117. Reduction to Improper Fractions. An integer may be re duced to an improper fraction. In a similar manner a mixed number may be reduced to an improper fraction. = 1, 6p = 2, = 4. Also, = = 1층, 글 = = fraction may be reduced to an integer or to a mixed number. 31. Thus, To reduce an improper fraction to an integer, or to a mixed number, divide the numerator by the denominator. The quotient is the integral part of the mixed number, and the remainder is the numerator of the fractional part, while the divisor is its denominator. 119. Numbers in the Simplest Form. A number in which every fraction is a proper fraction reduced to its lowest terms is said to be in its simplest form. Numbers representing final results should always be put into the simplest form. Reduce each of the following to integers or to mixed numbers: 27. 174 29. 81 37. If the numerator of a fraction is a multiple of the denominator, to what kind of number may the fraction be reduced? 38. Change 2, 4, and to other fractions. 120. Cancellation. The property of a fraction by which its numerator and denominator may both be multiplied or divided by the same number without changing its value may be used in simplifying many examples in division. Example. Divide 34 × 45 × 64 by 96 Solution. (34 × 45 × 64) ÷ (96 × 84 × 16) 17 5 15 48 28 12 4 = = 17 X 5 85 = × 84 × 16. 34 × 45 × 64. 96 x 84 x 16 First divide 64 and 16 by 16, 45 and 84 by 3, and 34 and 96 by 2. Then divide 4 and 48 by 4 and finally 15 and 12 by 3. In the numerator the factors 17 and 5 are left and in the denominator the factors 4 and 28 are left. The method used in simplifying this fraction is called cancellation. EXERCISES By means of cancellation simplify each of the following as much as possible and then reduce the result to the simplest form: 1. 36 X 34 × 72 × 12 ÷ 48 × 27 × 14 × 18. 2. 108 X 90 × 84 × 64 ÷ 32 × 26 × 45. 3. 16 X 18 × 32 × 42 × 54 ÷ 24 × 14 × 28. 4. 42 X 19 X 9 X 12 X 16 ÷ 35 X 24 X 12. 5. 76 X 43 X 36 X 98 ÷ 22 X 18X 12. 6. 64 X 36 X 94 X 18 ÷ 12 X 34 X 46. 7. 34 X 97 X 105 X 16 ÷ 12 X 28 X 46. 8. 82 × 68 × 72 × 96 ÷ 48 × 16 × 27. 9. 94 × 36 × 48 × 56 ÷ 18 X 24 X 14. 10. 12 X 15 X 25 × 36 ÷ 8 × 18 X 46. 11. 72 × 81 × 90 × 32 ÷ 54 × 27 × 12. 12. 154 X 124 × 360 ÷ 105 X 18 X 6. 13. 15 X 24 × 32 ÷ 30 × 18 × 12. 14. 48 × 36 × 27 ÷ 32 X 16 X 42. 15. 14 X 17 X 21 X 6 ÷ 9 × 24 × 42. CHAPTER IX ADDITION AND SUBTRACTION OF FRACTIONS 121. Addition and Subtraction of Fractions in Business. — The addition and subtraction of simple fractions is of very common occurrence in business. Sometimes even more complicated fractions occur. 122. Reducing Fractions to a Common Denominator. Even such simple fractions as and cannot be added until they are changed into fractions having a common denominator. The first step in adding fractions not having a common denominator is to change them into equal fractions having a common denominator. A fraction can always be changed into other fractions whose denominators are multiples of the given fraction. Thus, can be changed into fractions whose denominators are multiples of 4, and to no other fractions. That is, can be changed into §, 12, 1, but not into 6ths, or 9ths, or 10ths. If fractions such as and are to be changed into fractions having a common denominator, the new denominator must be a common multiple of 3 and 4. Hence the first step in changing fractions into fractions having a common denominator is to find a common multiple of the denominators of the given fractions. That is, to change,, into fractions having a common denominator we must first find a common multiple of 2, 3, 4. In practice we select the L. C. M. We see at once that the L. C. M. of 2, 3, 4 is 12. The process is then as follows: Since both terms of each fraction must be multiplied by the same number, it follows that both terms of must be multiplied by 6, both terms of by 4, and both terms of by 3. WRITTEN EXERCISES Find the L. C. M. of each of the following sets of numbers. (See We see at once that the L. C. M. of the denominators is 12. The nu In practice we omit writing the denominator. merators then appear in a column and are readily added. WRITTEN EXERCISES 123. Adding Mixed Numbers. To add mixed numbers, add the integers separately, and the fractions separately. Thus, 3+4+5=12+12+2+2 = 122. The common denominator is 3 x 7 and the numerators are 3 and 7. Hence The common denominator is the product of the denominators and the numerator of the sum is the sum of the denominators. Again the denominator of the sum is the product of the denominators and the numerator is twice the sum of the numerators. |