To reduce fractions to a common denominator, without altering their values. RULE I. Multiply each numerator by all the denominators except its own, for a new numerator; and multiply all the denominators together for a new denominator. RULE II. (Which gives the least common denominator.) Take the least common multiple of all the denominators for a new denominator. Then divide the new denominator by the denominator of each fraction, and multiply the quotient by the numerator, and the product will be the new numerator of the fraction. The reason of the first rule is, that the numerator and denominator of each fraction is multiplied by the product of all the other denominators, so that the values of the fractions remain unchanged. And the reason of the second rule is much the same; for the numerator and denominator of each fraction is multiplied by the quotient obtained from the division of the new (or common) denominator by the denominator of the fraction. By Rule I., multiply a and b by the product df, and we adf get bdf we multiply c and d by the product bf, we get bdf for the for the first new fraction; and in the same way, if bef bed second new fraction; and in a similar way we get for adf bef bde bdf are the new the third new fraction; so that bdfbdfbdf fractions, or the fractions when reduced to a common denominator. The least common multiple of the denominators is easily found to be cfkx, which is to be the common denominator of the new fraction, or of the fractions when reduced to a common denominator. Dividing (by Rule II.) ef ka1 by cx2, the denominator of the first fraction, we get fk for the quotient, and multiplying this quotient by ab, the numerator of the first fraction, we get abfka for the new numerator of the first new fraction; in like manner, the new numerator of the second new fraction is found to be cdekx, and that of the third fraction is cfgh, so that the new fractions are abfkx2 of ka which are easily seen to be equivalent to the Ans. 30a3g 26a g 126abg 135be 162a3g' 162a'g' 162a3g' 162a3g* 3a2 + 5b2 ac be 17 11m2 to a common denomi 935m3 1122m2 m2(33a2+55b2) 17ac17be 187m2 187m2' 187m2 187m2 CASE V. To add fractional quantities together. RULE. Reduce the fractions (when necessary) to a common denominator; then write the sum of the numerators (added according to their signs) over the common denominator, and the result will be the sum required. To show the correctness of the rule, we shall take the (a + b −c)m m am bm we get (by Addition and Ax. IV.) + m m ; or, omitting m, which is clearly a α b с m common factor of these equals, we have + a + b − c m agreeably to the rule; and it is clear that a simi lar proof is applicable in all cases of the addition of fractions, after they have been reduced to a common denomi first fractions, and 3 × 5 × 7 = 105 = their common denom a × 6 = 6a a × 5 = 5a) fractions, and 5 × 6 = 30 = their common denominator. Hence the sum is expressed by + = } are the new numerators of the second ва 5a 11a = the sum 30 30 30 as required. a a a a a 87a да 2. Add the fractions Ans. = a + 60 20° half difference (supposed positive) of s and d, and since their sum gives & the greater quantity, we see that the half sum of two quantities, when added to their half difference, gives Let the fractions be brought to a common denominator, as in Addition. Then subtract the numerator of the fraction to be subtracted, from the numerator of the other fraction; and the result, when written over the common denominator, will give the required difference. |