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The question involves calculating retirement contributions based on different starting ages and an assumed APR (annual percentage rate) of 6%, compounded monthly. Here’s a breakdown of each part of the question and the steps for calculating the answers:

### 1. Monthly Contribution for a $1,000,000 Goal Starting at Age 25
- **Goal**: Accumulate $1,000,000 by age 65, starting at age 25.
- **Parameters**:
  - APR: 6%, compounded monthly.
  - Time horizon: 65 - 25 = 40 years (or 480 months).
- **Formula**: The future value of an annuity formula can be used:
  \[
  FV = P \times \frac{(1 + r)^n - 1}{r}
  \]
  where:
  - $FV = 1,000,000$,
  - $r = \frac{0.06}{12}$ (monthly interest rate),
  - $n = 480$.
  - Solve for $P$ (monthly contribution).

### 2. Total Personal Contribution by Age 65 (Starting at Age 25)
This is the total amount personally contributed over 40 years (480 months):
\[
\text{Total Contribution} = P \times 480
\]

### 3. Monthly Contribution Starting at Age 45
- **Goal**: Accumulate $1,000,000 by age 65, starting at age 45.
- **Parameters**:
  - Time horizon: 20 years (or 240 months).
  - Monthly interest rate: Same as above.
- Use the same annuity formula with $n = 240$ to solve for $P$.

### 4. Total Personal Contribution by Age 65 (Starting at Age 45)
Calculate the total amount contributed over 20 years (240 months) by:
\[
\text{Total Contribution} = P \times 240
\]

### 5. Difference in Total Contribution Between Starting at Age 25 and Age 45
Calculate the difference in total contributions between the two scenarios.

If you’d like, I can run these calculations for precise answers or provide more guidance on each formula. 

Would you like the step-by-step calculations for each part, or do you have any specific questions?

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### Expanding Questions:
1. How would a change in APR affect the monthly contribution needed?
2. What impact does compounding frequency have on retirement savings?
3. How does starting earlier with smaller contributions compare to larger contributions later?
4. How do different retirement age goals (e.g., 60 vs. 70) affect the contribution calculations?
5. How could inflation-adjusted returns change the contribution required?

**Tip:** Starting contributions early helps leverage compound interest more effectively, resulting in lower required contributions.

We are considering the effects of starting early or late to save for retirement. Assume that each account considered has an APR of 6% compounded monthly.

Following expert advice, you begin your retirement program as soon as you graduate from college at age 25. You plan to retire at the age of 65. What monthly contributions do you need to make to have a retirement account worth $1,000,000? (Round your answer to the nearest cent.)

What will your total personal contribution be by the time you retire if you start saving after graduation?

Against expert advice, you begin your retirement program at age 45. You plan to retire at the age of 65. What monthly contributions do you need to make to have a retirement account worth $1,000,000? (Round your answer to the nearest cent.)

What will your total personal contribution be by the time you retire if you start saving at age 45?

How much more will you personally contribute by the time you retire if you start saving at age 45 instead of starting right after graduation?

Explore how to calculate retirement savings contributions with compound interest for different starting ages. Learn how monthly contributions vary if you start at age 25 or 45 to reach a $1,000,000 retirement goal. This problem covers key finance concepts, including the future value of an annuity formula and compound interest.

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