Bailey starts an IRA (Individual Retirement Account) at the age of 32 to save for retirement. She deposits each month. The IRA has an average annual interest rate of
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

Question

Bailey starts an IRA (Individual Retirement Account) at the age of 32 to save for retirement. She deposits

each month. The IRA has an average annual interest rate of

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

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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Solución

Known Conditions:

Future Value of an Annuity Formula:

Step-by-Step Calculation:

Step 1: Calculate the Monthly Interest Rate

Step 2: Calculate the Future Value

Final Answer

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