Suppose you wish to retire at the age of 6565 with $80,000$80,000 in savings. Determine your monthly payment into an IRA if the APR is 7.5%7.5% compounded monthly and you begin making payments at 2020 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)FV=PMT⋅[(1+rn)nt−1](rn)

To solve this problem, we need to determine the monthly payment (PMT) into an IRA (Individual Retirement Account) in order to have $80,000 by the time you retire at age 65. We know the APR (Annual Percentage Rate) is 7.5% compounded monthly, and you begin making payments at age 20.

We'll use the **Annuity Payment Formula**:

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

Where:
- $ FV = 80,000 $ (future value you want in the account),
- $ r = 7.5\% = 0.075 $ (annual interest rate),
- $ n = 12 $ (number of compounding periods per year, since the interest is compounded monthly),
- $ t = 65 - 20 = 45 $ (number of years you are contributing).

### Step 1: Plug in the values

\[
PMT = \frac{80,000 \cdot \left(\frac{0.075}{12}\right)}{\left[\left(1 + \frac{0.075}{12}\right)^{12 \cdot 45} - 1\right]}
\]

Now let's calculate this step by step.

### Step 2: Calculate $\frac{r}{n}$ and $\left(1 + \frac{r}{n}\right)^{nt}$

We can now compute this expression for the monthly payment. I'll calculate it now.The monthly payment you need to make into the IRA to accumulate $80,000 by age 65, with an APR of 7.5% compounded monthly, is approximately **$17.91**.

Would you like further details or explanations? Here are some related questions to explore:

1. How would the monthly payment change if the APR was 6% instead of 7.5%?
2. What happens if you start saving at age 30 instead of 20?
3. How much interest would you earn by retirement in total?
4. What if you want $100,000 instead of $80,000 by retirement?
5. How does the payment change if the interest is compounded quarterly instead of monthly?

**Tip:** Compounding more frequently increases the amount you accumulate due to interest growing on interest!

Suppose you wish to retire at the age of 65 with $80,000 in savings. Determine your monthly payment into an IRA if the APR is 7.5% compounded monthly and you begin making payments at 20 years old. Round your answer to the nearest cent, if necessary.

Learn how to calculate the monthly payment into an IRA needed to retire with $80,000 in savings. This detailed guide covers the annuity payment formula, the effect of 7.5% APR compounded monthly, and contributions starting at age 20. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation