Problem 59 - Algebra for College Students

Question

Use an algebraic approach to solve each problem. Angelo is paid double time for each hour he works over 40 hours in a week. Last week he worked 46 hours and earned \(\$ 572\). What is his normal hourly rate?

Step-by-Step Solution

Answer

Angelo's normal hourly rate is $11 per hour.

Step 1: Identify Normal and Overtime Hours

Angelo works a total of 46 hours. Since overtime is any time worked over 40 hours, his overtime hours are calculated as: \[ 46 - 40 = 6 \] So, he worked 40 normal hours and 6 overtime hours.

Step 2: Set Up the Equation for Total Pay

Let Angelo's normal hourly rate be \( x \) dollars. According to the problem, he is paid double time for his overtime hours. Therefore, his total earnings can be represented by the equation:\[ 40x + 2(6)x = 572 \] This equation accounts for both his normal and overtime pay.

Step 3: Simplify the Equation

Simplify the equation from Step 2 to make it easier to solve:\[ 40x + 12x = 572 \]Combine like terms:\[ 52x = 572 \]

Step 4: Solve for the Normal Hourly Rate

To find \( x \), divide both sides of the equation by 52:\[ x = \frac{572}{52} \]Calculate:\[ x = 11 \]

Key Concepts

Overtime Pay Calculations

Overtime pay is the additional amount employees earn when they work beyond regular working hours. In most scenarios, this threshold is set at 40 hours per week. Let's take a closer look at how overtime pay works. When an employee works overtime, they typically earn 1.5 to 2 times their usual hourly wage for every extra hour. In our example with Angelo, he earns double his regular rate for any hours worked over 40 hours.
To calculate overtime pay properly, first, identify the number of overtime hours worked. This requires subtracting the total hours worked from the regular threshold:

• Total hours worked - Standard hours = Overtime hours

Next, multiply the overtime rate by the number of overtime hours. This figure will then be added to the remuneration for regular hours, providing a complete picture of total pay for the week.

Linear Equations

A linear equation is an essential part of solving word problems involving multiple conditions, like those around work hours and pay. In mathematics, a linear equation represents a straight line and takes the form of ax + b = c. It is an equation where each term is either a constant or the product of a constant and a single variable.
For Angelo's problem, the linear equation helps encapsulate all parts of his pay structure by combining his normal and overtime hours in a single equation:

• Regular pay: 40 hours at rate \( x \)
• Overtime pay: 6 hours at double rate \( 2x \)

This gives us the equation: \( 40x + 2(6)x = 572 \). By simplifying, it becomes \( 52x = 572 \), which is easy to solve. Linear equations are powerful tools for quantifying real-world scenarios in a mathematical framework.

Hourly Rate Calculation

Calculating the hourly rate of an employee involves using basic algebraic principles to break down a broader equation that includes various types of pay. It is crucial to isolate the variable representing the hourly rate to find its value. For example, in Angelo's case, after setting up the equation with both regular and overtime pay combined, the next important step is simplifying it:

• Combine like terms: \( 40x + 12x = 52x \)
• Rearrange the equation: \( 52x = 572 \)
• Solve for \( x \) by dividing both sides by 52: \( x = \frac{572}{52} \)
• Calculate to find \( x = 11 \), which represents Angelo's regular hourly rate.

This process reveals essential techniques for tackling similar scenarios, emphasizing the need to simplify and isolate terms to extract the hourly rate from more complex equations.