RULE 1. Divide each term by any factor common to them; then divide these quotients by any factor common to THEM, and so proceed till the quotients are mutually prime. Or, RULE 2. Divide each term by their greatest common divisor. 2. Reduce & to its lowest terms. Ans. Ans. . Ans. Ans. 8. Reduce 9. Reduce 95. 10. Reduce 33. 13. Reduce 142. To multiply a fraction by a whole number. Ex. 1. Multiply by 3. FIRST OPERATION. X3=fs, Ans. Ans. for . It is just as evident that 3 times are as that 3 times 2 cents are 6 cents, or that 3 times 2 are 6; i. e. when the numerator is multiplied by 3 the fraction represents 3 times as many parts as before, and each part continues of the same size; .. the fraction is multiplied by 3. SECOND OPERATION. 1 x 3 = 3, Ans. If the denominator is divided by 3, the fraction represents just as many parts as before, but each part is three times as great, and .. the whole fraction is three times as great. Hence, RULE 1. Multiply the numerator by the whole number. Or, RULE 2. Divide the denominator by the whole number. NOTE 1. The correctness of Rule 1 is also evident from Art. 83 (a), and Art. 131. Rule 2 also depends on Art. 83 (d). 141. First rule for reducing a fraction to its lowest terms? Second rule? Reason? 142. First rule for multiplying a fraction by a whole number? Why? Second rule? Why? Another reason? 2. Multiply by 3. Ans. for NOTE 2. The second rule is preferable in this and all similar examples, because it gives the fraction in smaller terms. NOTE 3. The first rule is preferable for this and all similar examples, be cause the second gives a complex fraction. 3 14. Multiply by 15. Ans. 1 or 42 = X5; and & X 324, Ans. NOTE 4. We may here, as in whole numbers (Art. 61), use the factors of the multiplier, and in using these factors we may apply the 1st or the 24 (a) If we multiply a fraction by its denominator, the product will be the numerator. 19. Multiply by 8. 20. Multiply by 44. Ans. X8}=7, by Rule 2. 142. May the factors of the multiplier be used? What is the product if a fraotion is multiplied by its denominator? (b) To multiply a mixed number by an integer: Multiply the fractional part and the entire part separately, and add the products together; or, reduce the mixed number to an improper fraction (Art. 139), and then multiply. 21. Multiply 3 by 5. Ans. 19. First multiply by 5 and the product is 4; then multiply 3 by 5 and the product is 15. These partial products added give 15419 for the true product. Or, first reduce 3 to and then multiply by 5 and the product is 19, as before. 22. Multiply 8 by 9. × 9=34; 8 × 9=72; and 72 +3§=75§., Ans. 23. Multiply 9 by 12. Ans. 113. 24. Multiply 18 by 20. 25. Multiply 231⁄2 by 7. PROBLEM 5. 143. To divide a fraction by a whole number. Ex. 1. Divide § by 4. FIRST OPERATION. $43, Ans. = Ans.or It is just as evident that one fourth of is as that one fourth of 8 cents is 2 cents, or that one fourth of 8 is 2; i. e. when the numerator is divided by 4 the fraction represents only one fourth as many parts as before, and each part continues of the same size; .. the fraction is divided by 4. SECOND OPERATION. 4 36, Ans. If the denominator is multipled by 4, the fraction represents just as many parts as before, but each part is only one fourth as great, and.. the whole fraction is only one fourth as great. Hence, RULE 1. Divide the numerator by the whole number. Or, RULE 2. Multiply the denominator by the whole number. NOTE 1. These rules may also be explained by Art. 83 (b) and (c). 142. How is a mixed number multiplied by an integer? Another way? 143. First rule for dividing a fraction by a whole number? Why? Second rule? Why? Another explanation? 2. Divide 1 by 2. Ans. by Rule 1; 14 by Rule 2. NOTE 2. The 1st rule is preferable in this example. Why? Ans. 3. Divide by 6. 4. Divide by 11. 5. Divide by 25. 6. Divide by 12. 7. Divide 28448, by Rule 2. NOTE 3. The 2d rule is preferable in this example. Why? 8. Divide by 5. Ans. Ans. 63 9. Divide by 11. 10. Divide by 6. 11. Divide by 4. 12. Divide NOTE 4. See Art. 142, Note 4. 13. Divide by 35. 35 = 5 X 7. +5, and = 7, Ans, 14. Divide 15. Divide by 14. 16. Divide 64 by 44. (a) To divide a mixed number by a whole number. tient, 5, and we have 5 for the true quotient. 143. May the factors of the divisor be used separately? A mixed number, how divided by an integer? NOTE 5. In Ex. 21, the dividend is less than the divisor; hence the quo tient is a proper fraction. 22. Divide 7 by 9. 23. Divide 5 by 11. 24. Divide $63 equally between 9 boys. PROBLEM 6. 144. To multiply a fraction by a fraction. Ex. 1. Multiply 4 by 3. Ans. $. Ans. the multiplier, 3, is 5 times 3,.. the product, , is 5 times the product sought; hence, 2d, ÷ 5 = (Art. 143, Rule 2) is the product sought; i. e. X=. Hence, RULE. Multiply the numerators together for a new numerator, and the denominators for a new denominator. (a) To multiply by a fraction is only to multiply by the numerator, and then divide the product by the denominator. In Ex. 7 we multiply by 5, and obtain 12 (Art. 142, Rule 2), and then divided by 6 gives (Art. 143, Rule 1), the result sought. 144. Rule for multiplying one fraction by another? Reason? To multiply by a fraction, what is it? What principles in the operation in Ex. 7? |