EXAMPLES. 1. Reduce the complex fraction to a simple fraction. Now, if we multiply the numerator and denominator of this fraction by any number whatever, the value of the fraction will not be altered (Art. 102). Let us then multiply them by the denominator with it terms inverted. This will give, 36 14 36 It is plain that when the denominator is multiplied by the fraction which arises from inverting its terms, the product will be equal to unity. Hence, the required simple fraction will always be equal to the numerator of the given fraction multiplied by the denominator with its terms inverted. All the cases in the reduction of fractions of this class are embraced in the following eight forms. 117. 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 dénominator of each fraction by the product of the denominators of all the others. EXAMPLES. 1. Reduce, and to a common denominator. 1 × 3 × 5 = 15 the new numerator of the 1st. and 2 x 3 x 5 = 30, the common denominator. 2d. 3d. Therefore, 5, 38, and 34 are the equivalent fractions. 15 70 30 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). QUEST.-117. 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? 2. When the numbers are small the work may be performed mentally. Thus,, = 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. 4. Reduce 5, of 3, and 4 to a common denominator. 5. Reduce, 75, 135 and 37 to a common denominator. 6. Reduce 4, 3, 62 to a common denominator. 7. Reduce 71, 11, 61 to a common denominator. 8. Reduce 41, 81, and 21 of 5 to a common denomi nator. 5 9. Reduce,, §, and 11 to a common denominator. 10. Reduce of of and of of to a common denominator. 11. Reduce 5, 35, 4, and 65 to fractions having a common denominator. 12. Reduce,,, and to a common denominator. 13. Reduce, 9, 8, 5 3 7, and 19 to a common denominator. 433 속 3층 9 음을 4 6 9 14. Reduce' 24' 33' 5' 8' 1, and 11 to simple frac tions having a common denominator. 118. It is often convenient to reduce fractions to a common denominator by multiplying the numerator and denominator in each by such a number as shall make the denominators the same in all. QUEST.-When the numbers are small, how may the work be performed? 118. By what second method may fractions be reduced to a common denominator? 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, they will have a common denominator. The two fractions will be reduced to 3 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 8 3. Reduce, and to a common denominator. 4. Reduce 3. 28 14 to a common denominator. 5. Reduce, 35, and to a common denominator. 6. Reduce 62, 8%, and 54 to a common denominator. 7. Reduce 75, 2, and to a common denominator. 3 8 119. To reduce fractions to their least common denominator. I. Find the least common multiple of the denominators as in Art. 107, 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 multiple. 3 and 3 x 4 X 7 X 2 = 168 the least common denominator. QUEST.-119. How do you reduce fractions to their least common denominator? Does this reduction affect the values of the fractions? 3. Reduce 145, 63, and 51⁄2 to their least common de nominator. 4. Reduce, 24, and to their least common denominator. 5. Reduce 67 6 120' 40' to their least common denominator. 6. Reduce, 391, 41, 7. Reduce 31, 412, 88, nominator. and 8 to a common denominator. 1476 to their least common de 8. Reduce,,, and to fractions having the least common denominator. 9. Reduce, 4, 5, and to fractions having the least common denominator. 10. Reduce, 3, 5, 7, 1, and to equivalent fractions having the least common denominator. REDUCTION OF DENOMINATE FRACTIONS. 120. We have seen (Art. 14), that a denominate number is one in which the kind of unit is denominated or expressed. For the same reason, a denominate fraction is one which expresses the kind of unit that has been divided. Such unit is called the unit of the fraction. Thus, of a £ is a denominate fraction. It expresses that one £ is the unit which has been divided. QUEST.-120. What is a denominate number? What is a denominate fraction? What is the unit called? In two-thirds of a pound, what is the unit of the fraction? |