A financier obtains a loan at 4% annual simple interest and lends it out at 6% annual compound interest, compounded semi-annually. If their annual profit is ₹104.50, calculate the principal amount borrowed.

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Accepted Answer

Let the principal amount be $$ P $$.

1. **Interest paid by the financier (simple interest):**  
   $$
   	ext{Interest Paid} = 0.04P \quad [4\% 	ext{ annual rate}]
   $$

2. **Interest earned (compound interest, half-yearly compounding):**  
   - Semi-annual rate = $$ \frac{6\%}{2} = 3\% $$  
   - Total amount after 1 year:  
     $$
     P 	imes (1 + 0.03)^2 = P 	imes 1.0609
     $$  
   - Compound interest earned:  
     $$
     1.0609P - P = 0.0609P
     $$

3. **Annual profit equation:**  
   $$
   0.0609P - 0.04P = 104.50
   $$  
   $$
   0.0209P = 104.50 \implies P = \frac{104.50}{0.0209} = 5,\!000
   $$

Thus, the borrowed principal is **₹5,000**.

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