A certain sum of money becomes amoung at the certain rate of interest of CI to Rs. 7350 in 2 years & to Rs. 8575 in 3 years. Find the sum of money & rate of interest.

Answer

To solve the problem, we need to find the principal amount (P) and the rate of interest (R) given that the amount becomes Rs. 7350 in 2 years and Rs. 8575 in 3 years under compound interest.

### Step-by-Step Solution:

1. **Identify the Interest for One Year:**
- The amount after 2 years (A2) is Rs. 7350.
- The amount after 3 years (A3) is Rs. 8575.
- The interest for the third year (I) can be calculated as:
\[ I = A3 - A2 = 8575 - 7350 = 1225 \]

2. **Calculate the Rate of Interest:**
- We know that the interest for the first year is the same as the interest for the second year in compound interest.
- The amount after 2 years can be expressed in terms of the principal (P) and the rate of interest (R):
\[ A2 = P \left(1 + \frac{R}{100}\right)^2 \]
- The amount after 3 years can be expressed as:
\[ A3 = P \left(1 + \frac{R}{100}\right)^3 \]
- The interest for the first year can be derived from the amount after the first year:
\[ I = A2 - P \]
- Since we know the interest for the third year is Rs. 1225, we can set up the equation:
\[ I = P \left(1 + \frac{R}{100}\right)^2 - P \]
- Rearranging gives:
\[ 1225 = P \left( \left(1 + \frac{R}{100}\right)^2 - 1 \right) \]

3. **Using the Amount after 2 Years to Find Principal:**
- We can express the amount after 2 years as:
\[ 7350 = P \left(1 + \frac{R}{100}\right)^2 \]
- From the previous step, we can express \(P\) in terms of \(R\):
\[ P = \frac{7350}{\left(1 + \frac{R}{100}\right)^2} \]

4. **Substituting Back to Find R:**
- Substitute \(P\) into the interest equation:
\[ 1225 = \frac{7350}{\left(1 + \frac{R}{100}\right)^2} \left( \left(1 + \frac{R}{100}\right)^2 - 1 \right) \]
- This simplifies to:
\[ 1225 = 7350 \left(1 - \frac{1}{\left(1 + \frac{R}{100}\right)^2}\right) \]
- Rearranging gives:
\[ \frac{1225}{7350} = 1 - \frac{1}{\left(1 + \frac{R}{100}\right)^2} \]
- Solving for \(R\) gives:
\[ \left(1 + \frac{R}{100}\right)^2 = \frac{7350}{6125} \]
- Taking the square root and solving for \(R\) yields:
\[ R \approx 16.67\% \]

5. **Finding the Principal Amount:**
- Substitute \(R\) back into the equation for \(P\):
\[ P = \frac{7350}{\left(1 + \frac{16.67}{100}\right)^2} \]
- Calculate \(P\):
\[ P \approx 5400 \]

### Final Results:
- **Principal Amount (P)**: Rs. 5400
- **Rate of Interest (R)**: 16.67%