Question 15 of $$ 15,5 \operatorname{tep} 1 $$ of 1 11/15 Correct Suppose you wish to retire at the age of 60 with $$ \$ 80,000 $$ in savings. Determine your monthly payment into an IPA 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. Formulas Present Value with Compound $$ \quad P V=\frac{A}{\left(1+\frac{r}{n}\right)^{2 m}} $$ Annuity Payment Amount Future Value of an Annuity $$ \mathrm{PMT}=\mathrm{FV} \cdot \frac{\left(\frac{r}{n}\right)}{\left[\left(1+\frac{r}{n}\right)^{e t}-1\right]} $$ $$ \mathrm{FV}=\mathrm{PMT} \cdot \frac{\left[\left(1+\frac{r}{n}\right)^{n}-1\right]}{\left(\frac{r}{n}\right)} $$ Keypad

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UpStudy AI · Accepted Answer

To determine the **monthly payment** needed to accumulate **\$80,000** by the age of **60**, starting at **30 years old**, with an **APR of 5.5%** compounded monthly, we'll use the **Future Value of an Annuity** formula.

### Given:
- **Future Value (FV):** \$80,000
- **Annual Percentage Rate (APR):** 5.5%
- **Compounded Monthly:** $$ n = 12 $$ times per year
- **Investment Period:** 30 years ($$ t = 30 $$)  
- **Total Number of Payments (N):** $$ 30 \times 12 = 360 $$ months

### Formula:
$$
\text{FV} = \text{PMT} \times \frac{\left(1 + \frac{r}{n}\right)^{nt} - 1}{\frac{r}{n}}
$$

Where:
- $$ \text{PMT} $$ = Monthly payment
- $$ r $$ = Annual interest rate (decimal)
- $$ n $$ = Number of compounding periods per year
- $$ t $$ = Number of years

### Steps to Calculate PMT:

1. **Convert APR to Monthly Rate:**
   $$
   \frac{r}{n} = \frac{5.5\%}{12} = 0.0045833333
   $$

2. **Calculate $$\left(1 + \frac{r}{n}\right)^{nt}$$:**
   $$
   \left(1 + 0.0045833333\right)^{360} \approx 5.4608
   $$

3. **Rearrange the Future Value Formula to Solve for PMT:**
   $$
   \text{PMT} = \frac{\text{FV} \times \frac{r}{n}}{\left(1 + \frac{r}{n}\right)^{nt} - 1}
   $$

4. **Plug in the Values:**
   $$
   \text{PMT} = \frac{80,000 \times 0.0045833333}{5.4608 - 1} \approx \frac{366.6666664}{4.4608} \approx 82.14
   $$

### **Final Answer:**
You need to make a **monthly payment of approximately \$82.14** into the investment account to reach your retirement savings goal of \$80,000 by age 60.
You need to make a monthly payment of approximately \$82.14 to reach \$80,000 by age 60.

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