At the age of 

3030​, 

to save for​ retirement, you decide to deposit 

​$7070 

at the end of each month in an IRA that pays 

55​% 

compounded monthly.

a.a.

Use the following formula to determine how much you will have in the IRA when you retire at age 65.

Upper A equals StartFraction Upper P left bracket left parenthesis 1 plus r right parenthesis Superscript t Baseline minus 1 right bracket Over r EndFractionA=P(1+r)t−1r    

or

Upper A equals StartStartFraction Upper P left bracket left parenthesis 1 plus StartFraction r Over n EndFraction right parenthesis Superscript nt Baseline minus 1 right bracket OverOver left parenthesis StartFraction r Over n EndFraction right parenthesis EndEndFractionA=P1+rnnt−1rn

b.b.

Find the interest.

...

Question content area right

Part 1

a.a. 

You will have approximately 

​$enter your response here 

in the IRA when you retire.

​(Do not round until the final answer. Then round to the nearest dollar as​ needed.)

We are given the following:

- You plan to deposit $70 at the end of each month.
- The interest rate is 5% compounded monthly.
- You are currently 30 years old and plan to retire at age 65, so you have 35 years to save.

We will use the following formula for the future value of an annuity:

\[
A = P \cdot \frac{\left( \left(1 + \frac{r}{n} \right)^{nt} - 1 \right)}{\frac{r}{n}}
\]

Where:
- $A$ is the future value of the investment/IRA.
- $P = 70$ is the monthly deposit.
- $r = 0.05$ (5%) is the annual interest rate.
- $n = 12$ is the number of times the interest is compounded per year (monthly).
- $t = 35$ is the number of years.

### Step-by-Step Solution

1. First, calculate the monthly interest rate:
   \[
   \frac{r}{n} = \frac{0.05}{12} = 0.0041667
   \]

2. Then, calculate $nt$, which represents the total number of payments:
   \[
   nt = 12 \times 35 = 420
   \]

3. Now, we substitute these values into the formula and calculate $A$:

\[
A = 70 \cdot \frac{\left( \left(1 + 0.0041667 \right)^{420} - 1 \right)}{0.0041667}
\]

Let me calculate the result for you.The future value of the IRA when you retire will be approximately **$79,526**.

### Part b: Find the interest earned.
The total interest earned is the difference between the future value of the IRA and the total amount deposited.

Let me calculate that for you.The total interest earned over 35 years will be approximately **$50,126**.

### Summary:
- The amount in your IRA when you retire: **$79,526**.
- Total interest earned: **$50,126**.

Would you like more details or have any questions about this solution?

Here are 5 related questions to explore further:
1. How would the future value change if the interest rate increased to 6%?
2. What happens if you make quarterly contributions instead of monthly ones?
3. How much would you have if you started saving 5 years earlier?
4. What would be the impact of increasing the monthly deposit to $100?
5. How does compounding frequency affect the total interest earned?

**Tip:** Starting retirement savings earlier significantly increases the final amount due to compounding over a longer period.

At the age of 30, to save for retirement, you decide to deposit $70 at the end of each month in an IRA that pays 5% compounded monthly. Use the formula to determine how much you will have in the IRA when you retire at age 65. Find the interest.

Learn how to calculate the future value of an IRA using monthly contributions of $70 and a 5% interest rate compounded monthly. This solution covers compound interest and annuity formulas, offering a step-by-step guide to find how much you'll save by age 65. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation