If the compound interest on a certain sum for 2 years at 4% per annum is Rs. 102 the simple interest at the same rate of interest for two years would be

Answer

To solve the problem step by step, we need to find the simple interest for 2 years at a rate of 4% per annum, given that the compound interest for the same period is Rs. 102.

### Step 1: Understand the relationship between Compound Interest (CI) and Principal (P)
The formula for Compound Interest is:
\[ CI = A - P \]
Where \( A \) is the amount after interest, and \( P \) is the principal.

### Step 2: Use the formula for Compound Amount
The amount \( A \) can be expressed as:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Where:
- \( R \) is the rate of interest (4%),
- \( T \) is the time (2 years).

### Step 3: Substitute the known values into the formula
Given that the compound interest \( CI = 102 \):
\[ 102 = P \left(1 + \frac{4}{100}\right)^2 - P \]

### Step 4: Simplify the expression
First, calculate \( 1 + \frac{4}{100} \):
\[ 1 + \frac{4}{100} = 1 + 0.04 = 1.04 \]
Now, calculate \( (1.04)^2 \):
\[ (1.04)^2 = 1.0816 \]
So, we can rewrite the equation:
\[ 102 = P(1.0816 - 1) \]
\[ 102 = P(0.0816) \]

### Step 5: Solve for Principal (P)
To find \( P \):
\[ P = \frac{102}{0.0816} \]
Calculating this gives:
\[ P = 1250 \text{ Rs.} \]

### Step 6: Calculate Simple Interest (SI)
Now, we can calculate the simple interest using the formula:
\[ SI = \frac{P \times R \times T}{100} \]
Substituting the values:
\[ SI = \frac{1250 \times 4 \times 2}{100} \]

### Step 7: Simplify the calculation
Calculating this step-by-step:
1. \( 1250 \times 4 = 5000 \)
2. \( 5000 \times 2 = 10000 \)
3. \( \frac{10000}{100} = 100 \)

Thus, the simple interest for 2 years at 4% per annum is:
\[ SI = 100 \text{ Rs.} \]

### Final Answer
The simple interest at the same rate of interest for two years would be Rs. 100.

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