Question

A fraction becomes $\frac{1}{3}$ when 2 is subtracted from the numerator and it becomes $\frac{1}{2}$ when 1 is subtracted from the denominator. Find the fraction.

Answer 1

Let the numerator of the fraction be x and the denominator be y.

According to the question,

$$\begin{gathered} \frac{x-2}{y}=\frac{1}{3}...\left(\mathrm{i}\right) \\ \mathrm{and}\frac{x}{y-1}=\frac{1}{2}...\left(\mathrm{ii}\right) \\ \mathrm{Solving}\mathrm{equation}\left(\mathrm{i}\right),\mathrm{we}\mathrm{get} \\ 3\left(x-2\right)=y \\ \Rightarrow 3x-6=y...\left(\mathrm{iii}\right) \\ \mathrm{Substituting}\mathrm{the}\mathrm{value}\mathrm{of}y\mathrm{in}\mathrm{equation}\left(\mathrm{ii}\right),\mathrm{we}\mathrm{get} \\ \frac{x}{3x-6-1}=\frac{1}{2} \\ \Rightarrow 2\left(x\right)=3x-6-1 \\ \Rightarrow 2x=3x-7 \\ \Rightarrow 2x-3x=-7 \\ \Rightarrow -x=-7 \\ \Rightarrow x=7...\left(\mathrm{iv}\right) \\ \mathrm{From}\left(\mathrm{iii}\right)\mathrm{and}\left(\mathrm{iv}\right),\mathrm{we}\mathrm{get} \\ x=7\mathrm{and}y=15 \\ \mathrm{Hence},\mathrm{the}\mathrm{fraction}\mathrm{is}\frac{7}{15}. \end{gathered}$$