Problem 56 - Algebra 1

Question

You earn 9 dollar per hour. On major holidays, such as Thanksgiving, you earn twice as much per hour. You earned a total of \(\$ 405\) for the week including Thanksgiving. Write and solve an equation to find how many hours you worked on Thanksgiving if you worked 35 hours during the rest of the week.

Step-by-Step Solution

Answer

You worked 5 hours on Thanksgiving.

Step 1: Define the Variables

Let's define \( x \) as the number of hours worked on Thanksgiving. So, the total amount of money earned for the week is \( 9 × 35 + 2 × 9 × x = 405 \).

Step 2: Set Up the Equation

We know the total amount earned which includes normal work hours and Thanksgiving hours. Now, set up the equation like this: 315 (which is \( 9 × 35 \)) + 18x = 405.

Step 3: Solve the Equation

To solve for \( x \), first isolate the term with \( x \) by subtracting 315 from both sides of the equation: 18x = 90. Then, divide both sides by 18 to get \( x \): \( x = 90 / 18 = 5 \). Therefore, you worked 5 hours on Thanksgiving.

Key Concepts

Solving Equations

When it comes to solving equations, the primary goal is to find the value of the variable that makes the equation true. Equations often represent real-life situations like calculating income or expenses. In our example, the equation combines earnings from regular hours and holiday hours.
First, observe that the equation is composed of two parts: one part accounts for regular hours worked at the standard rate, while the second part corresponds to the additional holiday earnings rate.
When you set up the equation, you need to ensure every term is correctly accounted for, like the \(315 for regular hours and the \)18 for each stormy holiday hour. The total, $405, is what you'll aim to balance the equation to.
Solving equations can involve techniques such as:

• Combining like terms: Gather similar terms to simplify the equation.
• Isolating the variable: Use operations like addition, subtraction, multiplication, or division to get the variable by itself on one side.

In this exercise, you isolate the terms with \( x \) and follow arithmetic operations to find the value of \( x \), which are crucial steps in solving equations effectively.

Defining Variables

Defining variables is an important first step in solving algebra word problems. A variable represents an unknown value that we are aiming to determine. It's like a placeholder for the number we need to find.
In the context of the exercise, we have two distinct earning conditions: regular pay and holiday pay. To navigate this scenario:

• Choose a variable: Here, \( x \) is chosen to represent the unknown hours worked on Thanksgiving, which frames the problem more clearly.
• Relate the variable to the conditions: Since holiday pay is twice the regular pay, define the total earnings as a function of \( x \).

The clear definition makes it easier to write and solve the equation. Assigning a variable essentially transforms a word problem into an algebraic equation, giving you a structured path to follow. By establishing that \( x \) stands for the Thanksgiving hours worked, every step afterwards is guided and knows where it fits into the scenario.

Step-by-Step Problem Solving

Approaching a problem step-by-step ensures a thorough understanding and an organized way to tackle complex situations. Let's look at how each step facilitates an efficient solution.

• **Start by defining variables**: As we defined earlier, \( x \) is the number of Thanksgiving hours worked. Always start by understanding the problem's components and what you're solving for.
• **Set up the equation correctly**: Translate the word problem into a mathematical form. The earnings are separated into standard pay and holiday pay, creating the expression \( 9 \times 35 + 2 \times 9 \times x \).

Once your equation reflects the problem context, you can proceed to solve it with certainty.

• **Isolate \( x \)**: Subtract the constant part of the total from both sides to bring focus to the variable part.
• **Solve for the variable**: Division helps you find \( x = 5 \), the number of hours worked on the special rate day.

This structured approach not only helps in solving the current problem but builds a strong foundation for tackling future problems methodically.