III. Multiply the numerators together for a new numer ator, and the denominators together for a new denominator. 152. To reduce fractions of different denominators to equiv. alent fractions that shall have a common denominator. 1. Reduce, and to a common denominator. OPERATION. 2 X 5 X 4 = 40 1st num. 4 X 3 X 4 = 48 2d num. 3 × 5 × 3 = 45 3d num. 3 X 5 X 4 = 60 denom. ANALYSIS.-Multiplying both terms of the first fraction by 20, the product of the other denominators, givce 40 Multiplying both terms of the second fraction by 12, the product of the other denominators, gives. Multiplying both terms of the third by 15, the product. of the other denominators, gives . In each case, both terms of the fraction have been multiplied by the same number; therefore, the value is not changed (Art. 143): hence, Rule.-I. Reduce mixed numbers to improper fractions, and compound to simple fractions, when necessary : II. Multiply the numerator of each fraction by all the denominators except its own, for the new numerators, and all the denominators together for a common denominator. NOTE-When the numbers are small, the work may be performed mentally; thus, 20 15 24 20 ,,become, 28, 18, 33; and,, become, 18, 28, 18 NOTE.-We may often shorten the work of multiplying the numerator and denominator of each fraction by such a number as will make the denominators the same in all. Reduce the following fractions to common denominators. 1. Reduce,1⁄2, 1, and 1⁄2 to a common denominator. 5. Reduce 6, 6. Reduce, 5 6' 6' 7 141, and 3 to a common denominator. 8' 7. Reduce, 25, and 13 to a common denominator. 8. Reduce,, ži, 3 12, and to a common denominator. 9. Reduce, 4, 22, 10. Reduce 21, 51, 1%, and 45 to a common denominator. 152. How do you reduce fractions to a common denominator? 153. To reduce fractions to their least common denominator. The least common denominator is the least common multiple of the denominators. 1. Reduce,, and, to their least common denominator. ANALYSIS.-If there are mixed numbers or compound fractions, they must be reduced. We then find the least common multiple of the denominators 4, 6, and 9, which is 36. This number is divided by each denominator, to ascertain by what the terms of the fraction must be multiplied to reduce it to 36ths. Therefore, the fractions, reduced to their least common denominator, are 6) X X X 5 = 30 2d numerator. 4 = 16 3d numerator. I. Find the least common multiple of the denominators : this will be the least common denominator of the fractions: II. Divide the least common denominator by the denomina tor of each fraction, separately; multiply the quotient by the numerator and place the product over the least common denominator; the results will be the new and equivalent fractrons. 153. How do you reduce fractions to their least common denomina tor? 3 and Examples. 10. Reduce 41, 72, and 11. Reduce 63, 87, and 2% to their least common denominator 12. Reduce, 233, and 15 to their least common denominator 13. Reduce 5, 65, 6, and to their least denominator. to their least common denominator DENOMINATE FRACTIONS. 154. A DENOMINATE FRACTION is one whose unit is denominate Thus, of a yard is a denominate fraction. CASE VIII. 155. To change a denominate fraction from a greater unit to a less. 1. In of a yard, how many inches? ANALYSIS. Since 3 feet make a yard, yd. of feet; and since 12 inches make one foot, 7 yard inches = 03 = 311 inches. of of 12 Rule. 7182 OPERATION. 3 12 = 311 1 63 2 = 311. Multiply the fraction by the units of the scale, in succession, till you reach the required unit. 154. What is a denominate fraction?-155. How do you change a denominate fraction from a greater to a less unit? CASE IX. 156. To change a denominate fraction from a less unit to a greater. 1. Reduce of a pound to a fraction of a ton. ANALYSIS. Since one pound is of a quarter, and since one quarter is of a cwt., since one cwt. is of a ton, lb. of qr.; lb. of of cwt.; and 1 = 1 ton. of of of = 5 Rule.-Divide the fraction, that is, multiply the denominator by the units of the scale, in succession, till the required 157. To find the value of a denominate fraction in integers of lower denominations. 1. What is the value of lb. Troy? OPERATION. 7 12 156 How do you change a denominate fraction from a less to a greater unit?-157. How do you find the value of a denominate frac tion in integers of lower denominations. pwt. 6..6 24 9)144 gr. 16 |