Simple interest on a sum of money for 2 years at 4% is Rs 450. Find compound interest on the same sum and at the same rate for 2 years.

Answer

To solve the problem step by step, we need to find the compound interest on a sum of money for 2 years at a rate of 4%, given that the simple interest for the same period and rate is Rs 450.

### Step 1: Find the Principal Amount (P)
We know that the formula for Simple Interest (SI) is given by:

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

Where:
- \( SI = 450 \) (given)
- \( R = 4\% \) (rate)
- \( T = 2 \) years (time)

Substituting the values into the formula:

\[ 450 = \frac{P \times 4 \times 2}{100} \]

### Step 2: Simplify the Equation
Rearranging the equation to solve for \( P \):

\[ 450 = \frac{8P}{100} \]

Multiplying both sides by 100 to eliminate the fraction:

\[ 45000 = 8P \]

### Step 3: Solve for P
Now, divide both sides by 8:

\[ P = \frac{45000}{8} = 5625 \]

So, the principal amount \( P \) is Rs 5625.

### Step 4: Calculate the Amount (A) using Compound Interest Formula
The formula for the amount in compound interest is:

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

Substituting the values we have:

\[ A = 5625 \left(1 + \frac{4}{100}\right)^2 \]

### Step 5: Simplify the Expression
Calculating \( 1 + \frac{4}{100} = 1 + 0.04 = 1.04 \):

\[ A = 5625 \times (1.04)^2 \]

Calculating \( (1.04)^2 \):

\[ (1.04)^2 = 1.0816 \]

So now we can calculate:

\[ A = 5625 \times 1.0816 \]

### Step 6: Calculate the Final Amount
Now, performing the multiplication:

\[ A = 5625 \times 1.0816 = 6084 \]

### Step 7: Calculate Compound Interest (CI)
Now, we can find the compound interest using the formula:

\[ CI = A - P \]

Substituting the values:

\[ CI = 6084 - 5625 = 459 \]

Thus, the compound interest for 2 years at 4% on the principal amount of Rs 5625 is Rs 459.

### Final Answer
The compound interest is Rs 459.

---