At what rate per cent per annum will Rs 4,800 amount to Rs 5,547 in two years if the interest is compounded annually ?

Options

A. `7.5%`
B. `7%`
C. `8.5%`
D. `8%`

Answer

To find the rate of interest per annum that will cause Rs. 4,800 to amount to Rs. 5,547 in two years with compound interest, we can use the formula for compound interest:

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

Where:
- \( A \) = Amount after time \( t \) (Rs. 5,547)
- \( P \) = Principal amount (Rs. 4,800)
- \( r \) = Rate of interest per annum (unknown)
- \( t \) = Time in years (2 years)

### Step-by-Step Solution:

1. **Identify the given values**:
- Principal \( P = 4800 \)
- Amount \( A = 5547 \)
- Time \( t = 2 \)

2. **Substitute the values into the compound interest formula**:
\[ 5547 = 4800 \left(1 + \frac{r}{100}\right)^2 \]

3. **Divide both sides by 4800** to isolate the term with \( r \):
\[ \frac{5547}{4800} = \left(1 + \frac{r}{100}\right)^2 \]

4. **Calculate \( \frac{5547}{4800} \)**:
\[ \frac{5547}{4800} \approx 1.1546875 \]

5. **Take the square root of both sides** to solve for \( 1 + \frac{r}{100} \):
\[ \sqrt{1.1546875} = 1 + \frac{r}{100} \]

6. **Calculate the square root**:
\[ \sqrt{1.1546875} \approx 1.078 \]

7. **Subtract 1 from both sides**:
\[ 1.078 - 1 = \frac{r}{100} \]
\[ 0.078 = \frac{r}{100} \]

8. **Multiply both sides by 100 to find \( r \)**:
\[ r = 0.078 \times 100 = 7.8 \]

9. **Final answer**:
\[ r \approx 7.8\% \]