Problem 41 - Introductory Algebra for College Students

Question

This year's salary, \(\$ 50,220,\) is an \(8 \%\) increase over last year's salary. What was last year's salary?

Step-by-Step Solution

Answer

The last year's salary was approximately $46,500.

Step 1: Interpret the problem in terms of percentage

It is given that this year's salary is 8% more than last year's. Thus, this year's salary represents 108% of last year's salary.

Step 2: Express the percentage as a decimal

To be able to perform relevant calculations, it would be helpful to express 108% as a decimal. Thus, 108% becomes 1.08.

Step 3: Calculate last year's salary

We can express the relationship between this year's and last year's salary as an equation. If we let X represent last year's salary, the equation is 1.08X = $50,220. Solving for X will give us last year's salary. So, X equals $50,220 divided by 1.08.

Key Concepts

Salary Calculations

Understanding salary calculations involves determining how changes in salary, such as percentage increases, affect the overall income. In our example, we have a current salary of \(\\(50,220\), which is 8% more than last year's salary. This means last year's salary was less. To find last year's salary, we need to identify how much the salary has increased by.

The problem states that the amount \(\\)50,220\) is 108% of the previous year's amount. Why 108%? This is because the new salary not only includes the original amount (100%) but also the additional 8%. By understanding this percentage increase, you can more easily calculate what last year's salary was before it increased.

• Current salary includes the increase: Original + Increase = New
• The increase is 8% of the previous year's salary
• Thus, this year's total percentage is 108% compared to last year's 100%

Decimal Conversion

When dealing with percentages, converting them to decimals is crucial to simplify calculations. Percentages are a way to express a number out of 100, which can be cumbersome to work with directly in equations.

In this exercise, you are given 108%, which you need to convert to a decimal to solve for last year's salary. To convert a percentage to a decimal, you simply divide by 100. So, 108% becomes \( \frac{108}{100} = 1.08 \).

• Remove the % sign
• Divide by 100
• Use the decimal 1.08 in your calculations

This conversion helps in setting up the salary equation, making it easier to calculate the unknown value.

Equation Solving

Solving equations is a fundamental part of finding unknown values, like last year's salary in this scenario. Once you have the decimal conversion, you can set up the equation representing the relationship between this year's and last year's salary.

Here's how the equation is formed: Let \( X \) be last year's salary. This year's salary, represented by \( 1.08X \), equals \(\\(50,220\). To find \( X \), you solve the equation \[ 1.08X = 50,220 \].

• Divide \(\\)50,220\) by 1.08 to isolate \( X \)
• This gives you \( X = \frac{50,220}{1.08} \)

By calculating this division, you'll obtain the correct value for last year's salary. This straightforward approach highlights the steps needed to solve many problems involving percentage changes.