The time in which `Rs. 80,000` amounts to `Rs. 92,610` at `10%` p.a. compound interest, interest being compounded semi annually is

Options

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

Answer

To solve the problem step by step, we need to find the time it takes for Rs. 80,000 to amount to Rs. 92,610 at a compound interest rate of 10% per annum, compounded semi-annually.

### Step 1: Identify the given values
- Principal (P) = Rs. 80,000
- Amount (A) = Rs. 92,610
- Rate of interest (R) = 10% per annum
- Compounding frequency = Semi-annually

### Step 2: Convert the annual interest rate to the semi-annual rate
Since the interest is compounded semi-annually, we need to divide the annual interest rate by 2.
- Semi-annual interest rate (r) = R / 2 = 10% / 2 = 5%

### Step 3: Use the compound interest formula
The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- A = Amount after time n
- P = Principal amount
- r = Rate of interest per period
- n = Number of compounding periods

### Step 4: Rearrange the formula to solve for n
We need to rearrange the formula to find n:
\[ n = \frac{\log(A/P)}{\log(1 + r/100)} \]

### Step 5: Substitute the values into the formula
Substituting the known values:
- A = 92,610
- P = 80,000
- r = 5%

\[ n = \frac{\log(92610/80000)}{\log(1 + 5/100)} \]

### Step 6: Calculate the values
1. Calculate \( \frac{92610}{80000} \):
\[ \frac{92610}{80000} = 1.157625 \]

2. Calculate \( \log(1.157625) \):
\[ \log(1.157625) \approx 0.0697 \]

3. Calculate \( \log(1 + 5/100) = \log(1.05) \):
\[ \log(1.05) \approx 0.0212 \]

### Step 7: Calculate n
Now substitute these values back into the equation for n:
\[ n = \frac{0.0697}{0.0212} \approx 3.28 \]

### Step 8: Convert n to years
Since n represents the number of semi-annual periods, we need to convert it to years:
- Each year has 2 semi-annual periods, so:
\[ \text{Time in years} = \frac{3.28}{2} \approx 1.64 \text{ years} \]

### Final Answer
The time in which Rs. 80,000 amounts to Rs. 92,610 at 10% p.a. compound interest, compounded semi-annually, is approximately **1.64 years**.

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