A sum of money lent out at compound interest increases in value by 50% in 5 years. A person wants t lend three different sum x, y and z for 10, 15 and 20 years respectively at the above rate in such way that he gets back equal sums at the end of their respective periods. The ratio x : y : z is = ?

Explanation:-

$$\eqalign{ & P{\left( {1 + \frac{R}{{100}}} \right)^5} = 150\% \,{\text{of }}P = \frac{3}{2}P \cr & \Rightarrow {\left( {1 + \frac{R}{{100}}} \right)^5} = \frac{3}{2} \cr & x{\left( {1 + \frac{R}{{100}}} \right)^{10}} = y{\left( {1 + \frac{R}{{100}}} \right)^{15}} = z{\left( {1 + \frac{R}{{100}}} \right)^{20}} \cr & \Rightarrow x{\left\{ {{{\left( {1 + \frac{R}{{100}}} \right)}^5}} \right\}^2} = y{\left\{ {{{\left( {1 + \frac{R}{{100}}} \right)}^5}} \right\}^3} = z{\left\{ {{{\left( {1 + \frac{R}{{100}}} \right)}^5}} \right\}^4} \cr & \Rightarrow x \times {\left( {\frac{3}{2}} \right)^2} = y \times {\left( {\frac{3}{2}} \right)^3} = z \times {\left( {\frac{3}{2}} \right)^4} \cr & \Rightarrow \frac{{9x}}{4} = \frac{{7y}}{8} = \frac{{81z}}{{16}} = k({\text{say}}) \cr & \Rightarrow x = \frac{{4k}}{9},y = \frac{{8k}}{{27}},z = \frac{{16k}}{{81}} \cr & \Rightarrow x:y:z = \frac{{4k}}{9}:\frac{{8k}}{{27}}:\frac{{16k}}{{81}} \cr & \Rightarrow x:y:z = 36:24:16 \cr & \Rightarrow x:y:z = 9:6:4 \cr}$$