83. Fractions may be reduced to a common numerator by multiplying the terms of each fraction by the product of the numerators of all the other fractions, or by employing the least common multiple of the numerators in the same manner as the least common multiple of the denominators is employed in Art. 80. If, for example, it be required to reduce, 4, and 6, to a common numerator, the reduction may be made in either of the two following ways, Whence the proposed fractions are respectively equal to 36, 36, 88, or to IS, TT, TT. 6 6 a. Of all fractions having the same denominator, that is the greatest which has the greatest numerator; since it contains the greatest number of those equal parts into which unity, represented by the common denominator, is divided. Hence, by reducing any proposed fractions to the same denominator their numerical relations to each other can be determined. For instance, the fractions,, and are equal; to 50, 1 to 56. and 3. Therefore is the least, and the greatest of the fractions,, and . 63 70 9 to b. Of all fractions having the same numerator that is the greatest which has the least denominator: for, in this fraction, the equal parts into which unity is divided are the greatest, and the same number of parts is taken in all the fractions. Ex. Of the three fractions,, and, the greatest is and the least. ADDITION OF VULGAR FRACTIONS. 84. In Addition of Fractions, two or more fractional expressions are given, and it is required to find a number equal in value to all these expressions. If the proposed fractions have the same denominator, the numerators are different collections of the same part of unity. (Art. 78.) To form their sum it is therefore evidently necessary to combine into one number the different collections of these equal parts, and to express the value of the parts or their common relation to unity; that is, it is necessary to add the numerators together, and to write their sum over the common denominator. Thus the sum of 2 3 11 11' and is 2+3+4 9 or Results which are not expressed in the lowest terms, or which have the form of improper fractions, ought to be reduced, in the first case, to equivalent fractions expressed in the lowest terms; and, in the second, to whole or mixed numbers. (Art. 71. and 74.) a. Fractions having different denominators are reduced to equivalent fractions having a common denominator either by Art. 78. or Art. 80. The addition of the fractions is then effected as in the preceding case. As an example of the addition of fractions having different denominators, let it be required to find the sum of,,, 53 and 12. 7 The reduction of the fractions to the same denominator, and addition of the reduced fractions, may be made in the following form. b. Mixed numbers may be reduced to improper fractions (Art. 74. a.), and whole numbers to the form of fractions (Art. 69.). The reduced expressions may then be brought to the same denominator and added together, as in the preceding case. Ex. Let it be required to add together, 24, 5, and 6. 2; 5; 672-79 The fractional expressions which are to be added together are therefore, }, ,, and 79. c. Since a mixed number is the sum of a whole number and a fraction (Art. 74.), if it be required to find the sum of any whole numbers, mixed numbers, and fractions, the whole numbers and the integral parts of the mixed numbers may be added into one sum; the fractions and the fractional parts of the mixed numbers into another sum, and these two sums added together. This, since the whole is equal to the sum of all its parts, is the sum of the proposed numbers. Ex. Add together 37, 253, 165, and 14. 253=25+; 163=16+3. .. 37 +258 +165 +17=37 + 25 +16+++ 17. And+++48 +1=18=14=133; ' ... 37 +258 +163 +17=7933. d. General Rule for the Addition of fractional numbers: Reduce mixed numbers to improper fractions, whole numbers to the form of fractions, and all the fractions to a common denominator; add the numerators together, and place the sum over the common denominator. The result is the sum of the given fractional numbers. When the whole numbers or the integral parts of the mixed numbers are large, it is convenient to add the integers and fractions separately, and then to add the two sums together. 85. Add together the following fractions. 86. The addition of fractional, like that of integer literal expressions, is indicated by connecting the fractions by the symbol. Thus, the sum of the fractional expressions α + denotes also the sum of the α b represents a times the с fractions and But since 1th b part of 1, and represents c times the th ith part of 1, the sum b of these expressions is (a+c) times the th part of 1. There Now, a, c may represent the numerators of any two fractions, and b the common denominator of these fractions. Whence the sum of literal, like the sum of numerical fractions, having the same denominator, is obtained by adding the numerators together, and dividing the result by the common denominator. The same reasoning may be employed in the case of a greater number of fractions than two. Fractions having different denominators are reduced to the same denominator as in Art. 79. or 81. SUBTRACTION OF VULGAR FRACTIONS. 87. In Subtraction of Fractions, two unequal fractions or fractional expressions are given, and it is required to find their difference. If the proposed fractions have not the same denominator, they may be reduced to equivalent fractions having the same denominator by Art. 78. a., or 80. Then that which has the greater numerator is the greater fraction (Art. 83. a.). The fractions having the same denominator, their numerators express certain repetitions of the same part of unity. The difference between the repetitions of this common number is equal to the excess of the number of parts expressed by the minuend over that expressed by the subtrahend; that is, it is equal to the difference of the numerators divided by the common denominator. a. If a whole or mixed number is found either in the minuend or subtrahend, the whole number may be reduced to the form of a fraction, the mixed number to an improper fraction, and both expressions to the same denominator. The difference is then found as in the preceding case. b. The general Rule for the Subtraction of fractional num |