Problem 68 - Elementary and Intermediate Algebra

Question

Translate each problem to an equation. Do not solve. Hours Worked. A carpenter charges \(\$ 35\) an hour. How many hours did she work if she billed a total of \(\$ 3150 ?\)

Step-by-Step Solution

Answer

35h = 3150

Step 1: Understanding the Problem

Identify the variables. Let the number of hours worked be represented by the variable, say, h. We know the carpenter charges \(35 per hour and she billed a total of \)3150.

Step 2: Set Up the Equation

Write the equation that represents the given information. Since she charges \(35 per hour and the total cost is \)3150, the equation would be: oodles 35h = 3150

Key Concepts

Variables in Algebra

Variables are symbols used to represent unknown values in mathematical equations. In algebra, variables are often denoted by letters such as x, y, or h. Using variables allows us to create equations that can model real-world scenarios,
making the problem-solving process easier.

For example, in our exercise, the number of hours the carpenter worked is unknown. To represent this unknown value, we choose a variable, such as 'h'
. This variable 'h' stands for the number of hours worked by the carpenter.

• Variables simplify the expression of complex problems
• They enable easy identification and manipulation of unknowns

By identifying variables, we can transform word problems into algebraic equations. This step is crucial in algebraic problem-solving.

Setting Up Equations

Setting up equations involves converting a word problem into a mathematical statement with variables and constants. This process starts with understanding the problem: recognizing what is given and what needs to be found.

Let's follow the exercise example: the carpenter charges \(\$ 35\) per hour and the total bill is \(\$ 3150\). We need to find out how many hours 'h' the carpenter worked.

1. Identify the quantities:

• Carpenter's hourly rate: \(\$ 35\)
• Total bill: \(\$ 3150\)

2. Relate the quantities using an equation: Multiply the hourly rate by the number of hours 'h' to get the total bill. This translates to:

• \[ 35h = 3150 \]

By setting up this equation, we create a clear mathematical representation of the problem, ready to be solved if needed.

Word Problems in Algebra

Word problems in algebra are practical scenarios that require translation into equations before solving. Mastering this skill involves understanding the language of mathematics and knowing how to identify relevant information.

For instance, in our exercise, we turned the word problem about a carpenter's billing into an algebraic equation.
Here are steps to tackle word problems effectively:

• Read the problem carefully
• Identify the unknowns and assign variables
• Determine the relationships and operations that connect the known values to the unknowns
• Write the equation representing the problem

Practicing these steps can make translating word problems into algebraic equations second nature. This process bridges real-world situations with mathematical solutions, enhancing problem-solving skills.