If the compound interest on a certain sum for two years at 12% per annum is ₹ 2,544, the simple interest on it at the same rate for 2 years will be

Options

A. ₹ 2,400
B. ₹ 2,500
C. ₹ 2,480
D. ₹ 2,440

Answer

To find the simple interest on a certain sum for 2 years at 12% per annum, given that the compound interest for the same period is ₹2,544, we can follow these steps:

### Step 1: Understand the relationship between compound interest (CI) and simple interest (SI)
The compound interest for 2 years can be expressed in terms of the principal (P) and the rate of interest (R):
\[ CI = P \left( (1 + \frac{R}{100})^n - 1 \right) \]
Where:
- \( R \) is the rate of interest (12% in this case),
- \( n \) is the number of years (2 years here).

### Step 2: Calculate the total amount after 2 years using the compound interest formula
Given that \( CI = ₹2,544 \) and \( R = 12\% \), we can express the total amount (A) after 2 years as:
\[ A = P \left(1 + \frac{R}{100}\right)^n = P \left(1 + \frac{12}{100}\right)^2 = P \left(1.12\right)^2 = P \times 1.2544 \]
Thus, we can express the compound interest as:
\[ CI = A - P = P \times 1.2544 - P = P(1.2544 - 1) = P \times 0.2544 \]
Setting this equal to the given CI:
\[ P \times 0.2544 = 2544 \]

### Step 3: Solve for the principal (P)
To find \( P \):
\[ P = \frac{2544}{0.2544} = 10000 \]

### Step 4: Calculate the simple interest (SI)
The simple interest for 2 years at the same rate can be calculated using the formula:
\[ SI = \frac{P \times R \times n}{100} \]
Substituting the values:
\[ SI = \frac{10000 \times 12 \times 2}{100} = \frac{240000}{100} = 2400 \]

### Conclusion
The simple interest on the sum for 2 years at 12% per annum is ₹2,400.

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