To solve the problem of how long it will take for Rs. 10,000 to amount to Rs. 13,310 at an interest rate of 20% per annum compounded half-yearly, we can follow these steps:
### Step 1: Understand the Compounding Frequency
Since the interest is compounded half-yearly, we need to adjust the interest rate and the time period accordingly. The annual interest rate is 20%, so the half-yearly interest rate will be:
\[
\text{Half-yearly rate} = \frac{20\%}{2} = 10\%
\]
### Step 2: Set Up the Compound Interest Formula
The formula for compound interest is given by:
\[
A = P \left(1 + \frac{r}{100}\right)^n
\]
where:
- \(A\) is the amount of money accumulated after n years, including interest.
- \(P\) is the principal amount (the initial amount of money).
- \(r\) is the annual interest rate (decimal).
- \(n\) is the number of compounding periods.
### Step 3: Substitute Known Values
From the problem, we know:
- \(A = 13,310\)
- \(P = 10,000\)
- \(r = 10\%\) (half-yearly rate)
Substituting these values into the formula gives:
\[
13,310 = 10,000 \left(1 + \frac{10}{100}\right)^n
\]
This simplifies to:
\[
13,310 = 10,000 \left(1.1\right)^n
\]
### Step 4: Divide Both Sides by 10,000
To isolate the term with \(n\), we divide both sides by 10,000:
\[
\frac{13,310}{10,000} = (1.1)^n
\]
This simplifies to:
\[
1.331 = (1.1)^n
\]
### Step 5: Take Logarithm of Both Sides
To solve for \(n\), we take the logarithm of both sides:
\[
\log(1.331) = n \cdot \log(1.1)
\]
### Step 6: Solve for \(n\)
Now we can solve for \(n\):
\[
n = \frac{\log(1.331)}{\log(1.1)}
\]
### Step 7: 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 8: Convert \(n\) to Years
Since \(n\) represents the number of half-year periods, we need to convert this into years. Since each period is half a year:
\[
\text{Time in years} = \frac{n}{2} \approx \frac{3.02}{2} \approx 1.51 \text{ years}
\]
### Final Answer
Thus, the time it will take for Rs. 10,000 to amount to Rs. 13,310 at 20% per annum compounded half-yearly is approximately **1.5 years**.
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