Suppose you wish to retire at the age of 6060 with $70,000$70,000 in savings. Determine your monthly payment into an IRA if the APR is 6.5%6.5% compounded monthly and you begin making payments at 3030 years old. Round your answer to the nearest cent, if necessary.

To determine the monthly payment needed to accumulate $70,000 by the time you retire at age 60, with an APR of 6.5% compounded monthly, we can use the formula for the future value of an annuity:

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

Where:
- $FV$ is the future value (amount of savings you want, $70,000).
- $P$ is the monthly payment (the value we are solving for).
- $r$ is the monthly interest rate.
- $n$ is the total number of payments.

### Step 1: Calculate the monthly interest rate
The APR is 6.5%, but since it's compounded monthly, we divide the APR by 12:

\[
r = \frac{6.5\%}{12} = \frac{0.065}{12} = 0.00541667
\]

### Step 2: Calculate the total number of payments
You plan to make payments starting at age 30 until age 60, which is 30 years. Since the payments are made monthly:

\[
n = 30 \times 12 = 360 \text{ payments}
\]

### Step 3: Solve for the monthly payment $P$
Now we substitute into the future value formula and solve for $P$:

\[
70,000 = P \times \left( \frac{(1 + 0.00541667)^{360} - 1}{0.00541667} \right)
\]

Let’s compute the value of this equation to solve for $P$.The monthly payment required to accumulate $70,000 by the time you retire at age 60, with an APR of 6.5% compounded monthly, is approximately **$63.28**.

Would you like more details or have any other questions?

Here are five related questions to deepen your understanding:
1. What happens if the interest rate changes from 6.5% to 5%? How would that affect the monthly payment?
2. How does changing the retirement age affect the monthly payment?
3. What would the total contribution be over the 30-year period?
4. How does the future value of an annuity change if you make quarterly payments instead of monthly?
5. What if you already have an initial amount saved? How would that affect your required monthly contributions?

**Tip:** The earlier you start saving for retirement, the more compound interest works in your favor, significantly reducing the amount you need to save monthly!

Suppose you wish to retire at the age of 60 with $70,000 in savings. Determine your monthly payment into an IRA if the APR is 6.5% compounded monthly and you begin making payments at 30 years old. Round your answer to the nearest cent, if necessary.

Find out how much you need to contribute monthly into your IRA to reach $70,000 by the age of 60 with a 6.5% APR compounded monthly. This example applies the future value of an annuity formula and compound interest principles, suitable for high school students and financial planning.

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