Can ratios really be manipulated as fractions?

Question

In high-school Maths, we were taught that it was possible to manipulate ratios as fractions. For example, $$ 1 : 7 = 3 : x \\\\ \\frac{1}{7} = \\frac{3}{x} \\\\ \\frac{x}{7} = 3\\\\ x = 3 \ imes 7\\\\ x = 21\\\\ $$ My question relates to the second line of working, where we represent $1:7$ as $\\frac{1}{7}$. As I understand it, $1:7$ is better represented as one part of 8 total parts, i.e. $\\frac{1}{8}$. My question, then, is this: Is there a simple way of showing (or better still, a rigorous proof!) that a ratio-equation of the form $a:n = b:m$ can in all cases be rewritten as $\\frac{a}{n} = \\frac{b}{m}$ ?

Answer 1

LOL. You don't need an elaborate proof. By definition a fraction is always a ratio. 1:1,000,000 always = 1/1,000,000. 3/2 can always be expressed as 3:2.

Also, by definition, all ratios can be expressed as fractions. The only difference is the form. The use of the Fraction form in most mathematical operations is a matter of convention.