The simple interest on a certain sum of money for 2 years at` 5%` is `Rs. 1600`. The compound interest at the same rate after 3 years interest compound annually, is

Options

A. `Rs. 2520`
B. `Rs.2522 `
C. `Rs.2555 `
D. `Rs.2535`

Answer

To solve the problem, we need to find the compound interest after 3 years, given the simple interest for 2 years.

### Step 1: Calculate the Principal Amount
We know that the formula for Simple Interest (SI) is:

\[ SI = \frac{P \times R \times T}{100} \]

Where:
- \( SI \) = Simple Interest
- \( P \) = Principal amount
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

From the question, we have:
- \( SI = 1600 \)
- \( R = 5\% \)
- \( T = 2 \) years

Substituting these values into the formula, we get:

\[ 1600 = \frac{P \times 5 \times 2}{100} \]

### Step 2: Solve for Principal (P)
Rearranging the equation to solve for \( P \):

\[ 1600 = \frac{10P}{100} \]

\[ 1600 = \frac{P}{10} \]

Multiplying both sides by 10:

\[ P = 16000 \]

### Step 3: Calculate Compound Interest for 3 Years
Now that we have the principal amount, we can calculate the Compound Interest (CI) using the formula:

\[ CI = P \left(1 + \frac{R}{100}\right)^T - P \]

Where:
- \( CI \) = Compound Interest
- \( P \) = Principal amount (which we found to be 16000)
- \( R \) = Rate of interest per annum (5%)
- \( T \) = Time in years (3 years)

Substituting the values into the formula:

\[ CI = 16000 \left(1 + \frac{5}{100}\right)^3 - 16000 \]

\[ CI = 16000 \left(1 + 0.05\right)^3 - 16000 \]

\[ CI = 16000 \left(1.05\right)^3 - 16000 \]

Calculating \( (1.05)^3 \):

\[ (1.05)^3 = 1.157625 \]

Now substituting back into the equation:

\[ CI = 16000 \times 1.157625 - 16000 \]

\[ CI = 18521 - 16000 \]

\[ CI = 2521 \]

### Final Answer
The compound interest after 3 years is **Rs. 2521**.

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