Select The Correct Answer.It Takes Greg And Nikki A Total Of 40 Minutes To Paint A Room If They Work Together. If Greg Works Alone, He Takes 18 Minutes Longer To Paint The Room Than Nikki Takes. When $x$ Represents The Number Of Minutes It
Solving a Real-World Problem: A Math Puzzle
In this article, we will delve into a real-world problem that involves solving a system of equations. The problem revolves around Greg and Nikki, who are painting a room together. We are given that it takes them a total of 40 minutes to paint the room if they work together. Additionally, we know that Greg takes 18 minutes longer to paint the room alone than Nikki takes. Our goal is to determine the number of minutes it takes each of them to paint the room individually.
Let's break down the problem and understand what information we have been given. We know that Greg and Nikki can paint the room together in 40 minutes. This means that their combined rate of work is 1 room per 40 minutes. We also know that Greg takes 18 minutes longer to paint the room alone than Nikki takes. Let's represent the number of minutes it takes Nikki to paint the room alone as x. Then, the number of minutes it takes Greg to paint the room alone is x + 18.
To solve this problem, we need to set up a system of equations. We know that the combined rate of work of Greg and Nikki is 1 room per 40 minutes. This can be represented by the equation:
1/40 = 1/Greg's rate + 1/Nikki's rate
We also know that Greg takes x + 18 minutes to paint the room alone, and Nikki takes x minutes to paint the room alone. This can be represented by the equation:
Greg's rate = 1/(x + 18) Nikki's rate = 1/x
Now that we have set up the system of equations, we can solve for x. We can start by substituting the expressions for Greg's rate and Nikki's rate into the first equation:
1/40 = 1/(x + 18) + 1/x
To simplify this equation, we can find a common denominator:
1/40 = (x + 18 + x) / (x(x + 18))
We can then cross-multiply:
x(x + 18) = 40(x + 18 + x)
Expanding the right-hand side of the equation, we get:
x^2 + 18x = 80x + 720
Subtracting 80x from both sides, we get:
x^2 - 62x - 720 = 0
This is a quadratic equation, and we can solve it using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = -62, and c = -720. Plugging these values into the quadratic formula, we get:
x = (62 ± √((-62)^2 - 4(1)(-720))) / 2(1)
Simplifying the expression under the square root, we get:
x = (62 ± √(3844 + 2880)) / 2
x = (62 ± √6724) / 2
x = (62 ± 82) / 2
We have two possible solutions for x:
x = (62 + 82) / 2 = 72 x = (62 - 82) / 2 = -10
Since the number of minutes it takes to paint the room cannot be negative, we can discard the solution x = -10. Therefore, the number of minutes it takes Nikki to paint the room alone is x = 72.
Now that we know the number of minutes it takes Nikki to paint the room alone, we can find the number of minutes it takes Greg to paint the room alone. We know that Greg takes 18 minutes longer to paint the room alone than Nikki takes. Therefore, the number of minutes it takes Greg to paint the room alone is x + 18 = 72 + 18 = 90.
In this article, we solved a real-world problem that involved solving a system of equations. We were given that it takes Greg and Nikki a total of 40 minutes to paint a room if they work together, and that Greg takes 18 minutes longer to paint the room alone than Nikki takes. We set up a system of equations and solved for the number of minutes it takes each of them to paint the room individually. We found that the number of minutes it takes Nikki to paint the room alone is 72, and the number of minutes it takes Greg to paint the room alone is 90.
The final answer is: Nikki takes 72 minutes to paint the room alone, and Greg takes 90 minutes to paint the room alone.
Q&A: Solving the Math Puzzle
In our previous article, we solved a real-world problem that involved solving a system of equations. We were given that it takes Greg and Nikki a total of 40 minutes to paint a room if they work together, and that Greg takes 18 minutes longer to paint the room alone than Nikki takes. We set up a system of equations and solved for the number of minutes it takes each of them to paint the room individually. In this article, we will answer some frequently asked questions about the problem and its solution.
A: The combined rate of work of Greg and Nikki is 1 room per 40 minutes.
A: To set up the system of equations, we need to represent the combined rate of work of Greg and Nikki as an equation. We know that their combined rate of work is 1 room per 40 minutes, so we can write the equation:
1/40 = 1/Greg's rate + 1/Nikki's rate
We also know that Greg takes x + 18 minutes to paint the room alone, and Nikki takes x minutes to paint the room alone. This can be represented by the equation:
Greg's rate = 1/(x + 18) Nikki's rate = 1/x
A: To solve the system of equations, we can substitute the expressions for Greg's rate and Nikki's rate into the first equation:
1/40 = 1/(x + 18) + 1/x
We can then simplify the equation and solve for x.
A: The value of x is 72. This represents the number of minutes it takes Nikki to paint the room alone.
A: To find Greg's time, we can add 18 minutes to Nikki's time. Since Nikki takes 72 minutes to paint the room alone, Greg takes 72 + 18 = 90 minutes to paint the room alone.
A: The final answer is that Nikki takes 72 minutes to paint the room alone, and Greg takes 90 minutes to paint the room alone.
Q: What if Greg and Nikki work at different rates?
A: If Greg and Nikki work at different rates, we would need to set up a different system of equations. However, the basic steps of solving the system of equations would remain the same.
Q: Can we use other methods to solve the problem?
A: Yes, we can use other methods to solve the problem, such as using a graphing calculator or a computer algebra system. However, the method we used in this article is a straightforward and easy-to-understand approach.
Q: What if the problem is more complex?
A: If the problem is more complex, we may need to use more advanced mathematical techniques, such as using matrices or differential equations. However, the basic principles of solving a system of equations remain the same.
In this article, we answered some frequently asked questions about the math puzzle we solved in our previous article. We covered topics such as setting up the system of equations, solving the system of equations, and finding the value of x. We also discussed some frequently asked questions and provided additional information and resources for further learning.