Question

Solve. Denzel received a \(9 \%\) raise. His new salary is \(\$ 39,828.60 .\) What was his former salary?

Short answer

Answer: Denzel's original salary was approximately $36,540.

Step-by-step solution

Set up equation

Let x be Denzel's original salary. His new salary is the original salary increased by 9%, so we can express this as: $$x + 0.09x = 39828.60$$

Combine like terms

Combine the x terms on the left side of the equation: $$1x + 0.09x = 1.09x$$ So, the equation becomes: $$1.09x = 39828.60$$

Solve for x

Now, we can solve for x by dividing both sides of the equation by 1.09: $$x = \frac{39828.60}{1.09}$$ Calculating the value of x: $$x \approx 36540$$

Write the answer

Denzel's former salary was approximately \( \$ 36,540 \).

Key concepts

Percentage Increase

Understanding a percentage increase is essential when dealing with questions about salary raises, inflation, or profit margins. Essentially, a percentage increase represents how much a quantity grows relative to its original value. It's calculated by finding the difference between the new and old values, dividing that difference by the original value, and then multiplying the result by 100 to get a percentage.

For example, if Denzel's salary is increased by 9%, this means that his new salary is 109% of his original salary. Algebraically, if his original salary is represented by the variable 'x', then a 9% increase can be represented as '0.09x'. Adding that increase to the original salary, Denzel's new salary becomes 'x + 0.09x'.

Algebraic Equations

An algebraic equation is like a balance scale; whatever you do to one side of the equation, you must do to the other side to maintain equality. Equations are made of terms which are variables, numbers, or variables and numbers multiplied together. In this scenario with Denzel's salary, the algebraic equation formulated to find his original salary is rooted in the concept that his new salary consists of his original salary plus the percentage increase.

So, if we let his original salary be represented by 'x', and acknowledge a 9% raise, we can set up an equation like this: \(x + 0.09x = 39828.60\). Solving this type of equation involves steps like combining like terms and isolating the variable.

Combining Like Terms

In algebra, combining like terms is a technique used to simplify an equation or expression. Like terms are terms that have the same variables raised to the same power. You can add or subtract like terms just as you do with regular numbers.

When we see 'x + 0.09x', we are looking at two like terms. Both terms have 'x' and can be added together. This is an application of the distributive property, where you can factor out the 'x' and it becomes '1x + 0.09x = 1.09x'. By combining like terms effectively, we reduce the equation to a simpler form, making it easier to solve for the variable.

Simple Algebra

Simple algebra involves the basics of working with variables, constants, and arithmetical operations (addition, subtraction, multiplication, and division) to find an unknown quantity. After combining like terms in an algebraic equation, we often arrive at a point where we must isolate the variable, which means getting the variable on one side of the equation and everything else on the other side.

For example, in Denzel's problem, after combining like terms, the next step is to isolate 'x' by dividing both sides of the equation by 1.09, giving us \(x = \frac{39828.60}{1.09}\). This provides the solution for 'x', which represents Denzel's original salary. The steps of simple algebra are the building blocks required for nearly all algebraic problem-solving.