Problem 42 - Introductory and Intermediate Algebra for College Students 4th

Question

This year's salary, 42,074 dollar is a \(9 \%\) increase over last year's salary. What was last year's salary?

Step-by-Step Solution

Answer

Last year's salary was around $38,600.

Step 1: Convert the percentage increase to a decimal

To convert a percentage to a decimal, divide the percentage by 100. Here, the percentage is \(9\%\). So convert it by dividing by 100, which provides a decimal of \(0.09\).

Step 2: Determine the multiplication factor for the increase

An increase of \(9\%\) signifies this year's salary is 109% of last year's salary, because the salary is 100% of itself plus the additional 9%. In decimal form, this becomes \(1 + 0.09 = 1.09\). The total salary is0due to a 9% increment is 1.09 times the previous year's salary.

Step 3: Calculate last year's salary

We know from step 2 that this year's salary represents 1.09 times of last year's salary. We will, therefore, divide the current salary by 1.09 to figure out the previous year's wage. So last year's salary was \(42,074 / 1.09\). Perform this division to find last year's salary.

Step 4: Round the result

Lastly, it's unlikely that the salary was a number with decimals. So, to get the most probable number, round down the result, if the decimal part is less than 50 cents, otherwise round up.

Key Concepts

salary calculation

Calculating salaries often involves understanding how percentage changes impact the original amount. When you know this year's salary after an increase and need to find out last year's salary, you are doing a reverse calculation.
The method involves first identifying the current salary, in our case, it is $42,074. The challenge lies in reversing the percentage increase that led to this amount from a lesser amount the previous year. By understanding this principle of calculating backward, you can determine how last year's salary compares to the current figure after going through the increment.
To handle such calculations, follow these steps:

• Identify the rate of increase in percentage terms.
• Convert this percentage into a multiplication factor (more on this later).
• Use the multiplication factor to determine the original salary.

Ultimately, salary calculation is about dissecting how each part of a percentage affects the total and figuring out how to reconstruct the original from the modified total.

convert percentage to decimal

To manipulate percentage figures within mathematical equations, you often need to convert them into a decimal format. This is particularly important in calculations involving increases or decreases, where percentages need to be expressed in a way that complements mathematical operations like multiplication or division.
For instance, to convert a percentage like 9% into a decimal, divide the number by 100. Hence, 9% becomes 0.09. This transformation is crucial because it allows the percentage to be used within a formula or equation.
By converting percentages to decimals, you can efficiently apply further mathematical operations, knowing that you have a unified numerical format. For example, when calculating salary changes, using decimals means you can straightforwardly incorporate increases by multiplying or dividing by these decimal figures.
Here’s a quick reference:

• Divide the percentage value by 100.
• The result is your decimal representation.
• Use this decimal in further calculations.

This conversion makes it easier to work with percentages in equations, ensuring accuracy and simplifying complex calculations.

multiplication factor

The multiplication factor is a pivotal concept when calculating changes in financial figures, such as salaries. It serves as a convenient tool to express the total percentage in decimal form, necessary for deriving one number as a percentage of another.
Let's break down how the multiplication factor works in the context of a salary increase. When salary increases by a certain percentage, the current salary becomes a percentage more of the previous salary. Use the multiplication factor to pinpoint exactly what portion of the original the current salary represents. It's formulated by adding the percentage change as a decimal to 1.
For a 9% raise, convert the percentage first: 9% becomes 0.09. Then, the multiplication factor is calculated as:\[1 + 0.09 = 1.09\]This factor indicates that this year's salary is 109% of last year’s salary. To find the original amount, use this factor to adjust the new salary backwards.
Remember:

• Convert percentage to decimal first.
• Add to 1 for the multiplication factor.
• Use this factor to determine the relationship between old and new amounts.

Understanding the multiplication factor helps visualize how changes in percentages affect numerical values directly and indirectly.