A company will need ​$30 comma 000 in 7 years for a new addition. To meet this​ goal, the company deposits money in an account today that pays 11​% annual interest compounded quarterly. Find the amount that should be invested to total ​$30 comma 000 in 7 years.
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Part 1
The company should invest ​$
  
enter your response here.
​(Do not round until the final answer. Then round to the nearest dollar as​ needed.)

To find the amount that should be invested today (present value), we can use the formula for compound interest:

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

Where:
- $A$ is the amount of money in the future (the total after interest, which is $30,000),
- $P$ is the present value (the amount to invest today),
- $r$ is the annual interest rate (11% or 0.11),
- $n$ is the number of times the interest is compounded per year (quarterly, so $n = 4$),
- $t$ is the number of years (7 years).

Rearranging the formula to solve for $P$:

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

Now plug in the values:

\[
P = \frac{30,000}{\left(1 + \frac{0.11}{4}\right)^{4 \times 7}}
\]

First, calculate the term inside the parentheses:

\[
\frac{0.11}{4} = 0.0275
\]
\[
1 + 0.0275 = 1.0275
\]

Next, calculate the exponent $4 \times 7 = 28$, so we raise 1.0275 to the power of 28:

\[
1.0275^{28} \approx 2.109
\]

Now, divide the future value by this result:

\[
P = \frac{30,000}{2.109} \approx 14,220
\]

So, the company should invest **$14,220** today to meet their goal of $30,000 in 7 years.

Would you like more details on any step? Let me know if you have any questions.

### Related questions:
1. What happens if the interest rate changes to 12%?
2. How would the result change if the company needed the money in 10 years?
3. What is the formula for continuous compounding?
4. How would monthly compounding affect the investment amount?
5. How much interest would be earned over the 7 years?

### Tip:
When solving problems involving compound interest, be sure to match the compounding frequency with the given time period for accurate results.

A company will need $30,000 in 7 years for a new addition. To meet this goal, the company deposits money in an account today that pays 11% annual interest compounded quarterly. Find the amount that should be invested to total $30,000 in 7 years. The company should invest $____. (Do not round until the final answer. Then round to the nearest dollar as needed.)

Learn how to calculate the present value of an investment to meet a future goal of $30,000 in 7 years. This problem uses the compound interest formula with quarterly compounding at 11% interest and provides a detailed step-by-step solution. Suitable for high school students in Grades 10-12.

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