In what time will `Rs. 64,000` amount to `Rs. 68,921` at `5%` per annum, interest being compounded half yearly?

Options

A. 3 years
B. `2(1)/3` years
C. 2 years
D. `1(1)/2` years

Answer

To solve the problem of finding the time it takes for Rs. 64,000 to amount to Rs. 68,921 at a 5% annual interest rate compounded half-yearly, we can follow these steps:

### Step 1: Identify the variables
- Principal (P) = Rs. 64,000
- Amount (A) = Rs. 68,921
- Rate of interest (R) = 5% per annum
- Compounding frequency = half-yearly

### Step 2: Adjust the interest rate for half-yearly compounding
Since the interest is compounded half-yearly, we need to divide the annual interest rate by 2 and convert it into a percentage for each half-year:
- Half-yearly interest rate = R / 2 = 5% / 2 = 2.5%

### Step 3: Determine the number of compounding periods
Let \( n \) be the number of years. Since the interest is compounded half-yearly, the total number of compounding periods (t) will be:
- Total compounding periods = 2n (because there are 2 periods in a year)

### Step 4: Use the compound interest formula
The formula for compound interest is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^{nt} \]
Where:
- A = Amount
- P = Principal
- r = Rate of interest per period
- n = Number of compounding periods

Substituting the values we have:
\[ 68,921 = 64,000 \left(1 + \frac{2.5}{100}\right)^{2n} \]

### Step 5: Simplify the equation
Calculating \( 1 + \frac{2.5}{100} \):
\[ 1 + \frac{2.5}{100} = 1 + 0.025 = 1.025 \]
Thus, the equation becomes:
\[ 68,921 = 64,000 \times (1.025)^{2n} \]

### Step 6: Divide both sides by the principal
\[ \frac{68,921}{64,000} = (1.025)^{2n} \]
Calculating the left side:
\[ \frac{68,921}{64,000} \approx 1.078515625 \]

### Step 7: Take logarithm on both sides
Taking logarithm (base 10 or natural logarithm) on both sides:
\[ \log(1.078515625) = 2n \cdot \log(1.025) \]

### Step 8: Solve for n
Now we can calculate:
- \( \log(1.078515625) \) and \( \log(1.025) \)
Using a calculator:
- \( \log(1.078515625) \approx 0.0337 \)
- \( \log(1.025) \approx 0.0107 \)

Now substituting these values:
\[ 0.0337 = 2n \cdot 0.0107 \]
\[ n = \frac{0.0337}{2 \cdot 0.0107} \]
\[ n \approx \frac{0.0337}{0.0214} \approx 1.57 \text{ years} \]

### Step 9: Convert to years and months
Since \( n \approx 1.57 \) years, we can convert the decimal into months:
- 0.57 years = 0.57 * 12 months ≈ 6.84 months, which is approximately 7 months.

### Final Answer
Thus, the time required for Rs. 64,000 to amount to Rs. 68,921 at 5% per annum compounded half-yearly is approximately **1 year and 7 months**.

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