ILLUSTRATION.-Dividing both terms of the fraction by 2, it gives ; in which there are one-half as many parts, but each part is twice the size. Dividing both terms of by 3, gives, onethird as many parts, each part three times the size. Hence, PROP. VI. Dividing both terms by the same number, changes its form, but does not alter its value. To TEACHERS.-By considering the numerator a dividend, the denominator a divisor, and the value of the fraction the quotient (Art. 125), the preceding propositions may be regarded as inferences from Art. 57, 58, 59. This short method is not best.adapted to young pupils. ART. 137. REDUCTION OF FRACTIONS Is changing their form without altering their value. CASE I. ART. 138. To reduce a fraction to its lowest terms. A fraction is in its lowest terms, when the numerator and denominator are prime to each other. Art. 110, Def. 5. Thus, is in its lowest terms, while is not. 1. Reduce to its lowest terms. SOLUTION.-Since the value of a fraction is not altered by dividing both terms by the same number, (Art. 136), and, as two is a common 2) FIRST OPERATION. 24 3)12 4 30 = 155 Ans. factor, divide both terms by it; the fraction then becomes 13. Again, since 3 is a factor of 12 and 15, divide them both by it; the result,, can not be reduced lower. Instead of dividing by 2, and then by 3, divide at once by 6, the greatest com. divisor of the two terms, and the result is the same. SECOND OPERATION. 6) 24 4 Solve the two following Examples by both methods. NOTE.-All subsequent Examples having a star, *, are intended to illustrate the principles on which the next succeeding rule is founded. The pupil should solve them and explain the operation, referring, at the conclusion of the cxercise, to the rule which follows. Rule for Case I.-Divide the numerator and denominator by any common factor; divide the resulting fraction in the same manner, and so on till no number greater than 1 will exactly divide both terms. Or, Divide the numerator and denominator by their greatest common divisor; the resulting fraction will be in its lowest terms. REM. When the terms of a fraction are small, the first method is most convenient; when large, the second method. REDUCE TO THEIR LOWEST TERMS, ART. 139. To reduce an improper fraction to a whole or mixed number. 1. In 4 halves (4) of an apple, how many apples? in 6 thirds ()? in 8 fourths ()? in ?? in 42 ? 2. In 8 pecks, that is, in of 3 a bushel, how many bushels? in 2? in 40? in 11? in 13? REVIEW.-186. What is the effect of dividing both terms of a fraction by the same number? 137. What is Reduction of Fractions? 138. When is a fraction in its lowest terms? is a fraction reduced to its lowest terms, Rule? Give an example. How OPERATION. 3. In 9 fourths (2) of a dollar, how many dollars? SOLUTION. Since 4 fourths make one dollar, there are as many dollars as there are times 4 fourths in 9 fourths; that is, 21 dollars. 4. Reduce to a mixed number. SOLUTION. Since 5 fifths make 1 (unit), there will be as many ones as there are times 5 in 17; that is, 33. *5. In 3 of a dollar, how many dollars? *6. Reduce 25 to a mixed number. 4)9 Ans. $21. OPERATION. 5)17 Ans. 3. Ans. 2. Ans. 81. Rule for Case II.-Divide the numerator by the denomi nator: the quotient will be the whole or mixed number. ART. 140. To reduce a whole or mixed number to an REVIEW.-188. Why is the value of a fraction not altered by being reduced to its lowest terms? 189. How is an improper fraction reduced to a whole or mixed number, Rule? 6. In 5 dollars, how many fourths? Or, reduce 5 to an improper fraction. Ans. 35. Ans. 3. *7. In 83 apples, how many fourths? *8. Reduce 123 to an improper fraction. Rule for Case III.—Multiply the whole number by the denominator of the fraction; to the product add the numerator, and write the sum over the denominator. REM. The analysis of question 6, shows that the whole number is really the multiplier, and the denominator the multiplicand; but the result will be the same (Art. 30), if the denominator be taken as the multiplier. 9. In 5 dollars, how many tenths? REDUCE TO IMPROPER FRACTIONS, 53 8. Ans. Ans. 637 24 10. In 15 yards, 11. In 261 days, ART. 141. To reduce a whole number to a fraction having a given denominator. 1. Reduce 3 to a fraction whose denominator is 4. SOLUTION. Since there are 4 fourths in 1, in 3 there will be 3 times 4 fourths = OPERATION. 4 fourths in 1. = 3 Rule.-Multiply together the whole number and the denomi nator; beneath the product write the denominator. 2. Reduce 4 to a Frac. whose denom'r is 7. Ans. 28. Ans. 3. Ans. Ans. 851. 5. 37 to a fraction whose denom. is 23. CASE IV. ART. 142. To reduce compound to simple fractions. 1. Reduce of to a simple fraction. ANALYSIS.- of is 2 times as OPERATION. is 4 times 2 4 2 X 4 8 as much as of }; but of (Art. 130); and hence, of 4 = times (Art. 131), and of In this operation, the numerators are multiplied also are the denominators. *2. Reduce of to a simple fraction. Ans. 1. Ans. . Rule for Case IV.-Multiply the numerators together for a new numerator, and the denominators together for a new denominator. If mixed numbers occur, reduce them to improper fractions. 4. Reduce of of 2 to a simple fraction. SOL.-23, and of of =23 5. Reduce of to a simple fraction. REVIEW.-140. How is a mixed number reduced to an improper fraction, Rule? 141. How is a whole number reduced to a fraction having a given denominator, Rule? |