The SI on a certain sum of money for 2 years at 10% per annum is Rs. 400, find Cl at the same rate and for the same time.

Answer

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

### Step 1: Understand the formula for Simple Interest (SI)
The formula for Simple Interest 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)

### Step 2: Substitute the known values into the SI formula
Given:
- \( SI = 400 \) (for 2 years)
- \( R = 10\% \)
- \( T = 2 \) years

Substituting these values into the formula:
\[ 400 = \frac{P \times 10 \times 2}{100} \]

### Step 3: Simplify the equation
First, simplify the right side:
\[ 400 = \frac{20P}{100} \]
This simplifies to:
\[ 400 = \frac{P}{5} \]

### Step 4: Solve for Principal (P)
To find \( P \), multiply both sides by 5:
\[ P = 400 \times 5 = 2000 \]
So, the principal amount \( P \) is Rs. 2000.

### Step 5: Calculate the Compound Interest (CI)
The formula for Compound Interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Where \( A \) is the total amount after time \( T \).

Substituting the known values:
\[ A = 2000 \left(1 + \frac{10}{100}\right)^2 \]
This simplifies to:
\[ A = 2000 \left(1 + 0.1\right)^2 = 2000 \left(1.1\right)^2 \]

### Step 6: Calculate \( (1.1)^2 \)
Calculating \( (1.1)^2 \):
\[ (1.1)^2 = 1.21 \]
So,
\[ A = 2000 \times 1.21 = 2420 \]

### Step 7: Find Compound Interest (CI)
Now, to find the Compound Interest:
\[ CI = A - P = 2420 - 2000 = 420 \]

### Final Answer
The Compound Interest (CI) at the same rate and for the same time is Rs. 420.

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