Sum of money invested at compound interest amounts to Rs. 4624 in 2 years and Rs. 4913 in 3 years. The sum of money is-

Options

A. Rs.4086
B. Rs.4260
C. Rs.4335
D. None of these

Answer

To find the sum of money (the principal) invested at compound interest, we can use the information given about the amounts after 2 years and 3 years.

### Step-by-Step Solution:

1. **Identify the amounts after 2 and 3 years:**
- Amount after 2 years (A2) = Rs. 4624
- Amount after 3 years (A3) = Rs. 4913

2. **Calculate the increase in amount from the 2nd year to the 3rd year:**
- Increase = A3 - A2
- Increase = 4913 - 4624 = Rs. 289

3. **Understand that this increase (Rs. 289) is the interest earned on the amount after 2 years (A2) for one year at the compound interest rate (R%).**
- Let the principal amount be P, and the rate of interest be R%.

4. **Using the formula for compound interest:**
- Amount after 2 years:
\[ A2 = P \left(1 + \frac{R}{100}\right)^2 \]
- Amount after 3 years:
\[ A3 = P \left(1 + \frac{R}{100}\right)^3 \]

5. **Express the increase in terms of the principal and the rate:**
- The increase from A2 to A3 can also be expressed as:
\[ A3 - A2 = A2 \cdot \frac{R}{100} \]
- Therefore:
\[ 289 = 4624 \cdot \frac{R}{100} \]

6. **Solve for R:**
- Rearranging gives:
\[ \frac{R}{100} = \frac{289}{4624} \]
- Calculate:
\[ R = \frac{289 \times 100}{4624} \]
- Simplifying this gives:
\[ R = 6.25\% \]

7. **Now, use the amount after 2 years to find the principal:**
- We know:
\[ A2 = P \left(1 + \frac{R}{100}\right)^2 \]
- Substitute R:
\[ 4624 = P \left(1 + \frac{6.25}{100}\right)^2 \]
- This simplifies to:
\[ 4624 = P \left(1.0625\right)^2 \]
- Calculate \((1.0625)^2\):
\[ (1.0625)^2 = 1.12890625 \]
- Therefore:
\[ 4624 = P \cdot 1.12890625 \]

8. **Solve for P:**
- Rearranging gives:
\[ P = \frac{4624}{1.12890625} \]
- Calculate:
\[ P \approx 4096 \]

### Final Answer:
The sum of money (the principal) is Rs. 4096.