Suggestion.-First find the least common multiple of the denominators, which is 2 × 3 × 2=12. Now Operation. 12÷3=4, and 4 × 2=8, the 1st numerator. 2)3"4"6 73. Reduce,, and 3. 75. Reduce 2, 8, g, and 2. 77. Reduce, 5, and 17. 79. Reduce, 1, and 81. Reduce 83. Reduce of 14, and 3. of, and 163. 15. 74. Reduce,, and . 85. Reduce, 12, 11, and 2% to the common denomi nator 120. 0 86. Reduce, 12, 14, and 24 to the common denomi nator 144. 87. Reduce 1, 2, and 35 to the common denomi QUEST.-Obs. When the example contains compound fractions, whole or mixed numbers, how proceed? |