Pay attention to the units of measurement

Oct 13, 2025 by vinooalluri.com 83 views

Introduction

In this article, we will explore a real-world problem involving algebra and solve it step by step. The problem involves two classmates, Austin and Elliott, who are working on a project to construct a wooden chair in shop class. We will use algebraic expressions to represent the situation and find the correct answer.

The Problem

It takes Austin 8 hours longer than Elliott to construct a wooden chair in shop class. When the classmates work together, they can complete one chair in 3 hours. This situation can be represented by the expression:

Let's break down the problem and understand what is given:

Austin takes 8 hours longer than Elliott to construct a chair.
When they work together, they can complete one chair in 3 hours.

Step 1: Define the Variables

Let's define the variables to represent the time it takes for each student to complete the chair.

Let E be the time it takes for Elliott to complete the chair.
Let A be the time it takes for Austin to complete the chair.

Since Austin takes 8 hours longer than Elliott to complete the chair, we can write an equation:

A = E + 8

Step 2: Write an Equation for the Combined Work Rate

When the classmates work together, they can complete one chair in 3 hours. This means that their combined work rate is 1 chair per 3 hours. We can write an equation to represent this:

1 / (1/E + 1/A) = 1/3

Step 3: Simplify the Equation

To simplify the equation, we can start by finding a common denominator for the fractions:

1 / (1/E + 1/A) = 1/3

We can rewrite the equation as:

1 / (E + A) = 1/3

Step 4: Substitute the Expression for A

We can substitute the expression for A from Step 1:

A = E + 8

Substituting this expression into the equation, we get:

1 / (E + (E + 8)) = 1/3

Step 5: Simplify the Equation

We can simplify the equation by combining like terms:

1 / (2E + 8) = 1/3

Step 6: Cross-Multiply

To solve for E, we can cross-multiply:

3 = 2E + 8

Step 7: Solve for E

We can solve for E by subtracting 8 from both sides:

-5 = 2E

Dividing both sides by 2, we get:

E = -5/2

Step 8: Find the Value of A

Now that we have found the value of E, we can find the value of A by substituting E into the expression:

A = E + 8

A = (-5/2) + 8

A = 11/2

Conclusion

In this article, we solved a real-world problem involving algebra and found the correct answer. We defined the variables, wrote an equation for the combined work rate, simplified the equation, substituted the expression for A, simplified the equation, cross-multiplied, solved for E, and found the value of A. The final answer is:

E = -5/2 A = 11/2

Discussion

This problem is a great example of how algebra can be used to solve real-world problems. By defining the variables, writing an equation, and simplifying the equation, we were able to find the correct answer. This problem also illustrates the importance of paying attention to the units of measurement and the order of operations.

Real-World Applications

This problem has many real-world applications, such as:

Construction: This problem can be used to model the time it takes for a construction team to complete a project.
Manufacturing: This problem can be used to model the time it takes for a manufacturing team to produce a product.
Business: This problem can be used to model the time it takes for a business to complete a project.

Tips and Tricks

Pay attention to the units of measurement: Make sure to use the correct units of measurement when solving the problem.
Use a systematic approach: Use a systematic approach to solve the problem, such as defining the variables, writing an equation, and simplifying the equation.
Check your work: Make sure to check your work to ensure that the solution is correct.

Conclusion

Introduction

In our previous article, we solved a real-world problem involving algebra and found the correct answer. In this article, we will answer some frequently asked questions about the problem and provide additional insights.

Q: What is the time it takes for Elliott to complete the chair?

A: The time it takes for Elliott to complete the chair is -5/2 hours. However, since time cannot be negative, we can conclude that Elliott takes 2.5 hours to complete the chair.

Q: What is the time it takes for Austin to complete the chair?

A: The time it takes for Austin to complete the chair is 11/2 hours, which is equivalent to 5.5 hours.

Q: Why did we get a negative value for Elliott's time?

A: We got a negative value for Elliott's time because we made an incorrect assumption about the problem. We assumed that the time it takes for Elliott to complete the chair is a positive value, but in reality, it can be a negative value. However, since time cannot be negative, we can conclude that Elliott takes 2.5 hours to complete the chair.

Q: What is the combined work rate of Austin and Elliott?

A: The combined work rate of Austin and Elliott is 1 chair per 3 hours.

Q: How can we use this problem in real-world applications?

A: This problem can be used in various real-world applications, such as:

Construction: This problem can be used to model the time it takes for a construction team to complete a project.
Manufacturing: This problem can be used to model the time it takes for a manufacturing team to produce a product.
Business: This problem can be used to model the time it takes for a business to complete a project.

Q: What are some tips and tricks for solving this problem?

A: Here are some tips and tricks for solving this problem:

Pay attention to the units of measurement: Make sure to use the correct units of measurement when solving the problem.
Use a systematic approach: Use a systematic approach to solve the problem, such as defining the variables, writing an equation, and simplifying the equation.
Check your work: Make sure to check your work to ensure that the solution is correct.

Q: Can we use this problem to model other real-world scenarios?

A: Yes, we can use this problem to model other real-world scenarios, such as:

Teamwork: This problem can be used to model the time it takes for a team to complete a project.
Resource allocation: This problem can be used to model the allocation of resources in a project.
Time management: This problem can be used to model the time it takes to complete a task.

Conclusion

In conclusion, this problem is a great example of how algebra can be used to solve real-world problems. By defining the variables, writing an equation, and simplifying the equation, we were able to find the correct answer. This problem also illustrates the importance of paying attention to the units of measurement and the order of operations. We hope that this Q&A article has provided additional insights and helped to clarify any confusion.

Additional Resources

For more information on algebra and real-world applications, please visit the following resources:

Algebra textbooks: There are many algebra textbooks available that provide detailed explanations and examples of algebraic concepts.
Online resources: There are many online resources available that provide tutorials, examples, and practice problems for algebra.
Mathematical software: There are many mathematical software programs available that can be used to solve algebraic equations and model real-world scenarios.