EXAMPLES FOR PRACTICE. 1. Reduce to a fraction having 24 for a denominator. 2. Reduce to a fraction whose denominator is 96. Ans. 5. 3. Reduce 1 to a fraction whose denominator is 51. 13 375 Ans. 49. 3000' 6. Change 73 to a fraction whose denominator is 8. 28 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 to a common denominator. OPERATION. 3 × 9 5 × 9 7 x 5 = 9 × 5 RULE. 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 Multiply the terms of each fraction by the denominators of all the other fractions. NOTE.-Mixed numbers must first be reduced to improper fractions. 1. Reduce and 2. Reduce and EXAMPLES FOR PRACTICE. Ans. 18, T. 3. Reduce, and to a common denominator. Ans. 120, 120, 120. 72 50 60 4. Reduce, 5 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, and to their least common denominator. 12 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. 1. Reduce 2. Reduce, EXAMPLES FOR PRACTICE. and to their least common denominator. 12 10 and 1⁄2 to their least common denominator. 3. Reduce 3 7 and 11 to their least common denominator. 4. Reduce, and to their least common denominator. 5. Reduce 6 14 and 13 to their least common denominator. 9 147 24 6. Reduce,, 25 and to their least common denominator. 39 52 24 75 78 78 Ans. 3, 4, 8, 78. 7. Reduce 23, 7 5 3, 15, 24 and 87 to their least common denomi nator. 8. Reduce, 56 9. Reduce 25 and to their least common denominator. 25 and 14 to their least common denominator 40' 120 Ans. §8, 38, 3 10. Reduce 38 7 and 129 to their least common denomi Ans. 34, 111, al 389 and 117 to their least common denomi Ans. 37, 301, 713. 14 1219 12199 1219' 12. Reduce 25, and 17% to their least common denominator. 13. Reduce 931 3127 and 5133 to their least common denom 1829 5723 inator. 8 Ans. 180614 193874 323733 14. Reduce, 1, 2, 2, and 17 to their least common ΤΣ 13, 33 Ans. 5400 7560 16. Reduce, and 4 to their least common denomi nator. 28 60 Ans. 3, 173, 10%, 158. 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 suin, it follows, III. That fractions can be added only when they have a common denominator. have the ANALYSIS. We first reduce the given fractions to a common denominator, (III). And as the resulting fractions, 3, 25, and same fractional unit, (I), we add them by uniting their numerators into one sum, making 45 = 3, the answer. 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 8? 20. Add 41, 105, 300, 241 and 4721. 21. Add 41, 21, 116, 254, 576, 73, 4 and 65. Ans. 188. 99 Ans. 116138. 22. Four cheeses weighed respectively 365, 423, 39,7% and 514 pounds; what was their entire weight? Ans. 16947 pounds. 23. What number is that from which if 44 be taken, the remainder will be 3??? 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, 64 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. 1873 yards, for $66618. |