Select The Correct Answer.Sean And Coleen 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 Coleen Working
Introduction
Mathematics is an integral part of our daily lives, and problem-solving is an essential skill that we all need to develop. In this article, we will delve into a mathematical problem that requires critical thinking and analytical skills to arrive at the correct solution. The problem revolves around Sean and Coleen, who are raking leaves in their yard, and we need to determine how long it would take each of them to clear the yard working alone.
The Problem
Sean and Coleen 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 Coleen working alone. We need to find out how long it would take each of them to clear the yard working alone.
Step 1: Define the Variables
Let's define the variables:
- x = time it would take Coleen to clear the yard working alone
- x + 20 = time it would take Sean to clear the yard working alone
Step 2: Write the Equation
Since Sean and Coleen can clear the yard together in 24 minutes, we can write the equation:
1/x + 1/(x + 20) = 1/24
Step 3: Solve the Equation
To solve the equation, we can start by finding a common denominator:
((x + 20) + x)/((x + 20)x) = 1/24
Combine like terms:
(2x + 20)/((x + 20)x) = 1/24
Cross-multiply:
24(2x + 20) = (x + 20)x
Expand and simplify:
48x + 480 = x^2 + 20x
Rearrange the equation:
x^2 - 28x - 480 = 0
Step 4: Solve the Quadratic Equation
We can solve the quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = -28, and c = -480:
x = (28 ± √((-28)^2 - 4(1)(-480))) / 2(1)
x = (28 ± √(784 + 1920)) / 2
x = (28 ± √(2704)) / 2
x = (28 ± 52) / 2
We have two possible solutions:
x = (28 + 52) / 2 = 40
x = (28 - 52) / 2 = -12
Since time cannot be negative, we discard the solution x = -12.
Step 5: Find the Time it Would Take Sean to Clear the Yard Working Alone
Now that we have found the time it would take Coleen to clear the yard working alone (x = 40 minutes), we can find the time it would take Sean to clear the yard working alone:
x + 20 = 40 + 20 = 60 minutes
Conclusion
In this article, we have solved a mathematical problem that required critical thinking and analytical skills. We defined the variables, wrote the equation, solved the equation, and finally found the time it would take each of them to clear the yard working alone. The time it would take Coleen to clear the yard working alone is 40 minutes, and the time it would take Sean to clear the yard working alone is 60 minutes.
Key Takeaways
- Problem-solving is an essential skill that we all need to develop.
- Mathematics is an integral part of our daily lives.
- Critical thinking and analytical skills are necessary to arrive at the correct solution.
- We can solve quadratic equations using the quadratic formula.
- Time cannot be negative.
Final Thoughts
Q: What is the problem about?
A: The problem is about Sean and Coleen, who are raking leaves in their yard. We need to determine how long it would take each of them to clear the yard working alone.
Q: What is the given information?
A: The given information is that Sean and Coleen can clear the yard together in 24 minutes. Additionally, it would take Sean 20 minutes longer to clear the yard than it would take Coleen working alone.
Q: How do we solve the problem?
A: We solve the problem by defining the variables, writing the equation, solving the equation, and finally finding the time it would take each of them to clear the yard working alone.
Q: What are the steps to solve the problem?
A: The steps to solve the problem are:
- Define the variables
- Write the equation
- Solve the equation
- Solve the quadratic equation
- Find the time it would take Sean to clear the yard working alone
Q: What is the time it would take Coleen to clear the yard working alone?
A: The time it would take Coleen to clear the yard working alone is 40 minutes.
Q: What is the time it would take Sean to clear the yard working alone?
A: The time it would take Sean to clear the yard working alone is 60 minutes.
Q: Why do we need to solve the quadratic equation?
A: We need to solve the quadratic equation because it represents the time it would take Coleen to clear the yard working alone. The quadratic equation is x^2 - 28x - 480 = 0, and we solve it using the quadratic formula.
Q: What is the quadratic formula?
A: The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation.
Q: Why do we discard the negative solution?
A: We discard the negative solution because time cannot be negative.
Q: What is the key takeaway from this problem?
A: The key takeaway from this problem is that problem-solving requires critical thinking and analytical skills. We need to define the variables, write the equation, solve the equation, and finally find the solution.
Q: What is the final thought?
A: The final thought is that mathematics is a fascinating subject that requires critical thinking and analytical skills. We hope that this article has inspired you to develop your problem-solving skills and to appreciate the beauty of mathematics.
Additional Questions and Answers
Q: What if the time it would take Sean to clear the yard working alone was 30 minutes?
A: If the time it would take Sean to clear the yard working alone was 30 minutes, then the time it would take Coleen to clear the yard working alone would be 10 minutes.
Q: What if the time it would take Sean and Coleen to clear the yard together was 30 minutes?
A: If the time it would take Sean and Coleen to clear the yard together was 30 minutes, then the time it would take Coleen to clear the yard working alone would be 20 minutes, and the time it would take Sean to clear the yard working alone would be 40 minutes.
Q: What if the time it would take Sean to clear the yard working alone was 50 minutes?
A: If the time it would take Sean to clear the yard working alone was 50 minutes, then the time it would take Coleen to clear the yard working alone would be 30 minutes.
Conclusion
In this article, we have answered frequently asked questions about the problem of Sean and Coleen raking leaves in their yard. We have provided step-by-step solutions to the problem and have discussed the key takeaways and final thoughts. We hope that this article has inspired you to develop your problem-solving skills and to appreciate the beauty of mathematics.