If 2000 becomes 2420 at the rate of 10% com pound interest in certain time period then find the time period?

Options

A. 2.5 year
B. 2 year
C. 1.5 year
D. 3 year

Answer

To find the time period for which an amount of 2000 becomes 2420 at a rate of 10% compound interest, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given values:**
- Principal (P) = 2000
- Amount (A) = 2420
- Rate (R) = 10%

2. **Use the formula for compound interest:**
The formula for the amount in compound interest is given by:
\[ A = P \left(1 + \frac{R}{100}\right)^t \]
where:
- \( A \) = final amount
- \( P \) = principal amount
- \( R \) = rate of interest
- \( t \) = time in years

3. **Substitute the known values into the formula:**
\[ 2420 = 2000 \left(1 + \frac{10}{100}\right)^t \]
Simplifying the term inside the parentheses:
\[ 2420 = 2000 \left(1 + 0.1\right)^t \]
\[ 2420 = 2000 \left(1.1\right)^t \]

4. **Divide both sides by 2000:**
\[ \frac{2420}{2000} = (1.1)^t \]
\[ 1.21 = (1.1)^t \]

5. **Take the logarithm of both sides:**
\[ \log(1.21) = t \cdot \log(1.1) \]

6. **Solve for \( t \):**
\[ t = \frac{\log(1.21)}{\log(1.1)} \]

7. **Calculate the logarithms:**
Using a calculator:
\[ \log(1.21) \approx 0.0827 \quad \text{and} \quad \log(1.1) \approx 0.0414 \]
\[ t \approx \frac{0.0827}{0.0414} \approx 1.996 \]

8. **Round the result:**
Since \( t \approx 1.996 \), we can round it to approximately 2 years.

### Conclusion:
The time period is approximately **2 years**.