Problem 42 - Introductory Algebra for College Students

Question

This year's salary, \(\$ 42,074,\) 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 approximately \$38599.08.

Step 1: Understand the Problem and What We're Solving For

The problem tells us that this year's salary, $42,074, is a result of a 9% increase from last year's salary, which we don't know. We're trying to find the value of that initial salary before the increase.

Step 2: Set Up the Equation

We can represent percentage increase with the equation \(Final Amount = Original Amount + (Original Amount \times Percentage Increase)\). We know the 'Final Amount' (this year's salary) and the 'Percentage Increase' (9% or 0.09 as a decimal), but we need to find the 'Original Amount' (last year's salary). Our equation is \(42074 = Original Amount + (Original Amount \times 0.09)\).

Step 3: Rearrange the Equation to Solve for the Original Amount

To isolate 'Original Amount' on one side of the equation, we can rewrite our equation as \(42074 = Original Amount × (1 + 0.09)\), which simplifies further to \(42074 = Original Amount × 1.09\). Then divide both sides by 1.09 to solve for 'Original Amount': \(Original Amount = 42074 ÷ 1.09\).

Step 4: Solve the Equation

Dividing 42074 by 1.09 gives us approximately 38599.08. This should be the original salary before the increase.

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that use algebra to express relationships between numbers or variables. In the context of percentage increase, it helps us represent the relation between the original amount and the final amount after a percentage change.
For instance, in our exercise, we learn that this year's salary is derived from last year's salary with an increase of 9%. To capture this relationship, we use an algebraic equation:

• This year's salary equals last year's salary plus 9% of last year's salary.
• This is expressed mathematically as: \[42074 = x + (x \times 0.09)\]where \(x\) is last year's salary.

Such equations are useful in many real-life scenarios, such as calculating interest in finance or adjusting recipes in cooking. They allow for a straightforward way of solving problems involving changes in quantity.

Problem Solving

Problem solving involves breaking down a problem into manageable steps and using logical reasoning to reach a conclusion. In this exercise, the problem is determining last year's salary given this year's salary and the percentage increase.
The process begins with understanding the task: finding the original value before a percentage increase. Next, setting up the equation becomes crucial, representing the relation through algebra:

• Express the known and unknown variables clearly.
• Structure the equation around the total amount known, and the equation form: \[42074 = Original Amount + (Original Amount \times 0.09)\]

Then, rearrange the equation for solving: make the unknown, 'Original Amount', the focus. This systematic approach ensures each step logically follows from the last.

Mathematical Calculations

Mathematical calculations involve the basic arithmetic operations needed to solve equations and understand numerical relationships. Once we set up the correct equation, accurately calculating the solution is essential.
To isolate 'Original Amount' in our problem, we simplify the equation:

• Combine terms to form:\[42074 = Original Amount \times 1.09\]
• Solve for 'Original Amount' by dividing both sides of the equation by 1.09:\[Original Amount = \frac{42074}{1.09}\]
• This division results in approximately 38599.08.

Such arithmetic operations are fundamental in algebra, where solving for an unknown requires manipulating numbers precisely. Consistency and accuracy in these calculations ensure that the solution, such as determining last year’s salary, is reliable and verifiable.