Find the rate of interest, if the S.I. for 3 years is Rs. 1200 and C.I. for 2 years is Rs. 860.

Answer

To find the rate of interest given the simple interest (SI) for 3 years and the compound interest (CI) for 2 years, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Simple Interest for 3 years (SI) = Rs. 1200
- Compound Interest for 2 years (CI) = Rs. 860

2. **Calculate Simple Interest for 1 Year**:
- Since SI for 3 years = 1200, we can find SI for 1 year by dividing by 3:
\[ \text{SI for 1 year} = \frac{1200}{3} = 400 \text{ Rs.} \]

3. **Calculate Simple Interest for 2 Years**:
- SI for 2 years = 2 × SI for 1 year:
\[ \text{SI for 2 years} = 2 \times 400 = 800 \text{ Rs.} \]

4. **Find the Difference Between CI and SI for 2 Years**:
- The difference between the compound interest and simple interest for 2 years gives us the interest earned on the interest:
\[ \text{Difference} = \text{CI} - \text{SI for 2 years} = 860 - 800 = 60 \text{ Rs.} \]

5. **Use the Formula for Compound Interest**:
- The formula for compound interest for 2 years can be expressed as:
\[ \text{CI} = P \left(1 + \frac{r}{100}\right)^2 - P \]
- The difference we calculated (60 Rs.) can also be expressed in terms of principal (P) and rate (r):
\[ \text{Difference} = P \left( \left(1 + \frac{r}{100}\right)^2 - 1 - \frac{2Pr}{100} \right) \]
- Setting this equal to 60, we simplify it to:
\[ P \left( \frac{r^2}{10000} \right) = 60 \]

6. **Relate SI for 1 Year to Principal and Rate**:
- We know that:
\[ \text{SI for 1 year} = \frac{P \cdot r}{100} = 400 \]
- Rearranging gives us:
\[ P \cdot r = 40000 \]

7. **Substituting in the Difference Equation**:
- From \( P \cdot r = 40000 \):
\[ P = \frac{40000}{r} \]
- Substitute \( P \) in the difference equation:
\[ \frac{40000}{r} \cdot \frac{r^2}{10000} = 60 \]
- Simplifying gives:
\[ \frac{40000r}{10000} = 60 \implies 4r = 60 \implies r = 15 \]

8. **Conclusion**:
- The rate of interest \( r \) is 15% per annum.