REDUCTION OF FRACTIONS, THE method of changing fractions from one form to another, without altering their value, is called Reduction; 2=4===. Reduction serves to prepare fractions for Addition, Subtraction, Multiplication, and Divi sion. CASE I. To reduce fractions to their least terms. RULE. Divide the terms of the given fraction by any number which will divide them both without a remainder, and the quotients will be the terms of a new fraction, equal in value to the given fraction. Repeat the operation, till the terms of the reduced fraction are divisible only by 1. Ex. Reduce 3136 to its lowest terms. 3584 32 208 136 156 Reduce the following fractions to their lowest terms. Ex. 1. Ex. 2. Ex. 3. Ex. 4. Ex. 5. Ex. 6. Ex. 7. Ex. 8. 360 384 5194 2475 In all compound fractions, if there be the same figure in the denominator as there is in the numerator, they may be omitted in the work; thus we have no further concern with the 3 and 4, be Reduce 3 X 8 X 9 X 2 Ex. 11. = cause they occur in both terms of the fraction. CASE II. To find the greatest common measure of a fraction. RULE. Divide the greater term by the less, and this divisor by the remainder, then the last divisor will be the greatest common measure of both terms of the fraction. * Ex. What is the greatest common measure of the fraction 218? 1998 What is the greatest common measure of the following fractions? CASE HI. To reduce an improper fraction to an equivalent, whole, or mixed number. RULE. Divide the numerator by the denominator, and the quotient will be the integer, or mixed number required: thus 43, and 45 =5. 3 Reduce the following improper fractions to their proper terms. * A number, ending with an even figure, or a cypher, can be divided by 2, without a remainder. A number ending with 5 or 0, is divisible by 5. If a fraction has a cypher, or cyphers, at the right-hand of both iss terms, it may be abbreviated by cutting off the cyphers. CASE IV. To reduce a mixed number to an equivalent improper fraction. RULE. Multiply the whole number by the denominator of the fraction, to the product add the numerator, for a new numerator, under which place the denominator; Thus 435, and 296. CASE V. To reduce a compound fraction to an equivalent simple one. RULE (1). If any of the proposed quantities be integers, or mixed numbers, reduce them to their proper terms. (2). Multiply all the numerators together for a new numerator, and all the denominators for a new denominator, and then reduce the fraction to its lowest terms. Reduce of 3 of 71⁄2 to a simple fraction. Operation 5 4 3 47 2 X 2 X 3 X 47 94 1 6 5 X 1 X 2 X 3 5. The fraction is already in its lowest terms, because no figure higher than unit will divide both terms of the fraction without a remainder. Under this case a whole number may be reduced to an equivalent fraction, having any given denominator: thus, 5. For a whole number may be converted into a fraction, by placing under it an unit; and therefore it may be reduced to an improper fraction with any given denominator, by multiplying it by the denominator, and the product will be the numerator required. Thus, to reduce 15 to an equivalent frac15 X 6 90 tion, having the donominator 6, I say, 6 = 6 + Here, as has already been observed, 2 and 3 being found in the nu merator and denominator, are dropped in the work. CASE VI. To reduce fractions of different denominators to others of equal value, having a common denominator. RULE. (1). Multiply each numerator into all the denominators, except its own, for a new numerator, and all the denominators for a common denominator. Reduce 3, 3, 3, and 3, to a common denominator. New numerators. 3 X 9 X 3 X 1 = 81 7 X 5 X 3 X 1 105 11 X 5 X 9 X 1 = 495 3 X 5 X 9 X 3 405 New denom, 5 X 9 X 3 X 1135 I multiply the numerator of the first fraction 3, by 9, and 3, and 1: and then 7, the numerator of the second fraction, by 5, and 3, and 1, and so of the others; and for a new denominator. I multiply the 5, and 9, and 3, and 1, together. 495 405 Answer,, 185, 183, 193. Ex. 1. Reduce 3, 4, and 2, to a common denominator.* 2. Reduce, §, †, and 7, to a common denominator. 3. Reduce,,, and 3, to a common denominator.† 4. Reduce, To, 8, and 11, to a common denominator. 5. Reduce, 1, 4, 4, and 2}, to a common denominator. 6. Reduce,,, and 15, to a common denominator. 7. Reduce,, 4, %, and 7, to a common denominator. 8. Reduce,,, and, to a common denominator. 9. Reduce,,, and of 9, to a common denominator. (2). To find the least common denominator. Set down the denominators of the given fractions in a line, and divide as many of them as possible, by any number which will leave no remainder, and set down the quotients, NOTES. * If the products of the denominators are divided by their greatest common measure, the answer will be in the least common denominator, as in this example. See also next rule. New numerators. 2 X 4 X 4 5 X 3 X 4 3 X 3 X 4 Denominator. 3 X 4 X 4 Here 4 being common to each new numerator, and to the denominator, may be omitted, and the answer will be In the work there is no need to put down the units as multipliers. and the undivided numbers below. Repeat the operation till there be no two numbers which can be divided without a remainder. Then the product of all the divisors, and the quotients in the last lines will give the least common denominator. Divide this least common denominator by each of the given denominators separately, and multiply the quotients by their several numerators, their products will be the new numerators. Reduce, 3)5, 9, 3, 1 7 11 ,, to the least common denominator. then 3 X5 X 3 X 1 X 1 45, is the common deno. 5, 3, 1, 1 minator, and 45 divided by the given denominators, 3, 9, 3, 1, give 9, 5, 15, 45; these multiplied by the given numerators, give 27, 35, 165, 135, for new numerators, and the fractions will stand 3, 3, 5, 335 45 35 165 Reduce 3, 4, 3, 4, and, to the least common denomi nator. 3)3, 4, 5, 6, 8 4)1, 4, 5, 2, 8 2)1, 1, 5, 2,2 1, 1, 5, 1, 1 The least denominator is, acccordingly, 3 X 4 X 2 X 5120; 1208, 4, 5, 6, 8 40, 30, 24, 20, 15. 40 X 2; 30 X 3; 24 X 2; 20 X 4; 15 X 3, for new numerators; therefore the fractions rerequired are09 120 120 120 120• 20 20 48 20 45 The learner may now reduce to the least common denominator, the ten examples given under the first part of the Rule.* NOTE. *To find the least common multiple of two or more given numbers. RULE. Find the greatest common measure, by inspection, of Two of the numbers, and divide the product of them by the common measure so found; multiply this quotient by the third number, and divide the product by the common measure of the multiplier and multiplicand, and so proceed to the last number; the last quotient will be the least common multiple. Ex. Find the least number that can be divided by 2, 3, 4, 5, 6, and 7, without remainders. The greatest common measure of 2 and 3 is 1, and 2 × 3, divided by 1 is 6: the greatest com→ mon measure of the 6 just found, and 4, the next given number, is 2, and 6 X 4 divided by 2 12: the greatest common measure of 12 so found, and 5, the next figure, is 1; and 60 divided by 160, and so on. It is found that 420 is the least number that can be divided by 2, 3, 4, 5, 6, and 7. Ex. 1. What |
