The simple interest on a sum deposited at `8%` p.a for 8 years is Rs. 16,000 .What will be the compound interest on the same sum at one fourth of the above rate of interest for 2 years ?

Options

A. Rs. 1,020
B. Rs. 980
C. Rs. 1,010
D. Rs. 1,015

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the Simple Interest Formula
The formula for calculating Simple Interest (SI) is given by:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
Where:
- \( P \) = Principal amount
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

### Step 2: Calculate the Principal Amount
We know from the question that the Simple Interest for 8 years at 8% is Rs. 16,000. We can rearrange the formula to find \( P \):
\[ 16000 = \frac{P \times 8 \times 8}{100} \]
Now, simplifying this:
\[ 16000 = \frac{64P}{100} \]
Multiplying both sides by 100:
\[ 1600000 = 64P \]
Now, divide both sides by 64:
\[ P = \frac{1600000}{64} = 25000 \]

### Step 3: Determine the Rate for Compound Interest
The problem states that we need to find the Compound Interest (CI) at one fourth of the above rate. The rate for CI is:
\[ R_{CI} = \frac{8}{4} = 2\% \]

### Step 4: Use the Compound Interest Formula
The formula for Compound Interest (CI) is:
\[ \text{CI} = P \left(1 + \frac{R}{100}\right)^T - P \]
Where:
- \( P \) = Principal amount (which we calculated as Rs. 25,000)
- \( R \) = Rate of interest (which we found to be 2%)
- \( T \) = Time in years (which is 2 years)

Substituting the values into the formula:
\[ \text{CI} = 25000 \left(1 + \frac{2}{100}\right)^2 - 25000 \]
\[ = 25000 \left(1 + 0.02\right)^2 - 25000 \]
\[ = 25000 \left(1.02\right)^2 - 25000 \]

### Step 5: Calculate \( (1.02)^2 \)
Now, calculate \( (1.02)^2 \):
\[ (1.02)^2 = 1.0404 \]

### Step 6: Substitute Back to Find CI
Now substitute back into the CI formula:
\[ \text{CI} = 25000 \times 1.0404 - 25000 \]
\[ = 26010 - 25000 \]
\[ = 1010 \]

### Final Answer
The Compound Interest on the same sum at one fourth of the above rate of interest for 2 years is Rs. 1010.

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