An individual earns an extra $2000 each year and places this money at the end of each year into an Individual Retirement Account (IRA) in which both the original earnings and the interest in the account are not subject to taxation. If the account has an annual interest rate of 9.3% compounded annually, how much is in the account at the end of 35 years? (Round your answer to the nearest cent.)

This problem can be solved using the formula for the future value of an annuity due to regular payments with compound interest. The formula is:

\[
A = P \times \frac{(1 + r)^t - 1}{r}
\]

Where:
- $ A $ is the future value of the account (what we want to find),
- $ P $ is the annual payment ($2000),
- $ r $ is the annual interest rate (9.3% or 0.093),
- $ t $ is the number of years (35).

Let's calculate the future value of the retirement account:

\[
A = 2000 \times \frac{(1 + 0.093)^{35} - 1}{0.093}
\]

First, calculate $ (1 + 0.093)^{35} $:

\[
1.093^{35} \approx 18.9205
\]

Now subtract 1 from this value:

\[
18.9205 - 1 = 17.9205
\]

Next, divide by the interest rate $ r = 0.093 $:

\[
\frac{17.9205}{0.093} \approx 192.6946
\]

Finally, multiply by the annual payment $ P = 2000 $:

\[
A = 2000 \times 192.6946 \approx 385,389.20
\]

So, the total amount in the account at the end of 35 years is approximately **$385,389.20**.

Would you like further details or clarifications?

---

Here are 5 related questions to expand on this topic:
1. How would the total amount change if the interest rate was compounded monthly instead of annually?
2. What if the individual decided to increase their contribution each year by 5%?
3. How much would be in the account after 40 years instead of 35 years?
4. What happens to the total if the individual stops contributing after 20 years but leaves the money in the account?
5. How would inflation affect the purchasing power of the retirement savings over time?

**Tip:** The earlier you start saving for retirement, the more you benefit from compound interest!

Find out how much an IRA will be worth after 35 years of annual $2000 contributions at a 9.3% interest rate, compounded annually. This detailed solution uses the future value of annuity formula and explores the benefits of compound interest, making it perfect for high school students studying finance and math.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation