Problem 45 - Introductory Algebra for College Students

Question

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I can clean my house in 3 hours and my sloppy friend can completely mess it up in 6 hours, so if we both "work" together, the time, \(x,\) it takes to clean the house can be modeled by \(\frac{x}{3}-\frac{x}{6}=1\)

Step-by-Step Solution

Answer

The statement 'makes sense', because the equation models the situation accurately. It will take 6 hours for the house to be cleaned when both of you 'work' together.

Step 1: Understanding rates of work

The phrase 'I can clean my house in 3 hours' means that your work rate is \(1/3\) house per hour. Similarly, the sloppy friend's rate of work is \(1/6\) house per hour. However, unlike the friend is messing up the house while it's being cleaned.

Step 2: Understanding how work rates interact

When you are cleaning the house and the friend is messing it up at the same time, the friend's work rate opposes yours. So, instead of just adding the two rates together, the friend's rate should be subtracted from your's. Thus, \(\frac{1}{3} - \frac{1}{6} = \frac{1}{6}\) house per hour. This means when both of you 'work' together, \(1/6\) of the house will be cleaned in an hour.

Step 3: Verifying the equation

The equation \(\frac{x}{3} - \frac{x}{6} = 1\) represents this relationship correctly. On the left, \(\frac{x}{3}\) represents your work, and \(\frac{x}{6}\) represents the sloppy friend's work as time passes. The 1 on the right side of the equation represents 1 house. Solving the equation, we get \(x = 6\) hours, which means it will take 6 hours for the house to be cleaned.

Key Concepts

Understanding Rates of Work

When we talk about rates of work, we're essentially looking at how quickly a person or machine can complete a task. It's a measure of productivity and efficiency and is usually expressed as the amount of work done per unit of time. For instance, if you can clean your house in 3 hours, your work rate is the completion of 1 house in the span of 3 hours, or mathematically speaking, \( \frac{1}{3} \) house per hour.

It's vital to grasp this concept because it sets the stage for more complex scenarios where multiple rates are interacting, as often happens in real-world problems. Understanding that these rates can either aid each other (when two people work together on the same task) or oppose each other (as with the sloppy friend) is crucial for correctly modeling the situation.

Interacting Work Rates

Work rates can interact in different ways depending on the situation. In the case where two people are doing the same job, the rates add together. However, if their efforts oppose one another, as with a person cleaning and another making a mess, their rates will effectively subtract.

For example, your cleaning rate is \( \frac{1}{3} \) house per hour, and your friend's 'messing up' rate is \( \frac{1}{6} \) house per hour. These rates are not cumulative; your friend's actions are counterproductive to your goal. Therefore, the combined effective work rate is the difference of the two rates, giving us \( \frac{1}{3} - \frac{1}{6} = \frac{1}{6} \) house per hour.

Understanding this concept is essential for solving problems where multiple entities contribute to or impede the completion of a task.

Solving Work Rate Problems

To solve work rate problems, it's critical to set up an equation representing the relationship between the work rates and the task completion. Given that your work rate is \( \frac{1}{3} \) and your friend's negative impact is \( \frac{1}{6} \) per hour, the collective work rate is modeled as \( \frac{x}{3} - \frac{x}{6} \) for the time, \( x \) , it takes to clean the house. This equation can be solved to find the value of \( x \) that corresponds to the completion of the task.

Solving \( \frac{x}{3} - \frac{x}{6} = 1 \) yields \( x = 6 \) hours, meaning that, considering the opposing rates, it will take 6 hours to clean the house. These algebraic models can precisely predict the outcome of such collaborative or oppositional work scenarios.

Algebraic Modeling of Real-World Problems

Algebraic modeling is a powerful tool to translate real-world problems into equations that can be manipulated and solved. It bridges the gap between abstract mathematical concepts and practical applications. In the case of work rates, constructing an algebraic model involves defining the variables, setting up an equation based on the rates of work, and understanding how these rates interact within the context of the given situation.

For instance, in our example, the algebraic model takes into account that one person is cleaning \( \frac{1}{3} \) of a house per hour while the other messes up \( \frac{1}{6} \) of it per hour. By embedding this model within an equation, we can find actionable solutions to questions like, 'How long will it take to clean the house if both people are 'working' together?' This is invaluable for predicting outcomes, allocating resources, and planning efficiently in various fields, from project management to operations research.