​(Related to Checkpoint​ 5.4) ​ (Present value)  Sarah Wiggum would like to make a single investment and have ​$1.7 million at the time of her retirement in 30 years.  She has found a mutual fund that will earn 8 percent annually.  How much will Sarah have to invest​ today?  If Sarah earned an annual return of 17 ​percent, how soon could she then​ retire?  
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Part 1
a.  If Sarah can earn 8 percent annually for the next 30 ​years, the amount of money she will have to invest today is ​$
  
enter your response here.  ​(Round to the nearest​ cent.)

To calculate how much Sarah Wiggum needs to invest today (Present Value) to have $1.7 million at retirement in 30 years, we use the formula for Present Value (PV) in compound interest:

\[
PV = \dfrac{FV}{(1 + r)^n}
\]

Where:
- $FV$ is the future value, which is $1,700,000.
- $r$ is the annual interest rate (8% or 0.08).
- $n$ is the number of years (30 years).

Now, let's calculate:

\[
PV = \dfrac{1,700,000}{(1 + 0.08)^{30}}
\]

\[
PV = \dfrac{1,700,000}{(1.08)^{30}}
\]

I will calculate this next.The amount Sarah Wiggum needs to invest today is approximately **$168,941.47** to reach $1.7 million in 30 years at an 8% annual return.

Would you like me to calculate how soon she could retire with a 17% annual return? Let me know if you have any questions.

Here are 5 related questions to explore:
1. What would happen if Sarah could only invest for 20 years instead of 30?
2. How would different interest rates impact the required initial investment?
3. What if Sarah wanted $2 million instead of $1.7 million?
4. How does compound interest differ from simple interest in these calculations?
5. What if Sarah could make monthly investments instead of a single one?

**Tip:** Small changes in the interest rate have a large effect on long-term investments due to compounding.

(Related to Checkpoint​ 5.4) ​Sarah Wiggum would like to make a single investment and have ​$1.7 million at the time of her retirement in 30 years. She has found a mutual fund that will earn 8 percent annually. How much will Sarah have to invest​ today? If Sarah earned an annual return of 17 ​percent, how soon could she then​ retire?

Learn how to calculate the present value required to reach $1.7 million in 30 years with an 8% annual return. This problem covers key finance concepts, including compound interest, present value, and future value calculations. Suitable for college-level finance or mathematics students.

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