76. We are now prepared to reduce fractions to their least common denominator by the following RULE. Reduce the fractions to their simplest form; then find the least common multiple of their, denominators, (by Rule under Art. 75,) which will be their least common denominator. Divide this denominator by the respective denominators of the given fractions, and multiply the quotients by their respective numerators, and the respective products will be the new numerators. Repeat the Rule for reducing fractions to their least common denominator. 5 11 EXAMPLES. 1. Reduce 2,76,, to equivalent fractions having the least common denominator. The least common multiple of the denominators 12, 16, 24, is 48 common denominator. = New numerator of first fraction 48 X 5=20. X 7-21. Hence, the fractions, when reduced to their least common denominator, become 20, 21, 23. 48' 48" 48° 3 7 2. Reduce of of 12, 2015, to equivalent fractions having the least common denominator. Ans. 120, 120, 120 3. Reduce 3, 43, 8, to equivalent fractions having the least common denominator. 5 309 30 30 Ans. 105 130 36 5. Reduce, IT, 622, to equivalent fractions having the least common denominator. Ans. 88 150 2025 330 330 330 6. Reduce 1, 3, 34, and }, to equivalent fractions having the least common denominator. Ans. 30 40 195 12 60 60 60 60 7. Reduce,,,, to equivalent fractions having the least common denominator. 70 30 40 Ans. 16, 210, 210. 1o, K 9 8. Reduce,,, 5, 25, to equivalent fractions having the least common denominator. Ans. 40, 45, 48, 88, 23. ADDITION OF FRACTIONS. 50 27 77. Suppose we wish to add and . We know that so long as these fractions have different denominators, they can not be added; we will therefore reduce them to a common denominator. We thus obtain Reduce the fractions to a common denominator, and take the sum of their numerators, under which place the common denominator, and it will give the sum required. NOTE. The labor will be the least when we reduce the fractions to their least common denominator. EXAMPLES. 1. What is the sum of 1, 3, 1, and } ? 6 4 These fractions, when reduced to their least common denominator, are,,, and, the sum of whose numerators is 6+4+3+2=15. Hence we have }+{+1+}=}{={=14. 2. What is the sum of and ? 3. What is the sum of, 10, 20 1 3 ? 48' 12 4. What is the sum of 1, 3, 5. What is the sum of 45 5 3 15' 12' 10 6. What is the sum of 2, 4 ? NOTE. If any of the fractions are compound, they must first be reduced to simple fractions (by Ruleunder ART. 73). 7. What is the sum of of of, of, and ? These fractions, when reduced to their simplest forms, are,, and 3, which, when reduced to their least common denominator, become 3 4 9 13. What is the sum of 1, 2, 3, fo, and 18? 14. What is the sum of 3, of }, 504 Ans. 11=33. Ans. 18=310 of 2 of, and I's? Ans. 20-460 of, 3 of 3, 5 of 8, and of 1 1 1 1 47 Ans. 253=2105. 111? 9 95 6 7 8 9 4609 Ans. 1998-130202520 SUBTRACTION OF FRACTIONS. 2520 78. To subtract one fraction from another we have this RULE. Reduce the fractions to a common denominator, and sub truct the numerator of the subtrahend from that of the mirend; place the common denominator under the difference. Repeat this Rule. 1. From subtract . EXAMPLES.. Reducing these fractions to a common denominator, they become and subtracting the numerators we Therefore we have find 5-2-3. 480 120. 360 Ans. 53 105 Ans. 13 40 NOTE. As in addition, if either of the fractions is compound, it must first be reduced to its simplest form. 140' Ans. 23. subtract of. 14. From of of of, subtract of of of &. 45 MULTIPLICATION OF FRACTIONS. 79. Multiply by . We know (ART. 73), that multiplied by is the same as of. Hence, we must use the same rule as for reducing compound fractions. Therefore, to multiply fractions, we have this RULE. Multiply all the numerators together for a new numerator, and all the denominators together for a new denominator : always observing to reject, or cancel, such factors as are common to both numerators and denominators. If any of the factors are whole numbers, they may be made to take the form of a fraction by giving to them a denominator of 1; (see ART. 30,) and then the general rule will apply. What is the Rule for multiplying fractions? In this example, we cancelled the 4 of the numerator against a part of the 16 of the denominator; and 5 of the denominator against a part of the 10 in the numerator. Thus : Finally, cancelling the 2 in the numerator against a part of the 4 in the denominator, we find NOTE.-A little practice will enable the student to perform these operations of cancelling with great ease and rapidity. And since, as was remarked under ART. 73, it is immaterial which factors are first cancelled, the sim |