Find the balance if $27000 is invested in an account paying 6.5 % interest compounded semi-annually for 17 years.

To find the balance of an investment with compound interest, we use the formula:

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

Where:
- $A$ is the final amount (balance),
- $P$ is the principal (initial investment),
- $r$ is the annual interest rate (in decimal form),
- $n$ is the number of times the interest is compounded per year,
- $t$ is the time the money is invested for in years.

In this case:
- $P = 27000$,
- $r = 6.5\% = 0.065$,
- $n = 2$ (because the interest is compounded semi-annually),
- $t = 17$.

Now, plug in these values:

\[
A = 27000 \left( 1 + \frac{0.065}{2} \right)^{2 \times 17}
\]

Let's calculate this step by step:

\[
A = 27000 \left( 1 + 0.0325 \right)^{34}
\]
\[
A = 27000 \left( 1.0325 \right)^{34}
\]

Now, calculate $1.0325^{34}$, then multiply by 27000 to find the balance.

Let me calculate that.The balance after 17 years of investing $27,000 at an interest rate of 6.5% compounded semi-annually is approximately **$80,098.76**.

Would you like further details or have any questions?

Here are some related questions for practice:
1. What would the balance be if the interest was compounded quarterly?
2. How does changing the compounding frequency affect the final balance?
3. What if the interest rate were 5% instead of 6.5%?
4. How long would it take for the investment to double at 6.5% interest compounded semi-annually?
5. What would the balance be after 10 years instead of 17?

**Tip:** The more frequently interest is compounded, the larger the final balance will be due to compounding effects.

Learn how to calculate the compound interest balance for an investment of $27,000 at 6.5% interest compounded semi-annually over 17 years. Using the compound interest formula, this problem demonstrates the effects of semi-annual compounding. Suitable for high school students (Grades 10-12) and beyond.

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