Problem 316 - Elementary Algebra

Question

In the following exercises, solve. Ryan charges his neighbors \(\$ 17.50\) to wash their car. How many cars must he wash next summer if his goal is to earn at least \(\$ 1,500 ?\)

Step-by-Step Solution

Answer

Ryan must wash at least 86 cars.

Step 1: Identify the Variables

Let the number of cars Ryan needs to wash be represented by the variable \( x \).

Step 2: Set Up the Inequality

Ryan charges \( \$17.50 \) per car, and his goal is to earn at least \( \$1,500 \). Therefore, the inequality can be set up as: \( 17.50x \geq 1500 \).

Step 3: Solve the Inequality

To find \( x \), divide both sides of the inequality by \( 17.50 \): \[ x \geq \frac{1500}{17.50} \]

Step 4: Calculate the Solution

Perform the division to find the number of cars: \[ x \geq 85.71 \]. Since Ryan cannot wash a fraction of a car, he needs to wash at least 86 cars.

Step 5: Final Answer

Therefore, Ryan needs to wash at least 86 cars to meet his goal.

Key Concepts

Understanding Inequalities

An inequality is a relationship between two expressions that are not necessarily equal. It tells us how one quantity compares to another. In Ryan's car wash problem, the goal is to determine the number of cars he needs to wash to earn at least \(1,500. This means Ryan's earnings should be greater than or equal to \)1,500. Therefore, we use the inequality symbol \( \geq \) to represent this condition. Writing and solving inequalities is a powerful tool in math to represent real-world situations where values are not fixed. This helps us make informed decisions based on different outcomes.

Identifying Variables

Variables are symbols that represent unknown values. In our exercise, we need to find how many cars Ryan must wash. We introduce a variable, typically denoted as \( x \), to represent this unknown number. Identifying the variable correctly is crucial because it sets the stage for forming the inequality. For Ryan's problem, the variable \( x \) stands for the number of cars. By identifying this, we can then relate it to the other given values and constraints in the problem. This makes it easier to translate the word problem into a mathematical statement.

Breaking Down Step-by-Step Solutions

Step-by-step solutions are essential in solving math problems as they guide you through the entire process methodically. Let's revisit the steps in solving this exercise:

- First, we identified our variable, \( x \), which represents the number of cars.
- Next, we set up the inequality based on Ryan's goal: \( 17.50x \geq 1500 \).
- To solve for \( x \), we divided both sides by 17.50: \[ x \geq \frac{1500}{17.50} \].
- Finally, performing the division gives us \[ x \geq 85.71 \].
Because Ryan can't wash a fraction of a car, we round up to the next whole number, meaning he needs to wash at least 86 cars. Each step builds upon the previous one, leading to the final solution.

Setting Goals in Math

Just as Ryan has a goal to earn at least $1,500, setting goals in math helps give direction and purpose to problem-solving. When tackling a math problem:

- Start by understanding what the end goal is, just like Ryan's earning target.
- Break down the problem into smaller, manageable parts.
- Identify what you need to find out, i.e., your variable.
- Formulate equations or inequalities based on the given information and your goal.

Setting clear objectives and having a step-by-step plan can significantly improve your problem-solving skills and confidence in math. It makes complex problems more approachable and manageable, leading to successful outcomes.