Question

A fraction becomes 4/5 , if 1 is added to both numerator and denominator . If 5 is subtracted from both numerator and denominator , the fraction becomes 1/2 . What is the fraction?

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{numerator}=\mathrm{x} \\ \mathrm{Let}\mathrm{the}\mathrm{denominator}=\mathrm{y} \\ \mathrm{Now},\mathrm{fraction}=\frac{\mathrm{x}}{\mathrm{y}} \\ \mathrm{According}\mathrm{to}\mathrm{first}\mathrm{condition}: \\ \frac{\mathrm{x}+1}{\mathrm{y}+1}=\frac{4}{5} \\ \Rightarrow 5\mathrm{x}+5=4\mathrm{y}+4 \\ \Rightarrow 5\mathrm{x}-4\mathrm{y}=-1....\left(1\right) \\ \mathrm{According}\mathrm{to}\mathrm{second}\mathrm{condition}: \\ \frac{\mathrm{x}-5}{\mathrm{y}-5}=\frac{1}{2} \\ \Rightarrow 2\mathrm{x}-10=\mathrm{y}-5 \\ \Rightarrow 2\mathrm{x}-\mathrm{y}=5....\left(2\right) \\ \mathrm{Multiply}\left(2\right)\mathrm{by}4,\mathrm{we}\mathrm{get} \\ 8\mathrm{x}-4\mathrm{y}=20....\left(3\right) \\ \mathrm{Subtracting}\left(1\right)\mathrm{from}\left(3\right),\mathrm{we}\mathrm{get} \\ 3\mathrm{x}=21 \\ \Rightarrow \mathrm{x}=7 \\ \mathrm{Now},\mathrm{from}\left(2\right),\mathrm{we}\mathrm{get} \\ 2\times 7-\mathrm{y}=5 \\ \Rightarrow -\mathrm{y}=5-14 \\ \Rightarrow \mathrm{y}=9 \\ \mathrm{Hence},\mathrm{required}\mathrm{fraction}=\frac{7}{9} \end{gathered}$$