In how many months will₹ 8,000 yield ₹ 2,648 as compound interest at 20% per annum compounded semi-annually?

Options

A. 18
B. 12
C. 30
D. 24

Answer

To solve the problem of how many months ₹ 8,000 will yield ₹ 2,648 as compound interest at 20% per annum compounded semi-annually, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Principal (P) = ₹ 8,000
- Compound Interest (CI) = ₹ 2,648
- Rate of Interest (R) = 20% per annum
- Total Amount (A) = Principal + Compound Interest = ₹ 8,000 + ₹ 2,648 = ₹ 10,648

2. **Convert the Rate for Semi-Annual Compounding**:
- Since the interest is compounded semi-annually, we need to divide the annual rate by 2.
- Semi-annual rate (r) = 20% / 2 = 10% = 0.10

3. **Use the Compound Interest Formula**:
The formula for compound interest is:
\[ A = P \left(1 + r\right)^{nt} \]
Where:
- A = Total amount after time t
- P = Principal amount
- r = Rate of interest per period
- n = Number of times interest is compounded per year
- t = Time in years

Here, n = 2 (since it is compounded semi-annually).

4. **Substituting the Values into the Formula**:
\[ 10,648 = 8,000 \left(1 + 0.10\right)^{2t} \]
Simplifying:
\[ 10,648 = 8,000 \left(1.10\right)^{2t} \]

5. **Divide Both Sides by 8,000**:
\[ \frac{10,648}{8,000} = \left(1.10\right)^{2t} \]
\[ 1.331 = \left(1.10\right)^{2t} \]

6. **Taking Logarithm on Both Sides**:
To solve for \(2t\), we can take logarithm:
\[ \log(1.331) = 2t \cdot \log(1.10) \]

7. **Calculating the Values**:
Using a calculator:
- \(\log(1.331) \approx 0.1249\)
- \(\log(1.10) \approx 0.0414\)

Now substituting these values:
\[ 0.1249 = 2t \cdot 0.0414 \]
\[ 2t = \frac{0.1249}{0.0414} \approx 3.02 \]

8. **Finding t**:
\[ t \approx \frac{3.02}{2} \approx 1.51 \text{ years} \]

9. **Convert Years to Months**:
Since \(1 \text{ year} = 12 \text{ months}\):
\[ t \approx 1.51 \times 12 \approx 18.12 \text{ months} \]

Rounding off, we find that it will take approximately **18 months**.

### Final Answer:
The time it will take for ₹ 8,000 to yield ₹ 2,648 as compound interest at 20% per annum compounded semi-annually is **18 months**.