Find the amount with interest if sum of 8000 is kept in bank for 3 years with 12% p.a, compounded half yearly 

To find the amount with interest when a sum of $ 8,000 $ is kept in a bank for 3 years at an annual interest rate of $ 12\% $, compounded half-yearly, we use the compound interest formula:

\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]

where:
- $ A $ is the amount of money accumulated after $ t $ 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 times interest is compounded per year.
- $ t $ is the time the money is invested for in years.

Given:
- $ P = 8,000 $
- $ r = 12\% = 0.12 $
- $ n = 2 $ (since the interest is compounded half-yearly)
- $ t = 3 $ years

Plugging in these values:

\[ A = 8000 \left(1 + \frac{0.12}{2}\right)^{2 \times 3} \]

Simplify inside the parentheses:

\[ A = 8000 \left(1 + 0.06\right)^{6} \]

\[ A = 8000 \left(1.06\right)^{6} \]

Now, calculate $ (1.06)^6 $:

\[ (1.06)^6 \approx 1.41852 \]

Therefore:

\[ A = 8000 \times 1.41852 \]

\[ A \approx 11348.16 \]

So, the amount with interest after 3 years is approximately $ 11,348.16 $.

Do you want details or have any questions? Here are some questions you might want to ask:

1. How does compounding frequency affect the total amount of interest earned?
2. Can you explain the process of converting an annual interest rate to a rate for different compounding periods?
3. What is the difference between nominal and effective interest rates?
4. How do you calculate the future value of an investment with continuous compounding?
5. What are some examples of investments that use half-yearly compounding?

**Tip:** When dealing with compound interest problems, always pay attention to the compounding frequency as it significantly affects the final amount.

Learn how to calculate compound interest when $8,000 is invested at a 12% annual rate compounded half-yearly over 3 years. This problem uses the compound interest formula and involves exponential growth concepts. Suitable for advanced high school students.

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