The compound interest on a certain sum for 2 years at 12% per annum compounded annually is Rs. 1272. The simple interest for that sum at the same rate and for the same period will be?

Options

A. Rs. 1296
B. Rs. 1196
C. Rs. 1220
D. Rs. 1200

Answer

To solve the problem step by step, we need to find the simple interest for a certain sum given that the compound interest for 2 years at 12% per annum is Rs. 1272.

### Step 1: Understand the given information
We know:
- Compound Interest (CI) for 2 years = Rs. 1272
- Rate of interest (R) = 12% per annum
- Time (T) = 2 years

### Step 2: Use the formula for Compound Interest
The formula for compound interest is:
\[ CI = P \left(1 + \frac{R}{100}\right)^T - P \]
Where:
- \( P \) = Principal amount
- \( R \) = Rate of interest
- \( T \) = Time in years

### Step 3: Rearranging the formula
We can rearrange the formula to find the principal \( P \):
\[ CI + P = P \left(1 + \frac{R}{100}\right)^T \]
This simplifies to:
\[ P = \frac{CI}{\left(1 + \frac{R}{100}\right)^T - 1} \]

### Step 4: Substitute the known values
Substituting the known values into the formula:
\[ P = \frac{1272}{\left(1 + \frac{12}{100}\right)^2 - 1} \]
Calculating \( \left(1 + \frac{12}{100}\right)^2 \):
\[ = \left(1 + 0.12\right)^2 = (1.12)^2 = 1.2544 \]
Now, substituting this back into the equation:
\[ P = \frac{1272}{1.2544 - 1} = \frac{1272}{0.2544} \]

### Step 5: Calculate the principal
Calculating the principal:
\[ P = \frac{1272}{0.2544} \approx 5000 \]

### Step 6: Calculate the Simple Interest
Now that we have the principal, we can calculate the simple interest using the formula:
\[ SI = \frac{P \times R \times T}{100} \]
Substituting the values:
\[ SI = \frac{5000 \times 12 \times 2}{100} \]
Calculating:
\[ SI = \frac{120000}{100} = 1200 \]

### Final Answer
The simple interest for that sum at the same rate and for the same period will be Rs. 1200.