A sum of money is paid back in two annual instalments of `Rs.17, 640` each, allowing `5%` compound interest compounded annually. The sum borrowed was

Options

A. `RS. 32,800`
B. `Rs. 32,200 `
C. `Rs. 32,000 `
D. `Rs. 32,400`

Answer

To find the sum of money borrowed that is paid back in two annual installments of Rs. 17,640 each with a compound interest rate of 5% compounded annually, we can follow these steps:

### Step 1: Understand the Installments
The total amount paid back in two installments is:
- First installment: Rs. 17,640
- Second installment: Rs. 17,640

### Step 2: Calculate the Present Value of Each Installment
Since the installments are paid at different times, we need to calculate the present value of each installment.

1. **Present Value of the First Installment**:
The first installment is paid at the end of the first year. The formula for the present value (PV) is:
\[ PV = \frac{C}{(1 + r)^n} \]
Where:
- \( C = 17,640 \)
- \( r = 0.05 \) (5% interest)
- \( n = 1 \) (1 year)

So,
\[ PV_1 = \frac{17,640}{(1 + 0.05)^1} = \frac{17,640}{1.05} = 16,800 \]

2. **Present Value of the Second Installment**:
The second installment is paid at the end of the second year. Using the same formula:
\[ PV_2 = \frac{17,640}{(1 + 0.05)^2} = \frac{17,640}{1.1025} \approx 16,000 \]

### Step 3: Calculate the Total Present Value
Now, we add the present values of both installments to find the total sum borrowed:
\[ \text{Total Present Value} = PV_1 + PV_2 = 16,800 + 16,000 = 32,800 \]

### Conclusion
The sum borrowed was Rs. 32,800.

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