8. Reduce of off to a simple fraction. SOLUTION. After indicating the operation, the numerator of the result will be 2X3X4; the denominator, 3X4 X5. The value of a fraction not being altered by dividing both terms by the OPERATION. 1 3 4 X Ans. same number (Art. 136), Cancel the factors (3 and 4,) common to both terms. As 3=3X1, and 4-4X1, 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 of to a simple fraction. Ans. 11 *10. Reduce of of to a simple fraction. Ans. f. 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 of 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 OPERATION. 2 3 7 18 2 X Ans. 123 25 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. OPERATION. 1X3X4_12 new numer. 2×3×4 24 new denom. ART. 144. To reduce fractions of different denominators to equivalent fractions having a common denominator. 1. Reduce,, and to a common denominator. 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. = 2×2×4_16 new numer. 3×2×4 24 new denom. 3X2X3_18 new numer. 4X2X3 24 new denom. *2. Reduce and to a com. denom. *3.,, and to a com. denom. Ans. 18, 18. 75 36 Ans. 38, 38, 38. 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 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 1 by §=3, Ans. and . reduces it to OPERATION. 1x3 2x3 1 6 By similar process, Reduce to Com. Denominators, 3. 6 ANSWERS. 7 8 9 1.7. 4 9 14 3 5 8, 10.9. 2, 1, 3 10 16 12, 18, 11. 4 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,, and 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. OPERATION. 2)2 2 4 1 3 2 2×3×2=12, least com. mul. 2)12 3)12 4)12 1X6 6 2x6 12 2X4 8 = Ans. 3X3 9 4X3 12 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? of With regard to this Operation, the pupil will notice, 1st. When a fraction is in its lowest terms, the denominator fraction to which it can be reduced must be a multiple of the denominator of the given fraction: hence, any Any denominator common to two or more fractions must be a common multiple of their denominators: therefore, The least common multiple of their denominators, is the least com. denominator to which two or more fractions can be reduced. 2d. The values of the fractions are not altered, for both terms are multiplied by the same number (Art. 135.). Rule for Case VI.-Find the least common multiple of the denominators of the given fractions, and multiply both terms of each fraction by the quotient of the least common multiple, divided by the denominator of the fraction. NOTES.-1. Before commencing the operation, each fraction must be in its lowest terms. 2. Compound must be reduced to simple fractions, and whole or mixed numbers to improper fractions. 3. After the pupil is well acquainted with the nature of the operation, the multiplication of the denominators may be omitted, as the new denominators will be equal to the L. C. M. 4. The object of reducing fractions to a common denominator, is to prepare them for addition and subtraction, which can be performed only when the numbers are of the same unit value. REVIEW.-146. What is the least denominator common to the fractions whose denominators are 2, 3, and 4? How reduce one-half to twelfths? Two-thirds Three fourths? Why is the L. C. M. of the denominators, the least common denominator? Why are the values of fractions not altered by the operation? What is the Rule for Case VI? |