d or The quotient of unity by any number is termed the re Whence division by a fractional divisor is effected by multiplying the dividend by the reciprocal of the divisor. a. In deducing the general formula for division by a fractional divisor it has been found necessary to divide the dividend α by c, and to multiply the result by d. The division may be effected either by multiplying 6 by c, or dividing a by c: also α bc the product of by d, may be found by either multiplying á by d, or dividing b by d. (Art. 70.) Now when a is a multiple of c, and b a multiple of d, the division of a by c and of b by d is possible: therefore if the terms of the dividend are multiples of the terms of the divisor, the quotient is obtained by dividing the terms of the dividend by the corresponding terms of the divisor. b. As in the general method for multiplication of fractions, whole numbers must be reduced to the form of fractions and mixed numbers to improper fractions; also factors common to the numerator and denominator of the quotient may be cancelled in both terms. (Art. 92. b, c.) c. Hence, to divide one fractional expression by another,Rule,- Reduce mixed numbers to improper fractions and whole numbers to the form of fractions: then if the terms of the dividend are multiples of the corresponding terms of the divisor, divide the numerator of the dividend by the numerator of the divisor, and the denominator of the dividend by the denominator of the divisor. But if the terms of the dividend are not multiples of the terms of the divisor, multiply the dividend by the reciprocal of the divisor; the result is, in either case, the quotient required. d. Note. When the divisor is a proper fraction, the quotient is greater than the dividend. For the divisor being less than unity, its reciprocal is greater than unity; and the quotient is the product of the dividend by the reciprocal of the divisor. 99. In certain cases the quotient may be more easily obtained without the reduction of whole or mixed numbers to the fractional form. For instance, the quotient of a fraction by an integral divisor is obtained by dividing the numerator or multiplying the denominator of the fraction by the divisor. (Art. 70.) a. The quotient of a large whole number by a fractional divisor is obtained by multiplying the whole number by the reciprocal of the fraction; and this multiplication may be effected by multiplying the dividend by the denominator of the divisor and dividing the product by the numerator. (Art. 94.) b. The quotient of a large mixed number by an integral divisor less than the dividend may be obtained by decomposing the dividend into a multiple of the divisor, and a remainder, which may be either a mixed number less than the divisor, or a proper fraction; then dividing the two parts of the dividend by the divisor, and taking the sum of the partial quotients for the complete quotient. In practice, these and similar calculations may be made as follows. 1. 7)78543 1122+ rem. ÷7=* ... 78543÷7=11223. 2. 9)13271 147 +4 rem2. 41÷9=}÷9=}. ... 13271÷9=1471. c. The quotient of a large mixed number by a fraction may be obtained by combining the processes of Arts. 94. a. and 99. b. For example, the quotient of 15627 by is found by multiplying 15627} by 5 and dividing the product by 4, thus, 156271 5 4)781363 19534+rem'. and ÷4==}. ... 15627}÷=195341. If the divisor is a mixed number, it may be reduced to an improper fraction, and the division made as in the preceding example. d. These abridged processes of division are described in the following rules. I. To divide a fraction by a whole number:-Divide the numerator or multiply the denominator of the dividend by the divisor. II. To divide a whole number by a fraction: — Multiply the dividend by the denominator of the divisor and divide the product by the numerator of the divisor. III. To divide a mixed number by a whole number :-Divide the integral part of the dividend by the divisor; to the remainder, if there be one, annex the fractional part of the dividend; divide this mixed number or fraction by the divisor and annex the second partial quotient (which is a proper fraction) to the first. The mixed number thus formed is the required quotient. IV. To divide a mixed number by a fraction or a mixed number (having reduced the divisor, if a mixed number, to an improper fraction):- Multiply the dividend by the denominator of the divisor and divide this product by the numerator of the divisor. 100. Additional examples in Division of vulgar fractions. It is required to divide – by the divisor and dividend Divide Ans. mp 102. The fractional expressions hitherto fractions of unity: thus, of 1. considered are But a fractional part or certain fractional parts may also be taken of any number either less or greater than unity: as, 3 of ; of 3; of 4. A fraction of unity is called a simple fraction; and a fraction of any other number, a complex or compound fraction. If the sum or difference of any complex fractions or of complex and simple fractions is required, it becomes necessary to reduce the complex to simple fractions as a preparation for their reduction to a common denominator with each other or with simple fractions. Now the fraction, or of 1, signifies that 1 is divided into 8 equal parts, and that 7 of these equal parts compose the fraction. In like manner, the fraction of, or 3 of 7 of 1, signifies that of 1 are divided into 5 equal parts, and that 3 of these equal parts compose the fraction 3 of 7. Whence in these instances (and the method is general) the reduction of a fraction of any other fractional expression to a = 5 21 3 x 9 = 7 |