Question

The numerator of a rational number is 3 less than the denominator. If the denominator is increased by 5 and the numerator by 2, we get the rational number 1/2. Find the rational number.

Answer 1

$$\begin{gathered} \mathrm{Let},\mathrm{the}\mathrm{denominator}\mathrm{of}\mathrm{the}\mathrm{rational}\mathrm{number}\mathrm{be}\mathrm{x}. \\ \therefore \text{T}\mathrm{he}\mathrm{numerator}\mathrm{of}\mathrm{the}\mathrm{rational}\mathrm{number}\mathrm{will}\mathrm{be}\mathrm{x}-3. \\ \therefore \text{T}\mathrm{he}\mathrm{rational}\mathrm{number}=\frac{\mathrm{x}-3}{\mathrm{x}} \\ \mathrm{According}\mathrm{to}\mathrm{the}\mathrm{question}, \\ \frac{\mathrm{x}-3+2}{\mathrm{x}+5}=\frac{1}{2} \\ \mathrm{or}\frac{\mathrm{x}-1}{\mathrm{x}+5}=\frac{1}{2} \\ \mathrm{or}2\mathrm{x}-2=\mathrm{x}+5 \\ \mathrm{or}2\mathrm{x}-\mathrm{x}=5+2 \\ \mathrm{or}\mathrm{x}=7 \\ \therefore \text{T}\mathrm{he}\mathrm{rational}\mathrm{number}=\frac{7-3}{7}=\frac{4}{7} \end{gathered}$$