EXAMPLES FOR PRACTICE. 2. Reduce and to common denominators. Ans. 1, or 2, 12. 3. Reduce 7, †, and to a common denominator. Ans. 78, 73, 15 4. Reduce, g, and to a common denominator. 5. Reduce, 1, and to a common denominator. 6. Reduce,, 7, and Ans. 1,318, or 38, 18, 38. Ans. 188, 38, 9609 48 141. To reduce fractions to their least common denominator. Ex. 1. Reduce 3, §, and to the least common denominator. 12, least common multiple, and common denominator. Having first obtained the least common multiple of all the denomi nators of the given fractions, we assume this to be their least common denominator. We then take such a part of it as is expressed by each of the fractions separately for their respective new numerators. Thus, to get a new numerator for , we take of 12, the least common denominator, by dividing it by 8, and multiplying the quotient 4 by 2. We proceed in like manner with each of the fractions, and write the numerators thus obtained over the least common denominator. In this process the value of each fraction remains unchanged, as both terms are multiplied by the same number. (Art. 140.) RULE. 1. Find the least common multiple of the denominators for the least common denominator. 2. Divide the least common denominator by each given denominator, and multiply the quotient by the corresponding numerator, for the new numerators. NOTE. Compound fractions must be reduced to simple ones, whole and 141. How do you find the least common denominator of two or more fractions? Upon what principle does this process depend? What is the rule for reducing fractions to their least common denominator? What must be done with compound fractions, whole numbers, and mixed numbers? mixed numbers to improper fractions, and all to their lowest terms before finding the least common denominator. EXAMPLES FOR PRACTICE. 2. Reduce,,, and to the least common denominator. 3. Reduce,,, and to the least common denominator. 4. Reduce, ro, and 7 to the least common denominator. Ans. 28, 38, 0. 9 5. Reduce, 1, and 5% to the least common denominator. Ans.,, 1, 22. 6. Reduce,, 8, §, 7, and 11⁄2 to the least common denomiAns. 1, 8, 8, 18, 31, 18. nator. 7. Reduce,, 3, 4, 4, and 1⁄2 to the least common denomiAns. 18, 3, 1, 36, 36, 336• nator. 8. Reduce,, and to the least common denominator. Ans. 38, 18, 3. 9. Reduce 7, 5, 7, and 8 to the least common denominator. Ans. 341, 244, 308, 352. 앞, 숲, 10. Reduce, 4, 5, 7, and 9 to the least common denomi Ans. §, 16, 20, 28, 36. nator. ADDITION. 142. Addition of Fractions is the process of finding the sum of two or more fractions. Fractions can only be added when expressing fractional units of the same kind. 143. To add fractions having a common denominator. Ex. 1. Add 4, 4, 4, 4, and §. OPERATION. 2 4 5 +=+ 7 7 7; and thus obtain 6 = 7 Ans. 24. The fractions all being sevenths, we, add their numer 18: 24 ators, and write their sum, 18, over the common denominator, = 24, the required sum. That is, we Write the sum of the numerators over the common denominator. 142. What is addition of fractions? - 143. How are fractions having a common denominator added ? Give the reason. EXAMPLES FOR PRACTICE. 2. Add A, T, 71, fr, fr, and 19. Ans. 31. Ans. 217 Ans. 2. Ans. 219. Ans. 111. Ans. 11991. 2 12 2 X 7=14) Sum of numerators, 43 2×2×2×3=24. Com. denominator, 24 119, Ans. We reduce the given fractions to equivalent ones having a common denominator, that they may express fractional units of the same kind; and then we add the numerators, and write their sum over the common denominator, and reduce the fraction. RULE. Reduce the given fractions to a common denominator. Add the numerators, and write their sum over the common denominator. NOTE 1. First reduce mixed numbers to improper fractions, and compound fractions to simple fractions, and each fraction to its lowest terms. NOTE 2. - In adding mixed numbers, the fractional parts may be added separately, and their sum added to the amount of the whole numbers. EXAMPLES FOR PRACTICE. 2. What is the sum of §, 1, and 18? 3. What is the sum of, 14, and †? 4. What is the sum of and 34? 144. The rule for adding fractions not having a common denominator ? How may mixed numbers be added? 145. To add two fractions having 1 for their numerator. product of the denominators, which is 20, and then their sum, Sum of the denominators, which is 9, and write the former for the denominator of the required fraction, and the latter for the numerator. By this process we reduce the fractions to a common denominator, and then add their numerators. Hence, to add two fractions of this kind, Write the sum of the given denominators over their product. 2. Add EXAMPLES FOR PRACTICE to,to,to,to,to. 3. Add to, to, to, to, too, tot. 4. Add to, to 1⁄2,to,to,to, TT to T 5. Add to, to Tz, to fo, to, to, to f. 6. Add to,to,to,to,to,to rir. 7. Add to, to, to, too, to TT, to Th∙ SUBTRACTION. 146. Subtraction of Fractions is the process of finding the difference between two fractions. NOTE.-One fraction can be subtracted from another only when both express fractional units of the same kind. 145. How can you add two fractions when the numerators are a unit? The reason for this? - 146. What is subtraction of fractions? 147. To subtract fractions having a common denominator. Ex. 1. From 3 take . OPERATION. Ans. §. The fractions both being ninths, we subtract the less numerator from the greater, and write the difference, 5, over the common denominator, 9; and thus obtain as the required difference. That is, we Write the difference of their numerators over the common denominator. 148. To subtract fractions not having a common denominator. We reduce the given fractions to equivalent ones having a common denominator, that they may express fractional units of the same kind, and then we subtract the less numerator from the greater, and place the difference over the common denominator. RULE. Reduce the fractions to a common denominator, then write the difference of the numerators over the common denominator. NOTE. If the minuend or subtrahend, or both, are compound fractions, they must be reduced to simple ones. 147. How do you subtract fractions having a common denominator? 148. The rule for subtracting fractions not having a common denominator? If the minuend or subtrahend is a compound fraction, what must be done? |