CHAPTER VIII FRACTIONS, CANCELLATION 92. Use of Fractions in Business. Simple fractions occur frequently in business, and no one is competent to do the figuring required in everyday transactions unless he understands fractions. 93. Fractions. A number in the form is called a fraction. There are several definitions of fractions in general use. Thus may be regarded as one of the 3 equal parts of 2, or as 2 of the 3 equal parts of 1. In more advanced works on arithmetic, it is usual to define as a number such that 3 × 3 = 2. 94. Numerator, Denominator. A fraction is always written by means of two numbers (usually integers), one above a horizontal line, and the other below it. The number above the line is called. the numerator, and the one below it, the denominator. Every fraction is, therefore of the form: numerator denominator The denominator states into how many equal parts a unit has been divided; and the numerator states how many of these parts are represented by the fraction. Thus, in the fraction the denominator 5 states that a unit has been divided into five equal parts and the numerator 4 states that four of these parts are represented by the fraction. The numerator and denominator of a fraction are called the terms of the fraction. A fraction may also be regarded as an indicated quotient, the numerator being the dividend, and the denominator, the divisor. Thus the fraction may be regarded as the quotient when 4 is divided by 5. A problem in division is often represented in the form of a fraction. To deal effectively with fractions it is now necessary to study them more in detail. 95. A General Property of Fractions. From the figure it is 1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/4 1/4 1/2 1/4 1/4 1/2 1 These are instances of a general property of fractions: Multiplying both terms of a fraction by the same number does not change its value. Reading the above equations backward, we have } = }, } = 1, 4 = † = }. The above property may then be stated: Dividing both terms of a fraction by the same number does not change its value. These two rules may be stated in one as follows: Multiplying or dividing both terms of a fraction by the same number does not change its value. 96. Fractions in Lowest Terms. When the numerator and de nominator of a fraction have no common factor except 1, the fraction is said to be in its lowest terms. 97. Proper Fractions. A fraction whose numerator is less than its denominator is called a proper fraction. 98. Improper Fractions. A fraction whose numerator is not less than its denominator is called an improper fraction. Thus,,,are proper fractions, while,, are improper fractions. 99. Mixed Numbers. - A number consisting of an integer and a fraction is called a mixed number. 100. Reduction to Lowest Terms. - A fraction not in its lowest terms may be reduced to its lowest terms by dividing both numerator and denominator by their greatest common factor. 101. Reduction to Improper Fractions. An integer may be re In a similar manner a mixed number may be reduced to an im 102. Reduction to Integers or Mixed Numbers. An improper fraction may be reduced to an integer or to a mixed number. 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. 103. 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: 26. 25991 28. 8493 30. 15051 32. 71247 34. 91452 36. 62051 27 37 33 45 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, 4, and to other fractions, 104. 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 × 84 X 16. Solution. (34 × 45 × 64) ÷ (96 x 84 x 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 × 19 × 9 × 12 × 16 ÷ 35 × 24 × 12. 5. 76 × 43 × 36 × 98 ÷ 22 × 18× 12. 6. 64 × 36 × 94 × 18 ÷ 12 × 34 × 46. 7. 34 × 97 × 105 × 16 ÷ 12 × 28 × 46. 8. 82 × 68 × 72 × 96 ÷ 48 × 16 × 27. 9. 94 × 36 × 48 × 56 ÷ 18 × 24 × 14. 10. 12 X 15 × 25 × 36 ÷ 8 × 18 X 46. 11. 72 X 81 X 90 X 32 ÷ 54 × 27 X 12. 12. 154 X 124 X 360 ÷ 105 X 18 X 6. 13. 15 X 24 X 32 ÷ 30 X 18 X 12. 14. 48 X 36 X 27 ÷ 32 X 16 X 42. 15. 14 X 17 X 21 × 6 ÷ 9 × 24 × 42. CHAPTER IX ADDITION AND SUBTRACTION OF FRACTIONS The 105. Addition and Subtraction of Fractions in Business. addition and subtraction of simple fractions is of very common occurrence in business. Sometimes even more complicated fractions occur. 106. 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. This, however, is easily done, since = = 2. 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 §, 1, 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 1, 1, 2 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. |