Stuart Daniels estimates that he will need $28,000 to set up a small business in 7 years.
(a)
How much (in $) must Stuart invest now at 8% interest compounded quarterly to achieve his goal?
$
Incorrect: Your answer is incorrect.
(b)
How much compound interest (in $) will he earn on the investment?
$

To solve this, we need to use the compound interest formula:

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

Where:
- $A$ is the future value (the amount of money needed in the future, $28,000 in this case),
- $P$ is the present value (the amount to invest now),
- $r$ is the annual interest rate (8% or 0.08),
- $n$ is the number of times the interest is compounded per year (quarterly means 4 times per year),
- $t$ is the time in years (7 years).

### Part (a) Find the amount Stuart needs to invest now:

We are solving for $P$, so rearrange the formula:

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

Substitute the known values:

\[
P = \frac{28000}{\left(1 + \frac{0.08}{4}\right)^{4 \times 7}}
\]

Now let's calculate this.The amount Stuart must invest now is approximately **$16,082.49**.

### Part (b) Find the compound interest he will earn:

The compound interest $I$ is the difference between the future value $A$ and the present value $P$:

\[
I = A - P
\]

Let’s calculate this next.The compound interest Stuart will earn on the investment is approximately **$11,917.51**.

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. How does compounding frequency affect the amount to be invested?
2. What would the required investment be if the interest rate was 6% instead of 8%?
3. How much interest would be earned if compounding was done annually instead of quarterly?
4. What happens if Stuart wants to reach his goal in 5 years instead of 7?
5. How does continuous compounding change the investment needed?

**Tip:** The more frequently interest is compounded, the less initial investment is needed to reach a future value goal.

Stuart Daniels estimates that he will need $28,000 to set up a small business in 7 years. (a) How much (in $) must Stuart invest now at 8% interest compounded quarterly to achieve his goal? (b) How much compound interest (in $) will he earn on the investment?

Learn how to calculate the amount Stuart Daniels must invest today to reach a $28,000 goal in 7 years with 8% interest compounded quarterly. This problem explores key financial concepts such as compound interest, investment growth, and interest earned. Suitable for high school students in Grades 10-12.

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