17 times a month, William stops in Mahalo Coffee Roasters to drink a $2.50 coffee. After studying annuities, he decides to forgo the coffee and, at the end of each month, invest the amount he had been spending on coffee in an annuity account paying 4.8% per year compounded monthly. If he continues this for 25 years, how much will the annuity be worth?

To determine the future value of the annuity that William will accumulate, we can use the future value of an annuity formula:

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

Where:
- $FV$ is the future value of the annuity.
- $P$ is the monthly payment.
- $r$ is the monthly interest rate.
- $n$ is the total number of payments.

### Step 1: Determine the monthly payment, $P$
William buys 17 coffees per month at $2.50 each, so his monthly investment will be:

\[
P = 17 \times 2.50 = 42.50 \text{ dollars}
\]

### Step 2: Determine the monthly interest rate, $r$
The annual interest rate is 4.8%, compounded monthly. Therefore, the monthly interest rate is:

\[
r = \frac{4.8\%}{12} = \frac{0.048}{12} = 0.004 \text{ (or 0.4% per month)}
\]

### Step 3: Determine the total number of payments, $n$
William plans to invest for 25 years. Since payments are made monthly:

\[
n = 25 \times 12 = 300 \text{ payments}
\]

### Step 4: Calculate the future value, $FV$

Now, substitute the values into the formula:

\[
FV = 42.50 \times \frac{(1 + 0.004)^{300} - 1}{0.004}
\]

Let's calculate this.

\[
FV = 42.50 \times \frac{(1.004)^{300} - 1}{0.004}
\]

\[
FV = 42.50 \times \frac{3.2434 - 1}{0.004}
\]

\[
FV = 42.50 \times \frac{2.2434}{0.004}
\]

\[
FV = 42.50 \times 560.85 = 23856.125 \text{ dollars}
\]

### Final Answer:
The future value of the annuity will be approximately **$23,856.13**.

Would you like more details or have any questions?

Here are 5 related questions to consider:
1. What would be the future value if the interest rate was 6% instead of 4.8%?
2. How would the future value change if William invested the money for 30 years instead of 25 years?
3. What would the future value be if William invested only $30 per month?
4. How much interest would William earn over the 25 years?
5. What would the future value be if the interest was compounded quarterly instead of monthly?

**Tip:** Always double-check the compounding period when calculating interest, as it can significantly affect the future value.

Learn how to calculate the future value of an annuity with monthly compounding using the annuity formula. William decides to invest his coffee money in an annuity account with 4.8% annual interest compounded monthly for 25 years, replacing his $2.50 daily coffee expense. Find out the future value with a step-by-step solution.

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