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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.

Correct Answer: Rs. 2000, 3.5 years and 4 years
$$\eqalign{ & {\text{Let each sum}} = {\text{Rs}}{\text{. }}x. \cr & {\text{Let the first sum be invested for}} \cr & \left( {T - \frac{1}{2}} \right){\text{years and}} \cr & {\text{the second sum for }}T{\text{ years}}{\text{.}} \cr & {\text{Then,}} \cr & x + \frac{{x \times 8 \times \left( {T - \frac{1}{2}} \right)}}{{100}} = 2560 \cr & \Rightarrow 100x + 8xT - 4x = 256000 \cr & \Rightarrow 96x + 8xT = 256000....(i) \cr & {\text{And,}} \cr & x + \frac{{x \times 7 \times T}}{{100}} = 2560 \cr & \Rightarrow 100x + 7xT = 256000....(ii) \cr & {\text{From(i) and (ii), we get:}} \cr & 96x + 8xT = 100x + 7xT \cr & \Rightarrow 4x = xT \cr & \Rightarrow T = 4 \cr & {\text{Putting }}T = {\text{4 in (i),we get:}} \cr & 96x + 32x = 256000 \cr & \Rightarrow 128x = 256000 \cr & \Rightarrow x = 2000 \cr & {\text{Hence,}} \cr & {\text{each sum}} = {\text{Rs}}{\text{. 2000}} \cr & {\text{time periods}} = \cr & {\text{4 years and }}3\frac{1}{2}{\text{years}} \cr} $$

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