The compound interest on a certain sum at 4% per annum (compounded annually) for 2 years is ₹ 102. On the same principal at the same rate for the sametime, the simple interest will be:

Options

A. ₹100
B. ₹99
C. ₹98
D. ₹101

Answer

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

### Step 1: Understand the formula for Compound Interest
The formula for Compound Interest (CI) is:
\[ CI = A - P \]
where \( A \) is the amount after time \( t \), and \( P \) is the principal amount.

The formula for the amount \( A \) when compounded annually is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
where \( r \) is the rate of interest and \( n \) is the number of years.

### Step 2: Set up the equation using the given data
Given that the compound interest for 2 years at 4% per annum is ₹102, we can express this as:
\[ CI = A - P = 102 \]
So,
\[ A = P + 102 \]

### Step 3: Substitute the amount formula into the equation
Using the amount formula:
\[ A = P \left(1 + \frac{4}{100}\right)^2 \]
Substituting this into our equation gives:
\[ P + 102 = P \left(1 + \frac{4}{100}\right)^2 \]
Calculating \( \left(1 + \frac{4}{100}\right)^2 \):
\[ = \left(1 + 0.04\right)^2 = (1.04)^2 = 1.0816 \]
So, we have:
\[ P + 102 = P \cdot 1.0816 \]

### Step 4: Rearranging the equation to find \( P \)
Rearranging gives:
\[ P \cdot 1.0816 - P = 102 \]
Factoring out \( P \):
\[ P(1.0816 - 1) = 102 \]
\[ P(0.0816) = 102 \]
Now, solving for \( P \):
\[ P = \frac{102}{0.0816} \]
Calculating this gives:
\[ P = 1250 \]

### Step 5: Calculate Simple Interest (SI)
The formula for Simple Interest (SI) is:
\[ SI = \frac{P \cdot r \cdot t}{100} \]
Substituting the values \( P = 1250 \), \( r = 4 \), and \( t = 2 \):
\[ SI = \frac{1250 \cdot 4 \cdot 2}{100} \]
Calculating this gives:
\[ SI = \frac{10000}{100} = 100 \]

### Conclusion
The simple interest for the same principal at the same rate for the same time is ₹100.

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