In what time will Rs. 6250 amount to Rs. 6632·55 at `4%` compound interest payable half yearly ?

Options

A. 3 years
B. `3/2 `years
C. 1 years
D. `5/2 `years

Answer

To solve the problem of finding the time it takes for Rs. 6250 to amount to Rs. 6632.55 at a compound interest rate of 4% payable half-yearly, we can follow these steps:

### Step 1: Identify the given values
- Principal (P) = Rs. 6250
- Amount (A) = Rs. 6632.55
- Rate of interest (R) = 4% per annum

### Step 2: Adjust the rate for half-yearly compounding
Since the interest is compounded half-yearly, we divide the annual rate by 2:
- Rate per half-year = \( \frac{4\%}{2} = 2\% \)

### Step 3: Convert the rate into decimal form
- Rate per half-year in decimal = \( \frac{2}{100} = 0.02 \)

### Step 4: Set up the formula for compound interest
The formula for the amount in compound interest is given by:
\[ A = P \left(1 + \frac{R}{100}\right)^{nt} \]
Where:
- \( A \) = Amount
- \( P \) = Principal
- \( R \) = Rate of interest
- \( n \) = Number of times interest is compounded per year
- \( t \) = Time in years

### Step 5: Substitute the known values into the formula
Here, \( n = 2 \) (since it is compounded half-yearly):
\[ 6632.55 = 6250 \left(1 + 0.02\right)^{2t} \]

### Step 6: Simplify the equation
\[ 6632.55 = 6250 \left(1.02\right)^{2t} \]

### Step 7: Divide both sides by 6250
\[ \frac{6632.55}{6250} = (1.02)^{2t} \]

### Step 8: Calculate the left side
Calculating \( \frac{6632.55}{6250} \):
\[ \frac{6632.55}{6250} = 1.060008 \]

### Step 9: Set up the logarithmic equation
Now we have:
\[ 1.060008 = (1.02)^{2t} \]

### Step 10: Take logarithm on both sides
Taking logarithm:
\[ \log(1.060008) = 2t \cdot \log(1.02) \]

### Step 11: Solve for \( t \)
Rearranging gives:
\[ t = \frac{\log(1.060008)}{2 \cdot \log(1.02)} \]

### Step 12: Calculate the values
Using a calculator:
- \( \log(1.060008) \approx 0.0257 \)
- \( \log(1.02) \approx 0.0086 \)

Now substitute these values:
\[ t = \frac{0.0257}{2 \cdot 0.0086} \]
\[ t = \frac{0.0257}{0.0172} \approx 1.493 \text{ half-years} \]

### Step 13: Convert half-years to years
Since \( t \) is in half-years, we divide by 2:
\[ t \approx \frac{1.493}{2} \approx 0.7465 \text{ years} \]

### Step 14: Final result
To express this in years and months, we can say:
- \( 0.7465 \) years is approximately \( 9 \) months.

Thus, the time required for Rs. 6250 to amount to Rs. 6632.55 at 4% compound interest payable half-yearly is approximately **0.75 years or 9 months**.