Bailey starts an IRA (Individual Retirement Account) at the age of 32 to save for retirement. She deposits $$ \$ 450 $$ each month. The IRA has an average annual interest rate of $$ 5 \% $$ compounded monthly. How much money will she have saved when she retires at the age of 65 ? Round your answer to the nearest cent, if necessary. Formulas

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UpStudy AI · Accepted Answer

To solve this problem, we will use the future value of an annuity formula, which is used to calculate the future value of a series of equal payments made at regular intervals, compounded at a certain interest rate.

### Known Conditions:
- Monthly deposit ($$ P $$): $$ 450 $$
- Annual interest rate ($$ r $$): $$ 5\% $$ or $$ 0.05 $$
- Compounding frequency: Monthly
- Total number of years until retirement: $$ 65 - 32 = 33 $$ years
- Total number of deposits ($$ n $$): $$ 33 \times 12 = 396 $$ months

### Future Value of an Annuity Formula:
The future value ($$ FV $$) of an annuity can be calculated using the formula:

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

Where:
- $$ i $$ is the monthly interest rate, calculated as $$ \frac{r}{12} $$
- $$ n $$ is the total number of deposits

### Step-by-Step Calculation:
1. Calculate the monthly interest rate ($$ i $$):
   $$
   i = \frac{0.05}{12}
   $$

2. Substitute $$ P $$, $$ i $$, and $$ n $$ into the future value formula to find $$ FV $$.

Let's perform these calculations.

First, we will calculate $$ i $$ and then use it to find $$ FV $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{0.05}{12}$$
- step1: Convert the expressions:
  $$\frac{\frac{1}{20}}{12}$$
- step2: Multiply by the reciprocal:
  $$\frac{1}{20}\times \frac{1}{12}$$
- step3: Multiply the fractions:
  $$\frac{1}{20\times 12}$$
- step4: Multiply:
  $$\frac{1}{240}$$
Calculate or simplify the expression $$ 450*((1+(0.05/12))^(33*12)-1)/(0.05/12) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{450\left(\left(1+\left(\frac{0.05}{12}\right)\right)^{33\times 12}-1\right)}{\left(\frac{0.05}{12}\right)}$$
- step1: Remove the parentheses:
  $$\frac{450\left(\left(1+\left(\frac{0.05}{12}\right)\right)^{33\times 12}-1\right)}{\frac{0.05}{12}}$$
- step2: Divide the terms:
  $$\frac{450\left(\left(1+\frac{1}{240}\right)^{33\times 12}-1\right)}{\frac{0.05}{12}}$$
- step3: Add the numbers:
  $$\frac{450\left(\left(\frac{241}{240}\right)^{33\times 12}-1\right)}{\frac{0.05}{12}}$$
- step4: Multiply the numbers:
  $$\frac{450\left(\left(\frac{241}{240}\right)^{396}-1\right)}{\frac{0.05}{12}}$$
- step5: Subtract the numbers:
  $$\frac{450\times \frac{241^{396}-240^{396}}{240^{396}}}{\frac{0.05}{12}}$$
- step6: Divide the terms:
  $$\frac{450\times \frac{241^{396}-240^{396}}{240^{396}}}{\frac{1}{240}}$$
- step7: Multiply the numbers:
  $$\frac{\frac{241^{396}-240^{396}}{15^{394}\times 2^{1583}}}{\frac{1}{240}}$$
- step8: Multiply by the reciprocal:
  $$\frac{241^{396}-240^{396}}{15^{394}\times 2^{1583}}\times 240$$
- step9: Rewrite the expression:
  $$\frac{241^{396}-240^{396}}{15^{394}\times 2^{1583}}\times 15\times 16$$
- step10: Reduce the numbers:
  $$\frac{241^{396}-240^{396}}{15^{393}\times 2^{1583}}\times 16$$
- step11: Rewrite the expression:
  $$\frac{241^{396}-240^{396}}{15^{393}\times 2^{1583}}\times 2^{4}$$
- step12: Reduce the numbers:
  $$\frac{241^{396}-240^{396}}{15^{393}\times 2^{1579}}\times 1$$
- step13: Multiply the fractions:
  $$\frac{241^{396}-240^{396}}{15^{393}\times 2^{1579}}$$
### Step 1: Calculate the Monthly Interest Rate
The monthly interest rate ($$ i $$) is calculated as:
$$
i = \frac{0.05}{12} \approx 0.00416667
$$

### Step 2: Calculate the Future Value
Using the future value formula:
$$
FV = 450 \times \frac{(1 + 0.00416667)^{396} - 1}{0.00416667}
$$

Calculating this gives:
$$
FV \approx 452429.38
$$

### Final Answer
Bailey will have approximately **$452,429.38** saved when she retires at the age of 65.
Bailey will have approximately \$452,429.38 saved when she retires at age 65.

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