In what time will Rs. 1000 amount to Rs. 1331 at 20% per annum, compounded half yearly?

Answer

To solve the problem of finding the time it takes for Rs. 1000 to amount to Rs. 1331 at a 20% per annum interest rate compounded half-yearly, we can follow these steps:

### Step 1: Identify the variables
- Principal (P) = Rs. 1000
- Amount (A) = Rs. 1331
- Rate of interest (R) = 20% per annum
- Since the interest is compounded half-yearly, we need to adjust the rate and time accordingly.

### Step 2: Calculate the half-yearly interest rate
The annual interest rate is 20%, so the half-yearly interest rate (r) is:
\[ r = \frac{20}{2} = 10\% \]

### Step 3: Determine the number of compounding periods
Since the interest is compounded half-yearly, we need to determine how many half-year periods (n) it takes for the amount to reach Rs. 1331.

### Step 4: Use the compound interest formula
The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Substituting the known values:
\[ 1331 = 1000 \left(1 + \frac{10}{100}\right)^n \]
This simplifies to:
\[ 1331 = 1000 \left(1.1\right)^n \]

### Step 5: Divide both sides by 1000
\[ \frac{1331}{1000} = (1.1)^n \]
\[ 1.331 = (1.1)^n \]

### Step 6: Take the logarithm of both sides
Taking the logarithm (base 10 or natural logarithm) of both sides:
\[ \log(1.331) = n \cdot \log(1.1) \]

### Step 7: Solve for n
Now, we can solve for n:
\[ n = \frac{\log(1.331)}{\log(1.1)} \]

### Step 8: Calculate the values
Using a calculator:
- \(\log(1.331) \approx 0.1249\)
- \(\log(1.1) \approx 0.0414\)

Now substituting these values:
\[ n \approx \frac{0.1249}{0.0414} \approx 3.02 \]

### Step 9: Convert n to years
Since n represents the number of half-year periods, we convert this to years:
\[ \text{Time in years} = \frac{n}{2} \approx \frac{3.02}{2} \approx 1.51 \text{ years} \]

### Final Answer
Thus, the time required for Rs. 1000 to amount to Rs. 1331 at 20% per annum compounded half-yearly is approximately **1.51 years**.

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