If the simple interest for a certain sum for 2 yr at 5% pa is ? 200, then what will be the compound interest for same sum for same period and at the same rate of interest ?

Answer

To find the compound interest for the same sum, period, and rate of interest, we first need to determine the principal amount using the information given about the simple interest.

### Step-by-Step Solution:

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

2. **Given Values**:
From the question, we know:
- \( SI = 200 \)
- \( R = 5\% \)
- \( T = 2 \) years

3. **Substitute the Values into the SI Formula**:
We can rearrange the formula to find \( P \):
\[ 200 = \frac{P \times 5 \times 2}{100} \]

4. **Simplify the Equation**:
\[ 200 = \frac{10P}{100} \]
\[ 200 = \frac{P}{10} \]

5. **Solve for \( P \)**:
\[ P = 200 \times 10 = 2000 \]

6. **Calculate Compound Interest (CI)**:
The formula for Compound Interest (CI) is:
\[ CI = P \left(1 + \frac{R}{100}\right)^T - P \]
Substituting the values we found:
\[ CI = 2000 \left(1 + \frac{5}{100}\right)^2 - 2000 \]
\[ CI = 2000 \left(1 + 0.05\right)^2 - 2000 \]
\[ CI = 2000 \left(1.05\right)^2 - 2000 \]

7. **Calculate \( (1.05)^2 \)**:
\[ (1.05)^2 = 1.1025 \]

8. **Substitute Back to Find CI**:
\[ CI = 2000 \times 1.1025 - 2000 \]
\[ CI = 2205 - 2000 \]
\[ CI = 205 \]

### Final Answer:
The compound interest for the same sum for the same period and at the same rate of interest is **205**.

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