Suppose you wish to retire at the age of 70 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 25 years old. Round your answer to the nearest cent, if necessary.

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To determine the monthly payment into an IRA needed to accumulate $$ \$80,000 $$ by the age of 70, we can use the future value of an annuity formula. The formula for the future value of an annuity is:

$$
FV = P \times \frac{(1 + r)^n - 1}{r}
$$

Where:
- $$ FV $$ is the future value of the annuity (the amount you want to have at retirement, which is $$ \$80,000 $$).
- $$ P $$ is the monthly payment (the value we want to find).
- $$ r $$ is the monthly interest rate (annual interest rate divided by 12).
- $$ n $$ is the total number of payments (number of years until retirement multiplied by 12).

### Step 1: Identify the known values
- Future Value, $$ FV = 80000 $$
- Annual Interest Rate, $$ APR = 5.5\% = 0.055 $$
- Monthly Interest Rate, $$ r = \frac{0.055}{12} $$
- Current Age = 25
- Retirement Age = 70
- Total Years until Retirement = $$ 70 - 25 = 45 $$
- Total Payments, $$ n = 45 \times 12 $$

### Step 2: Calculate $$ r $$ and $$ n $$
Now, let's calculate $$ r $$ and $$ n $$:

$$
r = \frac{0.055}{12}
$$
$$
n = 45 \times 12
$$

### Step 3: Substitute into the formula and solve for $$ P $$
Rearranging the future value formula to solve for $$ P $$:

$$
P = \frac{FV \times r}{(1 + r)^n - 1}
$$

Now, let's calculate $$ r $$, $$ n $$, and then $$ P $$.

I'll perform these calculations now.
Calculate the value by following steps:
- step0: Calculate:
  $$45\times 12$$
- step1: Multiply the numbers:
  $$540$$
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:
  $$\frac{11}{2400}$$
The calculations yield the following results:

- Monthly Interest Rate, $$ r \approx 0.00458333 $$
- Total Payments, $$ n = 540 $$

### Step 4: Substitute into the formula to find $$ P $$

Now we can substitute $$ FV = 80000 $$, $$ r \approx 0.00458333 $$, and $$ n = 540 $$ into the rearranged formula for $$ P $$:

$$
P = \frac{80000 \times 0.00458333}{(1 + 0.00458333)^{540} - 1}
$$

Let's calculate $$ P $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{80000\times 0.00458333}{\left(\left(1+0.00458333\right)^{540}-1\right)}$$
- step1: Remove the parentheses:
  $$\frac{80000\times 0.00458333}{\left(1+0.00458333\right)^{540}-1}$$
- step2: Add the numbers:
  $$\frac{80000\times 0.00458333}{1.00458333^{540}-1}$$
- step3: Convert the expressions:
  $$\frac{80000\times 0.00458333}{\left(\frac{100458333}{100000000}\right)^{540}-1}$$
- step4: Multiply the numbers:
  $$\frac{366.6664}{\left(\frac{100458333}{100000000}\right)^{540}-1}$$
- step5: Subtract the numbers:
  $$\frac{366.6664}{\frac{100458333^{540}-100000000^{540}}{100000000^{540}}}$$
- step6: Convert the expressions:
  $$\frac{\frac{458333}{1250}}{\frac{100458333^{540}-100000000^{540}}{100000000^{540}}}$$
- step7: Multiply by the reciprocal:
  $$\frac{458333}{1250}\times \frac{100000000^{540}}{100458333^{540}-100000000^{540}}$$
- step8: Rewrite the expression:
  $$\frac{458333}{1250}\times \frac{1562500^{540}\times 64^{540}}{100458333^{540}-100000000^{540}}$$
- step9: Rewrite the expression:
  $$\frac{458333}{1250}\times \frac{1250^{1080}\times 64^{540}}{100458333^{540}-100000000^{540}}$$
- step10: Reduce the numbers:
  $$458333\times \frac{1250^{1079}\times 64^{540}}{100458333^{540}-100000000^{540}}$$
- step11: Multiply:
  $$\frac{458333\times 1250^{1079}\times 64^{540}}{100458333^{540}-100000000^{540}}$$
The calculation gives us:

$$
P \approx 33.90
$$

### Conclusion
The monthly payment into the IRA needed to accumulate $$ \$80,000 $$ by the age of 70, with an APR of $$ 5.5\% $$ compounded monthly, is approximately **\$33.90**.
The monthly payment needed is approximately \$33.90.

Suppose you wish to retire at the age of 70 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 25 years old. Round your answer to the nearest cent, if necessary.

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