A certain sum borrowed at 5% per annum com pound interest and paid back in two equal installmentseach₹882. What was the sum borrowed?

Options

A. 1600
B. 1680
C. 1640
D. 1700

Answer

To find the sum borrowed, we will use the formula for compound interest and the information given about the installments. Let's break down the solution step by step.

### Step 1: Understand the Problem
We know that a certain sum is borrowed at a compound interest rate of 5% per annum, and it is paid back in two equal installments of ₹882 each. We need to find the original sum borrowed (let's denote it as \( P \)).

### Step 2: Set Up the Formula
For the first installment, we can use the formula for the present value of a future amount:
\[ P_1 = \frac{I}{(1 + r/100)^k} \]
where:
- \( I \) is the installment amount (₹882),
- \( r \) is the rate of interest (5%),
- \( k \) is the number of years until the payment (1 year for the first installment).

For the second installment, the formula will be:
\[ P_2 = \frac{I}{(1 + r/100)^{k+1}} \]
where \( k \) is now 2 years for the second installment.

### Step 3: Calculate \( P_1 \) and \( P_2 \)
1. **Calculate \( P_1 \)**:
\[ P_1 = \frac{882}{(1 + 5/100)^1} = \frac{882}{1.05} = \frac{882 \times 20}{21} = 840 \]

2. **Calculate \( P_2 \)**:
\[ P_2 = \frac{882}{(1 + 5/100)^2} = \frac{882}{(1.05)^2} = \frac{882}{1.1025} = \frac{882 \times 400}{441} = 800 \]

### Step 4: Find the Total Sum Borrowed
Now, we can find the total sum borrowed by adding \( P_1 \) and \( P_2 \):
\[ P = P_1 + P_2 = 840 + 800 = 1640 \]

### Conclusion
Thus, the sum borrowed is ₹1640.