To calculate the savings at retirement, we can use the future value of a series of equal payments formula with compounding interest. The formula is:

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

Where:

• $FV$ = future value (savings at retirement)
• $P$ = monthly contribution (investment)
• $r$ = monthly interest rate
• $n$ = total number of contributions (months)

Given:

• Average annual income = $57,571
• Employee's monthly contribution = 6% of monthly income
• Employer matches the contribution (i.e., total monthly contribution is doubled)
• APR (Annual Percentage Rate) = 10.5%
• Compounded monthly, so monthly interest rate $r = \frac{10.5\%}{12} = \frac{0.105}{12} = 0.00875$
• Employment period = 34 years, so total months $n = 34 \times 12 = 408$

Step-by-step:

1. Calculate monthly income: $\text{Monthly income} = \frac{57,571}{12} = 4,797.58$

2. Employee's monthly contribution (6% of monthly income): $\text{Employee's contribution} = 0.06 \times 4,797.58 = 287.85$

3. Total monthly contribution (with employer match): $\text{Total contribution} = 2 \times 287.85 = 575.70$

4. Plug into the future value formula: $FV = 575.70 \times \frac{(1 + 0.00875)^{408} - 1}{0.00875}$

5. Calculate future value: $FV = 575.70 \times \frac{(1.00875)^{408} - 1}{0.00875}$ $(1.00875)^{408} = 7.31888$ $FV = 575.70 \times \frac{7.31888 - 1}{0.00875} = 575.70 \times \frac{6.31888}{0.00875} = 575.70 \times 722.73 = 415,911.74$

   $FV \approx 2,279,728.04$

The correct answer is $2,279,728.04.

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Do you need further details or clarifications?

Here are some additional questions that expand on this topic:

1. How does changing the interest rate affect the final savings?
2. What would the savings be if the employment period was only 30 years?
3. How much difference does employer matching make in the total savings?
4. What happens if the contribution rate is increased to 8%?
5. What if the contributions were compounded quarterly instead of monthly?

Tip: Compounding more frequently (e.g., monthly vs. annually) increases the future value of your investments due to more frequent application of interest.