CASE VI. 131. To reduce two or more fractions to a common denominator. A Common Denominator is a denominator common to two or more fractions. 1. Reduce and to a common denominator. OPERATION. 3 × 5 = 18 4 x 5 2 x 4 5 x 4 ANALYSIS. Multiply the terms of the first fraction by the denominator of the second, and the terms of the second fraction by the denominator of the first, (124.) This must reduce each fraction to the same denominator, for each new denominator will be the product of the given denominators. RULE. Multiply the terms of each fraction by the denominators of all the other fractions. Mixed numbers must first be reduced to improper fractions. EXAMPLES FOR PRACTICE. 2. Reduce,, and to a common denominator. 5. Reduce,,, and to a common denominator. Ans. 1, H, HI, H. 6. Reduce, and to a common denominator. 168 9336 7. Reduce §, 24, 4, and to a common denominator. Ans. H, H, lft, fa 8. Reduce 13, o, and 4 to a common denominator. 48 Ans. 150, ff, see. Case VI is what? What is a common denominator? Give analysis. Rule. CASE VII. 132. To reduce fractions to the least common denominator. The Least Common Denominator of two or more fractions is the least denominator to which they can all be reduced, and it must be the least common multiple of the lowest denominators. 1. Reduce,, and to the least common denominator. duce the fraction to the denominator, 24. Reducing each fraction to this denominator, by Case V, we have the answer. Since the common denominator is already determined, it is only necessary to multiply the numerators by the multipliers. RULE. I. Find the least common multiple of the given denominators, for the least common denominator. II. Divide this common denominator by each of the given denominators, and multiply each numerator by the corres-ponding quotient. The products will be the new numerators. EXAMPLES FOR PRACTICE. 2. Reduce,, H, and 4 to their least common denominator. Ans. 18%, 15%, ttt, 180. 3. Reduce, 4, 6, to their least common denominaAns. HI, HI, 33, 336· tor. 32 What is Case VII? What must be the least common denominator? Give analysis. Rule, first step. Second. 56 2529 4. Reduce,,, and 6 to their least common denominator. Ans. 1, v2. 5. Reduce 51, 21, and 1 to their least common denominator. Ans. 44, 18, 11. 6. Reduce,, 4, and † to their least common denomiAns. it, it, tit, tit. nator. 60 7. Reduce, 1, 4, 25, and to their least common denominator. Ans. HH, s, 18, 178, f. 8. Change,, 3, 9, and to equivalent fractions having the least common denominator. 9. Change, 14, 3, 4, and 6 to equivalent fractions having the least common denominator. 10. Change 2, 17, 4, 14, 11, and to equivalent fractions having the least common denominator. 11. Reduce, †, †, and to a common denominator. 12. Reduce, 4, 24, and 1⁄2 to a common denominator. 13. Reduce 1,,, and 31 to equivalent fractions having a common denominator. Ans. 8, 1, 8, 18. 360 405 864 14. Change,, and to equivalent fractions having a common denominator. Ans. 10%, 10. 15. Change, 7, 8, and 5 to equivalent fractions havAns. 4, 45, 48, 330. 16. Change, 64, 1%, 7, 8, and 14 to equivalent fractions having a common denominator. ing a common denominator. ADDITION. 133. 1. What is the sum of 1, †, §, and J? OPERATION. }+} + {+}== 2, Ans. ANALYSIS. Since the given fractions have a common denominator, 8, their sum may be found by adding their numerators, 1, 3, 5, and 7, and placing the sum, 16, over the common denominator. We thus obtain 1-2, the required sum. 63 13 ? 4. What is the sum of,, 2, 1, 1, and 134. 1. What is the sum of and †? OPERATION. 1209 Ans. 24. ANALYSIS. In whole numbers we can add like numbers only, or those having the same unit value; so in fractions we can add the numerators when they have a common denominator, but not otherwise. As § and have not a common denominator, first reduce them to a common denominator, and then add the numerators, 27+10=37, the same as whole numbers, and place the sum over the common denominator. RULE. I. When necessary, reduce the fractions to their least common denominator. II. Add the numerators, and place the sum over the common denominator. If the amount be an improper fraction, reduce it to a whole or a mixed number. R. P. Ans. 231 ANALYSIS. tions,, and Ans. 317. 252 The sum of the frac is 11; the sum of the integers, 7, 5, and 10, is 22; and the sum of both fractions and integers is 2311. Hence, Give second explanation? Rule, first step. Second. 5 To add mixed numbers, add the fractions and integers separately, and then add their sums. If the mixed numbers are small, they may be reduced to improper fractions, and then added after the usual method. 10. What is the sum of 144, 3, 1, and 18? Ans. 2148. 11. What is the sum of 3, 1, 10%, and 5? Ans. 184. 12. What is the sum of 17, 185, and 26,4? 13. What is the sum of,, 11, 3, and 11? 14. What is the sum of 1254, 327, and 251? Ans. 478 15. What is the sum of 48, 7, 116, 18, and 188? 160 Ans. 317. 16. What is the sum of 3%, 218, 404, and 10? 17. Bought 3 pieces of cloth containing 1253, 961, and 48 yards; how many yards in the 3 pieces ? 18. If it take 5 yards of cloth for a coat, 34 yards for a pair of pantaloons, and of a yard for a vest, how many yards will it take for all? Ans. 91. 19. A farmer divides his farm into 5 fields; the first contains 26 acres, the second 4014 acres, the third 514 acres, the fourth 594 acres, and the fifth 62 acres ; how many acres in the farm? Ans. 2411. 20. A speculator bought 1753 bushels of wheat for 2051 dollars, 3254 bushels of barley for 2964 dollars, 2701 bushels of corn for 20011 dollars, and 437 bushels of oats for 156 dollars; how many bushels of grain did he buy, and how much did he pay for the whole? Ans. (1209 bushels. by subtracting 3, the less numerator, from 7, the greater, and write How are mixed numbers added? Give note. |