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?

Question

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?

anonymous · Answer

@imer, I haave the equation but I don't know how to solve because there are 2 variables... 

1/11+1/14+1/m=1/x

mathmale · Answer

I'd prefer you define your variables, m and x, beforehand, so that we'll have a better idea whether or not your results make sense.

But, focusing on your 1/11+1/14+1/m=1/x, or $$\frac{ 1 }{ 11 }+\frac{ 1 }{ 14 }+\frac{ 1 }{ m }=\frac{ 1 }{ x }$$ you'll see that you have four different fractions with four different denominators.  Remember the concept of "lowest common denominator?"  What is your LCD here?  How are you going to use it to solve this equation for either m or x?

Note that you'll need 2 equations to solve for 2 unknowns.

anonymous · Answer

the lcd is 154?

anonymous · Answer

$$\frac{ 1 }{ M }+\frac{ 25 }{ 154 }=\frac{ 1 }{ x }$$ ?

anonymous · Answer

What do I do now? @mathmale

anonymous · Answer

it says its possible.. :/

mathmale · Answer

Regarding "the lcd is 154?"  Your LCD has to include all four quantities from those four denominators (at least in this case).  We can't set aside the m and x because they don't yet have numeric values.

Since the four denoms. are all different, the LCD is simply their product:  154mx.

anonymous · Answer

oh ok so all the denominators have 154mx?

mathmale · Answer

I'd suggest you multiply your entire equation (enclosed in parentheses) by this LCD:  154mx:$$154mx(\frac{ 1 }{ 11 }+\frac{ 1 }{ 14 }+\frac{ 1 }{ m }=\frac{ 1 }{ x })$$

mathmale · Answer

Please give that a try.  Be certain you know what you're solving for:  m or x or ... ??

anonymous · Answer

? how would I do that? sorry im not very good with this

mathmale · Answer

Multiply each term within the parentheses by 154mx.  (If you do that you're applying the "distributive property of multiplication.")

$$154mx(\frac{ 1 }{ 11 }+\frac{ 1 }{ 14 }+\frac{ 1 }{ m }=\frac{ 1 }{ x })$$

mathmale · Answer

becomes...$$\frac{ 154mx }{ 11 }=14mx$$...and three more terms from that equation inside parentheses.

anonymous · Answer

$$\frac{ 154mx }{ 11 }+\frac{ 154mx }{ 14 }+\frac{ 154mx }{ m }=\frac{ 154mx }{ x }$$

mathmale · Answer

Then you'd simplify each of the four terms:

14mx + 11mx +   ?    =   ?

anonymous · Answer

14mx+11mx+m=x?

mathmale · Answer

so far, so good.  However (as I've mentioned before, this one equation is in two variables, and so, without further info, it's hard or impossible to determine the values of both m and x.  

I've gone forward with this because you've said, "I have the equation."

anonymous · Answer

I have the equation $$\frac{ 1 }{ 11 }+\frac{ 1 }{ 14 }+\frac{ 1 }{ m }=\frac{ 1 }{ x }$$

mathmale · Answer

We could let ' m ' be a ' parameter '.  ' m ' represents the # of hours it'd take Matthew to wash all of the cars, working alone.  You could choose various values for ' m ' and then calculate the number of hours it'd take all 3 young people working together to wash all of the cars.  

OR, you could leave ' m ' as is and solve the resulting equation for x (the number of hours it'd take all three young people to wash all of the cars).

anonymous · Answer

the second choice :)

mathmale · Answer

All right.  How are you going to solve$$\frac{ 1 }{ 11 }+\frac{ 1 }{ 14 }+\frac{ 1 }{ m }=\frac{ 1 }{ x }$$ for x?  Hint:  You've already done most of the work:$$14mx + 11mx + m = x$$

anonymous · Answer

combine like terms so $$25mx+m=$$

anonymous · Answer

=x*

anonymous · Answer

I don't know what to do next though :/ @mathmale

mathmale · Answer

Group all the x-terms together on the left side of your equation and everything else on the right side.

mathmale · Answer

Show your result.

Then, factor x out of the terms on the left side.

Last, solve this equation for x.

anonymous · Answer

sorry sir, I was eating breakfast @mathmale

anonymous · Answer

@mathmale  how do I do this?

mathmale · Answer

Let's step back a bit and start over:

Solve $$\frac{ 1 }{ 11 }+\frac{ 1 }{ 14 }+\frac{ 1 }{ m }=\frac{ 1 }{ x }~for~x.$$Multiply every one of these four terms by the LCD (154mx):$$154mx(\frac{ 1 }{ 11 }+\frac{ 1 }{ 14 }+\frac{ 1 }{ m }=\frac{ 1 }{ x}),$$...obtaining$$14mx+11mx+154x=154m$$

mathmale · Answer

Factor x out of the left side:  x(25m+154)=154m.

mathmale · Answer

Now, please solve that latest equation for x.  x represents the number of hours it will take for all three young people to wash all of the cars, and will be measured in hours.

mathmale · Answer

sorry, but I need to get off the 'Net now.  Perhaps I'l see you later today.  Good luck!

anonymous · Answer

thank you sir :)