Question

A fraction reduces to 1/4 when 2 is subtracted from numerator and 3 is added to the denominator. But it reduces to 2/3 if 6 is added to the numerator and the denominator is multiplied by 3. Find the fraction.

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{numerator}=x \\ \mathrm{Let}\mathrm{the}\mathrm{denominator}=y \\ \mathrm{so},\mathrm{required}\mathrm{fraction}=\frac{x}{y} \\ \mathrm{according}\mathrm{to}\mathrm{the}\mathrm{first}\mathrm{condition}, \\ \frac{x-2}{y+3}=\frac{1}{4} \\ \Rightarrow 4x-8=y+3 \\ \Rightarrow y=4x-11........\left(1\right) \\ \mathrm{according}\mathrm{to}\mathrm{the}\mathrm{second}\mathrm{condition}, \\ \frac{x+6}{3y}=\frac{2}{3} \\ \Rightarrow x+6=2y \\ \Rightarrow x-2y+6=0......\left(2\right) \\ \mathrm{substituting}\mathrm{the}\mathrm{value}\mathrm{of}x\mathrm{from}\left(1\right)\mathrm{in}\left(2\right),\mathrm{we}\mathrm{get} \\ x-2\left(4x-11\right)+6=0 \\ \Rightarrow x-8x+22+6=0 \\ \Rightarrow -7x=-28 \\ \Rightarrow x=4 \\ \mathrm{from}\left(1\right),\mathrm{we}\mathrm{get} \\ y=4\times 4-11=16-11=5 \\ \mathrm{Hence}\mathrm{required}\mathrm{fraction}=\frac{x}{y}=\frac{4}{5} \end{gathered}$$