Simple interest on a sum of money for 4 yr at 7 p.c.p.a is ₹3584. what would be the compound interest (compounded annually) on the same amount of money for 2 yr at 4 pcpa?

Options

A. `1162.32`
B. `1044.48`
C. 1992.38
D. `1231.76`

Explanation

To solve the problem step by step, we will first find the principal amount using the simple interest formula, and then we will calculate the compound interest for the given conditions.

### Step 1: Find 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 Amount
- \( R \) = Rate of Interest (per annum)
- \( T \) = Time (in years)

Given:
- \( SI = 3584 \)
- \( R = 7\% \)
- \( T = 4 \) years

Substituting the values into the formula:

\[ 3584 = \frac{P \times 7 \times 4}{100} \]

Now, simplify the equation:

\[ 3584 = \frac{28P}{100} \]

Multiplying both sides by 100:

\[ 358400 = 28P \]

Now, divide both sides by 28 to find \( P \):

\[ P = \frac{358400}{28} = 12800 \]

So, the principal amount \( P \) is ₹12,800.

### Step 2: Calculate the Compound Interest

Now that we have the principal, we will calculate the compound interest for 2 years at a rate of 4% per annum.

The formula for Compound Interest (CI) is:

\[ A = P \left(1 + \frac{R}{100}\right)^n \]

Where:
- \( A \) = Amount after time \( n \)
- \( P \) = Principal Amount
- \( R \) = Rate of Interest (per annum)
- \( n \) = Time (in years)

Given:
- \( P = 12800 \)
- \( R = 4\% \)
- \( n = 2 \)

Substituting the values into the formula:

\[ A = 12800 \left(1 + \frac{4}{100}\right)^2 \]

Calculating \( 1 + \frac{4}{100} = 1.04 \):

\[ A = 12800 \times (1.04)^2 \]

Calculating \( (1.04)^2 = 1.0816 \):

\[ A = 12800 \times 1.0816 = 13884.48 \]

### Step 3: Calculate the Compound Interest

Now, we can find the compound interest by subtracting the principal from the amount:

\[ CI = A - P \]

Substituting the values:

\[ CI = 13884.48 - 12800 = 1084.48 \]

Thus, the compound interest is ₹1,084.48.

### Final Answer

The compound interest on the same amount of money for 2 years at 4% per annum is ₹1,084.48.

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