A sum amounts to Rs7,562 in 4 years and to Rs8,469 .44 in 5 years, at a certain rate per cent per annum when the interest is compounded yearly. lf Rs 10.000 at the same rate of interest is borrowed for two years, then what will be the compound interest (in Rs}?

Options

A. 1965
B. 1736
C. 2764
D. 2544

Answer

To solve the problem step-by-step, we need to find the compound interest on Rs 10,000 borrowed for two years at the same rate of interest that was used to calculate the amounts of Rs 7,562 and Rs 8,469.44 over four and five years respectively.

### Step 1: Determine the interest for the fifth year
We know:
- Amount after 4 years (A4) = Rs 7,562
- Amount after 5 years (A5) = Rs 8,469.44

The interest for the fifth year (I5) can be calculated as:
\[ I5 = A5 - A4 \]
\[ I5 = 8469.44 - 7562 = 907.44 \]

### Step 2: Calculate the principal amount after 4 years
The amount after 4 years is Rs 7,562, which is the principal (P) plus the interest earned over 4 years. We can denote the interest earned in the first four years as I4.

### Step 3: Calculate the rate of interest
The interest for the fourth year (I4) is equal to the interest for the fifth year (I5) because the interest is compounded yearly. Therefore, we can use the amount after 4 years to find the rate of interest.

Using the formula for the rate of interest:
\[ \text{Rate} = \frac{\text{Interest}}{\text{Principal}} \times 100 \]
Where:
- Interest = I5 = Rs 907.44
- Principal = A4 = Rs 7,562

Calculating the rate:
\[ \text{Rate} = \frac{907.44}{7562} \times 100 \approx 12\% \]

### Step 4: Calculate the compound interest for Rs 10,000 over 2 years
Now that we have the rate of interest (12%), we can calculate the compound interest for Rs 10,000 over 2 years.

1. **Calculate the amount after 1 year (A1)**:
\[ A1 = P(1 + r)^t \]
Where:
- P = Rs 10,000
- r = 12/100 = 0.12
- t = 1 year

\[ A1 = 10000 \times (1 + 0.12)^1 = 10000 \times 1.12 = 11200 \]

2. **Calculate the amount after 2 years (A2)**:
\[ A2 = A1(1 + r) \]
\[ A2 = 11200 \times (1 + 0.12) = 11200 \times 1.12 = 12544 \]

### Step 5: Calculate the compound interest
The compound interest (CI) for 2 years is given by:
\[ CI = A2 - P \]
\[ CI = 12544 - 10000 = 2544 \]

### Final Answer
The compound interest on Rs 10,000 borrowed for two years at the same rate of interest is Rs 2,544.

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