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 ₹ 104.50 a year. The amount of money be borrows, is
(a)₹ 4500
(b)₹ 5000
(c)₹ 5500
(d)₹ 6000
Answer
Answer (as printed): B
Explanation
Let the sum be ₹ $x$. Then C.I. when compounded half-yearly $=₹\left[x \times\left(1+\frac{3}{100}\right)^{2}-x\right]$ $$=₹\left(\frac{10609}{10000} x-x\right)=₹\left(\frac{609 x}{10000}\right) .$$ C.I. when compounded yearly $=₹\left[x\left(1+\frac{4}{100}\right)-x\right]$ $$\begin{aligned} & \quad=₹\left(\frac{26 x}{25}-x\right)=₹\left(\frac{x}{25}\right) . \\ & \therefore \frac{609 x}{10000}-\frac{x}{25}=104.50 \\ & \Rightarrow \frac{209 x}{10000}=104.50 \\ & \Rightarrow x=\left(\frac{104.50 \times 10000}{209}\right)=5000 . \end{aligned}$$
Explanation as extracted from the printed page; notation may be imperfect.