54. Reduce of 1⁄2 of 3 of 4 of to a simple fraction. of of to a simple fraction. of 4 of to a simple fraction. 60. Reduce of 3 of 3 of of to a simple fraction. Note. For reduction of complex fractions to simple ones, see Art. 239. CASE V. Ex. 61. Reduce and to a common denominator. Note.-Two or more fractions are said to have a common denominator, they have the same denominator. when Solution. If both terms of the first fraction, are multiplied by the denominator of the second, it becomes; and if both terms of the second fraction, are multiplied by the denominator of the first, it becomes. Thus the fractions and have a common denominator, and are respectively equal to the given fractions, viz:, and. (Art. 191.) Hence, 200. 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. 62. Reduce, 4, and to a common denominator. Operation. 1X4X6=24 3X3X6=54 the three numerators. 5X3X4=60 3X4×6=72 the common denominator. Ans.,, and 92. OBS. The reason that the process of reducing fractions to a common denominator does not alter their value, is because the numerator and denominator of each of the given fractions, are multiplied by the same numbers; and multiplying QUEST.-Note. What is meant by a common denominator? 200. How are fractions reduced to a common denominator? Obs. Does the process of reducing fractions to a com mon denominator alter their value? Why not? both the numerator and denominator of a fraction by the same number, does not alter its value. (Art. 191.) 63. Reduce, 3, 4, and to a common denominator. 65. Reduce, 1, 4, and . 70. Reduce, 70%, and 5. 69. Reduce 5, 53, and 43. 71. Reduce 28, 35, and 18. 72. Reduce f, §5, and 137. CASE VI. 73. Reduce, 2, and § to the least common denominator. Operation. 2)3 " 4" 8 2)3" 2" 4 3 1 / 2 Now 2×2×3×2=24, the least common denominator. Analysis.-We first find the least common multiple of all the given denominators, which is 24. (Art. 176.) The next 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 such a number as will make its denominator 24. (Art. 191.) Thus 3, the denominator of the first fraction, is contained in 24, 8 times; now, multiplying both terms of the fraction by 8, it becomes. The denominator 4, is contained in 24, 6 times; hence, multiplying the second fraction by 6, it becomes. The denominator 8, is contained in 24, 3 times; and multiplying the third fraction by 3, it becomes. Therefore, and are the fractions required. Hence, 201. 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. 176.) II. Divide the least common denominator by the denominator of each given fraction, and multiply the quotient by the numerator; the products will be the numerators of the fractions required. QUEST.-201. How are fractions reduced to the least common denominator? OBS. 1. This process, in effect, multiplies both the numerator and denominator of the given fractions by the same number, and consequently does not alter their value. (Art. 191.) 2. The rule supposes each of the given fractions to be reduced to its lowest terms; otherwise, the least common multiple of their denominators may not be the least common denominator to which the given fractions are capable of being reduced. Thus, the fractions, 2, and, when reduced to the least common denominator as they stand, become 31⁄2, 1, and. But it is obvious that these fractions are not reduced to their least common denominator; for, they can be reduced to 1, 2, and 3. Now, if the given fractions are reduced to the lowest terms, they become I, 1, and ‡, and the least common multiple of their denominators, is also 4. (Art. 176.) 3. By a moment's reflection the student will often discover the least common denominator of the given fractions, without going through the ordinary process of finding the least common multiple of their denominators. Take the fractions,, and; the least common denominator, it will be seen at a glance, is 4. Now if we multiply both terms of by 2, it becomes 2; and if we divide both terms of by 3, or reduce it to its lowest terms, it becomes . Thus the given fractions are equal to 2, 2, and 1, and are reduced to the least common denominator. 74. Reduce 2, 5, and 7 to the least common denominator. Now 2X2X3X2=24, the least com. denom. 24 6-4, and 4×5=20, the 2d 66 Ans. 1, 2, and 21. 75. Reduce and to the least common denominator. Reduce the following fractions to the least common denominator: QUEST.-Obs. Does this process alter the value of the given fractions? Why not? What does this rule suppose respecting the given fractions? ADDITION OF FRACTIONS. Ex. 1. A beggar meeting four persons, obtained of a dollar from the first, 3 from the second, from the third, and ÷ from the fourth how much did he receive from all? Solution. Since the several donations are all in the same parts of a dollar, viz: sixths, it is plain they may be added together in the same manner as whole dollars, whole yards, &c. Thus, 1 sixth and 3 sixths are 4 sixths, and 4 are 8 sixths, and 5 are 13 sixths. Ans. 13, or 24 dollars. Ex. 2. What is the sum of and 3? OBS. A difficulty here presents itself to the learner; for, it is evident, that 2 thirds and 3 fourths neither make 5 thirds, nor 5 fourths. (Art. 51.) This difficulty may be removed by reducing the given fractions to a common denominator. (Art. 200.) Thus, 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 ones, before attempting to reduce them to a common denominator. (Art. 198.) 2. Mixed numbers may be reduced to improper fractions, and then be added according to the rule; or, we may add the whole numbers and fractional parts separately, and then unite their sums. 3. In many instances the operation may be shortened by reducing the given fractions to une least common denominator. (Art. 201.) QUEST.-202. How are fractions added? Obs. What must be done with compound fractions? How are mixed numbers added? How may the operation frequently be shortened? EXAMPLES. Ans. 12=2. 3. What is the sum of, 4, and 1⁄2? 39 of 1, of, and ? of 9, of 1, and ? of of 7 of 4, and ? of 3, † of §, and § ? 81, 21, 64, and ? 19. What is the sum of 214, 35, 42, and of } ? 20. What is the sum of 4 of 3, 25, 64, 13, and § ? 21. What is the sum of and ? Note. It is obvious, if two fractions, each of whose numerators is 1, are reduced to a common denominator, the new numerators will be the same as the given denominators. (Art. 200.) Thus, if and are reduced to a common denominator, the new numerators will be 12 and 8, the same as the given denominators. Now, the sum of the new numerators, placed over the product of the denominators, will be the answer; (Art. 202;) that is 5 the answer required. Hence, 1248 20 12×8 96' or 203. To find the sum of any two fractions whose numerators are one. Add the denominators together, place this sum over their product, and the result will be the answer required. OBS. 1. The reason of this rule may be seen from the fact that the operation is the same as reducing the given fractions to a common denominator, then adding their numerators. 2. When the numerators of two fractions are the same, their sum may be found QUEST.-203. How is the sum of any two fractions found whose numerators are 1 ? Obs. How find the sum of two fractions whose numerators are the same? |