by 3: and these are the terms which would have been obtained from dividing at once by 72, the greatest common measure found by the rule. (2) Express in their simplest forms, the fractions, (3) Find the simplest fractions expressive of the values of and (4) Reduce as much as possible, the fractions, 8398 11050 109375 29393' 35581' 10000000 222210 135795 and 85. 7' 161' 640 18 Two or more fractions having different denominators, may be transformed into other equivalent fractions having a common denominator. Let it be required to reduce, and to a common denominator; then, since the continued product of the denominators is expressed by 2 × 5 × 7, we have so that 35 28 70, 70 30 and are the new equivalent fractions with the common denominator 70; and the steps taken manifestly lead to the same thing as the operations here subjoined: 35 28 wherefore the new equivalent fractions are and as above: and hence we derive the following rule. 30 70' RULE. Multiply each numerator by all the denominators except the one placed under it, and the product will be the corresponding new numerator: and multiply together the denominators of all the fractions for a common denominator. 86. If two or more of the denominators have a common measure, the equivalent fractions may be expressed in simpler terms than obtainable by the Rule, and still having a common denominator: thus, if the fractions be, and 2, we have from Article (56), the least common multiple of the denominators: also, with the least common denominator 12; and the new numerators are here obtained by multiplying those of the fractions proposed by the quotients arising from its division by their respective denominators. It need scarcely be observed that mixed quantities, compound and complex fractions, must all be reduced to the forms of simple fractions, before this and the subsequent rules can be applied: and that the magnitudes of fractional quantities may also be compared with each other by what is here done. Examples for Practice. 4 (1) Reduce and; and ; and respectively, to common denominators. (2) Reduce to common denominators,, and † ; also,, and . 3 4 (3) Reduce, 2 and 3 to fractions, having a common denominator. (4) Transform, and into equivalent fractions, with the least common denominator. I. ADDITION OF FRACTIONS. RULE. Reduce the proposed quantities, if need be, to equivalent fractions with a common denominator; add together the new numerators, and under their sum place the common denominator: and the resulting fraction, reduced when possible, will be the sum required. 51 For, let 7 and 4 be the proposed quantities, which reduced to improper fractions are and then, since addition can be performed only upon quantities of the same denominations, these fractions must first be reduced to a common denominator; and their sum will be This process may be rendered simpler as follows: for, the sum of the integers = 7 + 4 = 11 : 11 + 18% = 12 56 and therefore the entire sum = as before; and this is much shorter and easier, particularly when the numbers are large: also, each of these methods is evidently applicable, whatever be the number of quantities proposed. Examples for Practice. (1) Find the sums of and; of and; of and , and of and 1. 12 (2) Add together 1 and 7; 29 and 13; 5 and 12, and 37 and 24. Answers: 8, 16, 17% and 62. 41 38 (3) What are the sums of and; of 7 and 8; of 3 and 4, and of and . 49 71 3 2 (4) Add together, and;, and, and, 9 and 10. 45 (5) Add together, 5, and 7; 2, 3 and 5, and 84, 135 and 271. Answers: 54, 10287 and 49593. (6) Find the respective sums of, 3, and 7: of,,and, and of,, and . (7) Find the respective sums of 13, and of 3, 23, and 7. Answers: 52663 and 143. 3465 2 35 14 3 (8) Add together 5, 80, 100, 140 285, 394 and of 3704. 37 1320 and : also, 387, Answers 1 and 2548. 3 2800 (9) Required the respective sums of of of 8: of, 4 and 3 of 2 and of of +. 51, by 41 34 II. SUBTRACTION OF FRACTIONS. RULE. Transform the proposed quantities, if necessary, so as to have a common denominator; subtract the less numerator from the greater; under the remainder place the common denominator, and the result properly reduced, will be the required difference. For, taking the quantities 5 and 1, and reducing them to fractional forms, we have, for the reason mentioned in the last rule, the difference Like the last, this operation may frequently be performed in a more convenient form as follows: 12 the difference = 5-15-18: where, being |