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The problem provided is a retirement planning question involving Hal Thomas, who plans to make yearly deposits into an Individual Retirement Account (IRA) with a 12% annual return over a 35-year period. The questions ask for calculations involving different scenarios of deposit timings and their effects on future values. Let’s break down the problem and solve it step-by-step.

### Assumptions:
- Hal will retire at age 60 (35 years from now).
- Annual deposit of $2,200.
- Return rate (r) = 12% or 0.12 annually.
- The deposits are made either at the **end** or **beginning** of each year.

### 1. **Part (a) Calculation:**

If Hal makes **end-of-year deposits** of $2,200 for 35 years, the future value of the IRA is calculated using the future value of an annuity formula:

\[
FV = P \times \left(\frac{{(1 + r)^n - 1}}{r}\right)
\]
where:
- $P$ = $2,200 (annual deposit),
- $r$ = 0.12 (annual return),
- $n$ = 35 years.

Let me compute this first.For part (a), if Hal makes end-of-year deposits of $2,200 into the IRA, he will accumulate **$949,659.69** by the time he turns 60.

### 2. **Part (b) Calculation:**

If Hal waits until age 35 (10 years from now) to start making **end-of-year deposits**, he will only deposit for 25 years. Using the same formula, but with $ n = 25 $:

\[
FV = P \times \left(\frac{{(1 + r)^{25} - 1}}{r}\right)
\]

Let's calculate this now.For part (b), if Hal waits until age 35 to start making end-of-year deposits, he will accumulate **$293,334.51** by the time he turns 60.

### 3. **Part (c) Discussion:**

The difference in the accumulated amount between starting at age 25 versus age 35 can be calculated as:

\[
\text{Difference} = \text{Amount at 60 from (a)} - \text{Amount at 60 from (b)}
\]

Let me calculate that difference.The impact of delaying deposits by 10 years (starting at age 35 instead of age 25) results in **$656,325.18** less in accumulated funds by the time Hal turns 60. This demonstrates the significant effect of compounding interest over time.

### 4. **Part (d) Rework: Beginning-of-Year Deposits**

Now, if Hal makes deposits at the **beginning of each year** instead of the end, the future value formula changes slightly to account for one extra period of interest:

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

Let's recalculate for both scenarios:
- **Starting at age 25** (for 35 years),
- **Starting at age 35** (for 25 years).

I'll calculate these now.It seems like I can’t do more advanced data analysis right now. Please try again later. 

However, you can calculate the future value for part (d) using this formula:
\[
FV = P \times \left(\frac{{(1 + r)^n - 1}}{r}\right) \times (1 + r)
\]
where:
- $P$ = $2,200 (annual deposit),
- $r$ = 0.12 (return rate),
- $n$ = number of years (35 for part a, 25 for part b).

Feel free to reach out if you'd like me to assist further later!

Would you like more details on any part, or have any questions? Here are some related topics you might find interesting:
1. How does compounding interest affect retirement savings?
2. What's the difference between end-of-year and beginning-of-year deposit strategies?
3. How do changes in the interest rate affect future value?
4. What is the impact of delaying savings by 5, 10, or more years?
5. How can you calculate the present value of future retirement savings?

**Tip**: Starting early with retirement savings can significantly increase future value due to the power of compounding!

Hal Thomas, a 25-year-old college graduate, wishes to retire at age 60. To supplement other sources of retirement income, he can deposit $2,200 each year into a tax-deferred individual retirement arrangement (IRA). The IRA will earn a return of 12% over the next 35 years. Calculate the amount accumulated based on various scenarios of deposit timings.

Learn how to calculate the future value of an IRA with a 12% annual return for 35 years. The problem involves scenarios of end-of-year and beginning-of-year deposits, with detailed steps on how delaying deposits impacts the total accumulated amount. Suitable for college students and those interested in personal finance.

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