uppose you wish to retire at the age of 65
 with $70,000
 in savings. Determine your monthly payment into an IRA if the APR is 8.5%
 compounded monthly and you begin making payments at 25
 years old. Round your answer to the nearest cent, if necessary

To solve this problem, we need to determine the monthly payment into an IRA, given that you want to retire with $70,000 at the age of 65, starting at the age of 25. The interest is compounded monthly at an APR of 8.5%.

We will use the **future value of an annuity formula**, which is appropriate since you're making monthly payments and the interest is compounded monthly.

The formula for the future value of an annuity is:

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

Where:
- $FV$ is the future value (which is $70,000).
- $P$ is the monthly payment (what we are solving for).
- $r$ is the monthly interest rate (which is the annual interest rate divided by 12).
- $n$ is the number of payments (the total number of months you are contributing).
  
### Step 1: Identify the values
- Annual interest rate (APR) = 8.5% = 0.085
- Monthly interest rate $r = \frac{0.085}{12} = 0.00708333$
- You contribute for 40 years (from age 25 to 65), so $n = 40 \times 12 = 480$ months.
- Future value $FV = 70,000$.

### Step 2: Plug values into the formula
\[
70,000 = P \times \frac{(1 + 0.00708333)^{480} - 1}{0.00708333}
\]

### Step 3: Solve for $P$
1. First, calculate the compound factor:
\[
(1 + 0.00708333)^{480} = 26.5017
\]
2. Now calculate the annuity factor:
\[
\frac{26.5017 - 1}{0.00708333} = 3595.404
\]
3. Finally, solve for $P$:
\[
70,000 = P \times 3595.404
\]
\[
P = \frac{70,000}{3595.404} = 19.47
\]

Thus, the monthly payment $P$ is approximately **$19.47**.

---

Would you like more details or have any questions? Here are some related questions to expand your understanding:

1. How does increasing the interest rate affect the monthly payment?
2. What happens if you start saving at a later age, say 35 instead of 25?
3. How does the total amount saved change if you save for 45 years instead of 40?
4. What if the goal was $100,000 instead of $70,000? How would the payment change?
5. How would the formula change if the interest were compounded quarterly instead of monthly?

**Tip**: Starting your investment early allows compound interest to grow your savings significantly, even with smaller monthly contributions!

Suppose you wish to retire at the age of 65 with $70,000 in savings. Determine your monthly payment into an IRA if the APR is 8.5% compounded monthly and you begin making payments at 25 years old. Round your answer to the nearest cent, if necessary.

Find out how to calculate the monthly payments needed to reach $70,000 in savings by retirement at 65, starting at age 25. This solution uses the future value of an annuity formula with an 8.5% APR compounded monthly, covering compound interest and annuity concepts.

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