Question

The numerator of a fraction is 6 less than the denominator. If 3 is added to the numerator, the fraction is equal to $\frac{2}{3}$. What is the original fraction equal to?

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{denominator}\mathrm{of}\mathrm{the}\mathrm{fraction}\mathrm{be}\mathrm{x}. \\ \mathrm{Therefore},\mathrm{the}\mathrm{numerator}\mathrm{will}\mathrm{be}(\mathrm{x}-6). \\ \therefore \mathrm{Fraction}=\frac{\mathrm{x}-6}{\mathrm{x}} \\ \mathrm{According}\mathrm{to}\mathrm{the}\mathrm{question}, \\ \frac{\mathrm{x}-6+3}{\mathrm{x}}=\frac{2}{3} \\ \mathrm{or}\frac{\mathrm{x}-3}{\mathrm{x}}=\frac{2}{3} \\ \mathrm{or}3\mathrm{x}-9=2\mathrm{x}\left[\text{After cross multiplication}\right] \\ \mathrm{or}3\mathrm{x}-2\mathrm{x}=9 \\ \mathrm{or}\mathrm{x}=9 \\ \mathrm{Thus},\mathrm{the}\mathrm{original}\mathrm{fraction}=\frac{9-6}{9}=\frac{1}{3} \end{gathered}$$