Examples. 1. Divide both terms of 4. Divide both terms of 44 by 2 and by 4. by 3. by 2, 3, 4, 6, and 12. 5. Divide both terms of 12 by 2, 3, 4, 6, and 12. 6. Divide both terms of 36 by 2, 3, 4, 6, and 36. REDUCTION OF FRACTIONS. 145. REDUCTION OF FRACTIONS is the operation of changing a fractional number from one unit to another without altering its value. 146. THE LOWEST TERMS of a fraction are when the numerator and denominator are prime to each other. CASE I. 147. To reduce a whole number to a fraction having a given denominator. 1. Reduce 17 to a fraction whose denominator shall be 5. ANALYSIS.-To reduce 17 to such a fraction is the same as to reduce 17 to fifths. In 17 there are 17 times as many fifths as there are in 1. In 1 there are 5 fifths; therefore, in 17 there are 17 times 5 fifths, or, 85 fifths; hence, OPERATION. 17 x 5 = 85 17=85 Rule.-Multiply the whole number by the denominator, and write the product over the required denominator. 145. What is reduction of fractions? 146. What are the lowest terms of a fraction? 147. How do you reduce a whole number to a fraction having a given denominator? Examples. 1. Change 18 to a fraction whose denominator shall be 7. 2. Change 25 to a fraction whose denominator shall be 12. 3. Change 19 to a fraction whose denominator shall be 8. 4. Change 29 to a fraction whose denominator shall be 14. 5. Change 65 to a fraction whose denominator shall be 37. 6. Reduce 145 to a fraction having 9 for its denominator. 7. Reduce 450 to twelfths. 8. Reduce 327 to a fraction having 36 for its denominator. 9. Reduce 97 to a fraction having 128 for its denominator. 10. Reduce 167 to eighty-ninths. 11. Reduce 325 to a fraction whose denominator shall be 75. CASE II. 148. To reduce a mixed number to an equivalent improper fraction. 1. Reduce 12 to its equivalent improper fraction. ANALYSIS. Since in any number there are 7 times as many 7ths as units 1, there will be 84 sevenths in 12: To these add 5 sevenths, and the equivalent fraction becomes 89 sevenths. Hence, add OPERATION. 12 × 784 sevenths. 5 sevenths. = gives 8,9. 125 89 sevenths. Ans. = Rule.-Multiply the whole number by the denominator: to the product add the numerator, and place the sum over the given denominator. Examples. 1. Reduce 397 to its equivalent improper fraction. 2. Reduce 112 to its equivalent improper fraction. 148. How do you reduce a mixed number to an equivalent improper fraction? 3. Reduce 427 to its equivalent improper fraction. 9. How many 151ths in 1874? 10. Reduce 1495 to an improper fraction. 12. Reduce 17494,543 to an improper fraction. 15. In 125 16. In 375 17. In 464 18. In 96 11 64 hogsheads, how many sixty-thirds. acres, how many 640ths of an acre? 19. In 984 pounds, how many 112ths of a pound? 20. In 35,72 years, how many 366ths of a year? 21. How many one hundred and thirty-fifths are there in the mixed number 874? 22. Place 4 sevens in such a manner that they shall express the number 78. 23. By means of 5 threes write a number that is equal to 334. CASE III. 149. To reduce an improper fraction to an equivalent whole or mixed number. 1. In 27 how many entire units? 149. How do you reduce an improper fraction to an equivalent mixed number? ANALYSIS. Since there are 5 fifths in 1 unit, there will be, in 278 fifths, as many units 1 as 5 is contained times in 278, viz., 55 and times. Hence, the following OPERATION. 5)278 Rule.-Divide the numerator by the denominator, and the quotient will be the equivalent whole or mixed number. Examples. Reduce the following fractions to whole, or mixed numbers. 150. To reduce a fraction to its lowest terms. 70 1. Reduce to its lowest terms. ANALYSIS.-By inspection, it is seen that 5 is a common factor of the numerator and denominator. Dividing by it we have 14 We then see that 7 is a common factor of 14 and 35: dividing by it, we have. Now, 2 and 5 are prime to each other; therefore, the fraction is in its lowest terms. 2D OPERATION. 2d. The greatest common divisor of 70 and 175, is 85 (Art. 119); if we divide both terms of the fraction by it, we obtain . The value of the fraction is not changed in either operation, since the numerator and denominator are both divided by the same number (Art. 144): hence, Rule. I. Divide the numerator and denominator, successively, by all their common factors: Or, II. Divide the numerator and denominator by their greatest common divisor. 151. To reduce a compound fraction to a simple fraction. 1. What is the equivalent fraction of of? 35 ANALYSIS.-Three-fifths of is three times of: 1 fifth ofis (Art. 142); and 3 times is (Art. 139); hence, of = 1; hence, 35 Rule. I. If there are mixed numbers, reduce them to improper fractions: II. When there are common factors in the numerators and denominators, cancel them: 150. How do you reduce a fraction to its lowest terms? 151. How do you reduce a compound fraction to a simple one? |