Question 15 of 15, Step 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 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. Formulas Present Value with Compound Interest $$ \begin{array}{l} \mathrm{PV}=\frac{A}{\left(1+\frac{r}{n}\right)^{n t}} \ \mathrm{PMT}=\mathrm{FV} \cdot \frac{\left(\frac{r}{n}\right)}{\left[\left(1+\frac{r}{n}\right)^{n t}-1\right]} \ \mathrm{FV}=\mathrm{PMT} \cdot \frac{\left[\left(1+\frac{r}{n}\right)^{n t}-1\right]}{\left(\frac{r}{n}\right)} \end{array} $$ Future Value of an Annuity

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To determine the monthly payment into an IRA, we will use the formula for the Future Value of an Annuity (FVA). The formula is given as:

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

### Known Conditions:
- Future Value ($$ \mathrm{FV} $$): $$ 80,000 $$
- Annual Percentage Rate ($$ r $$): $$ 5.5\% = 0.055 $$
- Compounding frequency ($$ n $$): $$ 12 $$ (monthly)
- Time until retirement ($$ t $$): $$ 60 - 30 = 30 $$ years

### Step 1: Calculate the monthly interest rate
The monthly interest rate is calculated as:

$$
\frac{r}{n} = \frac{0.055}{12}
$$

### Step 2: Calculate the total number of payments
The total number of payments is:

$$
n \cdot t = 12 \cdot 30
$$

### Step 3: Substitute values into the PMT formula
Now we can substitute the values into the PMT formula to find the monthly payment.

Let's perform the calculations step by step.

1. Calculate $$ \frac{r}{n} $$.
2. Calculate $$ n \cdot t $$.
3. Substitute these values into the PMT formula.

Let's calculate these values.
Calculate the value by following steps:
- step0: Calculate:
  $$12\times 30$$
- step1: Multiply the numbers:
  $$360$$
Calculate or simplify the expression $$ 0.055/12 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{0.055}{12}$$
- step1: Convert the expressions:
  $$\frac{\frac{11}{200}}{12}$$
- step2: Multiply by the reciprocal:
  $$\frac{11}{200}\times \frac{1}{12}$$
- step3: Multiply the fractions:
  $$\frac{11}{200\times 12}$$
- step4: Multiply:
  $$\frac{11}{2400}$$
### Step 1: Calculate the monthly interest rate
The monthly interest rate is:

$$
\frac{r}{n} = 0.00458333
$$

### Step 2: Calculate the total number of payments
The total number of payments is:

$$
n \cdot t = 360
$$

### Step 3: Substitute values into the PMT formula
Now we can substitute these values into the PMT formula:

$$
\mathrm{PMT} = 80,000 \cdot \frac{0.00458333}{\left(1 + 0.00458333\right)^{360} - 1}
$$

Let's calculate this expression to find the monthly payment.
Calculate the value by following steps:
- step0: Calculate:
  $$80000\left(\frac{0.00458333}{\left(\left(1+0.00458333\right)^{360}-1\right)}\right)$$
- step1: Remove the parentheses:
  $$80000\left(\frac{0.00458333}{\left(1+0.00458333\right)^{360}-1}\right)$$
- step2: Add the numbers:
  $$80000\left(\frac{0.00458333}{1.00458333^{360}-1}\right)$$
- step3: Convert the expressions:
  $$80000\left(\frac{0.00458333}{\left(\frac{100458333}{100000000}\right)^{360}-1}\right)$$
- step4: Subtract the numbers:
  $$80000\left(\frac{0.00458333}{\frac{100458333^{360}-100000000^{360}}{100000000^{360}}}\right)$$
- step5: Divide the terms:
  $$80000\times \frac{458333\times 100000000^{359}}{100458333^{360}-100000000^{360}}$$
- step6: Multiply:
  $$\frac{80000\times 458333\times 100000000^{359}}{100458333^{360}-100000000^{360}}$$
- step7: Multiply:
  $$\frac{36666640000\times 100000000^{359}}{100458333^{360}-100000000^{360}}$$
The calculated monthly payment into the IRA is approximately:

$$
\mathrm{PMT} \approx 87.56
$$

### Final Answer
You need to make monthly payments of approximately **$87.56** into your IRA to reach your goal of $80,000 by the age of 60, assuming an APR of 5.5% compounded monthly.
You need to make monthly payments of approximately **$87.56** into your IRA to reach $80,000 by age 60 with a 5.5% APR compounded monthly.

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