Operation. 2)4" 6" 8 2)2 " 3" 4 1 " 3 2 least com. denom. First find the least common multiple of all the given denominators, (Art. 102,) and it will be the least common denominator required. The next step is to re 2×2×3×2=24, the duce the given fractions to twentyfourths without altering their value. This may evidently be done by multiplying both terms of each fraction by the number of times its denominator is contained in 24. Thus 4 the denominator of the first fraction, is contained in 24, 6 times; hence, multiplying both terms of the fraction by 6, it becomes 1. The denominator 6 is contained in 24, 4 times; hence, multiplying the second fraction by 4, it becomes The denominator 8 is contained in 24, 3 times; and multiplying the third fraction by 3, it becomes 14. Therefore 18 1,4, and are the fractions required. Hence, 249 18 24 8 126. To reduce fractions to their least common denominator. I. Find the least common multiple of all the denominators of the given fractions, and it will be the least common denominator. (Art. 102.) II. Multiply each given numerator by the number of times its denominator is contained in the least common denominator, and place the respective products over the least common denominator. QUEST. 126. How are fractions reduced to the least common denominator? OBS. Multiplying each numerator by the number of times its denominator is contained in the least common denominator, is, in effect, multiplying both terms of the given fractions by the same number. For it we multiply each denominator by the number of times it is contained in the least common denominator, the product will be equal to the least common denominator. Hence, the new fractions thus obtained must be of the same value as the given fractions. (Art. 116.) 14. Reduce, 2, and to the least com. denominator. 2×3×2=12, the least com. denominator. Now 12-3x2=8, numerator of 1st. Operation. 2)3 "4" 6 15. Reduce and to the least common denominator. Ans. 35 and 36. Reduce the following fractions to the least common denominator. Ex. 1. What is the sum of 1, 2, 3 and 5? Suggestion. Since all these fractions have the same denominator, it is plain their numerators may be added as well as so many pounds or bushels, and their sum placed over the common denominator, will be the answer required. Thus 1 eighth and 2 eighths are 3 eighths, and 3 are 6 eighths, and 5 are 11 eighths. Ans., or 1}. QUEST. Obs. Does this process alter the value of the given fractions? Why? 3 5 45 10. What is the sum of 100, 100, 100, and 180? EXERCISES FOR THE SLATE. 11. What is the sum of and? Suggestion. A difficulty presents itself here; for it is manifest that 1 half added to 1 third are neither 2 halves nor 2 thirds. (Art. 22.) This difficulty may be removed by reducing the given fractions to a common denominator. (Art. 125.) Thus, 1x3=31 1x2=25 the new numerators. 2x3=6, the common denominator. The fractions reduced are 3 and 2, and may now be added. Thus 3+2=5. 12. What is the sum of, 4, and ? Ans.. Ans. 12, or 12. 127. From these illustrations we deduce the following RULE FOR ADDITION OF FRACTIONS. Reduce the fractions to a common denominator; add their numerators, and place the sum over the common de nominator. OBS. 1. Compound fractions must, of course, be reduced to simple QUEST.-127 How are fractions added? Obs. What must be done with compound fractions? ones, before attempting to reduce the given fractions to a common denominator. (Art. 123.) 2. Mixed numbers may be reduced to improper fractions, then added according to the rule; or, we may add the whole numbers and fractional parts separately, and then unite their sums. : 19. What is the sum of,, and §? 20. What is the sum of 3, 3, and 1?? 21. What is the sum of, 1 2, and ? 3, 3, 6, and ? ? 3, 3, 7 of 4, and †? 24 81 30. What is the sum of 25, 6, 13, and §? Ex. 1. Henry had of a watermelon, and gave away of it: how much had he left? Solution. 3 sevenths from 5 sevenths leaves 2 sevenths. Ans. 2. 2. John had of a bushel of chestnuts, and gave away how many had he left? 3. If I own of an acre of land, and sell of it, how much shall I have left? QUEST. Obs. How are mixed numbers added? 13. Reduce, 2, and to the least common denomi nator. Operation. 2)4" 6 2)2 8 311 4 " 1 "3 2 2×2×3×2=24, the least com. denom. First find the least common multiple of all the given denominators, (Art. 102,) and it will be the least common denominator required. The next step is to reduce the given fractions to twentyfourths without altering their value. This may evidently be done by multiplying both terms of each fraction by the number of times its denominator is contained in 24. Thus 4 the denominator of the first fraction, is contained in 24, 6 times; hence, multiplying both terms of the fraction by 6, it becomes 14. The denominator 6 is contained in 24, 4 times; hence, multiplying the second fraction by 4, it becomes. The denominator 8 is contained in 24, 3 times; and multiplying the third fraction by 3, it becomes 15. Therefore 1,4, and are the fractions required. Hence, 18 8 24 126. To reduce fractions to their least common denominator. I. Find the least common multiple of all the denominators of the given fractions, and it will be the least common denominator. (Art. 102.) II. Multiply each given numerator by the number of times its denominator is contained in the least common denominator, and place the respective products over the least common denominator. QUEST.-126. How are fractions reduced to the least common denomi nator? |