Question

A fraction becomes 9/11 if 2 is added to both numerator and denominator. if 3 is added to both numerator and denominator it becomes 5/6 find the fraction by substitution method

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{numerator}=\mathrm{x} \\ \mathrm{Let}\mathrm{the}\mathrm{denominator}=\mathrm{y} \\ \mathrm{Now},\mathrm{fraction}=\frac{\mathrm{x}}{\mathrm{y}} \\ \mathrm{Now},\mathrm{according}\mathrm{to}\mathrm{first}\mathrm{condition}, \\ \frac{\mathrm{x}+2}{\mathrm{y}+2}=\frac{9}{11} \\ \Rightarrow 9\mathrm{y}+18=11\mathrm{x}+22 \\ \Rightarrow 9\mathrm{y}=11\mathrm{x}+4 \\ \Rightarrow \mathrm{y}=\frac{11\mathrm{x}+4}{9}......\left(1\right) \\ \mathrm{Now},\mathrm{according}\mathrm{to}\mathrm{second}\mathrm{condition}, \\ \frac{\mathrm{x}+3}{\mathrm{y}+3}=\frac{5}{6} \\ \Rightarrow 5\mathrm{y}+15=6\mathrm{x}+18 \\ \Rightarrow 5\mathrm{y}-6\mathrm{x}=3.......\left(2\right) \\ \mathrm{put}\mathrm{the}\mathrm{value}\mathrm{of}\mathrm{y}\mathrm{from}\left(1\right)\mathrm{in}\left(2\right),\mathrm{we}\mathrm{get} \\ 5\times \left(\frac{11\mathrm{x}+4}{9}\right)-6\mathrm{x}=3 \\ \Rightarrow \frac{55\mathrm{x}+20-54\mathrm{x}}{9}=3 \\ \Rightarrow \mathrm{x}=27-20=7 \\ \mathrm{Now},\mathrm{from}\left(1\right),\mathrm{we}\mathrm{get} \\ \mathrm{y}=\frac{11\times 7+4}{9}=\frac{81}{9}=9 \\ \mathrm{So},\mathrm{required}\mathrm{fraction}=\frac{7}{9} \end{gathered}$$