A certain sum amounts to Rs 5292 in two years and Rs 5556.60 in three years, interest being compounded annually. Find:
the original sum.

Answer

To find the original sum (principal) given the amounts after two and three years with compound interest, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the amounts:**
- Amount after 2 years (A2) = Rs 5292
- Amount after 3 years (A3) = Rs 5556.60

2. **Calculate the interest for the third year:**
- Interest for the third year (I3) = A3 - A2
- \( I3 = 5556.60 - 5292 = 264.60 \)

3. **Use the interest to find the rate of interest:**
- We know that the interest for the first year (I1) is equal to the compound interest for the first year.
- The formula for simple interest is:
\[ I = \frac{P \times R \times T}{100} \]
- Here, \( I = 264.60 \), \( P = 5292 \), \( T = 1 \) year.
- Rearranging the formula to find R:
\[ R = \frac{I \times 100}{P \times T} \]
- Substituting the values:
\[ R = \frac{264.60 \times 100}{5292 \times 1} = \frac{26460}{5292} \approx 5\% \]

4. **Use the rate to find the original sum (P):**
- We can use the formula for compound interest:
\[ A = P \left(1 + \frac{R}{100}\right)^n \]
- For the second year:
\[ 5292 = P \left(1 + \frac{5}{100}\right)^2 \]
- Simplifying:
\[ 5292 = P \left(\frac{105}{100}\right)^2 = P \left(\frac{11025}{10000}\right) \]
- Rearranging to find P:
\[ P = 5292 \times \frac{10000}{11025} \]

5. **Calculate P:**
- First, calculate \( \frac{10000}{11025} \):
\[ \frac{10000}{11025} = \frac{400}{441} \quad (\text{after simplifying}) \]
- Now, calculate P:
\[ P = 5292 \times \frac{400}{441} \approx 4800 \]

### Final Answer:
The original sum (principal) is Rs 4800.