Problem 89 - Prealgebra

Question

Compensation Employee A had an annual salary of \(\$ 42,000\), Employee B had an annual salary of \(\$ 48,000,\) and Employee C had an annual salary of \(\$ 46,000\) before each employee was given a \(5 \%\) raise. Which of the three employee's annual salary is now the highest? Explain how you arrived at your answer.

Step-by-Step Solution

Answer

Employee B now has the highest salary at \( \$50,400 \).

Step 1: Determine the Raise Amount

To calculate the raise each employee receives, multiply each annual salary by 5%. For Employee A, the raise is \( 0.05 \times 42,000 \). For Employee B, it is \( 0.05 \times 48,000 \). For Employee C, it's \( 0.05 \times 46,000 \).

Step 2: Calculate the New Salaries

Add the raise to the original salary for each employee to determine the new salary. Employee A's new salary is \( 42,000 + 0.05 \times 42,000 \). Employee B's new salary is \( 48,000 + 0.05 \times 48,000 \). Employee C's new salary is \( 46,000 + 0.05 \times 46,000 \).

Step 3: Compare the New Salaries

Compute the new salaries from Step 2: \( 42,000 + 2,100 = 44,100 \) for Employee A, \( 48,000 + 2,400 = 50,400 \) for Employee B, and \( 46,000 + 2,300 = 48,300 \) for Employee C. Now, compare these amounts to determine which is the highest.

Key Concepts

Percentage Increase

Understanding percentage increase is fundamental in solving salary-related problems. A percentage increase is a way to express the amount of raise or growth from a base value.
To calculate a percentage increase:

• Multiply the original amount by the percentage (expressed as a decimal).
• The formula used is: \( \text{Increase} = \text{Original Amount} \times \text{Percentage as a Decimal} \).

Applying this to the exercise: Employee A had a salary of \( \\(42,000 \) and received a 5% raise. You calculate the raise by multiplying \( \\)42,000 \times 0.05 \), which equals \( \\(2,100 \).
For Employee B with \( \\)48,000 \): \( \\(48,000 \times 0.05 = \\)2,400 \).
Lastly, Employee C with \( \\(46,000 \): \( \\)46,000 \times 0.05 = \$2,300 \).
Learning to determine the percentage increase helps grasp changes in salaries or other values efficiently.

Annual Salary Calculation

Once you know the percentage increase, you calculate the new annual salary by adding the increase to the original salary.

• This step involves simple addition: \( \text{New Salary} = \text{Original Salary} + \text{Increase} \).

Following our exercise, Employee A's new salary is calculated as follows:
\( \\(42,000 + \\)2,100 = \\(44,100 \).
For Employee B: \( \\)48,000 + \\(2,400 = \\)50,400 \).
And for Employee C: \( \\(46,000 + \\)2,300 = \$48,300 \).
This method shows you how even a small percentage affects the overall annual income, giving perspective into financial planning and salary expectations.

Comparison of Values

The last step is comparing values after the calculation to determine which is the highest or most favorable. It's a straightforward process but crucial in decision-making.

• After calculating, align the values clearly.
• Assess which number is the largest. This helps recognize the highest salary post-raise.

In our case, comparing the new salaries:
Employee A: \( \\(44,100 \), Employee B: \( \\)50,400 \), Employee C: \( \\(48,300 \).
Here, \( \\)50,400 \) is the highest, indicating Employee B has the highest adjusted annual salary.
This process teaches you how comparisons are essential not only for salary evaluations but for understanding financial positioning in various scenarios.