Let it be required to reduce to the least common de nominator the fractions , to, and 44. If we take the least common multiple of the denominators 12, 16, and 24, which is 48, and divide it in turn by these denominators, we shall obtain the respective quotients 4, 3, and 2. Hence, if we multiply the numerator and denominator of each fraction by 4, 3 and 2 respectively, they will become 48, and the These fractions are equivalent to the original ones, and have their least common denominator. Hence fractions may be reduced 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. 41, which will be their least common denominator. Divide this denominator by the respective denominators of the given fractions ; multiply the quotients thus obtained by the respective numerators, and the several products will be the new numerators. Repeat the Rule for reducing fractions to their least common denominator. EXAMPLES. 1. Reduce the 17, H4, to equivalent fractions having the least common denominator. The least common multiple of the denominators 12, 15 24, is 120=common denominator. New numerator of first fraction 420 x 5=50. Hence, the fractions, when reduced to their least common denominator, become 2. Reduce 4 of 4 of 7o, tu, 75, to equivalent fractions having the least common denominator. Ans. tu, The, pero 3. Reduce 37, 41, ., to equivalent fractions having the least common denominator. Ans. 100, 130, . 4. Reduce , 75, 7, to equivalent fractions having the least common denominator. Ans. 164, dt, 187. 5. Reduce t, f, 6., to equivalent fractions having the least common denominator. Ans. , 150, 2003 2002 6. Reduce }, \, 37, and }, to equiva.ent fractions having the least common denominator. Ans. 20, 46, 10, LT. 7. Reduce to, }, 7, , to equivalent fractions having the least common denominator. Ans. ito, 700, , %. 8. Reduce \, , , , , to equivalent fractions having the least common denominator. Ans. 46, 47, 48, 54, 37. 9. Reduce $, , , , , T o, to equivalent fractions having the least common denominator. Ans. So, ito, moto, 197, 198, Tho. 10. Reduce , }, 4, 5, 6, 7, }, }, to equivalent fractions having the least common denominator. Ans. 194, , , 20%, 3, %. ADDITION OF FRACTIONS. 43. Suppose we wish to add 4 and We know that Lo long as these fractions have different denominators, they cannot be added any more than pounds and yards can be added together. We will therefore reduce them to a common denominator. We thus obtain = Now, taking their sum, we obtain 3 4 15 28 15+28 =*=135 43 775-35735- 35 - 35 Hence, to add fractions, we have this RULE. 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 h, j, 4, and ? ? These fractions, when reduced to their least common denominator, are 12, 1, in, and , the sum of whose numerators is 6+4+3+2=15. Hence we have }+i+iti=i= =11. 2. What is the sum of 4 and } ? Ans. 7. 3. What is the sum of }, to, . Ans. . 4. What is the sum of 1, $, 7z? Ans. 44=10 5. What is the sum of 16, te, i? Ans. He 6. What is the sum of 1, 46, 17, 7%? Ans. 5=113. NOTE:-If any of the fractions are compound, they must first be reduced to simple fractions, (by Rule under Art. 39.) 7. What is the sum of 1 of of 4, of , and f? These fractions, when reduced to their simplest forms, are , ta, and }; which, when reduced to their least common denominator, become 1, 3, Hence, their sum is 4+2+9 15 5 2423 8. What is the sum of 1 of 6 of 19, of of 6, and of of 3? . ! Ans. 471. 9. What is the sum of of of 8, } of of 10, and } of 16? Ans. 81. 10. What is the sum of of off, of f of $, and % of ? Ans. To 11. What is the sum of 5, 4, 5, and ?? Ans. 136=2167. 12. What is the sum of 1, 3, 4, 5, and 11? Ans. H=3 13. What is the sum of 7, , , , and 18? Ans. H=315... 14. What is the sum of 37, 4 of 1, $ of of 4, and ? Ans. 2 =476. 15. What is the sum of £ of %, of 4, 5 of 4, and of po? Ans. 157=2105 16. What is the sum of 1, }, }, }, }, 7, i, }? Ans. 4628=1441. 17. What is the sum of 1, 1, 1, à, i ? Ans. 187=277. 18. What is the sum of , , , ? Ans. 347=577. 19. What is the sum of 1, }, }, {, i, d, to it, it is? Ans. 185333=1344373. 20. What is the sum of 5, 7, 4, 6, 10, ii, ii, ii, 14? Ans. 43688393=1147). color 5 SUBTRACTION OF FRACTIONS. 44. SUPPOSE we wish to subtract + from £ We know that so long as these fractions have different de. nominators, the one cannot be subtracted from the other any more than pounds can be subtracted from yards. Wo therefore reduce them to a common denominator, and obtain }=; = Now, taking their difference, we obtain - = - =is. Hence, to subtract one fraction from another we have this RULE. Reduce the fractions to a common denominator ; suturact the less numerator from the greater, and place the common denominator under the difference. Repeat this Rule. EXAMPLES 1. From subtract } = ; }=; 5—2=3. Therefore we have {-}=t. 2. From subtract. Ans. $. 3. From 1 subtract ž. Ans. A. 4. From subtract. Ans. The 5 From 4 subtract . Ans. $. 6 From to subtract do. Ans. st. 7. From 185 subtract 15. Ans. Por 8. From 364 subtract Bato Ans. 19. Note.-As in Addition, if either of the fractions is compound, it must first be reduced to its simplest form. 9. Froin ļ of of subtract too Ans. 4. 10. From subtract of .. Ans. z. 11. From I of 4 subtract 4 off. 12. From 3 subtract 21. Ans. 14 Ans. |