The simple interest accrued on a sum of certain principal is Rs. 6500 in eight years at the rate of 13 per cent per year. What would be the compound interest accrued on that principal at the rate of 8 per cent per year in 2 years?

Options

A. Rs. 1040
B. Rs. 1020
C. Rs. 1060
D. Rs. 1200

Answer

To solve the problem step by step, we need to find the compound interest accrued on a principal amount at a rate of 8% per year over 2 years, given that the simple interest accrued on the same principal at a rate of 13% per year over 8 years is Rs. 6500.

### Step 1: Calculate the Principal Amount
We know the formula for Simple Interest (SI):
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \(SI\) = Simple Interest
- \(P\) = Principal
- \(R\) = Rate of interest per annum
- \(T\) = Time in years

Given:
- \(SI = 6500\)
- \(R = 13\%\)
- \(T = 8\)

Substituting the values into the formula:
\[ 6500 = \frac{P \times 13 \times 8}{100} \]

### Step 2: Rearranging to Find Principal (P)
Rearranging the equation to solve for \(P\):
\[ P = \frac{6500 \times 100}{13 \times 8} \]

### Step 3: Calculate the Value of P
Calculating the denominator:
\[ 13 \times 8 = 104 \]
Now substituting back:
\[ P = \frac{6500 \times 100}{104} = \frac{650000}{104} \approx 6250 \]

### Step 4: Calculate Compound Interest (CI)
Now we need to calculate the compound interest for 2 years at a rate of 8%. The formula for Compound Interest (CI) is:
\[ CI = P \left(1 + \frac{R}{100}\right)^T - P \]

Where:
- \(R = 8\%\)
- \(T = 2\)

Substituting the values:
\[ CI = 6250 \left(1 + \frac{8}{100}\right)^2 - 6250 \]

### Step 5: Simplifying the Expression
Calculating \(1 + \frac{8}{100}\):
\[ 1 + 0.08 = 1.08 \]
Now raising it to the power of 2:
\[ 1.08^2 = 1.1664 \]

### Step 6: Calculate CI
Now substituting back into the CI formula:
\[ CI = 6250 \times 1.1664 - 6250 \]
Calculating:
\[ CI = 7290 - 6250 = 1040 \]

### Final Answer
The compound interest accrued on the principal at the rate of 8% per year in 2 years is **Rs. 1040**.

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