Since make a unit, there will be 4 times as many 4ths as units; .., in 3 units there will be 4 times 3 fourths 12, and in 3 there will be 12+1=4. Hence, To reduce a mixed number to an improper fraction, RULE.-Multiply the whole number by the denominator of the fraction; to the product add the numerator, and under the sum write the denominator. (a) To reduce an integer to a fraction having any given denominator:-Multiply the integer by the proposed denominator, and under the product write the denominator (60). 15. Reduce 9 to a fraction whose denominator is 5. Ans. 5. 16. Reduce 7 to a fraction whose denominator is 1. Ans. 7 (127, d). 17. Reduce 87 to a fraction whose denominator is 87. 18. Reduce 7345 to a fraction whose denominator is 372. 19. Reduce 47 to a fraction having 18 for a denominator. 20. Reduce 734 to a fraction having 173 for a denominator. The operation required is only to divide a fraction by a frac tion; thus, } 1÷÷4=1×2. Hence, To reduce a complex fraction to a simple one, RULE.-First, if necessary, reduce the numerator and denominator of the complex fraction each to a simple fraction; then divide the fractional numerator by the fractional denominator (128). (a) The above rule is always applicable, but examples like the 7th may also be reduced by Art. 126; thus, This example may also be reduced by multiplying both numerator and denominator of the complex fraction by 7 (60, fraction by multiplying the whole number by the denominator, and then dividing the product by the numerator. 9 CASE 9. 133. Fractions having like denominators are said to have a common denominator; thus, & and have a common denominator; so, also, have, and ; but and have not; however, and will be changed to equivalent fractions having a common denominator, if the terms of the 1st be multiplied by 7, and the terms of the 2d by 3; i. e. if the terms of each fraction be multiplied by the denominator of the other (60, Cor.); thus, }=}×7=11, and = X; i. e. 3 and § § and, fractions that have a common denominator. = Similar explanations may be given when there are more than 2 fractions. Hence, To reduce fractions to a common denominator, RULE 1.-Multiply all the denominators together for a common denominator, and multiply each numerator into the continued product of all the denominators except its own, for new numer ators. Ex. 1. Reduce 3, 4, 3 and 3 to a common denominator. 12540. 2. Reduce 12, 75, 13 and 14 to a common denominator. 7140 Ans. 19, 1980, 18388 and 18348. 3. Reduce, 4, and 4. Ans. 288, 315, 336 and 390. 420 420 420 420 (a) The above rule is always applicable, but it will not always give the least common denominator; this, however, may be effected by the following: RULE 2. Reduce each fraction, if necessary, to its lowest terms (129); find the least common multiple of the denominators (114) for a common denominator; and, having divided this multiple by each denominator, multiply the several quotients by the respective numerators, for new numerators. NOTE 1. - Each of these rules is founded on the principle that multiplying both terms of a fraction by the same number does not alter its value (60, Cor). 15. Reduce, 12, 1's and to their least common denomi 1, 3, 1 12 × 11 = 33, 4th numerator. ··· 3, 12, 78 and 11 = 41, 42, 43 and 33, Ans. |