Problem 30 - Algebra 2 and Trigonometry

Question

Keegan started a job that paid \(\$ 20,000\) a year. Each year after the first, he received a raise of \(\$ 600 .\) What was the total amount that Keegan earned in six years?

Step-by-Step Solution

Answer

Keegan earned \( \$129,000 \) in six years.

Step 1: Identify Initial Salary

Keegan's initial salary for the first year was given as \( \$20,000 \).

Step 2: Identify Raise Amount

Each year after the first year, Keegan received a raise of \( \$600 \).

Step 3: Calculate Salary for Each Year

Calculate Keegan's salary for each year by adding the raise to the previous year's salary.1. Year 1: \( \\(20,000 \)2. Year 2: \( \\)20,000 + \\(600 = \\)20,600 \)3. Year 3: \( \\(20,600 + \\)600 = \\(21,200 \)4. Year 4: \( \\)21,200 + \\(600 = \\)21,800 \)5. Year 5: \( \\(21,800 + \\)600 = \\(22,400 \)6. Year 6: \( \\)22,400 + \\(600 = \\)23,000 \)

Step 4: Calculate Total Earnings Over Six Years

Add up all the salaries from the six years to find the total earnings.Total earnings = Year 1 salary + Year 2 salary + Year 3 salary + Year 4 salary + Year 5 salary + Year 6 salary.\[\\(20,000 + \\)20,600 + \\(21,200 + \\)21,800 + \\(22,400 + \\)23,000 = \$129,000 \]

Step 5: Conclusion

Keegan's total earnings over the six-year period is \( \$129,000 \).

Key Concepts

Salary Progression

When we talk about salary progression, it's all about understanding how someone's salary increases over time. In Keegan's case, he starts with an initial salary of $20,000. This is his base pay before any raises take effect.

Each year, he receives a raise of $600. This means that with each passing year, his salary becomes a little more substantial. For example:

• In Year 1, Keegan earns $20,000.
• In Year 2, he earns $20,600 (that's $20,000 plus the $600 raise).
• By Year 6, this progression continues until his salary becomes $23,000.

Thus, understanding salary progression is like adding small increments to a starting amount, year after year.

It’s a simple but effective way of modeling salary increases, which often corresponds with annual raises in many jobs.

Total Earnings Calculation

Calculating total earnings means summing up the income received over a period. For Keegan, this involves adding his salary from each year together.

We can add the salary from each of the six years as follows:

• Year 1: \(20,000
• Year 2: \)20,600
• Year 3: \(21,200
• Year 4: \)21,800
• Year 5: \(22,400
• Year 6: \)23,000

When you put all these amounts together, you end up with:\[20,000 + 20,600 + 21,200 + 21,800 + 22,400 + 23,000 = 129,000\]Understanding this process shows how simple addition can help deduce total earnings over time. This knowledge helps in financial planning and expectations for future income.

Step-by-Step Solution

A step-by-step solution can demystify how we arrive at the total earnings over six years.

Here's a thorough breakdown: - **Identify Initial Salary**: Recognizing that the starting point is $20,000 is essential. - **Determine the Increment**: Keegan’s raise is $600 annually. This is crucial for knowing how the salary increases. - **Calculate Annual Salaries**: Using the initial salary and the raise, calculate each year's salary from Year 1 to Year 6. This involves adding $600 to the previous year's salary successively. - **Total the Salaries**: Sum the salaries for all six years to get the total earnings. Breaking down problems into smaller, manageable steps helps clarify the process. Each step logically follows from the prior, ensuring you don’t miss anything important. This method is useful beyond just mathematics—it's a valuable technique for solving complex problems in various areas.