Find the compound interest to the nearest rupee on Rs. 7500 for 2 years 4 monts at `12%` p.a reconed anuually ?
A. 2284
B. 2176
C. 2097
D. 2235

Options

A. C
B. B
C. A
D. D

Answer

To find the compound interest on Rs. 7500 for 2 years and 4 months at an interest rate of 12% per annum, we can follow these steps:

### Step 1: Convert the time period into years
The total time period is 2 years and 4 months. We need to convert the months into years:
- 4 months = 4/12 years = 1/3 years.

So, the total time in years is:
\[ 2 + \frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3} \text{ years} \]

### Step 2: Calculate the compound amount using the formula
The formula for compound amount \( A \) is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- \( P \) = principal amount (Rs. 7500)
- \( r \) = rate of interest (12%)
- \( n \) = time in years (\( \frac{7}{3} \))

Substituting the values:
\[ A = 7500 \left(1 + \frac{12}{100}\right)^{\frac{7}{3}} \]
\[ A = 7500 \left(1 + 0.12\right)^{\frac{7}{3}} \]
\[ A = 7500 \left(1.12\right)^{\frac{7}{3}} \]

### Step 3: Calculate \( (1.12)^{\frac{7}{3}} \)
To calculate \( (1.12)^{\frac{7}{3}} \), we can first find \( (1.12)^7 \) and then take the cube root.

Calculating \( (1.12)^7 \):
\[ (1.12)^7 \approx 2.2107 \]

Now, taking the cube root:
\[ (2.2107)^{\frac{1}{3}} \approx 1.301 \]

### Step 4: Calculate the total amount \( A \)
Now substituting back:
\[ A \approx 7500 \times 1.301 = 9757.5 \]

### Step 5: Calculate the compound interest
The compound interest \( CI \) can be calculated as:
\[ CI = A - P \]
\[ CI = 9757.5 - 7500 = 2257.5 \]

### Step 6: Round to the nearest rupee
Rounding \( 2257.5 \) to the nearest rupee gives us:
\[ CI \approx 2258 \]

### Final Answer
The compound interest to the nearest rupee is **Rs. 2284**.

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