Two equal sums of money are lent at the same time at 8% and 7% per annum simple interest. The former is recovered 6 months earlier than the latter and the amount in each case is Rs. 2560. The sum and the time for which the sums of money are lent out are.

Explanation:-

$$\eqalign{ & {\text{Let each sum}} = {\text{Rs}}{\text{. }}x. \cr & {\text{Let the first sum be invested for}} \cr & \left( {T - \frac{1}{2}} \right){\text{years and}} \cr & {\text{the second sum for }}T{\text{ years}}{\text{.}} \cr & {\text{Then,}} \cr & x + \frac{{x \times 8 \times \left( {T - \frac{1}{2}} \right)}}{{100}} = 2560 \cr & \Rightarrow 100x + 8xT - 4x = 256000. \cr & \Rightarrow 96x + 8xT = 256000....(i) \cr & {\text{And,}} \cr & x\frac{{x \times 7 \times T}}{{100}} = 2560 \cr & \Rightarrow 100x + 7xT = 256000....(ii) \cr & {\text{From(i) and (ii), we get:}} \cr & 96x + 8xT = 100x + 7xT \cr & \Rightarrow 4x = xT \cr & \Rightarrow T = 4. \cr & {\text{Putting }}T = {\text{4 in (i),we get:}} \cr & 96x + 32x = 256000 \cr & \Rightarrow 128x = 256000 \cr & \Rightarrow x = 2000. \cr & {\text{Hence,}} \cr & {\text{each sum}} = {\text{Rs}}{\text{. 2000}}; \cr & {\text{time periods}} = \cr & {\text{4 years and }}3\frac{1}{2}{\text{years}} \cr}$$