Question

The length of a rectangle exceeds its breadth by 9 cm. If length and breadth are each increased by 3 cm, the area of the new rectangle will be 84 cm² more than that of the given rectangle. Find the length and breath of the given rectangle.

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{breadth}\mathrm{of}\mathrm{the}\mathrm{rectangle}\mathrm{be}\mathrm{x}\mathrm{cm}. \\ \mathrm{Therefore},\mathrm{the}\mathrm{length}\mathrm{of}\mathrm{the}\mathrm{rectangle}\mathrm{will}\mathrm{be}(\mathrm{x}+9)\mathrm{cm}. \\ \therefore \mathrm{Area}\mathrm{of}\mathrm{the}\mathrm{rectangle}=\mathrm{x}(\mathrm{x}+9){\mathrm{cm}}^2. \\ \mathrm{If}\mathrm{the}\mathrm{length}\mathrm{and}\mathrm{breadth}\mathrm{are}\mathrm{increased}\mathrm{by}3\mathrm{cm}\mathrm{each}, \\ \text{a}\mathrm{rea}=(\mathrm{x}+3)(\mathrm{x}+9+3){\mathrm{cm}}^2. \\ \mathrm{Now}, \\ (\mathrm{x}+3)(\mathrm{x}+12)-\mathrm{x}(\mathrm{x}+9)=84 \\ \mathrm{or}{\mathrm{x}}^2+15\mathrm{x}+36-{\mathrm{x}}^2-9\mathrm{x}=84 \\ \mathrm{or}6\mathrm{x}=84-36 \\ \mathrm{or}\mathrm{x}=\frac{48}{6}=8. \\ \mathrm{Thus},\mathrm{brea}\text{d}\mathrm{th}\mathrm{of}\mathrm{the}\mathrm{rectangle}=8\mathrm{cm}. \\ \mathrm{Length}\mathrm{of}\mathrm{the}\mathrm{rectangle}=(8+9)=17\mathrm{cm}. \end{gathered}$$