EXERCISES. 1. Reduce and 18 to their lowest terms. Ans. and 3. 2. Reduce 38 and 125 to their lowest terms. Ans. and . 3. Reduce 45 and 273 to their lowest terms. 403 Ans. and 3. 4. Reduce and 99 to their lowest terms. 225 Ans. and 5. Reduce 32 and 180 to their lowest terms. 68 Ans. 13 and 3. 6. Reduce 384 and 6 to their lowest terms. 600 7 Reduce 345 and 46 to their lowest terms. 940 Ans. 69 and 199. Common Denominator. 160 (68.) Two or more fractions are said to have a common denominator, when they have the same number for a denominator. Thus, 4, and have a common denominator. Two or more fractions are reduced to their least common denominator, when their common denominator is made the smallest by which the value of each fraction can be expressed. The fractions and are reduced to their least common denominator, when these fractions are made and 19, respectively. This is done by multiplying both terms of the first by 3, and both terms of the second by 2 (65). The least common denominator, 12, is the least comman multiple of the given denominators 4 and 6. RULE XIII. (69.) To reduce two or more Fractions to a Common Denominator. 1. Multiply each numerator by all the denominators except its own, for the new numerators; and multiply all the denominators together, for a common denominator. 2. If the least common denominator be required,—take the least common multiple of the given denominators, for the ⚫ common denominator. Divide this multiple by each given denominator, and multiply the quotient by the corresponding numerator, for the new numerators. EXAMPLES. 1. To reduce, 5, and 7 to a common denominator. For the new numerators, we have 2 x 6 x 8= 96; 5 x 3 x 8 = 120; 7 × 6 × 3 =126, and for the common denominator, 3 x 6 x 8 = 144. The given fractions thus become 96 14412, and 125, respectively. 2. To reduce the same fractions,,, to their least common denominator. The least common multiple of the given denominators is 24 (56); then 24 is the common denominator required; and the new numerators are (243) x 216; (246) x 5 = 20; (248) × 7=21 The given fractions are thus reduced to ,, and 3, respectively. This method will not always find the least common denominator, unless each of the given fractions is in its lowest terms. Thus, in the preceding Example, if we take instead of its equal, the least common multiple will be 48, which, as has been seen, is not the least common denominator by which the equivalent of each fraction can be expressed. The values of the fractions are not altered in reducing them to a common denominator, because both terms of each fraction are, in the operation, multiplied by the same numbers, (65). This will be evident upon inspecting the preceding Examples. EXERCISES. 1. Reduce ,, and to a common denominator. Ans. 80 96 120 120, 120, and 45 , and to a common denominator. Ans. 35, 288, and 38. 260 585 585 360 585 and to a common denominator. Ans. 429, 579, and 199. 672 672' 4. Reduce, and to a common denominator. 168 672 5. Reduce, and to the least common denominator. 6. Reduce, and to the least common denominator. 7. Reduce,, and to the least common denominator. Ans. 24, 25, and 7. 8. Reduce, and to the least common denomi nator. Ans. 15, 18, and 28 60' ADDITION OF FRACTIONS. (70.) The Sum of two or more Fractions is found by means of a common denominator. For example, to find the Sum of and 3. These fractions are equal, respectively, to and; the sum of 8 twelfths and 9 twelfths is 1, equal to 1; then (71.) For the Addition of Fractions. 1. If the fractions have not a common denominator, reduce them to a common denominator. 2. Add all the numerators together, and place the Sum, as a numerator, over their common denominator. 3. Mixed numbers may be added under the form of improper fractions (64); or their fractional and integral parts may be added separately. EXAMPLE. To add together the mixed numbers 7, 85, 153. Reducing the fractional parts of the given numbers to their least common denominator (69), we find them equal to 1, and, respectively. These fractions added together make, which is 23 (61), or 21; and this sum added to the sum, 30, of the integrai parts of the given numbers, makes the entire sum 321. Instead of the preceding method, we might reduce the given mixed numbers to the improper fractions 31, 53, 47, (64), and then reduce these improper fractions to a common denominator, &c.; but this would not be so convenient a method. In all subsequent Exercises, improper fractions in the Answers are to be reduced to integral or mixed numbers (61); and proper fractions to their lowest terms (67). EXERCISES. 1. What sum should be paid for a vest at 42 dollars, and a hat at 5 dollars? Ans. 10 dollars. 2. What sum should be paid for a cord of wood at 31 dollars, a barrel of flour at 52 dollars, and a shote at 21 dollars? Ans. 11 dollars. dollars, a ton of hây dollars. What did Ans. 48 dollars. for 37 dollars, and 3. Bought a quantity of corn for 157 for 13 dollars, and a lot of pork for 19 the whole amount to? 4. Sold wheat for 275 dollars, oats rye for 27 dollars. What did the whole amount to ? Ans. 3398 dollars |