Select The Correct Answer.Sean And Colleen Are Raking Leaves In Their Yard. Working Together, They Can Clear The Yard Of Leaves In 24 Minutes. Working Alone, It Would Take Sean 20 Minutes Longer To Clear The Yard Than It Would Take Colleen Working

May 4, 2025 by ADMIN 248 views

**The Power of Collaboration: A Math Problem Involving Sean and Colleen**

Introduction

In this article, we will delve into a math problem involving Sean and Colleen, who are raking leaves in their yard. The problem requires us to determine how long it would take each of them to clear the yard alone, given that they can work together to complete the task in 24 minutes. We will use algebraic equations to solve this problem and understand the concept of collaboration and individual work rates.

Understanding the Problem

Sean and Colleen are raking leaves in their yard. When they work together, they can clear the yard in 24 minutes. However, if they were to work alone, it would take Sean 20 minutes longer to clear the yard than it would take Colleen. Let's assume that the time it takes Colleen to clear the yard alone is represented by the variable 'c'. Since Sean takes 20 minutes longer than Colleen, the time it takes Sean to clear the yard alone would be 'c + 20'.

Setting Up the Equation

When Sean and Colleen work together, their combined work rate is the sum of their individual work rates. We can represent their combined work rate as 1/24, since they can clear the yard together in 24 minutes. We can also represent their individual work rates as 1/c and 1/(c + 20), respectively. Using the formula for combined work rate, we can set up the following equation:

1/c + 1/(c + 20) = 1/24

Solving the Equation

To solve this equation, we can start by finding a common denominator for the fractions on the left-hand side. The common denominator would be c(c + 20). Multiplying both sides of the equation by c(c + 20), we get:

c(c + 20) + c(c + 20) = c(c + 20)/24

Simplifying the equation, we get:

2c^2 + 40c = c^2 + 20c

Subtracting c^2 + 20c from both sides, we get:

c^2 + 20c = 0

Factoring out c, we get:

c(c + 20) = 0

This equation has two possible solutions: c = 0 or c + 20 = 0. Since c cannot be 0 (as it represents the time it takes Colleen to clear the yard alone), we can discard this solution. Therefore, we are left with c + 20 = 0.

Finding the Value of c

Solving for c, we get:

c = -20

However, since time cannot be negative, we need to reconsider our approach. Let's go back to the original equation and try a different method.

Alternative Method

We can start by finding a common denominator for the fractions on the left-hand side. The common denominator would be 24. Multiplying both sides of the equation by 24, we get:

24(1/c + 1/(c + 20)) = 24(1/24)

Simplifying the equation, we get:

24/c + 24/(c + 20) = 1

Multiplying both sides of the equation by c(c + 20), we get:

24(c + 20 + 24c = c(c + 20)

Expanding and simplifying the equation, we get:

24c + 480 + 24c = c^2 + 20c

Rearranging the terms, we get:

c^2 - 44c + 480 = 0

Solving the Quadratic Equation

We can solve this quadratic equation using the quadratic formula:

c = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 1, b = -44, and c = 480. Plugging these values into the formula, we get:

c = (44 ± √((-44)^2 - 4(1)(480))) / 2(1)

Simplifying the equation, we get:

c = (44 ± √(1936 - 1920)) / 2

c = (44 ± √16) / 2

c = (44 ± 4) / 2

Therefore, we have two possible solutions: c = (44 + 4) / 2 and c = (44 - 4) / 2.

Finding the Value of c

Solving for c, we get:

c = 48/2 or c = 40/2

c = 24 or c = 20

Conclusion

In this article, we used algebraic equations to solve a math problem involving Sean and Colleen, who are raking leaves in their yard. We found that it would take Colleen 20 minutes to clear the yard alone, and it would take Sean 24 minutes to clear the yard alone. This problem demonstrates the concept of collaboration and individual work rates, and how we can use algebraic equations to solve real-world problems.

Key Takeaways

When working together, the combined work rate of two individuals is the sum of their individual work rates.
We can use algebraic equations to solve problems involving collaboration and individual work rates.
The quadratic formula can be used to solve quadratic equations.

Final Answer

Introduction

In our previous article, we solved a math problem involving Sean and Colleen, who are raking leaves in their yard. We found that it would take Colleen 20 minutes to clear the yard alone, and it would take Sean 24 minutes to clear the yard alone. In this article, we will answer some frequently asked questions related to this problem.

Q: What is the combined work rate of Sean and Colleen?

A: The combined work rate of Sean and Colleen is 1/24, since they can clear the yard together in 24 minutes.

Q: How do we calculate the combined work rate of two individuals?

A: To calculate the combined work rate of two individuals, we add their individual work rates. In this case, the individual work rates of Sean and Colleen are 1/20 and 1/24, respectively.

Q: What is the formula for combined work rate?

A: The formula for combined work rate is:

Combined Work Rate = Individual Work Rate 1 + Individual Work Rate 2

Q: How do we solve a quadratic equation?

A: To solve a quadratic equation, we can use the quadratic formula:

c = (-b ± √(b^2 - 4ac)) / 2a

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula used to solve quadratic equations. It is given by:

c = (-b ± √(b^2 - 4ac)) / 2a

Q: What is the significance of the quadratic formula?

A: The quadratic formula is significant because it allows us to solve quadratic equations, which are equations of the form ax^2 + bx + c = 0. Quadratic equations are used to model a wide range of real-world problems, including physics, engineering, and economics.

Q: Can we use the quadratic formula to solve all types of quadratic equations?

A: Yes, we can use the quadratic formula to solve all types of quadratic equations. However, we need to be careful when using the formula, as it may not always give us the correct solution.

Q: What are some common mistakes to avoid when using the quadratic formula?

A: Some common mistakes to avoid when using the quadratic formula include:

Not checking the discriminant (b^2 - 4ac) before using the formula
Not simplifying the expression before solving for c
Not checking the solutions for validity

Q: How do we check the solutions for validity?

A: To check the solutions for validity, we need to make sure that the solutions are real and make sense in the context of the problem. For example, if we are solving a problem involving time, we need to make sure that the solutions are positive and make sense in the context of the problem.

Conclusion

In this article, we answered some frequently asked questions related to Sean and Colleen's leaf raking problem. We discussed the combined work rate of two individuals, the formula for combined work rate, and the quadratic formula. We also discussed some mistakes to avoid when using the quadratic formula and how to check the solutions for validity.