Problem 30 - Introductory Algebra for College Students

Question

A car rental agency charges 180 dollars per week plus 0.25 dollars per mile to rent a car. How many miles can you travel in one week for 395 dollars ?

Step-by-Step Solution

Answer

The total number of miles that can be traveled for 395 dollars in one week is \( x = 215 / 0.25 = 860 \) miles.

Step 1: Define the Variables

Let's define \( x \) as the number of miles that can be driven. Hence, the total cost of renting a car and driving \( x \) miles is \( 180 + 0.25x \) dollars

Step 2: Formulate the Equation

You know the total cost for the week is 395 dollars. So, form an equation that equals this value: \( 180 + 0.25x = 395 \)

Step 3: Solve for \( x \)

Start by subtracting 180 from both sides of the equation to isolate the term with \( x \): \( 0.25x = 395 - 180\). Then, do the subtraction: \( 0.25x = 215 \). To solve for \( x \), divide both sides of this equation by 0.25: \( x = 215 / 0.25 \).

Key Concepts

Linear Equations

A linear equation is a mathematical expression that models a relationship between two variables, typically represented by two expressions set equal to each other. In the context of cost problems, it shows how a certain cost builds up based on fixed charges and variable costs. For example, in our car rental problem, the total cost is represented as a linear equation where the fixed weekly charge is $180, and the variable cost is $0.25 per mile.

• The general form of a linear equation in this context is: cost = fixed charge + (cost per mile × number of miles).
• Linear equations are used to find unknown values by substituting known values into the equation.
• By balancing both sides of the equation, we can solve for unknowns like the number of miles driven.

Understanding linear equations helps in forming connections between different parts of a mathematical problem, aiding in solving complex problems easily.

Variable Isolation

Variable isolation is a crucial mathematical concept that involves manipulating an equation to express one variable in terms of others. When solving linear equations, the goal is often to find the value of the unknown variable by isolating it on one side of the equation.

• Start by simplifying the equation by removing constants from the side with the variable. This is done by subtracting or adding from both sides.
• For multiplication or division factors attached to the variable, perform the opposite operation on both sides to further isolate the variable.
• In our exercise, for example, we begin by subtracting 180 from both sides, so the equation becomes: 0.25x = 215. The final step involves dividing by 0.25 to solve for x.

This method is effective in many algebra problems, making it easier to solve equations accurately.

Cost Analysis

Cost analysis is the process of evaluating the various components that contribute to the total cost of an item or service. It helps in understanding how different factors affect the final price, allowing for better financial planning and decision-making. For the car rental problem, we're analyzing how driving distance impacts the total rental cost.

• Break down costs into fixed and variable components. Here, $180 is the fixed weekly charge.
• The $0.25 per mile charge is the variable component, depending on the miles driven.
• Total cost can be calculated by adding these components as shown in the linear equation.

By understanding these elements, one can effectively gauge how different variables affect costs, critical for budget planning and cost control.

Word Problems

Word problems combine real-world scenarios with mathematical concepts to solve for unknowns. They are excellent for developing problem-solving skills and applying mathematics to real-life situations.

• Translate the problem into a mathematical form by identifying all relevant quantities and their relationships.
• In our example, the word problem describes a car rental situation. Translation involves recognizing the fixed cost and per-mile variable cost, then constructing a linear equation.
• Carefully read the problem to understand what is being asked, ensuring every part of the scenario is translated into appropriate mathematical actions.

By successfully solving word problems, students learn to think critically and apply mathematical techniques beyond mere computation.