Rule.-Multiply together the whole number and the denominator; beneath the product write the denominator. 7 2. Reduce 4 to a Frac. whose denom'r is 7. Ans. 28 3. Reduce 8 to ninths. 4. 19 to nineteenths. 5. 37 to a fraction whose denom. is 23. CASE IV. Ans.. ART. 142. To reduce compound to simple fractions. 1. Reduce of to a simple fraction. ANALYSIS. ofis 2 times as much as of, and of is 4 times as much as of; but of (Art. 130); and hence, of 2 of 3 4 = OPERATION. 4 = 2 X 4 5 3 X 5 times (Art. 131), and 3 of 2 times = 18%. 15 15 = 8 15 In this operation, the numerators are multiplied together, as also are the denominators. *2. Reduce of to a simple fraction. *3. Reduce of to a simple fraction. Ans. 10 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 3 of 23 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? 8. Reduce of of to a simple fraction. SOLUTION. After indicating the operation, the numerator of the result will be 2 X3 X4; the denominator, 3X4 X5. The value of a fraction not being altered by dividing both terms by the OPERATION. 1 1 X 5 Ans. same number (Art. 136), Cancel the factors (3 and 4,) common to both terms. As 3=3X1, and 4=4×1, the factors 1 and 1 will remain after canceling 3 and 4. Hence, the products of the remaining factors are 2X1X1, and 1X1X5, which give the terms of the required fraction in its simplest form. 24 9. Reduce of 3 of 5 to a simple fraction. Ans. 54 *10. Reduce of of to a simple fraction. Ans. §. ART. 143. Hence, to reduce compound to simple fractions by Cancellation, Indicate the operation; cancel all the factors common to both terms, and multiply together the factors remaining in each. REM. As all the factors common to both terms are canceled by the operation, the result will be in its simplest form. 11. Reduce of 4 of 7 of 18 to a simple fraction. SOLUTION. First, cancel the factors 3 and 4 in the numerator, and 12 in the denominator, as 4X3=12. Since 9 is a factor of 18, cancel the factor 9 in both terms, and 2 OPERATION. 3 7 18 2 X X X Ans. write the remaining factor, 2, above 18; as 7 is a factor of 35, cancel the factor 7 in both terms, and write the remaining factor, 5, below 35. Then multiply the remaining factors as before. REVIEW.-142. How are compound reduced to simple fractions, Rule? 143. How reduced by Cancellation? Why is the value of the fraction not altered? REM. Why is the result in its lowest terms? For method of reducing complex to simple fractions, see page 167. CASE V. ART. 144. To reduce fractions of different denominators to equivalent fractions having a common denominator. 3, and to a common denominator. 1. Reduce, 3, OPERATION. = 1X3X4_12 new numer. 2×3×4 24 new denom. 2×2×4_16 new numer. 3×2×4 24 new denom. SOLUTION. The value of a fraction not being altered by multiplying both terms by the same number (Art. 135), multiply the numerator and denominator of each by the denominators of the other fractions; this will render the new denominator of each the same; since, in each case, it will consist of the product of the same numbers, that is, of all the denominators. = 3×2×3_18 new numer. 4×2×3 24 new denom. and to a com, denom. , and to a com. denom. Ans. 16 15 40' 40° 36 75 30 Ans. 38, 38, 35 Rule for Case V.-Multiply both terms of each fraction by the product of all the denominators except its own. NOTE. First reduce compound to simple fractions, and whole or mixed numbers to improper fractions. 4. Reduce,, and 7 to a common denominator. SUGGESTION.-Since the denominator of each new fraction consists of the product of the same numbers, (all the denominators of the given fractions,) we multiply them together but once. OPERATION. 1X3X5=15 1st num. 4X2X5=40 2d num. 7X2X3=42 3d num. 2X3X5=30 denom. Observe, that, in each case, the result is obtained by multiplying the numerator and denominator by the same number. ART. 145. When the given fractions are expressed in small numbers, and the denominator of either fraction is a multiple of the denominators of the others, reduce them to a common denominator; thus, Multiply both terms of each fraction by such a number as will render its denominator the same as the largest denominator; obtain this number by dividing the largest denominator by the denominator of the fraction to be reduced. 1. Reduce and to a com. denom. SOLUTION. Since the largest denom., 6, is a multiple of 2, multiplying both terms of by 6=3, reduces it to 3. Ans. 3 and §. OPERATION. 3 1x3 2x3 6 = = By similar process, Reduce to Com. Denominators, 4 10 7 2 3 6. 1 5 7 9 89 2 3 8. 3 5 4 8 1 1 16 4 12 8 14 14 14: 12 10 11 16 16 16' 8 9 11 12' 12' 12' REVIEW.-144. How are two or more fractions reduced to a common denominator? Why is the value of each fraction not altered? Why does this operation render the new denominator of each the same? CASE VI. . ART. 146. To reduce fractions of different denominators, to equivalent fractions having the least com. denominator. 1. Reduce 1 2 3, and 21 37 to the least com. denominator. SOLUTION. Since multiplying both terms of a fraction by the same number, does not alter its value, a fraction may be reduced to another whose denominator is any multiple of the denominator of the given fraction. Thus, may be reduced to a fraction, whose denominator is either 4, 6, 8, 10, 12, 14, 16, &c. And, may be reduced to a fraction, whose denominator is either 6, 9, 12, 15, 18, 21, &c. And, may be reduced to a fraction whose denominator is either 8, 12, 16, 20, 24, &c. We find 12 to be the least denominator common to all of them. These denominators being multiples of the denominators of the given fractions, it follows, that 12, their least common multiple, is the least com. denominator to which the fractions can be reduced. It now remains to reduce the fractions to TWELFTHS. Thus, will be reduced to twelfths by multiplying both of its terms by 6, which is the quotient of the L. C. M., 12, divided by 2. And, will be reduced to twelfths, by multiplying both of its terms by 4, the quotient of 12 divided by 3. And, will be reduced to twelfths, by multiplying both of its terms by 3, the quotient of 12 divided by 4. REVIEW.-145. When the denominator of one of the fractions is a multiple of the others, how reduce them to a com. denominator? How is the multiplier of each fraction obtained? 146. What are the denominators of the fractions to which one-half may be reduced? Two-thirds? Three-fourths ? |