The simple interest accrued of an amount of Rs. 22,500 at the end of four years is Rs. 10,800. what would be the compound interest accrued on the same amount at the same rate at the end of two years ?

Options

A. Rs. 16,908
B. Rs. 5,724
C. Rs. 28,224
D. Rs 8,586

Answer

To solve the problem step by step, we will first determine the rate of interest using the simple interest formula, and then we will calculate the compound interest for two years.

### Step 1: Calculate the Simple Interest for 1 Year
Given:
- Total Simple Interest (SI) for 4 years = Rs. 10,800
- Principal (P) = Rs. 22,500

To find the Simple Interest for 1 year:
\[ \text{SI for 1 year} = \frac{\text{Total SI}}{\text{Number of years}} = \frac{10,800}{4} = 2,700 \]

### Step 2: Calculate the Rate of Interest
Using the formula for Simple Interest:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
Where:
- SI = Simple Interest for 1 year = Rs. 2,700
- P = Principal = Rs. 22,500
- R = Rate of interest (unknown)
- T = Time in years = 1

Rearranging the formula to find R:
\[ R = \frac{\text{SI} \times 100}{P \times T} = \frac{2,700 \times 100}{22,500 \times 1} \]
Calculating R:
\[ R = \frac{270,000}{22,500} = 12\% \]

### Step 3: Calculate the Compound Interest for 2 Years
The formula for Compound Interest (CI) is:
\[ \text{CI} = P \left(1 + \frac{R}{100}\right)^T - P \]
Where:
- P = Principal = Rs. 22,500
- R = Rate = 12%
- T = Time = 2 years

Calculating:
\[ \text{CI} = 22,500 \left(1 + \frac{12}{100}\right)^2 - 22,500 \]
\[ = 22,500 \left(1.12\right)^2 - 22,500 \]
Calculating \( (1.12)^2 \):
\[ (1.12)^2 = 1.2544 \]
Now substituting back:
\[ \text{CI} = 22,500 \times 1.2544 - 22,500 \]
Calculating:
\[ = 28,440 - 22,500 = 5,940 \]

### Step 4: Final Calculation of CI
The Compound Interest accrued on the same amount at the same rate at the end of two years is Rs. 5,940.

### Summary
The compound interest accrued on Rs. 22,500 at the end of two years at a rate of 12% is Rs. 5,940.

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