A sum of Rs 10000 becomes Rs 14641 when invested at compound interest at the yearly interest rate of 20% per annum. If the compounding is done half yearly, then for how many years was the sum invested?

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to determine how many years a sum of Rs 10,000 was invested to grow to Rs 14,641 at a compound interest rate of 20% per annum, compounded half-yearly.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Principal (P) = Rs 10,000
- Amount (A) = Rs 14,641
- Annual interest rate (R) = 20%

2. **Convert the Interest Rate for Half-Yearly Compounding:**
- Since the interest is compounded half-yearly, we need to divide the annual interest rate by 2.
- Half-yearly interest rate = R/2 = 20% / 2 = 10%

3. **Determine the Number of Compounding Periods:**
- Let the number of years be \( t \).
- Since compounding is done half-yearly, the total number of compounding periods (n) will be \( 2t \).

4. **Use the Compound Interest Formula:**
- The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
- Here, \( r \) is the half-yearly interest rate (10%), and \( n \) is the total number of compounding periods (2t).
- Plugging in the values, we get:
\[ 14,641 = 10,000 \left(1 + \frac{10}{100}\right)^{2t} \]

5. **Simplify the Equation:**
- This simplifies to:
\[ 14,641 = 10,000 \left(1.1\right)^{2t} \]
- Dividing both sides by 10,000:
\[ \frac{14,641}{10,000} = (1.1)^{2t} \]
- This gives:
\[ 1.4641 = (1.1)^{2t} \]

6. **Take Logarithm of Both Sides:**
- Taking logarithm (base 10) on both sides:
\[ \log(1.4641) = 2t \cdot \log(1.1) \]

7. **Calculate the Logarithms:**
- Using a calculator:
\[ \log(1.4641) \approx 0.1661 \]
\[ \log(1.1) \approx 0.0414 \]

8. **Solve for \( t \):**
- Plugging in the values:
\[ 0.1661 = 2t \cdot 0.0414 \]
- Rearranging gives:
\[ t = \frac{0.1661}{2 \cdot 0.0414} \approx \frac{0.1661}{0.0828} \approx 2 \]

9. **Conclusion:**
- Therefore, the sum was invested for **2 years**.