133. The value of a fraction is the quotient arising from the division of the numerator by the denominator. Thus, the value of §, or 62, is 3; and the value of †, or 3 ÷ 4, is 2. REDUCTION. 134. Reduction of Fractions is the process of changing their form without altering their value. A fraction is in its lowest terms, when its terms are prime to each other. (Art. 112.) 135. To reduce a fraction to its lowest terms. Ex. 1. Reduce OPERATION. to its lowest terms. Ans. = We divide the terms of the fraction by 2, a factor common to them both, and obtain . We divide, again, both terms of by 3, a factor common to 13 them, and obtain. Now, as 1 and 3 are numbers prime to each other, the fraction is in its lowest terins. The same result would have been produced, if we had divided the terms by 6, the greatest common divisor. 2) 18 3) = Since the numerator and denominator of a fraction correspond to the dividend and divisor in division (Art. 132), dividing both by the same number, or canceling equal factors in both (Art. 115), changes only the form of the fraction, while the value expressed remains the same. Therefore, Dividing the numerator and denominator of a fraction by the same number does not alter the value of the fraction. RULE. Divide the numerator and denominator by any number greater than 1, that will divide them both without a remainder, and thus proceed until they are prime to each other. Or, Divide both terms of the fraction by their greatest common divisor. EXAMPLES FOR PRACTICE. 2. Reduce to its lowest terms. 3. Reduce to its lowest terms. 4. Reduce to its lowest terms. 5. Reduce to its lowest terms. 6. Reduce 96 7. Reduce 8. Reduce to its lowest terms. to its lowest terms. Ans. Ans.. Ans. §. Ans.. Ans.. Ans. Ans.. 133. What is the value of a fraction?-134. What is reduction of fractions? When is a fraction in its lowest terms? 135. Why does dividing both terms of a fraction by the same number not alter the valuc? Has the same value as? Why? Repeat the rule. 35 fifths. 3 9. Reduce to its lowest terms. 10. What is the lowest expression of 848? 136. To reduce a mixed number to an improper frac tion. Ex. 1. In 7g how many fifths? Ans. 38. OPERATION. 7/ Since there are 5 fifths in 1 whole one, there will be 5 times as many fifths as whole ones; therefore, in 7 there are 35 fifths, and the 3 fifths being added make 38 fifths, which are expressed thus, 38. = 38. 38 fifths RULE. Multiply the whole number by the denominator of the fraction, and to the product add the numerator, and place the sum over the given denominator. Ans. iit. Ans. 133. NOTE. To reduce a whole number to a fraction of the same value, having a given denominator, we multiply the whole number by the given denominator, and make the product the numerator; thus 5, reduced to a frac tion, having 3 for a denominator, becomes 15. EXAMPLES FOR PRACTICE. 2. In 83 dollars how many sevenths? to an improper fraction. 6. Reduce 15 to an improper fraction. 7. In 187 how many ninths? 8. In 161 how many one hundred and seventeenths? Ans. 18848. 9. Change 43114 to an improper fraction. 10. What improper fraction will express 27? 11. Change 111 to an improper fraction. 12. Change 125 to an improper fraction. 13. Change 25 to an improper fraction, having nominator. 14. Reduce 75 to ninths. 15. Change 343 to the form of a fraction. 16. Reduce 84 to fifteenths. Ans. 59. Ans. 13. Ans. 103. Ans. 187. Ans. 169. Ans. 142. Ans. 360. Ans. 12322. Ans. 125. 6 for a de- Ans. £75. Ans. 1260. 136. What is the rule for reducing a mixed number to an improper fraction? The reason? How do you reduce a whole number to a fraction of the same value, having a given denominator? 137. To reduce improper fractions to whole or mixed numbers. Ex. 1. How many dollars in 7 dollars? OPERATION. 16) 37 (21% 32 5 RULE. ·Divide the numerator by the denominator, and the quotient will be the whole or mixed number. Ans. $25. Since 16 sixteenths make one dollar, there will be as many dollars in 37 sixteenths of a dollar as 37 contains times 16, or $215. EXAMPLES FOR PRACTICE. TT 76 2. Reduce 96 to a whole number. 3. Change 17 to a mixed number. 4. Change to a mixed number. 5. Change 1735 to a mixed number. 6. Reduce 1900 to a mixed number. 7. Reduce 37 to a whole number. 8. Change 567 to a whole number. 9. Reduce to a mixed number. 10. Reduce 1848 to a mixed number. 459 7 times as much. 138. To reduce a compound fraction to a simple frac tion. Ex. 1. Reduce of 7 to a simple fraction. OPERATION. Ans. 12. Ans. 10,7. Ans. 10 Ans.. parts, one of be, it is If be divided into 5 equal these parts is; and if of evident that of will be 7 times as much. is; and if of be, of 4 times are. will be 4 times Or, by multiplying the denominator of nator of, it is evident we obtain of√ into which the number or thing is divided are and consequently only as large as before. Ans. 188. Ans. 1429. Ans. 1. Ans. 567. Ans. 93. Ans. 43. by 5, the denomi , since the parts 5 times as many, Again, since t 137. What is the rule for reducing improper fractions to whole or mixed numbers? A reason for the rule. -138. How do you reduce a compound fraction to a simple one? The reason for the operation? of 1=35, of will be 4 times as much; and 4 times 55 = 28. This process will be seen to be precisely like the operation. Ex. 2. Reduce of of of of to a simple fraction. Ans. 3 RULE. Multiply all the numerators together for a new numerator, and all the denominators for a new denominator. The numerators and denominators which are common factors we cancel according to the principles of cancellation.~ (Art. 117.) NOTE 1.- All whole and mixed numbers in the compound fraction must be reduced to improper fractions, before multiplying the numerators and denominators. 3. What is 5. What is NOTE 2. When there are factors common to both numerator and denominator, they may be canceled in the operation. EXAMPLES FOR PRACTICE. of ‡ of § ? of of 7? Ans.. ? of 7 to a simple fraction. 7. Required the value of of of 1 of 1 of 58. of 1 of 3 of of of of Ans.. 8. Reduce of & of of § of to a simple fraction. Ans. 10. Reduce of of 7 to a simple fraction. 11. Reduce of 3 of 15 of 9g to a whole number. 12. Reduce of of of 8 of to a simple fraction. Ans. 1. Ans. 3. 138. When there are common factors in the numerator and denominator, how may the operation be shortened? The rule? What must be done with all whole and mixed numbers in the compound fraction? How may the operation be shortened by canceling ? A COMMON DENOMINATOR. 139. A Common Denominator of two or more fractions is a common multiple of their denominators. The Least Common Denominator of two or more fractions is the least common multiple of their denominators. NOTE. Fractions have a common denominator, when all their denomi nators are alike. 140. To reduce fractions to a common denominator. Ex. 1. Reduce 2, %, and 7 to a common denominator. OPERATION. 66 3 X 6 X 8 = 144, new numerator. 160, 66 = 168, 66 66 6 4 × 6 × 8 = 19 2, common denominator. Ans. 13, 182, 189. 349478 =111. 160 = 192. We first multiply the numerator of by the denominators 6 and 8, and obtain 144 for a new numerator. We next multiply the numerator of by the denominators 4 and 8, and obtain 160 for a new numerator; and then we multiply the numerator of by the denominators 4 and 6, and obtain 168 for a new numerator. Finally, we multiply all the denominators together for a common denominator, and write it under the several numerators, as in the operation. 168 - 192. By this process, since the numerator and denominator of each fraction are multiplied by the same numbers, only the form of the fraction is changed, while the quotient arising from dividing the numerator by the denominator, or the value of the fraction (Art. 133), remains the same. Therefore, Multiplying the numerator and denominator of a fraction by the same number does not alter the value of the fraction. RULE. Multiply each numerator by all the denominators except its own, for the new numerators; and all the denominators together for a common denominator. NOTE 1. Compound fractions, if any, must first be reduced to simple ones, and whole or mixed numbers to improper fractions. ― NOTE 2. Fractions may often be reduced to lower terms, without destroying their common denominator, by dividing all their numerators and denominators by a common divisor. 139. What is a common denominator of two or more fractions? What is the least common denominator ? When have fractions a common denominator? - 140. How do you find a common denominator of two or more fractions? Give the reason of the operation. What inference is drawn from it? What is the rule for finding a common denominator? How may frac tions having a common denominator be reduced to lower terms? |