Question
A money lender borrows money at 4% per annum and pays the interest at the end of the year. He lends it at 6% per annum compound interest compounded half yearly and receives the interest at the end of the year. In this way, he gains Rs. 104.50, a year. The amount of money be borrows, is ?
Answer: Option B
$$\eqalign{
& {\text{Let the sum Rs}}{\text{. }}x{\text{ }} \cr
& {\text{Then,}} \cr
& {\text{C}}{\text{.I}}{\text{. when compounded half yearly}} \cr
& {\text{ = Rs}}{\text{.}}\left[ {x \times {{\left( {1 + \frac{3}{{100}}} \right)}^2} - x} \right] \cr
& = {\text{Rs}}{\text{.}}\left( {\frac{{10609}}{{10000}}x - x} \right) \cr
& = {\text{Rs}}{\text{.}}\left( {\frac{{609x}}{{10000}}} \right) \cr
& {\text{C}}{\text{.I}}{\text{. when compounded yearly}} \cr
& {\text{ = Rs}}{\text{.}}\left[ {x \times \left( {1 + \frac{4}{{100}}} \right) - x} \right] \cr
& = {\text{Rs}}{\text{.}}\left( {\frac{{26x}}{{25}} - x} \right) \cr
& = {\text{Rs}}{\text{.}}\frac{x}{{25}} \cr
& \therefore \frac{{609x}}{{10000}} - \frac{x}{{25}} = 104.50 \cr
& \Rightarrow \frac{{209x}}{{10000}} = 104.50 \cr
& \Rightarrow x = \left( {\frac{{104.50 \times 10000}}{{209}}} \right) \cr
& \Rightarrow x = 5000 \cr} $$
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$$\eqalign{
& {\text{Let the sum Rs}}{\text{. }}x{\text{ }} \cr
& {\text{Then,}} \cr
& {\text{C}}{\text{.I}}{\text{. when compounded half yearly}} \cr
& {\text{ = Rs}}{\text{.}}\left[ {x \times {{\left( {1 + \frac{3}{{100}}} \right)}^2} - x} \right] \cr
& = {\text{Rs}}{\text{.}}\left( {\frac{{10609}}{{10000}}x - x} \right) \cr
& = {\text{Rs}}{\text{.}}\left( {\frac{{609x}}{{10000}}} \right) \cr
& {\text{C}}{\text{.I}}{\text{. when compounded yearly}} \cr
& {\text{ = Rs}}{\text{.}}\left[ {x \times \left( {1 + \frac{4}{{100}}} \right) - x} \right] \cr
& = {\text{Rs}}{\text{.}}\left( {\frac{{26x}}{{25}} - x} \right) \cr
& = {\text{Rs}}{\text{.}}\frac{x}{{25}} \cr
& \therefore \frac{{609x}}{{10000}} - \frac{x}{{25}} = 104.50 \cr
& \Rightarrow \frac{{209x}}{{10000}} = 104.50 \cr
& \Rightarrow x = \left( {\frac{{104.50 \times 10000}}{{209}}} \right) \cr
& \Rightarrow x = 5000 \cr} $$
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