(b) Suppose that Patty's husband delays starting an IRA for the first 10 years he works but then makes $2800 deposits at the end of each of the next 15 years. If the interest rate is 10%, compounded annually, and if he leaves the money in his account for 5 additional years, how much will be in his account at the end of the 30-year period? (Round your answer to the nearest cent.)

We can break down this problem into two parts: 

1. **Deposits for 15 years:**
   Patty's husband delays for the first 10 years and then deposits $2800 at the end of each year for the next 15 years. These deposits will grow for 5 additional years without any further contributions. This situation can be analyzed using the **Future Value of an Annuity** formula.

   The formula for the future value of an annuity is:

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

   Where:
   - $ P = 2800 $ (annual deposit)
   - $ r = 0.10 $ (interest rate, compounded annually)
   - $ n = 15 $ (number of deposits)

2. **Growth after the 15 deposits:**
   After the 15 years of depositing, the amount accumulated will remain in the account for 5 additional years, compounding at the same interest rate. To calculate this, we will use the **Future Value** formula for compounding:

   \[
   FV_{\text{additional}} = FV \times (1 + r)^5
   \]

### Step-by-step solution:

1. **Calculate the future value of the annuity:**
   \[
   FV = 2800 \times \frac{(1 + 0.10)^{15} - 1}{0.10}
   \]
   \[
   FV = 2800 \times \frac{(1.10)^{15} - 1}{0.10}
   \]
   \[
   FV = 2800 \times \frac{4.177248 - 1}{0.10}
   \]
   \[
   FV = 2800 \times 31.77248
   \]
   \[
   FV = 88,962.944
   \]

2. **Calculate the future value after 5 additional years of compounding:**
   \[
   FV_{\text{additional}} = 88,962.944 \times (1.10)^5
   \]
   \[
   FV_{\text{additional}} = 88,962.944 \times 1.61051
   \]
   \[
   FV_{\text{additional}} = 143,272.16
   \]

### Final Answer:
The amount in Patty's husband's IRA account at the end of the 30-year period is **$143,272.16**.

Would you like more details on any step or have questions?

Here are 5 related questions:
1. What would the balance be if the interest rate was compounded monthly instead of annually?
2. How would the amount change if Patty's husband started depositing right from the beginning?
3. How does the future value change if the annual contribution is increased by $500?
4. What would the balance be if the interest rate was 8% instead of 10%?
5. How does the final amount compare with simple interest calculations?

**Tip:** For long-term savings, the earlier you start investing, the more compound interest can work in your favor, significantly increasing the future value.

Suppose that Patty's husband delays starting an IRA for the first 10 years he works but then makes $2800 deposits at the end of each of the next 15 years. If the interest rate is 10%, compounded annually, and if he leaves the money in his account for 5 additional years, how much will be in his account at the end of the 30-year period? (Round your answer to the nearest cent.)

Calculate how much will be in an IRA account after 30 years when $2800 annual contributions are made for 15 years, with a 10% interest rate compounded annually. Includes detailed breakdown of future value of annuity and compound interest formulas.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation