98. Other changes in the Terms of a Fraction. 1. Reduce to an equivalent fraction having 25 for its nume rator, SOLUTION. Observing that the proposed numerator, 25, is 5 times the given numerator, 5, we multiply both terms of § by 5, which gives § = 45. 2. Reduce to an equivalent fraction having 15 for a numerator. SOLUTION.— Observing that the proposed numerator, 15, is 15, or § of the given numerator, 6, we multiply both terms of by §, or by 24, which gives 6 = 7 15 171 (a.) In the fractions written below substitute an appropriate denominator in place of x. SOLUTION. Observing that the proposed denominator, 24, is 3 times the given denominator, 8, we multiply both terms of § by 3, which gives = 15 10. Reduce to sixths. SOLUTION.-Observing that the proposed denominator, 6, is of the given denominator, 8, we multiply both terms of by which gives 5 3 32 6 NOTE. The following form of solution is sometimes adopted instead of the preceding: (a.) Fractions have a COMMON DENOMINATOR when their denominators are alike. ILLUSTRATIONS. and have the common denominator, 9. and do not have a common denominator. (b.) Fractions having different denominators can be reduced to equivalent fractions having a common denominator. This is called REDUCING THEM TO A COMMON DENOMINATOR, and is illustrated in the last six examples of the preceding article. (c.) In reducing fractions to a common denominator — 1st. Reduce the compound and complex fractions, if any, to simple ones. 2d. Reduce the simple fractions to their lowest terms, except when to do it would increase the labor of reducing to a common denominator. 3d. Select a convenient number for the common denominator. Usually it will be best to select the least common multiple of the denominators. 4th. Multiply both terms of each fraction by the number which gives the common denominator as the denominator of the resulting fraction. 1. Reduce, 1, 11, 18, and 18 to a common denominator. SOLUTION. The least common multiple of the given denominators, 4, 5, 12, 15, and 20, is 60. Hence, we select 60 as the common denominator. 15 times 4, we multiply both terms of by 15, which 12 times 5, we multiply both terms of 4 by 12, which, etc. NOTE. Many select the product of the denominators for a common denominator. This method usually involves larger numbers than the preceding, as will be seen in the following 2ND SOLUTION. - The product of the given denominators is 72000, which we select for the common denominator. To obtain this, we multiplied 4, the denominator of the first fraction, by all the other denominators, i. e. by 5, 12, 15, and 20. Hence, we must multiply the numerator by the same numbers, which gives 1888. We multiplied the denominator of the second fraction by all the other denominators, i. e. by 4, 12, 15, and 20; hence, we must multiply the numerator by the same numbers, which gives # = $1888, etc. = 100. Addition and Subtraction of Fractions. (a.) Fractions, like other numbers, must be of the same denomination in order to be added or subtracted. (b.) To be of the same denomination, they must be fractions of the same unit, and also have a common denominator. ILLUSTRATIONS. and cannot be added in their present form, any more than can 5 shillings and 3 pence. of a gallon and of a quart cannot be added, in their present form, any more than can 11 gallons and 11 quarts. (c.) Hence, to add or subtract fractions, it is necessary to reduce compound and complex fractions to simple ones, and fractions of different units (as, of a bushel and of a quart) to fractions of the same unit, and then to reduce the fractions thus obtained to a common denominator. The addition or subtraction can then be made by adding or subtracting the numerators, as illustrated in 77. Selecting 72 as the common denominator, and reducing the fractions to seventy-seconds, gives 45 +48 + $8 + 48 = W = 248. 2. What is the sum of† of £1 + 24 shillings? |
