A sum of ₹ 6250 at 8% per annum, compounded annually, after `2(3)/(4)` years amounts to

Options

A. ₹ 7165.60
B. ₹ 7872.60
C. ₹ 7581.40
D. ₹ 7727.40

Answer

To find the amount after a certain period when a sum is compounded annually, we can use the formula for compound interest:

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

Where:
- \( A \) is the amount after time \( t \),
- \( P \) is the principal amount (initial sum),
- \( r \) is the rate of interest per annum,
- \( t \) is the time in years.

### Step 1: Identify the values
- Principal \( P = 6250 \)
- Rate \( r = 8\% \)
- Time \( t = 2 \frac{3}{4} \) years = \( 2.75 \) years

### Step 2: Convert mixed fraction to improper fraction
- \( t = 2 \frac{3}{4} = 2 + \frac{3}{4} = \frac{8}{4} + \frac{3}{4} = \frac{11}{4} \) years

### Step 3: Substitute values into the formula
Now we can substitute the values into the compound interest formula:

\[ A = 6250 \left(1 + \frac{8}{100}\right)^{2.75} \]

### Step 4: Calculate \( 1 + \frac{8}{100} \)
- \( 1 + \frac{8}{100} = 1 + 0.08 = 1.08 \)

### Step 5: Calculate \( (1.08)^{2.75} \)
To calculate \( (1.08)^{2.75} \), we can break it down:
- First calculate \( (1.08)^2 \):
\[ (1.08)^2 = 1.1664 \]
- Then calculate \( (1.08)^{0.75} \):
\[ (1.08)^{0.75} \approx 1.0595 \] (using a calculator or logarithm)
- Now multiply these results:
\[ (1.08)^{2.75} = (1.08)^2 \times (1.08)^{0.75} \approx 1.1664 \times 1.0595 \approx 1.2377 \]

### Step 6: Calculate the final amount
Now substitute back into the formula:
\[ A = 6250 \times 1.2377 \approx 7735.625 \]

### Step 7: Round to two decimal places
The final amount rounded to two decimal places is:
\[ A \approx 7735.63 \]

### Final Answer:
The amount after \( 2 \frac{3}{4} \) years is approximately ₹ 7735.63.

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