To approach Dewey's retirement plan problem, let's break it down part by part:

a) Determine the number of years Dewey will be saving for his retirement.

The problem doesn't explicitly state when Dewey starts saving or his current age, but we can assume that he plans to save until the typical retirement age of 65.

If Dewey is currently 25 years old (a typical assumption for retirement problems), the number of years he will be saving is:

$\text{Number of years} = 65 - 25 = 40 \text{ years}$

So Dewey will be saving for 40 years.

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b) How much money will Dewey personally deposit into his retirement account?

Dewey deposits $1200 at the end of each year for 40 years. To find the total amount he personally deposits, we multiply the annual deposit by the number of years:

$\text{Total Deposits} = \$1200 \times 40 = \$48,000$

Dewey will personally deposit $48,000 into his retirement account.

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c) Find Dewey’s account balance when he retires at age 65.

To find the future value of Dewey's account, we can use the future value formula for an annuity since he makes regular yearly payments:

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

Where:

• $FV$ = Future value (what we are solving for)
• $PMT$ = Annual payment (deposit) = $1200
• $r$ = Annual interest rate (assuming, for example, 5% or 0.05 as the rate)
• $n$ = Number of years = 40

Assuming a 5% annual interest rate, we can calculate the future value. Let's plug in the numbers:

$FV = 1200 \times \frac{(1 + 0.05)^{40} - 1}{0.05}$

I'll compute this next:

$FV = 1200 \times \frac{(1.05)^{40} - 1}{0.05}$

I'll calculate that now:

$(1.05)^{40} \approx 7.04$

So,

$FV = 1200 \times \frac{7.04 - 1}{0.05} = 1200 \times 120.88 = 145,056$

Thus, Dewey’s account balance at retirement will be $145,056.

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How much interest will Dewey’s account earn?

To calculate the interest earned, we subtract the total amount Dewey deposited from the future value:

$\text{Interest Earned} = FV - \text{Total Deposits}$ $\text{Interest Earned} = 145,056 - 48,000 = 97,056$

Dewey's account will earn $97,056 in interest over 40 years.

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Summary:

• a) Dewey will be saving for 40 years.
• b) Dewey will deposit $48,000 into his retirement account.
• c) Dewey’s account balance when he retires will be $145,056.
• Dewey’s account will earn $97,056 in interest.

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Do you have any questions, or would you like more details on any part of the solution?

Here are 5 related questions to deepen your understanding:

1. What would happen to Dewey's retirement savings if the interest rate was higher, say 6%?
2. How would the future value change if Dewey started saving at age 30 instead of 25?
3. If Dewey increased his yearly contribution by $100 every year, how much more would he have at retirement?
4. What is the difference in interest earned between monthly payments and yearly payments for the same annual total?
5. How does compounding frequency (e.g., monthly vs. yearly) affect Dewey’s retirement balance?

Tip: The power of compound interest grows exponentially over time, so starting to save early—even with small amounts—can lead to significant growth by retirement.