Pete’s real estate is currently valued at $66,000. Pete feels the value of his business will increase at a rate of 8% per year compounded semi annually for the next four years. At a local fundraiser, a competitor offered Pete $71,000 for the business. If he sells, Pete plans to invest the money at 4% compounded quarterly. What price should he asked for?

To determine the price Pete should ask for, we need to compare two investment options over the next four years:

1. **Option 1:** Keep the business and let its value grow at 8% per year, compounded semi-annually.
2. **Option 2:** Sell the business now for $71,000 and invest the money at 4% per year, compounded quarterly.

We'll calculate the future value of both options after four years, and then determine what price Pete should ask for to make the first option (keeping the business) as attractive as the second option.

### Step 1: Calculate the future value of Pete's business
The formula for compound interest is:

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

Where:
- $A$ = future value
- $P$ = present value ($66,000)
- $r$ = annual interest rate (8% = 0.08)
- $n$ = number of times interest is compounded per year (semi-annually, $n = 2$)
- $t$ = number of years (4 years)

Substitute the values:

\[
A = 66,000 \left(1 + \frac{0.08}{2}\right)^{2 \times 4}
\]

### Step 2: Calculate the future value of the $71,000 investment
If Pete sells the business for $71,000 and invests the money at 4% compounded quarterly, the future value of the investment after 4 years can be calculated similarly:

\[
A = 71,000 \left(1 + \frac{0.04}{4}\right)^{4 \times 4}
\]

Now let's compute both future values.### Future Values:
- **Option 1 (Keep the business):** The future value of Pete's business after 4 years is approximately **$90,325.56**.
- **Option 2 (Sell and invest):** If Pete sells the business for $71,000 and invests it, the future value after 4 years would be approximately **$83,253.08**.

To make the first option (keeping the business) as appealing as the second, Pete should ask for an amount that, when invested at 4% compounded quarterly, grows to **$90,325.56** after 4 years.

Now, let's calculate the price Pete should ask for by reversing the compound interest formula:

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

Substituting the future value $A = 90,325.56$, $r = 0.04$, $n = 4$, and $t = 4$.Pete should ask for approximately **$77,031.56** for his business to match the future value of keeping the business, assuming he plans to invest the money at 4% compounded quarterly for 4 years.

Would you like further details or have any questions?

Here are 5 related questions to consider:
1. How would the required price change if the investment rate were higher than 4%?
2. What if the competitor offered Pete less than $71,000—how would that affect the calculation?
3. How does semi-annual compounding compare to annual compounding for the business valuation?
4. What happens if the business’s growth rate changes from 8% to another percentage?
5. How does the time period (e.g., 5 years instead of 4 years) impact Pete's decision?

**Tip:** Always compare future values of different investment options when making financial decisions, especially when compounding is involved.

Pete’s real estate is currently valued at $66,000. Pete feels the value of his business will increase at a rate of 8% per year compounded semi-annually for the next four years. At a local fundraiser, a competitor offered Pete $71,000 for the business. If he sells, Pete plans to invest the money at 4% compounded quarterly. What price should he ask for?

Learn how to calculate the future value of Pete's real estate investment using compound interest formulas. This problem explores the comparison between business growth at 8% compounded semi-annually and a 4% quarterly investment. Suitable for Grades 10-12 students interested in finance and compound interest applications.

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