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In the article titled ‘equivalent fractions’

Conceiving equivalents of any fraction is a simple exercise. For arithmetic fractions, we will refer to Mr Gary. S. Goldman’s Pre-Algebra book, in which he writes “Whatever you do to the numerator (or top of the fraction), you have to do the denominator (or bottom of the fraction).” This procedure works because essentially you are multiplying the fraction by a factor of 1 which does not change the value of the fraction’

Making Equal algebraic fractions

Mathematician Leonhard Euler writes: ‘To be convinced of the existence of equal algebraic fractions, we have only to write for the value of the fraction a certain letter c, representing by this letter c the quotient of the division of a by b; and to recollect that the multiplication of the quotient c by the divisor b must give the dividend. For since c multiplied by b gives a, it is evident that c multiplied by 2b will give 2a, that c multiplied by 3b will give 3a, and that, in general, c multiplied by mb will give ma. Now, changing this into an example of division, and dividing the product ma by mb, one of the factors, the quotient must be equal to the other factor c; but ma divided by mb gives also the fraction which is consequently equal to c; and this is what was to be proved: for c having been assumed as the value of the fraction it is evident that this fraction is equal to the fraction whatever be the value of m.’

Any fraction a/b can be assigned a literal k, such that the letter k