If the simple interest on a certain sum at 4% p.a. for 2 years is Rs. 80 then find the compound interest on the same sum for the same period ?

Answer

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

### Step 1: Calculate the Principal Amount
We know that the formula for Simple Interest (SI) is given by:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( SI \) = Simple Interest
- \( P \) = Principal amount
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

From the question, we have:
- \( SI = 80 \)
- \( R = 4\% \)
- \( T = 2 \) years

Substituting the values into the formula:
\[ 80 = \frac{P \times 4 \times 2}{100} \]
\[ 80 = \frac{8P}{100} \]
Multiplying both sides by 100:
\[ 8000 = 8P \]
Dividing both sides by 8:
\[ P = 1000 \]

### Step 2: Calculate the Compound Interest
Now that we have the principal amount \( P = 1000 \), we can calculate the Compound Interest (CI) using the formula:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Where:
- \( A \) = Amount after time \( T \)
- \( CI = A - P \)

Substituting the values:
\[ A = 1000 \left(1 + \frac{4}{100}\right)^2 \]
\[ A = 1000 \left(1 + 0.04\right)^2 \]
\[ A = 1000 \left(1.04\right)^2 \]
Calculating \( (1.04)^2 \):
\[ (1.04)^2 = 1.0816 \]
Now substituting back:
\[ A = 1000 \times 1.0816 = 1081.6 \]

Now, calculating the Compound Interest:
\[ CI = A - P = 1081.6 - 1000 = 81.6 \]

### Final Answer
The compound interest on the same sum for the same period is **Rs. 81.6**.

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