186. TO SUBTRACT COMMON FRACTIONS. RULE.-Reduce the fractions if necessary to equivalent fractions having a common denominator, find the difference of their numerators, and write it over the common denominator. Fractions. 187. To subtract one mixed number from another when the fraction in the subtrahend is greater than the fraction in the minuend. Operation. 58% = 582 32 = 3210 Difference 2517 EXAMPLE. From 58% take 32. Explanation. is greater than, therefore we take 1 unit from the 8 units, change it to 24ths, and add it to the 9 24ths. 31+2 = ?3• 33—1;= 17. 2 units from 7, (8 — 1), units = 5 units. 3 tens from 5 tens = 2 tens. (a) Find the sum of the ten differences. 1. Observe that by the first operation we obtain ; that in 12 there are 6 times as many parts as there are in and that the parts are of the same size as those in 7. 2. Observe that by the second operation we obtain 7; that in there are the same number of parts as there are in 71, and that the parts are 6 times as great as those in 7. Note 1.-The 7 of 7 may be regarded as a dividend; the 24, as a divisor, and itself, as a quotient. In 2, we have a dividend 6 times as great as that in, the divisor remaining unchanged. In, we have a divisor 1 sixth as great as that in , the dividend remaining unchanged. Multiplying the dividend or dividing the divisor by any number, multiplies the quotient by the same number. Note 2.-The 7 of may be regarded as the antecedent of a couplet; the 24, as itself as the ratio. Multiplying the antecedent or dividing the consequent of any couplet multiplies the ratio by the same number. the consequent, and RULE. To multiply a fraction by an integer, multiply its numerator or divide its denominator by the integer. (a) Find the sum of the twelve products. (b) Find the sum of the twelve products. 1. Observe that by the first operation we obtain ; that are 1 third as many parts as there are in § and that the pa the same size as those in §. 2. Observe that by the second operation we obtain ; t there are the same number of parts as there are in § and parts are 1 third as great as those in §. Note 1.-The 6 of may be regarded as a dividend; the 7 as a diviso itself as a quotient. In, we have a dividend 1 third as great as th divisor remaining unchanged. In, we have a divisor 3 times as grea , the dividend remaining unchanged. Dividing the dividend or multi divisor by any number, divides the quotient by the same number. RULE. To divide a fraction by an integer, di numerator or multiply its denominator by the integer. I. Find the quotient. (See p. 105, problems 15 and 16.) Fractions. 190. TO MULTIPLY BY A FRACTION. $6 multiplied by 3, means, take 3 times $6. $6 multiplied by 2, means, take 2 times $6. $6 multiplied by 2, means, take 2 + of $6. $6 × 24 = $15. $6 x 3 $18. = $6 x 2 = $12. times $6; or 2 times $6 TO THE TEACHER.-Require the pupil to examine the preceding statements and similar ones presented by the teacher or by himself, until he clearly understands that to multiply by a fraction is to take such part of the multiplicand as is indicated by the fraction. Thus: to multiply 48 by is to take three fourths of 48; that is, three times I fourth of 48. It will thus be clear that multiplication by a fraction involves both multiplication and division; hence the work on the preceding pages should be mastered by the pupil before attempting what follows. EXAMPLE I. Multiply 24 by 2. 1 fourth of 24 is 6. 3 fourths of 24 are 18. EXAMPLE III. Multiply 275% by 2. 1 fourth of 275 is 681%. 3 fourths of 275% are 2061%. RULE. To multiply by a fraction, divide the multiplicand by the denominator of the fraction and multiply the quotient thus obtained by the numerator of the fraction. Observe that in practice we may, if more convenient, multiply the multiplicand by the numerator of the fraction, and divide the product thus obtained by the denominator. To multiply 12 by 3 we may take 3 times 1 fourth of 12 or 1 fourth of 3 times 12, as we choose. *This means, take 2 times 3463 and 1 of 3463. |