Problem 56 - Elementary Algebra Within Reach

Question

In Exercises \(55-58\), write an algebraic equation. Do not solve the equation. A school science club conducts a car wash to raise money. The club spends \(\$ 12\) on supplies and charges \(\$ 5\) per car. After the car wash, the club has a profit of \(\$ 113\). How many cars did the members of the science club wash?

Step-by-Step Solution

Answer

The algebraic equation representing this situation is \(5c - 12 = 113\).

Step 1: Identifying Constants and Variables

First, recognize the constants given in the problem: the cost of supplies which is $12, the price charged per car which is $5, and the total profit which is $113. The variable to determine is the number of cars washed, represented as 'c'.

Step 2: Formulating the Profit Equation

Next, express the total profit as the total income subtracted by the cost, which gives the equation \(5c - 12 = 113\). Where \(5c\) represents the total charge from washing cars, \(-12\) represents the cost of supplies, and \(113\) is the total profit made.

Key Concepts

Mathematical Problem-Solving

Mathematical problem-solving involves using structured methods to find solutions to various problems, like the one in this algebraic equation example. Here, we are tasked to create an equation from a given scenario and not necessarily solve it. It's important to first understand the question fully and identify the key parts of the problem including all given information.
The process of solving mathematical problems usually involves several steps:

• Understanding the Problem: Carefully read the problem to identify what is being asked. Gather all information provided and think about what it represents.
• Devising a Plan: Formulate a strategy. Consider similar problems you’ve solved before and how they were approached.
• Carrying Out the Plan: Use algebraic operations to write an equation based on the problem statement.
• Reviewing: Examine the formed equation to ensure it correctly represents the problem.

These steps help in breaking down the problem into manageable parts, ensuring clarity and facilitating accurate equation formulation. Consistent practice with problem-solving enhances your ability to understand and form equations in various situations.

Variables and Constants

In math, variables and constants are fundamental concepts in forming algebraic equations. Let's explore what they mean in the context of our problem.
Constants are fixed values that do not change. In the given problem, the constants are:

• $12: This is the cost for the car wash supplies. This amount does not change regardless of the number of cars washed.
• $5: This is the charge per car. Each car washed is associated with this constant charge.
• $113: This is the total profit made by the club, which remains fixed.

Variable is a symbol that represents an unknown value, which can change depending on the situation. We use a letter like 'c' to represent the variable. In this case, 'c' stands for the number of cars washed.
The role of a variable is crucial as it lets us build an equation that helps us understand the relationship between these numbers. Recognizing which parts of the problem are constants and which are variables is key to setting up a correct equation.

Profit Calculation

Profit calculation is a practical application of algebraic equations usable in various real-life situations, such as determining how many cars need to be washed for a science club fundraiser.
To calculate profit, you need to understand the basic formula for profit:\[Profit = Total \; Income - Total \; Expenses \]Based on this fundamental relationship, the problem statement gives us clues on setting up a specific equation:

• **Total Income:** This is calculated as the charge per car multiplied by the number of cars washed, which is denoted as \(5c\).
• **Total Expenses:** This includes the cost of supplies, fixed at \(12.

The main equation derived from these factors is \(5c - 12 = 113\). Here:

• The term \(5c\) represents the money made from washing cars.
• Subtracting \)12 is necessary to account for the initial cost of supplies.
• The result 113 signifies the total profit after expenses have been deducted.

These steps show how algebraic equations can be powerful tools in profit calculation scenarios, making it easier to determine necessary conditions to achieve set financial goals. Understanding this concept can be beneficial in business, personal finance, and any situation involving sales and expenses.