A loan was repaid in two annual instalments of Rs. 3630 each. If the rate of interest be 10% per annum compounded annually, then find the sum that was borrowed.

Options

A. Rs. 5200
B. Rs. 6100
C. Rs. 6300
D. Rs. 5600

Answer

To find the sum that was borrowed, we can use the formula for the present value of an annuity since the loan is repaid in two equal annual installments. The formula for the present value \( PV \) of an annuity is given by:

\[ PV = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \]

Where:
- \( P \) is the amount of each installment (Rs. 3630)
- \( r \) is the rate of interest per period (10% or 0.10)
- \( n \) is the number of periods (2 years)

### Step 1: Identify the values
- \( P = 3630 \)
- \( r = \frac{10}{100} = 0.10 \)
- \( n = 2 \)

### Step 2: Substitute the values into the formula
\[ PV = 3630 \times \left( \frac{1 - (1 + 0.10)^{-2}}{0.10} \right) \]

### Step 3: Calculate \( (1 + r)^{-n} \)
\[ (1 + 0.10)^{-2} = (1.10)^{-2} = \frac{1}{(1.10)^2} = \frac{1}{1.21} \approx 0.8264 \]

### Step 4: Substitute back into the formula
\[ PV = 3630 \times \left( \frac{1 - 0.8264}{0.10} \right) \]

### Step 5: Calculate \( 1 - 0.8264 \)
\[ 1 - 0.8264 = 0.1736 \]

### Step 6: Substitute and calculate
\[ PV = 3630 \times \left( \frac{0.1736}{0.10} \right) = 3630 \times 1.736 \]

### Step 7: Final calculation
\[ PV \approx 6300.48 \]

Thus, the sum that was borrowed is approximately Rs. 6300.