Question 8 of 15, Step 1 of 1 Rachel starts an IRA (Individual Retirement Account) at the age of 26 to save for retirement. She deposits $$ \$ 350 $$ each month. The IRA has an average annual interest rate of $$ 7 \% $$ 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. Answer How to enter your answer (opens in new window)

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UpStudy AI · Accepted Answer

To solve this problem, we will use the future value of an annuity formula, which is appropriate for calculating the total amount saved in an IRA with regular monthly deposits and compounded interest.

### Known Conditions:
- Monthly deposit ($$ P $$): $$ 350 $$
- Annual interest rate ($$ r $$): $$ 7\% $$ or $$ 0.07 $$
- Compounding frequency: monthly
- Total number of years until retirement: $$ 65 - 26 = 39 $$ years
- Total number of deposits ($$ n $$): $$ 39 \times 12 = 468 $$ 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:
- $$ P $$ is the monthly deposit
- $$ i $$ is the monthly interest rate
- $$ n $$ is the total number of deposits

### Step-by-Step Calculation:
1. **Calculate the monthly interest rate ($$ i $$)**:
   $$
   i = \frac{r}{12} = \frac{0.07}{12}
   $$

2. **Calculate the future value ($$ FV $$)** using the formula above.

Now, let's perform the calculations.

First, we will calculate $$ i $$ and then use it to find $$ FV $$.

### Tool Calls:
1. Calculate $$ i $$.
2. Calculate $$ FV $$.

Let's proceed with these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{0.07}{12}$$
- step1: Convert the expressions:
  $$\frac{\frac{7}{100}}{12}$$
- step2: Multiply by the reciprocal:
  $$\frac{7}{100}\times \frac{1}{12}$$
- step3: Multiply the fractions:
  $$\frac{7}{100\times 12}$$
- step4: Multiply:
  $$\frac{7}{1200}$$
Calculate or simplify the expression $$ 350 * (((1 + (0.07/12))^(39*12)) - 1) / (0.07/12)) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{350\left(\left(1+\left(\frac{0.07}{12}\right)\right)^{39\times 12}-1\right)}{\left(\frac{0.07}{12}\right)}$$
- step1: Remove the parentheses:
  $$\frac{350\left(\left(1+\left(\frac{0.07}{12}\right)\right)^{39\times 12}-1\right)}{\frac{0.07}{12}}$$
- step2: Divide the terms:
  $$\frac{350\left(\left(1+\frac{7}{1200}\right)^{39\times 12}-1\right)}{\frac{0.07}{12}}$$
- step3: Add the numbers:
  $$\frac{350\left(\left(\frac{1207}{1200}\right)^{39\times 12}-1\right)}{\frac{0.07}{12}}$$
- step4: Multiply the numbers:
  $$\frac{350\left(\left(\frac{1207}{1200}\right)^{468}-1\right)}{\frac{0.07}{12}}$$
- step5: Subtract the numbers:
  $$\frac{350\times \frac{1207^{468}-1200^{468}}{1200^{468}}}{\frac{0.07}{12}}$$
- step6: Divide the terms:
  $$\frac{350\times \frac{1207^{468}-1200^{468}}{1200^{468}}}{\frac{7}{1200}}$$
- step7: Multiply the numbers:
  $$\frac{\frac{7\times 1207^{468}-7\times 1200^{468}}{50^{467}\times 24^{468}}}{\frac{7}{1200}}$$
- step8: Multiply by the reciprocal:
  $$\frac{7\times 1207^{468}-7\times 1200^{468}}{50^{467}\times 24^{468}}\times \frac{1200}{7}$$
- step9: Rewrite the expression:
  $$\frac{7\left(1207^{468}-1200^{468}\right)}{50^{467}\times 24^{468}}\times \frac{1200}{7}$$
- step10: Reduce the numbers:
  $$\frac{1207^{468}-1200^{468}}{50^{467}\times 24^{468}}\times 1200$$
- step11: Rewrite the expression:
  $$\frac{1207^{468}-1200^{468}}{50^{467}\times 24^{468}}\times 50\times 24$$
- step12: Reduce the numbers:
  $$\frac{1207^{468}-1200^{468}}{50^{466}\times 24^{468}}\times 24$$
- step13: Rewrite the expression:
  $$\frac{1207^{468}-1200^{468}}{50^{466}\times 24^{468}}\times 8\times 3$$
- step14: Rewrite the expression:
  $$\frac{1207^{468}-1200^{468}}{2^{466}\times 25^{466}\times 24^{468}}\times 8\times 3$$
- step15: Rewrite the expression:
  $$\frac{1207^{468}-1200^{468}}{2^{466}\times 25^{466}\times 24^{468}}\times 2^{3}\times 3$$
- step16: Reduce the numbers:
  $$\frac{1207^{468}-1200^{468}}{2^{463}\times 25^{466}\times 24^{468}}\times 3$$
- step17: Rewrite the expression:
  $$\frac{1207^{468}-1200^{468}}{2^{463}\times 25^{466}\times 3^{468}\times 8^{468}}\times 3$$
- step18: Reduce the numbers:
  $$\frac{1207^{468}-1200^{468}}{2^{463}\times 25^{466}\times 3^{467}\times 8^{468}}\times 1$$
- step19: Multiply the fractions:
  $$\frac{1207^{468}-1200^{468}}{2^{1867}\times 25^{466}\times 3^{467}}$$
### Step 1: Calculate the Monthly Interest Rate ($$ i $$)
The monthly interest rate is calculated as follows:

$$
i = \frac{0.07}{12} \approx 0.0058333
$$

### Step 2: Calculate the Future Value ($$ FV $$)
Using the future value formula for the annuity:

$$
FV = 350 \times \frac{(1 + 0.0058333)^{468} - 1}{0.0058333}
$$

Calculating this gives us:

$$
FV \approx 852705.16
$$

### Final Answer
When Rachel retires at the age of 65, she will have approximately **$852,705.16** saved in her IRA.
Rachel will have approximately \$852,705.16 saved in her IRA when she retires at age 65.

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