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Let's work through the problems shown in the image one by one.

### 1. Problem 1 (Effective Annual Rate of Return)
Juan invested \$25,000 in a mutual fund 5 years ago, and the investment is now worth \$36,733. We need to find the **effective annual rate of return** over the 5-year period.

We can use the formula for compound interest to solve this:

\[
A = P(1 + r)^t
\]

Where:
- $ A = 36,733 $ (the future value)
- $ P = 25,000 $ (the initial investment)
- $ r $ is the annual rate (what we're solving for)
- $ t = 5 $ years

Rearranging the formula to solve for $ r $:

\[
r = \left( \frac{A}{P} \right)^{\frac{1}{t}} - 1
\]

Plugging in the values:

\[
r = \left( \frac{36,733}{25,000} \right)^{\frac{1}{5}} - 1
\]

Let me calculate that for you.The effective annual rate of return for Juan's investment is **8.00%**.

---

### 2. Problem 2 (Retirement Account Value)
Mai Lee contributes \$200 at the end of each month to her retirement account, with her employer matching her contributions. The account earns an interest rate of 2% compounded monthly over a period of 10 years. We need to determine how much Mai Lee has in her retirement account after 10 years.

Since both Mai and her employer contribute the same amount each month, the total monthly contribution is \$400. The future value of these contributions can be calculated using the **future value of an annuity** formula:

\[
FV = P \times \frac{(1 + r/n)^{nt} - 1}{r/n}
\]

Where:
- $ P = 400 $ (total monthly contribution)
- $ r = 0.02 $ (annual interest rate)
- $ n = 12 $ (compounding periods per year)
- $ t = 10 $ (years)

Let's compute the future value.After 10 years, Mai Lee will have approximately **$53,087.86** in her retirement account.

---

### 3. Problem 3 (Price Range of Houses)
The Johnsons have a nest egg of \$50,000 for a down payment and plan to spend between \$2,700 and \$3,000 per month on mortgage payments. The mortgage rate is 3% per year, compounded monthly, over a 15-year period. We need to find the price range of houses they can afford.

We will use the formula for the present value of an annuity (mortgage):

\[
PV = P \times \frac{1 - (1 + r/n)^{-nt}}{r/n}
\]

Where:
- $ P $ is the monthly payment (we'll calculate this for both \$2,700 and \$3,000),
- $ r = 0.03 $ is the annual interest rate,
- $ n = 12 $ is the number of compounding periods per year,
- $ t = 15 $ years is the duration of the mortgage.

Finally, we need to add their \$50,000 down payment to the result to get the total house price.

Let's calculate the price range.The Johnsons can afford houses in the price range of approximately **$440,974.77** to **$484,416.41**, depending on their monthly mortgage payments.

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### Do you have any questions or would you like further details on these calculations?

Here are 5 related questions to consider:
1. How does compounding frequency affect the future value of an investment?
2. What is the difference between nominal and effective interest rates?
3. How would a change in the interest rate affect the future value of Mai Lee's retirement account?
4. What happens to the mortgage affordability if the Johnsons opt for a 30-year mortgage instead of a 15-year one?
5. How can inflation affect the long-term returns on investments like Juan's?

**Tip:** When making long-term financial decisions, always account for the effects of compounding interest and inflation to get a clearer picture of real returns.

1. Juan invested $25,000 in a mutual fund 5 years ago. Today his investment is worth $36,733. Find the effective annual rate of return on his investment over the 5-year period. 2. Mai Lee has contributed $200 at the end of each month for the past 10 years into her company's employee retirement account, and her employer has matched the contribution. The account earns interest at a rate of 2% per year compounded monthly. Determine how much Mai Lee has now. 3. The Johnsons plan to buy a house with a 15-year mortgage at 3% per year, compounded monthly, with a down payment of $50,000. They want to spend between $2,700 and $3,000 per month on the mortgage. Find the price range of houses they should consider.

This page covers three finance problems: 1. Calculating the effective annual rate of return on an investment, 2. Finding the future value of a retirement account with monthly contributions and matching, and 3. Determining the price range of houses based on a 15-year mortgage with fixed monthly payments. These problems require understanding of compound interest, annuities, and mortgage formulas.

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