The simple interst on a certain sum for 2 yr is ? 120 and compound interest is ? 129 . Find the rate of interest.

Options

A. 0.14
B. 0.15
C. 0.12
D. `12 ( 1)/( 2)`

Answer

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

### Step-by-Step Solution:

1. **Understand the Given Information:**
- Simple Interest (SI) for 2 years = ₹120
- Compound Interest (CI) for 2 years = ₹129

2. **Use the Formula for Simple Interest:**
The formula for Simple Interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( P \) = Principal amount (the initial sum of money)
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

From the problem, we know:
\[ 120 = \frac{P \times R \times 2}{100} \]

3. **Rearrange to Find Principal (P):**
Rearranging the above equation gives:
\[ P = \frac{120 \times 100}{2R} = \frac{12000}{2R} = \frac{6000}{R} \]

4. **Use the Formula for Compound Interest:**
The formula for Compound Interest is:
\[ CI = A - P \]
Where \( A \) is the amount after time \( T \):
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
For 2 years, we have:
\[ A = P \left(1 + \frac{R}{100}\right)^2 \]
Therefore, the Compound Interest can be expressed as:
\[ 129 = P \left(1 + \frac{R}{100}\right)^2 - P \]

5. **Substitute P from Step 3 into the CI Equation:**
Substitute \( P = \frac{6000}{R} \) into the CI equation:
\[ 129 = \frac{6000}{R} \left( \left(1 + \frac{R}{100}\right)^2 - 1 \right) \]

6. **Simplify the Equation:**
The expression \( \left(1 + \frac{R}{100}\right)^2 - 1 \) can be simplified:
\[ \left(1 + \frac{R}{100}\right)^2 - 1 = \frac{R^2}{10000} + \frac{2R}{100} \]
Thus, we have:
\[ 129 = \frac{6000}{R} \left( \frac{R^2}{10000} + \frac{2R}{100} \right) \]

7. **Multiply through by R to eliminate the fraction:**
\[ 129R = 6000 \left( \frac{R^2}{10000} + \frac{2R}{100} \right) \]

8. **Distribute and simplify:**
\[ 129R = \frac{6000R^2}{10000} + \frac{12000R}{100} \]
\[ 129R = \frac{3R^2}{5} + 120R \]

9. **Rearranging the equation:**
\[ 3R^2 - 45R = 0 \]
Factor out \( R \):
\[ R(3R - 45) = 0 \]
Thus, \( R = 0 \) or \( R = 15 \).

10. **Conclusion:**
The rate of interest \( R \) is \( 15\% \).