Liliana's Mom Can Do A Job In 6 Minutes Less Than Liliana. If Together They Can Complete A Job In A Certain Number Of Minutes, How Long Does It Take Liliana's Mom To Do The Job By Herself?
Introduction
In this article, we will delve into a fascinating mathematical problem that revolves around the concept of time and efficiency. We will explore the scenario where Liliana's mom can complete a job in 6 minutes less than Liliana, and together they can finish the job in a certain number of minutes. Our goal is to determine how long it takes Liliana's mom to do the job by herself.
The Problem
Let's assume that Liliana takes x minutes to complete the job. Since Liliana's mom can do the job in 6 minutes less than Liliana, she will take x - 6 minutes to complete the job.
When they work together, their combined work rate is the sum of their individual work rates. Let's denote the time it takes for them to complete the job together as t minutes. We can set up the following equation based on their combined work rate:
1/x + 1/(x - 6) = 1/t
Simplifying the Equation
To simplify the equation, we can find a common denominator for the fractions on the left-hand side. The common denominator is x(x - 6).
1/x + 1/(x - 6) = (x - 6 + x) / (x(x - 6)) = (2x - 6) / (x^2 - 6x)
Now, we can rewrite the equation as:
(2x - 6) / (x^2 - 6x) = 1/t
Cross-Multiplying
To eliminate the fractions, we can cross-multiply the equation:
2x - 6 = (x^2 - 6x) / t
Multiplying Both Sides by t
To get rid of the fraction on the right-hand side, we can multiply both sides of the equation by t:
2xt - 6t = x^2 - 6x
Rearranging the Terms
Now, let's rearrange the terms to get a quadratic equation in terms of x:
x^2 - 2xt - 6x + 6t = 0
Solving the Quadratic Equation
To solve the quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = -2t, and c = -6 + 6t.
x = (2t ± √((-2t)^2 - 4(1)(-6 + 6t))) / 2(1) = (2t ± √(4t^2 + 24 - 24t)) / 2 = (2t ± √(4t^2 - 24t + 24)) / 2
Simplifying the Expression
To simplify the expression under the square root, we can factor out a 4:
4t^2 - 24t + 24 = 4(t^2 - 6t + 6)
Now, we can rewrite the expression as:
x = (2t ± √(4(t^2 - 6t + 6))) / 2 = (2t ± 2√(t^2 - 6t + 6)) / 2 = t ± √(t^2 - 6t + 6)
Finding the Time it Takes for Liliana's Mom to Do the Job
Since we want to find the time it takes for Liliana's mom to do the job by herself, we can set x = x - 6. This means that Liliana's mom takes x - 6 minutes to complete the job.
Substituting x = x - 6 into the expression for x, we get:
x - 6 = t ± √(t^2 - 6t + 6)
Simplifying the Expression
To simplify the expression, we can substitute x = x - 6 into the expression for t:
t = x - 6
Now, we can rewrite the expression as:
x - 6 = (x - 6) ± √((x - 6)^2 - 6(x - 6) + 6)
Simplifying the Expression
To simplify the expression, we can expand the square and combine like terms:
(x - 6)^2 - 6(x - 6) + 6 = x^2 - 12x + 36 - 6x + 36 + 6 = x^2 - 18x + 78
Now, we can rewrite the expression as:
x - 6 = (x - 6) ± √(x^2 - 18x + 78)
Finding the Time it Takes for Liliana's Mom to Do the Job
Since we want to find the time it takes for Liliana's mom to do the job by herself, we can set x = x - 6. This means that Liliana's mom takes x - 6 minutes to complete the job.
Substituting x = x - 6 into the expression for x, we get:
x - 6 = (x - 6) ± √(x^2 - 18x + 78)
Simplifying the Expression
To simplify the expression, we can expand the square and combine like terms:
(x - 6)^2 - 6(x - 6) + 6 = x^2 - 12x + 36 - 6x + 36 + 6 = x^2 - 18x + 78
Now, we can rewrite the expression as:
x - 6 = (x - 6) ± √(x^2 - 18x + 78)
Conclusion
In this article, we explored a fascinating mathematical problem that revolves around the concept of time and efficiency. We found that the time it takes for Liliana's mom to do the job by herself is x - 6 minutes, where x is the time it takes for Liliana to complete the job.
We also derived an expression for the time it takes for Liliana's mom to do the job by herself, which is:
x - 6 = (x - 6) ± √(x^2 - 18x + 78)
This expression can be used to find the time it takes for Liliana's mom to do the job by herself, given the time it takes for Liliana to complete the job.
Final Answer
Introduction
In our previous article, we explored a fascinating mathematical problem that revolves around the concept of time and efficiency. We found that the time it takes for Liliana's mom to do the job by herself is x - 6 minutes, where x is the time it takes for Liliana to complete the job.
In this article, we will answer some of the most frequently asked questions about the problem and provide additional insights into the solution.
Q&A
Q: What is the time it takes for Liliana's mom to do the job by herself?
A: The time it takes for Liliana's mom to do the job by herself is x - 6 minutes, where x is the time it takes for Liliana to complete the job.
Q: How do we find the time it takes for Liliana's mom to do the job by herself?
A: To find the time it takes for Liliana's mom to do the job by herself, we can use the expression:
x - 6 = (x - 6) ± √(x^2 - 18x + 78)
Q: What is the significance of the expression √(x^2 - 18x + 78)?
A: The expression √(x^2 - 18x + 78) represents the difference in time between Liliana and her mom. When this expression is positive, it means that Liliana's mom is faster than Liliana. When this expression is negative, it means that Liliana is faster than her mom.
Q: Can we simplify the expression √(x^2 - 18x + 78)?
A: Yes, we can simplify the expression √(x^2 - 18x + 78) by factoring the quadratic expression:
x^2 - 18x + 78 = (x - 9)^2 - 9
Now, we can rewrite the expression as:
√(x^2 - 18x + 78) = √((x - 9)^2 - 9)
Q: What is the significance of the expression √((x - 9)^2 - 9)?
A: The expression √((x - 9)^2 - 9) represents the difference in time between Liliana and her mom, taking into account the fact that Liliana's mom is 6 minutes faster than Liliana.
Q: Can we find a specific value for the time it takes for Liliana's mom to do the job by herself?
A: Yes, we can find a specific value for the time it takes for Liliana's mom to do the job by herself by substituting a specific value for x into the expression:
x - 6 = (x - 6) ± √(x^2 - 18x + 78)
For example, if we substitute x = 12 into the expression, we get:
12 - 6 = (12 - 6) ± √(12^2 - 18(12) + 78) = 6 ± √(144 - 216 + 78) = 6 ± √6
Q: What is the final answer for the time it takes for Liliana's mom to do the job by herself?
A: The final answer for the time it takes for Liliana's mom to do the job by herself is 6 ± √6 minutes.
Conclusion
In this article, we answered some of the most frequently asked questions about the problem and provided additional insights into the solution. We found that the time it takes for Liliana's mom to do the job by herself is x - 6 minutes, where x is the time it takes for Liliana to complete the job.
We also derived an expression for the time it takes for Liliana's mom to do the job by herself, which is:
x - 6 = (x - 6) ± √(x^2 - 18x + 78)
This expression can be used to find the time it takes for Liliana's mom to do the job by herself, given the time it takes for Liliana to complete the job.
Final Answer
The final answer is: