The difference between compound interest and simple interest on an amount of Rs. 50000 for 2 years is Rs. 320. What is the rate of interest p.a.?

Options

A. A)0.12
B. B)0.08
C. C)0.1
D. D)0.06

Answer

To solve the problem, we need to find the rate of interest per annum (r) given that the difference between compound interest (CI) and simple interest (SI) on an amount of Rs. 50,000 for 2 years is Rs. 320.

### Step-by-Step Solution:

1. **Understand the formulas:**
- Simple Interest (SI) for 2 years:
\[ SI = \frac{P \times r \times t}{100} \]
- Compound Interest (CI) for 2 years:
\[ CI = P \left(1 + \frac{r}{100}\right)^t - P \]

2. **Calculate SI for 2 years:**
- Here, \( P = 50000 \), \( t = 2 \).
- Therefore,
\[ SI = \frac{50000 \times r \times 2}{100} = \frac{100000r}{100} = 1000r \]

3. **Calculate CI for 2 years:**
- Using the formula for CI:
\[ CI = 50000 \left(1 + \frac{r}{100}\right)^2 - 50000 \]
- Expanding \( \left(1 + \frac{r}{100}\right)^2 \):
\[ CI = 50000 \left(1 + \frac{2r}{100} + \frac{r^2}{10000}\right) - 50000 \]
- Simplifying:
\[ CI = 50000 \left(\frac{2r}{100} + \frac{r^2}{10000}\right) \]
\[ CI = 50000 \cdot \frac{2r}{100} + 50000 \cdot \frac{r^2}{10000} \]
\[ CI = 1000r + 5r^2 \]

4. **Set up the equation for the difference:**
- Given that the difference between CI and SI is Rs. 320:
\[ CI - SI = 320 \]
- Substituting the values:
\[ (1000r + 5r^2) - 1000r = 320 \]
- This simplifies to:
\[ 5r^2 = 320 \]

5. **Solve for r:**
- Dividing both sides by 5:
\[ r^2 = \frac{320}{5} = 64 \]
- Taking the square root:
\[ r = \sqrt{64} = 8 \]

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

### Final Answer:
The rate of interest per annum is **8%**.