Question

Can someone please check to see if I did this word problem correctly?

Hayden is a manager at a landscaping company. He has 3 workers to landscape an entire park, Cody, Kaitlyn, 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.

So far, I have an answer of (6x)/(48x) + (8x)/(48x) = (48)/(48x), which I've simplified down to (6x + 8x)/(48x) = (48)/(48x). Do I need to combine the numerator of the first part and then solve, or is there something I'm missing?

Answer 1

@ranga @phi @hero

Answer 2

I think you're on the right track, but doing it in an unnecessarily complicated fashion.

Answer 3

Figure out the rate for each worker:

Cody needs 8 hours, so he does 1 project/8 hours = 1/8 project/hour
Kaitlyn needs 6 hours, so she does 1/6 project/hour
Joseph is a mystery, so he does 1/x project/hour.

Now, is there some more information you didn't provide?

Answer 4

I could imagine the problem saying it takes some number of hours for the whole group to do the project, so how long would Joseph take to do it himself?

Answer 5

I do apologize, there is. It's my first time on the site and it keeps posting without me clicking the post button. It also says
"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."

Answer 6

I started with (1)/(8) + (1)/(6) = (1)/(x), but I'm not sure if that's right or not.

Answer 7

Okay, so here's what you do:

add the rates together:

\[\frac{1}{8} + \frac{1}{6} + \frac{1}{x}\]You have one project, so the time it will take them to do it is \[\frac{1}{\frac{1}{8} + \frac{1}{6} + \frac{1}{x}} = \text{ total time}\]

Answer 8

If you know the total time, you can solve the equation for \(x\) to find out how many hours Joseph would take to do it by himself.

Answer 9

The last fraction you typed I'm unsure of how to solve. I've not learned anything about how to solve something like that before.

Answer 10

Okay, pick a number for the total time, and we'll solve it.

Answer 11

10

Answer 12

No, it should be shorter than any of them will take to do it by themselves, unless Joseph is a really poor worker :-)

Answer 13

Oh, I thought you said for Joseph. I'm sorry about that!

Answer 14

Is 4 alright?

Answer 15

Just the two known workers together:

\[\frac{1}{\frac{1}{8} + \frac{1}{6}} = \frac{1}{\frac{6*1}{6*8} + \frac{8*1}{8*6}} = \frac{1}{\frac{14}{48}} = \frac{48}{14}*1 = \frac{24}{7} = 3\frac{3}{7}\]

Answer 16

How about 2 hours?

\[\frac{1}{\frac{1}{8}+\frac{1}{6}+\frac{1}x} = 2\]Why don't you multiply both sides by the denominator of the fraction?

Answer 17

that gives us \[1 = 2(\frac{1}{8} + \frac{1}{6} + \frac{1}{x})\]or\[\frac{1}{2} = \frac{1}{8}+\frac{1}{6}+\frac{1}{x}\]

Answer 18

Forgive me, but I'm not sure what you mean by that. I thought that I needed to multiply all parts of the denominator to find a LCD.

Answer 19

I calculated x = 24/7. Is that correct?

Answer 20

For the original problem, not the one you just typed.

Answer 21

Well, 24/7 was the total time it would take the two working together, I didn't use x anywhere in that problem.

Answer 22

Hmm, I'm not sure where I'm going wrong.

Answer 23

I think I know where I went wrong. Even when I re-assessed the problem I still had (1)/(x) as = (1)/(x) instead of + (1)/(x). Now I have an answer of (7)/(24)

Answer 24

Wait, what problem are you working? The original one you posted?

Answer 25

Yes. I need to find how long it would take Joseph to complete the project himself, which is why I'm running into issues.

Answer 26

I wasn't taught how to find separate times, only how to find one combined time.

Answer 27

Okay, but they didn't tell you how long the whole project takes when he works with them, so you can't actually find the answer.

Answer 28

I think the problem is asking you to describe HOW you would find how long it would take Joseph to do it himself, if you knew how long it took the whole group to do it.

Answer 29

"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."

This project will take a certain amount of time (call it T hours) for all three of them working together to complete. Using the known rates, and the new info of T hours, we can solve for how long it takes Joseph to do the job on his own.

Answer 30

Some background: we use
rate * time = 1 job where rate is measured in "jobs per hour"
For example, Cody can complete the project in 8 hours, means Cody works at a rate of 1 job per 8 hours, written ⅛ jobs/hour

using the formula rate * time, ⅛ * t = 1 job, and solving for t, we get t= 8 hours (just as we should)

Answer 31

If there are 3 people (and they don't slow each other down...) we can add up there rates

Cody's rate is ⅛, Kaitlyn's rate is 1/6, and Joe (who will take x hours to finish the job on his own), works at a rate of 1 job/x hours = 1/x

the total rate is
\[ \frac{1}{8}+ \frac{1}{6}+ \frac{1}{x} \]

and the equation, using T for how long it took them to do the job:
\[ \left(\frac{1}{8}+ \frac{1}{6}+ \frac{1}{x}\right)T=1 \]

Answer 32

The equation is a bit messy, but can be solved for x.
First, we can add ⅛ + 1/6 (using a common denominator of 24 (48 also works))
\[\left(\frac{1}{8}+ \frac{1}{6}+ \frac{1}{x}\right)T=1 \\
\left(\frac{3}{24}+\frac{4}{24}+ \frac{1}{x}\right)T=1 \\
\left(\frac{7}{24}+ \frac{1}{x}\right)T=1 \\
\]
use a common denominator of 24x to add the fractions
\[ \left(\frac{7}{24}\cdot \frac{x}{x}+ \frac{1}{x}\cdot\frac{24}{24}\right)T=1 \\
\left(\frac{7x+24}{24x}\right)T=1
\]
divide both sides by T
\[ \left(\frac{7x+24}{24x} \right)=\frac{1}{T} \]
multiply booth sides by 24x
\[ 7x +24= \frac{24x}{T} \]
subtract 7x from both sides
\[ 24=\frac{24x}{T} -7x \]
or
\[ \frac{24x}{T} -7x = 24\]
factor out x
\[ \left(\frac{24}{T} -7\right) x = 24 \\
\left(\frac{24}{T} -\frac{7T}{T}\right) x = 24 \\
\left(\frac{24-7T}{T} \right) x = 24
\]

Answer 33

finally, multiply by the inverse (flip the fraction) to get
\[ x= \frac{24T}{24-7T} \]

we can write this in different ways, but that is as good as any of them.

This equation says, "if we know that it takes T hours for all 3 to do the job, then Joe will take that many hours working alone"

Answer 34

@phi Nice writeup!