maining factors together, we have, which is the an swer required. Hence, 124. To reduce compound fractions to simple ones by CANCELATION. Cancel all the factors which are common to the numerators and denominators; then multiply the remaining terms together as before. (Art. 123.) OBS. This method not only shortens the operation of multiplying, but at the same time reduces the answer to its lowest terms. A little practice will give the learner great facility in its application. 51. Reduce of 12 of Operation. 2 ΧΟ $ 2 of of Ans. to a simple fraction. Final First, we cancel the 4 and 3 in the numerator, then the 12 in the denominator, which is equal to the factors 4 and 3. ly, we cancel the 5 in the denominator, and the factor 5 in the numerator 10, placing the other factor 2 above. We have 2 left in the numera-tor, and 7 in the denominator. 12 7-7 of § of 12 to a simple fraction. 52. Reduce Ans. . to a simple fraction. Note. For the method of reducing complex fractions to simple ones see Art. 143. CASE V. Ex. 1. Reduce and to a common denominator. Note.-Two or more fractions are said to have a common denom inator, when they have the same denominator. QUEST.-124. How by cancelation? How does it appear that this method does not alter the value of the fraction? Obs. What is the advantage of this method? Note. What is meant by a common denominator? Suggestion. The object of this example is to find two other fractions, which have the same denominator, and whose values are respectively equal to the values of the given fractions, and. Now, if both terms of the first fraction, are multiplied by the denominator of the second, it becomes 3, and if both terms of the second fraction , are multiplied by the denominator of the first, it becomes. But the fractions and have a common denominator, and are respectively equal to the given fractions, viz: 2, and 3. (Art. 116.) Hence, 125. To reduce fractions to a common denominator. Multiply each numerator into all the denominators except its own for a new numerator, and all the denominators together for a common denominator. 2. Reduce, 3, and 1⁄2 to a common denominator. Operation. 1x4×6=24 3×2×6=36 the three numerators. 5X2X4=40 2X4×6=43, the common denominator. 36 The fractions required are 24, 48, and 48. OBS. It is manifest that the process of reducing fractions to a common denominator, does not change their value; for, it is simply multiplying each numerator and denominator of the given fractions by the same number. (Art. 116.) 35 3. Reduce 3, 4, and to a common denominator. Ans. 120. 4. Reduce,, and to a common denominator. Reduce the following fractions to a common denomi nator: 5. Reduce, 1, 4, and . 6. Reduce, 4, §, and *. - 7. Reduce 3, 4, 1, and 7. 8. Reduce, †,†, and §. 6 QUEST.-125. How are fractions reduced to a common denominator? Obs. Does the process of reducing fractions to a common denominator alter their value? Why not? 9. Reduce 3, 3, and 27. 10. Reduce,, and . 11. Reduce, 35, and 18. 12. Reduce, 5, and 18. CASE VI. 13. Reduce 2, 2, and to the least common denomi nator. Operation. 2)4" 6" 8 2)2" 3" 4 1 // 3 // 2 We first find the least common multiple of all the given denominators, which is 24; (Art. 102;) and this is the least common de 2×2×3×2=24, the least nominator required. The next com. denom. step is to reduce the given fractions to twenty-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. (Art. 116.) Thus 4, the denominator of the first fraction, is contained in 24, 6 times; now multiplying both terms of the fraction by 6, it becomes. The denominator 6 is contained in 24, 4 times; and 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, and 14 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 denom inator. QUEST.-126. How are fractions reduced to the least common de nominator? 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 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. 2 1 15. Reduce and to the least common denominator. Reduce the following fractions to the least common denominator : . 16. 2, 5, and 7. 17. 4, 1, and H. 25. 1, 1, and 38. ADDITION OF FRACTIONS. MENTAL EXERCISES. 2 Ex. 1. What is the sum of 3, 4, 3, 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. 1, or 13. QUEST.-Obs. Does this process alter the value of the given fractions? Why not? 2. What is the sum of 1, 3, 7, and ? 30 10 EXERCISES FOR THE SLATE. 11. What is the sum of,, and ? Solution. +1+3=§, or 13. Ans. 12. What is the sum of, and †? and? ? Suggestion. A difficulty here presents itself; for it is manifest that 1 half added to 1 third will make 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=3 1x2=2 } the new numerators. 2x3=6, the common denominator. The fractions reduced are 3 and 2, and may now be added. Thus, 3+. Ans. 127. From these illustrations we deduce the following general RULE FOR ADDITION OF FRACTIONS. Reduce the fractions to a common denominator; all their numerators, and place the sum over the common denomi nator. OBS. 1. Compound fractions must, of course, be reduced to simple ones, before attempting to reduce the given fractions to a common denominator. (Art. 123.) QUEST.-127. How are fractions added? Obs. What must be done with compound fractions? |