The simple interest on a certain sum for `3(1)/(2)` years at 10% per annum in Rs.2,940. What will be the compound interest on the same sum for `2(1)/(2)` years at the same rate when interest in compounded yearly (nearest to a rupee) ?

Options

A. Rs. 2,272
B. Rs. 2,227
C. Rs. 2,327
D. Rs. 2,372

Answer

To solve the problem step by step, we will first calculate the principal amount using the given simple interest, and then we will calculate the compound interest for the specified period.

### Step 1: Calculate the Principal Amount (P)

The formula for Simple Interest (SI) is given by:

\[ SI = \frac{P \times R \times T}{100} \]

Where:
- \( SI = 2940 \) (given)
- \( R = 10\% \) (rate of interest)
- \( T = 3.5 \) years (which is \( 3 \frac{1}{2} \))

Substituting the values into the formula:

\[ 2940 = \frac{P \times 10 \times 3.5}{100} \]

### Step 2: Rearranging to find P

Rearranging the equation to solve for \( P \):

\[ P = \frac{2940 \times 100}{10 \times 3.5} \]

Calculating the denominator:

\[ 10 \times 3.5 = 35 \]

Now substituting back:

\[ P = \frac{294000}{35} \]

Calculating \( P \):

\[ P = 8400 \]

### Step 3: Calculate the Compound Interest (CI)

Now, we need to calculate the compound interest for \( 2.5 \) years at \( 10\% \) per annum. The formula for Compound Interest is:

\[ A = P \left(1 + \frac{R}{100}\right)^T \]

Where:
- \( P = 8400 \)
- \( R = 10\% \)
- \( T = 2.5 \)

Substituting the values into the formula:

\[ A = 8400 \left(1 + \frac{10}{100}\right)^{2.5} \]

This simplifies to:

\[ A = 8400 \left(1 + 0.1\right)^{2.5} \]
\[ A = 8400 \left(1.1\right)^{2.5} \]

### Step 4: Calculate \( (1.1)^{2.5} \)

Calculating \( (1.1)^{2.5} \):

Using a calculator, we find:

\[ (1.1)^{2.5} \approx 1.2763 \]

### Step 5: Calculate the Amount (A)

Now substituting back to find \( A \):

\[ A = 8400 \times 1.2763 \approx 10714.92 \]

### Step 6: Calculate the Compound Interest (CI)

The compound interest is given by:

\[ CI = A - P \]

Substituting the values:

\[ CI = 10714.92 - 8400 \approx 2314.92 \]

Rounding to the nearest rupee gives:

\[ CI \approx 2315 \]

### Final Answer

The compound interest on the same sum for \( 2.5 \) years at \( 10\% \) per annum, compounded yearly, is approximately **Rs. 2315**.

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