At what rate per cent per annum will Rs. 2304 amount to Rs. 2500 in 2 years at compound interest ?

Options

A. `4(1)/(2)%`
B. `4(1)/(5)%`
C. `4(1)/(6)%`
D. `4(1)/(3)%`

Answer

To find the rate per cent per annum at which Rs. 2304 will amount to Rs. 2500 in 2 years at compound interest, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Principal (P) = Rs. 2304
- Amount (A) = Rs. 2500
- Time (T) = 2 years

2. **Use the Compound Interest Formula:**
The formula for the amount in compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Substituting the known values:
\[ 2500 = 2304 \left(1 + \frac{R}{100}\right)^2 \]

3. **Rearranging the Equation:**
To isolate the term with R, divide both sides by 2304:
\[ \frac{2500}{2304} = \left(1 + \frac{R}{100}\right)^2 \]

4. **Calculate the Left Side:**
Calculate \(\frac{2500}{2304}\):
\[ \frac{2500}{2304} = 1.08464 \quad \text{(approximately)} \]

5. **Taking the Square Root:**
Now, take the square root of both sides:
\[ 1 + \frac{R}{100} = \sqrt{1.08464} \]
Calculate the square root:
\[ \sqrt{1.08464} \approx 1.042 \]

6. **Isolate R:**
Now, isolate R:
\[ \frac{R}{100} = 1.042 - 1 \]
\[ \frac{R}{100} = 0.042 \]

7. **Calculate R:**
Multiply both sides by 100 to find R:
\[ R = 0.042 \times 100 = 4.2 \]

### Final Answer:
The rate of interest is **4.2% per annum**.