EXAMPLES FOR PRACTICE. 1. Reduce to a fraction having 24 for a denominator. 2. Reduce to a fraction whose denominator is 96. 2 3. Reduce 1 to a fraction whose denominator is 51. 4. Reduce to a fraction whose denominator is 78. 9 13 5. Reduce 62 to a fraction whose denominator is 3000. 375 Ans. 496 3000 6. Change 73 to a fraction whose denominator is 8. 8 7. Change 16 to a fraction whose denominator is 176. 8. Change 53 to a fraction whose denominator is 363. 9. Change 365 to a fraction whose denominator is 42. Ans. 1542. CASE VI. 182. 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 OPERATION. 3 × 9 5 × 9 7 x 5 9 × 5 = = 27 to a common denominator. ANALYSIS. We 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, (174, III). This must reduce each fraction to the same denominator, for each new denominator will be the product of the given denominators. Hence the RULE. Multiply the terms of each fraction by the denominators of all the other fractions. NOTE.-Mixed numbers must first be reduced to improper fractions. 3. Reduce, and to a common denominator. 50 60 Ans. 720 120 120. 1209 4. Reduce, 53 and 13 to equivalent fractions having a com 183. To reduce fractions to their least common denominator. The Least Common Denominator of two or more fractions is the least denominator to which they can all be reduced. 184. We have seen that all higher terms of a fraction must be multiples of its lowest terms, (181, II). Hence, I. If two or more fractions be reduced to a common denominator, this common denominator will be a common multiple of the several denominators. II. The least common denominator must therefore be the least common multiple of the several denominators. 1. Reduce , 12 and to their least common denominator. 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 corresponding quotient. The products will be the new numerators. NOTES.-1. If the several fractions are not in their lowest terms, they should be reduced to their lowest terms before applying the rule. 2. When two or more fractions are reduced to their least common denominator, their numerators and the common denominator will be prime to each other. EXAMPLES FOR PRACTICE. 1. Reduce and to their least common denominator. 2. Reduce, 7 12 Ans. 28, 18. and 1⁄2 to their least common denominator. 5 3. Reduce, and to their least common denominator. 4. Reduce, and to their least common denominator. 5. Reduce 6 10 14, 14 and 1 to their least common denominator. 9 84 84 84 Ans. 36 35 26 6. Reduce 4 25 and to their least common denominator. 13' 26 39 Ans. 7, 78, 78, 78° 52 24 75 8 17 and 37 to their least common denomi 7. Reduce 23, 15, 24 56 25 74 120 120, Ans. 313, 58, 120, 120. to their least common denominator. to their least common denominator. Ans. 68, 38, 3. 60 20 21 10. Reduce 38 3,77% and 129 to their least common denomi nator. 143 60 Ans. 34, 11, 221. 11. Reduce 161 289 and 1147 to their least common denomi 354826 354826' 354826' and 17 to their least common Ans. 5400 6930 1008 2240 1944 3213 7560 7560' 7560' 7560' 7560' 7360 and 15 to their least common de 15. Reduce, 13, 28, 52 3 5 nominator. 182 5 45, 3, 3 and 4 to their least common denomi 16. Reduce, nator. 7 Ans. 28 60 455 105 105' 105' 105' ADDITION. 185. The denominator of a fraction determines the value of the fractional unit, (165); hence, I. If two or more fractions have the same denominator, their numerators express fractional units of the same value. II. If two or more fractions have different denominators, their numerators express fractional units of different values. And since units of the same value only can be united into one sum, it follows, III. That fractions can be added only when they have a common denominator. 1. What is the sum of, and ? 12 2 8 60 ANALYSIS. We first reduce the given fractions to a common denominator, (III). And as the resulting fractions, 13, 25, and have the same fractional unit, (I), we add them by uniting their numerators into one sum, making 45 3, the answer. 2. Add 53, 37 and 4. 186. From these principles and illustrations we derive the following general RULE. I. To add fractions. When necessary, reduce the fractions to their least common denominator; then add the numerators and place the sum over the common denominator. II. To add mixed numbers. Add the integers and fractions separately, and then add their sums. NOTE. All fractional results should be reduced to their lowest terms, and if improper fractions, to whole or mixed numbers. 5. What is the sum of 37, 1237, 1337 and 58? 20. Add 41, 1053, 3003, 2413 and 4721. 21. Add 41, 24, 116, 254, 56, 73, 4 and 65. 1 22. Four cheeses weighed respectively 36, 423, 39,7% and 511 pounds; what was their entire weight? Ans. 16947 pounds. 23. What number is that from which if 44 be taken, the remainder will be 332? 18 Ans. 83. 24. What fraction is that which exceeds by 54 ? 25. A beggar obtained of a dollar from one person, from another, from another, and from another; how much did he get from all? 26. A merchant sold 464 yards of cloth for $127,7, 641 yards for $2265, and 765 yards for $3123; how many yards of cloth did he sell, and how much did he receive for the whole? Ans. 1877 yards, for $66615. Ans. 1414. Ans. 37. Ans. 188. Ans. 116138. |