The proof is as follows:
(1) {a}{b} represents a equal parts, b of which make up the unit ; {ac}{b} represents ac equal parts, b of which make up the unit ; and the number of parts taken in the second fraction is c times the number taken in the first ;
that is, {a}{b} \times c = {ac}{b}
(2) {a}{bd} d = {ad}{bd} = {a}{b}. [Art. 136.1
Hence {a}{b} b = {ab}{b} = a; that is, the fraction {a}{b} is the quantity which must be multiplied by b in order to obtain a. Now the quantity which must be multiplied by b in order to obtain a is the quotient resulting from the division of a by b [Art. 53] ; therefore we may define a fraction thus : the fraction {a}{b} is the quotient of a divided by b.
141. Rule II. To divide a fraction by an integer. Divide the numerator, if it be divisible, by the integer ; or, if the numerator be not divisible, multiply the denominator by that integer.
The proof is as follows : (1) {ac}{b} represents ac equal parts, b of which make up the unit ; {a}{b} represents a equal parts, b of which make up the unit.
The number of parts taken in the first fraction is c times the number taken in the second. Therefore the second fraction is the quotient of the first fraction divided by c; that is, {ac}{b} \div c = {a}{b}
