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A man lent a part of his money at $10\%$ p.a. and the rest at $15\%$ p.a. His income at the end of the year is ₹1,900. If he had interchanged the rate of interest on the two sums, he would have earned ₹200 more. Find the amount lent in both cases.
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Let amount lent for $10\%$ p. a. = $\text{Rs}x$ and amount lent for $15\%$ p. a. = $$\begin{aligned}& \text{Rs}y \\ & text{ATQ, } \frac{10x}{100} + \frac{15y}{100} = 1900 \\ & text{or } 2x + 3y = 38000 \\ & text{and } \frac{15x}{100} + \frac{10y}{100} = 2100 \\ & text{or } 3x + 2y = 42000 \\ & text{On solving these equations, we get} \\ & x = 10000 \text{ and } y = 6000 \\ & therefore \text{Amount lent for } 10\% \text{ p. a. = } \text{Rs}10000 \text{ \& money lent for } 15\% \text{ p. a. = } \text{Rs}6000\end{aligned}$$