If some amount becomes 6000 in 2 years and 6615 in 4 years then find the rate of interest?

Options

A. `4%`
B. `6%`
C. `5%`
D. `7%`

Answer

To find the rate of interest given that an amount becomes Rs 6000 in 2 years and Rs 6615 in 4 years, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the amounts and time periods:**
- Let \( A_1 = 6000 \) (Amount after 2 years)
- Let \( A_2 = 6615 \) (Amount after 4 years)

2. **Calculate the ratio of the two amounts:**
\[ \frac{A_2}{A_1} = \frac{6615}{6000} \]

3. **Simplify the ratio:**
- Divide both the numerator and denominator by 15:
\[ \frac{6615 \div 15}{6000 \div 15} = \frac{441}{400} \]

4. **Use the formula for compound interest:**
- The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
- For \( A_1 \) (after 2 years):
\[ A_1 = P \left(1 + \frac{R}{100}\right)^2 \]
- For \( A_2 \) (after 4 years):
\[ A_2 = P \left(1 + \frac{R}{100}\right)^4 \]

5. **Set up the equation using the amounts:**
\[ \frac{A_2}{A_1} = \frac{P \left(1 + \frac{R}{100}\right)^4}{P \left(1 + \frac{R}{100}\right)^2} \]
- The \( P \) cancels out:
\[ \frac{A_2}{A_1} = \left(1 + \frac{R}{100}\right)^{4-2} = \left(1 + \frac{R}{100}\right)^2 \]

6. **Substitute the ratio we calculated:**
\[ \left(1 + \frac{R}{100}\right)^2 = \frac{441}{400} \]

7. **Take the square root of both sides:**
\[ 1 + \frac{R}{100} = \sqrt{\frac{441}{400}} = \frac{21}{20} \]

8. **Rearrange to find \( R \):**
\[ \frac{R}{100} = \frac{21}{20} - 1 = \frac{21 - 20}{20} = \frac{1}{20} \]
\[ R = 100 \times \frac{1}{20} = 5 \]

9. **Conclusion:**
- The rate of interest \( R \) is \( 5\% \).