To determine Carlos’s salary over time with each job offer, we need to calculate the annual salary for each specified year using the two different raise structures: one with a fixed dollar amount and the other with a percentage increase.

1. Fixed Dollar Raise ($3300 per year)

The salary progression for the fixed dollar raise job can be calculated using a linear function: $\text{Salary} = 55000 + (3300 \times t)$ where $t$ is the number of years.

The values provided so far are:

• Year 1: $55000 + 3300 \times 1 = 58300$
• Year 5: $55000 + 3300 \times 5 = 71500$
• Year 10: $55000 + 3300 \times 10 = 88000$
• Year 15: $55000 + 3300 \times 15 = 104500$
• Year 20: $55000 + 3300 \times 20 = 121000$

These are correct as you’ve confirmed.

2. Percentage Raise (5% per year)

The salary progression for the percentage raise job can be calculated using an exponential function: $\text{Salary} = 55000 \times (1.05)^t$

Calculations:

1. Year 1: $55000 \times (1.05)^1 = 57750$
2. Year 5: $55000 \times (1.05)^5 \approx 70131$
3. Year 10: $55000 \times (1.05)^{10} \approx 89597$
4. Year 15: $55000 \times (1.05)^{15} \approx 114456$
5. Year 20: $55000 \times (1.05)^{20} \approx 146879$

Summary Table

Years	Salary with $3300 raise per year	Salary with 5% raise per year
1	$58,300	$57,750
5	$71,500	$70,131
10	$88,000	$89,597
15	$104,500	$114,456
20	$121,000	$146,879

Would you like a detailed explanation of these calculations or have any specific questions?

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1. How does the salary difference grow over 20 years between the two raise options?
2. What would be the percentage difference between the salaries in year 20?
3. How much more does Carlos earn with a 5% raise after 10 years compared to the $3300 raise?
4. What is the break-even year where the percentage raise first surpasses the fixed raise?
5. What impact would a different initial salary have on the percentage increase?

Tip: Exponential growth (like the percentage raise) can eventually surpass linear growth if given enough time.