Question

Hayden is a manager at a landscaping company. He has 3 workers to landscape an entire park, Cody, Kaitlin, and Joseph. Cody can complete the project in 8 hours. Kaitlyn can complete the project in 6 hours. Joseph is new, so no one knows how long it will take him.

Hayden assigns all of them to complete the park together. Explain to Hayden how this project can tell him how long it would take Joseph to complete the project if he worked by himself. Use complete sentences.

Answer 1

Good morning, JackApple!
It helps me to indicate the appropriate units of measurement when I attempt a "rate" problem like this one.

The length of time required to do a given job can be found by dividing "1 job" by the rate, (1 job / 8 hours). So, if it were Cody doing the job alone, The length of time required for him to complete the job would be (1 job) / (1 job / 8 hours) = 8 hours, right?

Now, if Kaitlin comes along and offers to help, we add together their rates:

New, net rate of job completion:\[\frac{ 1 job }{ \frac{ 1 job }{ 8 hrs } +\frac{ 1 job }{ 6 hrs }}=\frac{ 1 job }{ (1/8 +1/6){ job/hours }{ } }=\frac{ 1 job }{ 14/48 job/hrs}=48/14 hours.\]

This is certainly not the "answer" to the question you 've posted, but does illustrate how to perform the calculations.

We don't know how many hours it takes our friend Joe to complete the whole job on his own. Let x=number of hours it requires him to do so).

Then the three people working together can complete the job in \[\frac{ 1 }{ \frac{ 1 }{ 8 }+\frac{ 1 }{ 6 }+\frac{ 1 }{ x} }hours.\]

Answer 2

Try to build upon this background material to explain how we'd determine "x hours."

Answer 3

Hint: We don't know how long it would take this team of 3 people to complete the job working together. To simplify matters, represent this length of time by h (hours) and then equate the expression immediately above to (1 job / h hours); treat h as a constant and then find x in terms of h.