Suppose you wish to retire at the age of 7070 with $70,000$70,000 in savings. Determine your monthly payment into an IRA if the APR is 5.5%5.5% compounded monthly and you begin making payments at 3030 years old. Round your answer to the nearest cent, if necessary.
 

Formulas

**Present Value with Compound Interest**

PV=A(1+rn)ntPV=A(1+rn)nt

**Annuity Payment Amount**

PMT=FV⋅(rn)[(1+rn)nt−1]PMT=FV⋅(rn)[(1+rn)nt−1]

**Future Value of an Annuity**

FV=PMT⋅[(1+rn)nt−1](rn)


We need to determine your monthly payment into an IRA based on a future goal of having $70,000 by age 70, starting at age 30. You plan to make payments for 40 years, with an APR of 5.5% compounded monthly. The relevant formula for calculating the monthly payment (PMT) is the **Annuity Payment Amount** formula:

\[
PMT = \frac{FV \cdot \left(\frac{r}{n}\right)}{\left( \left(1 + \frac{r}{n} \right)^{nt} - 1 \right)}
\]

Where:
- $ FV = 70,000 $ (the future value you want),
- $ r = 0.055 $ (the annual interest rate),
- $ n = 12 $ (since the interest is compounded monthly),
- $ t = 40 $ (the number of years of payments).

### Step 1: Plug the values into the formula

\[
PMT = \frac{70,000 \cdot \left(\frac{0.055}{12}\right)}{\left( \left(1 + \frac{0.055}{12} \right)^{12 \cdot 40} - 1 \right)}
\]

### Step 2: Calculate the values step by step
- Monthly interest rate $ \frac{r}{n} = \frac{0.055}{12} \approx 0.004583 $.
- Number of total payments $ nt = 12 \cdot 40 = 480 $.

Now, calculate:

1. $ \left(1 + 0.004583 \right)^{480} \approx 10.965 $,
2. Subtract 1 from this: $ 10.965 - 1 = 9.965 $,
3. Multiply $ FV \cdot \left( \frac{r}{n} \right) = 70,000 \cdot 0.004583 \approx 320.81 $,
4. Finally, divide: $ \frac{320.81}{9.965} \approx 32.20 $.

### Answer: 
Your monthly payment into the IRA needs to be **$32.20**.

Would you like more details or have any further questions?

Here are five related questions:
1. What if the interest rate was higher, say 6%? How would the payment change?
2. How does changing the compounding frequency affect the monthly payment?
3. What if you started saving 10 years later, at age 40? How would that impact your payments?
4. How would the formula change if the contributions were made annually instead of monthly?
5. What happens if the interest is compounded quarterly instead of monthly?

**Tip**: Regularly review your retirement goals to ensure your savings strategy aligns with your changing financial situation!

Suppose you wish to retire at the age of 70 with $70,000 in savings. Determine your monthly payment into an IRA if the APR is 5.5% compounded monthly and you begin making payments at 30 years old. Round your answer to the nearest cent, if necessary.

Find out how to calculate the monthly payment needed to reach $70,000 in retirement savings by age 70, starting at age 30, with an APR of 5.5% compounded monthly. This guide uses the annuity payment formula and future value of an annuity concept, ideal for students in Grades 11-12 or college-level financial mathematics courses.

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