Question

Help with rate of time, please?

Answer 1

Jordan is a manager of a car dealership. He has 3 professional car washers to clean the entire lot of cars, Jennifer, Arianna, and Matthew. Jennifer can wash all the cars in 14 hours. Arianna can wash all the cars in 11 hours. Matthew is new to the car dealership, so no one knows how long it will take him.

Jordan assigns all of them to wash the cars together. Explain to Jordan how this task can tell him how long it would take Matthew to complete the task if he worked by himself. Use complete sentences.

Answer 2

@johnweldon1993

Answer 3

This is as far as I've gotten...

As stated, Jennifer can was all the cars in 14 hours, so her work speed is 1/14 of the job per hour. Arianna can wash all the cars in 11 hours, so her work speed is 1/11 of the job per hour. So 1/14 + 1/11 + 1/M = 1/E (M is Matthew's hours, E is how long it will take for all of them to get the job done.).

Answer 4

Okay so that makes sense, sorry btw working on another question as well, so we have

\[\large \frac{1}{14} + \frac{1}{11} + \frac{1}{M} = \frac{1}{E}\]

Answer 5

Right! So don't I multiply 14 and 11?

Answer 6

So it looks like all we need to do is solve for M

So I would start by finding the least common denominator...yes 14*11 = ?

Answer 7

154!

Answer 8

Okay, so we have *putting them over the common denominator

\[\large \frac{11 + 14}{154} + \frac{1}{M} = \frac{1}{E}\]

Now, I would get that \(\large \frac{1}{M}\) by itself

Answer 9

So 25/154+1/M=1/E

Answer 10

Yes, and now subtract that \(\large \frac{25}{154}\) from both sides to isolate \(\large \frac{1}{M}\)

Answer 11

So 1/M = 1/E - (25/154)

Answer 12

(Sorry for not using the Equation function on here)

Answer 13

Right...now we need to, again, get a common denominator on that right side

*and no problem :)

Answer 14

What would that be?

Answer 15

Well if we have

\[\large \frac{1}{M} = \frac{1}{E} - \frac{25}{154}\]

Focusing on

\[\large \frac{1}{E} - \frac{25}{154}\]

What would be a common denominator? Well, if we multiply 154 and E, what do we get? We get 154E right? That is our common denominator

But how do we get it? It looks like if we multiply that first fraction by \(\large \frac{154}{154}\) and that second fraction by \(\large \frac{E}{E}\) we can get that result...what do we get after we do that?

Answer 16

Uhhh

Answer 17

Would that cancel things out?

Answer 18

So remember how we had before the

\[\large \frac{1}{14} + \frac{1}{11}\]

And if we multiplied 14 times 11 we got the 154...

\[\large (\frac{11}{11})\frac{1}{14} + (\frac{14}{14})\frac{1}{11} = \frac{11}{154 } + \frac{14}{154} = \frac{11 + 14}{154} = \frac{25}{154}\]

that was our whole process to find that common denominator before

Answer 19

Now here, the same thing

we have

\[\large \frac{1}{E} - \frac{25}{154}\]

When we multiply the E and the 154...we get 154E which is our common denominator, and to get it we

\[\large (\frac{154}{154})\frac{1}{E} - (\frac{E}{E})\frac{25}{154}\]

we then get

\[\large \frac{154}{154E} - \frac{25E}{154E}\]

so now we can putit over that common denominator

\[\large \frac{154 - 25E}{154E}\]

Answer 20

Did that make sense? :/

Answer 21

Yes!!

Answer 22

Okay good! :D

Now remember we ignored that \(\large \frac{1}{M}\) from before...so we have

\[\large \frac{1}{M} = \frac{154 - 25E}{154E}\] right?

How would we solve that for M?

Answer 23

Umm, cancel out 154?

Answer 24

Not quite, so hmm, think about it like this....if you flip both sides...flip both the fractions...what do we have then?

Answer 25

I'm so beyond brain dead right now.

Answer 26

If we want to ACTUALLY think about this correctly I guess, we can do it algebraically

\[\large \frac{1}{M} = \frac{154 - 25E}{154E}\]

We can cross multiply here

Answer 27

Okay so 154-25 / 154 E = 1/M