Review Rational functions

Question

When Amy and Shaylee work together, a job take 6 hours. When Shaylee does the same job alone, it takes 5 fewer hours than when Amy does the job alone. How long will it take Amy to do the job herself?

Let x=the number of hours for Amy to do the job alone.

Answer 1

Hey there! Let's set this up by letting [ Amy's time = A ] and [Shaylee's time = A - 5], to represent Shaylee taking 5 hours longer to complete the same task.

We'll set up the equation to represent the tag team completing the task together for a total of 6 hours

1/A + 1/(A-5) = 1/6 (Amy's contribution + Shaylee's contribution = 1 project consuming 6 hours)

We can multiply the entire equation by our Common Denominator, which is A*(A-5)*6, where we get:

6(A-5) + 6A = A(A-5)

Then, we distribute:

6A - 30 + 6A = A² -5A

Now, combine like terms:

12A - 30 = A² - 5A

Next, we can move our variables and constants to one side:

A²- 17A + 30 = 0

Since this is a quadratic, we can solve for the "zeros" by factoring:

(A -15)(A-2) = 0 , then set each factor for zero.

A=2, A=15.

Refer back to the original question, as you have two answers and in the context of the question, you only have one "TRUE" answer! Shaylee completes her work 5 fewer hours than Amy, so it doesn't make sense for Amy to complete her work in 2 hours.

Amy's time= 15, Cheers!

Answer 2

Let x = Amy's time. Then x - 5 = Shaylee's time

1/x + 1/(x-5) = 1/6

6(x-5) + 6x = x(x-5)

12x - 30 = x² - 5x

x² - 17x + 30 = 0

Solve for x.