by cancelling or striking out the 3's and 6's in the numerator and denominator. By cancelling or striking out the 3's we only divide the numerator and denominator of the fraction by 3; and in cancelling the 6's we divide by 6. Hence, the value of the fraction is not affected by striking out like factors, which is merely dividing the numerator and denominator by the same number. 2. Reduce of of to its simplest terms. Besides cancelling the like factors 8 and 8 and 9 and 9, we also cancel the factor 3 common to 6 and 15, and write the quotients 2 and 5 above and below the numbers. 3. Reduce of of of 2 of to its simplest terms. 4. Reduce of 9 of 41 of 2 to its simplest terms. 108 5. Reduce 35 of 5 of 327 of 49 to its simplest terms. 301 CASE VI. 134. To reduce complex fractions to simple ones. Reduce the numerator and denominator, when necessary, to simple fractions; then multiply both terms by the denominator, with its terms inverted, and the product will be the equivalent simple fraction. For, if we multiply the numerator and denominator of a fraction by any number whatever, the value of the fraction will not be altered (ART. 122). Let us then multiply both terms by the denominator with its terms inverted. 133. How may the work often be shortened? Ought it to be so abridged? 134. What is a complex fraction? How do you reduce a complex to a simple fraction? If you invert the terms of any fraction, and then multiply it by the new fraction so found, what will the product be equal to ? 135. To reduce fractions of different denominators to equivalent fractions having a common denominator. I. Reduce complex and compound fractions to simple ones, and all whole or mixed numbers to improper fractions. II. Then multiply the numerator and denominator of each fraction by the product of the denominators of all the others. 135. What is the first step in reducing fractions to a common denominator? What is the second? Does the reduction alter the values of the several fractions? Why not? When the numbers are small, how may the work be performed? EXAMPLES. 1. Reduce,, and to a common denominator. 1x3x5=15, the new numerator of the 1st. 7X2X5=70 4x3x2=24 and 2×3×5=30, the common denominator. 70 2d. 3d. Therefore, 15, 30, and 34, are the equivalent fractions. It is plain, that this reduction does not alter the values of the several fractions, since the numerator and denominator of each are multiplied by the same number (see Prop. V). 2. When the numbers are small the work may be performed mentally. Thus, 1, 1, 3=28, 18, 18. Here we find the first numerator by multiplying 1 by 4 and 5; the second, by multiplying 1 by 2 and 5; the third, by multiplying 2 by 4 and 2; and the common denominator by multiplying 2, 4, and 5 together. tor. 3. Reduce 21, and of to a common denominator. 2=; and 3 and 56 2 and 3 the answers. 4. Reduce 5, of 3, and 4, to a common denominaAns. 1,,. 5. Reduce, 135, and 37, to a common denominator. Ans. 525 1080, and 22200 600' 6009 6. Reduce 4, 3, 62, to a common denominator. 31 600 Ans. 200, 2, and 1550. 7. Reduce 7, 1, 61, to a common denominator. 50 144 Ans. 10, 4, and 200. 1080 144, 8. Reduce 41, 84, and 21 of 5, to a common denomi nator. Ans. 136. It is often convenient to reduce fractions to a common denominator by multiplying the numerator and denominator of each fraction by such a number as shall make the denominators the same in both. 136. By what second method may fractions be reduced to a common denominator? Will the value of either fraction be changed? EXAMPLES. 1. Let it be required to reduce and to a common denominator. We see at once that if we multiply the numerator and denominator of the first fraction by 3, and the numerator and denominator of the second by 2, that they will have a common denominator. The two fractions will be reduced to % and 2. 2. Reduce and to a common denominator. If we multiply both terms of the first fraction by 3 and both terms of the second by 5, we have tor. 3. Reduce, and to a common denominator. Ans. 17, 12, 12. 4. Reduce, 214, to a common denominator. 5. Reduce, 35, and to a common denominator. 8 Ans. 15, 92, 18. 6. Reduce 65, 91, and 57 to a common denomina 7. Reduce 75,, Ans. 154 228 127 24 24 24 4 1, and to a common denominator. Ans. 282, 18, 36, 56. 36 16 9 LEAST COMMON DENOMINATOR. 137. It is often necessary to reduce fractions to their least common denominator. For this purpose, I. Find the least common multiple of the denominators as in ART. 126, and it will be the least denominator sought. II. Multiply the numerator of each fraction by the quotient which arises from dividing the common multiple by the denominator, and the products will be the numerators of the required fractions; under which write the least common denominator. 137. How do you reduce fractions to their least common denominator? Does this reduction affect the values of the fractions? 2. Reduce,, and to their least common denominator. Ans. 36, 48, and. 3. Reduce 145, 63, and 51 to their least common denominator. Ans. 122, 5, 4. 8 51 4. Reduce, and to their least common denominator. Ans. 360, 360, 360. 67 72 320 5. Reduce 6, to their least common denomi nator. 18 Ans. 67 300 120 120' 120' 6. Reduce 1, 391, 41 and 8 to a common denominaAns. 100 100' 100' 100' tor. 82 605 450 800 7. Reduce 31, 41, 81, 14% to their least common denominator. Ans. 8. Reduce,, 2, and to fractions having the least common denominator. Ans. 12, 12, 12, 12. 6 8 9 9. Reduce,,, and to fractions having the least Ans. 36, 88, 88, 83. common denominator. 7 50 60 63 10. Reduce,,, 1, 1, and 17 to equivalent fractions having the least common denominator. |