70, and 9. Reduce 13,9%, and 27. 10. Reduce 2010 11. Reduce 1, 36, and S. 12. Reduce 70, 96, and 33. CASE VI. 13. Reduce , s, and to the least common denomi nator. 11 Operation. First find the least common 2)4 6" 8 multiple of all the given denom2)2 inators, (Art. 102,) and it will be 3 4 the least common denominator 1113 2 required. The next step is to re2x2x3x2=24, the duce the given fractions to twentyleast com. denom. fourths 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 3. 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 1 Therefore 24, 24, and á are the fractions required. Hence, 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 if 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 njust be of the same value as the given fractions. (Art. 116.) 14. Reduce , }, and to the least com. denominator. Operation. 2x3x2=12, the least com, denominator. 2)3" 4" 6 Now 12-3x2=8, numerator of Ist. 12-4*3=9, of 2d. 3)3 2 3 12-6x5=10, of 3d. 1" 2"1 Ans. ii, 12, and 12. 15. Reduces and so to the least common denominator. Ans. 36 and 20 Reduce the following fractions to the least common denominator. 16. , , and î, 62 į, and 11 3 17. 6 ADDITION OF FRACTIONS. MENTAL EXERCISES. Ex. 1. What is the sum of , , š, and ? 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 13. QUEST.-Obs. Does this process alter the value of the given fractions ? 8 6 II, II, II, ty, and ii ? 2. What is the sum of 4, 4, , and ? 3. What is the sum of ý, ž, g, 1, 1, and g? 4. What is the sum of is, is, is, and is ? 5. What is the sum of 6. What is the sum of 5, 25, 33, and ? ? 7. What is the sum of 15, 16, jo, ig, and 19? 8. What is the sum of 24, A6, and ? 9. What is the sum of 16, 18, 5, 5, and is? 10. What is the sum of 46, tio, 10, and 18? EXERCISES FOR THE SLATE. 11. What is the sum of land ? 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, 1x2=2} the new numerators. 2x3=6, the common denominator. The fractions reduced area and , and may now be added. Thus 3+2=5. 12. What is the sum of 3, 4, and d? Ans. 19, or 11 127. From these illustrations we deduce the following Ans. a RULE FOR ADDITION OF FRACTIONS. Reduce the fractions to a common denominator ; add their numerators, and place the sum over the common denominator. 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 ? 6 ? 13 ig? 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. 13. What is the sum of 4 and 5? Ans. :=1%, or 15. 14. What is the sum of 4 and ? 15. What is the sum of $, 3, and f? 16. What is the sum of 9, 1}, and }? 17. What is the sum of 3 12, s and 18. What is the sum of ž, , and 19. What is the sum of to, y, and [? 20. What is the sum of i, j, and ? 21. What is the sum of }, į, , and g? 22. What is the sum of $, Ž, , 23. What is the sum of ģ, of 1, and 12 ? 24. What is the sum of }, }, of , and ? 25. What is the sum of 1 of 3, of }, and į? 26. What is the sum of 21, 6}, and ? 27. What is the sum of of 2, 3), and 52 ? 28. What is the sum of 38, }, and 18. 29. What is the sum of 35}, íà, and of f? 30. What is the sum of 25, 65, 13, and ? 6, and , ? SUBTRACTION OF FRACTIONS. MENTAL EXERCISES. 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. John had of a bushel of chestnuts, and gave away many had he left? 3. If I own, of an acre of land, and sell of it, how much shall I have left ? ģ: how QUEST.-Obs. How are mixed numbers added ? 4. A man owning á of a ship, sold ģ: what part of the ship had he left? 5. William had io of a dollar, and spent is: how many tenths had he left ? 6. What is the difference between 1 and 13 ? 7. What is the difference between ji and 13? 8. What is the difference between it and 33 ? 9. What is the difference between and : 20 ? 10. What is the difference between Too 31 and EXERCISES FOR THE SLATE. 11. From take 4. Suggestion. A difficulty here meets the learner similar to that which occurred in the 12th example of addition of fractions, viz: that of subtracting a fraction of one denominator from a fraction of a different denominator. He must therefore reduce the fractions to a common denominator before the subtraction can be performed. , 20 And 3*6=18} the numerators. (Årt. 125.) and Also 6x4=24, the common denominator. The fractions are : Now 20-18=2. Ans. 12. From & take Ans 12: 128. From these illustrations we deduce the following RULE FOR SUBTRACTION OF FRACTIONS. Reduce the given fractions to a common denominator ; subtract the less numerator from the greater, and place the remainder over the common denominator. Obs. Compound fractions must be reduced to simple ones, as in addition of fractions. (Art. 123.) QUEST.-128. How is one fraction subtracted from another? Obs. What is to be done with compound fractions ? |