The simple interest accrued on an amount of Rs. 19800 at the end of 3 years is Rs. 7128. What would be the compound interest accrued on the same amount at the same rate in the same period?

Options

A. Rs. 8934.6784
B. Rs. 8017.5744
C. Rs. 7861.8754
D. Rs. 6871.6734

Answer

To solve the problem step by step, we need to find the compound interest accrued on an amount of Rs. 19800 at the same rate and for the same period where the simple interest is given.

### Step 1: Calculate the Rate of Interest
We know the formula for Simple Interest (SI):
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( SI \) = Simple Interest (Rs. 7128)
- \( P \) = Principal amount (Rs. 19800)
- \( R \) = Rate of interest (unknown)
- \( T \) = Time (3 years)

Substituting the known values into the formula:
\[ 7128 = \frac{19800 \times R \times 3}{100} \]

### Step 2: Rearranging the Equation
To find \( R \), we can rearrange the equation:
\[ 7128 \times 100 = 19800 \times R \times 3 \]
\[ 712800 = 59400R \]

### Step 3: Solve for R
Now, divide both sides by 59400:
\[ R = \frac{712800}{59400} \]
Calculating this gives:
\[ R = 12\% \]

### Step 4: Calculate Compound Interest
Now that we have the rate of interest, we can calculate the Compound Interest (CI) using the formula:
\[ CI = P \left(1 + \frac{R}{100}\right)^T - P \]
Substituting the known values:
\[ CI = 19800 \left(1 + \frac{12}{100}\right)^3 - 19800 \]
\[ CI = 19800 \left(1 + 0.12\right)^3 - 19800 \]
\[ CI = 19800 \left(1.12\right)^3 - 19800 \]

### Step 5: Calculate \( (1.12)^3 \)
Calculating \( (1.12)^3 \):
\[ (1.12)^3 = 1.404928 \]

### Step 6: Substitute Back to Find CI
Now substitute this value back into the CI formula:
\[ CI = 19800 \times 1.404928 - 19800 \]
Calculating this gives:
\[ CI = 27868.5 - 19800 \]
\[ CI = 8068.5 \]

### Step 7: Final Result
Thus, the compound interest accrued on the same amount at the same rate in the same period is approximately Rs. 8068.5.