Question

Jordan is a manager of a car dealership. He has two professional car washers, Matthew and Arianna, to clean the entire lot of cars. Matthew can wash all the cars in 14 hours. Arianna can wash all the cars in 11 hours. Jordan wants to know how long it will take them to wash all the cars in the lot if they work together.

Write an equation and solve for the time it will take Matthew and Arianna to wash all the cars together. Explain each step.

Answer 1

This might help http://openstudy.com/study#/updates/55b95fc1e4b0adef802b9a16

Answer 2

\[\frac{1}{14}+\frac{1}{11}=\frac{1}{x} \]

Answer 3

1 can be interpreted as the job, the project, the whole thing or all of the cars.

Answer 4

one way to think about this problem is use the idea of
rate* time = amount of work
(a bit like rate*time = distance for figuring how far you go at a certain speed)

in these problems, we call the "amount of work" 1 job
i.e. 1 without the label.

rate is the ratio (or fraction) jobs per hour

for example,
Matthew can wash all the cars in 14 hours.
means Matt does the job in 14 hours or (as a fraction) 1 job per 14 hours or 1 job/14 hours

Answer 5

Say we wanted to know how long it takes Matt to do 1 job. Using
rate * time = 1 job
\[ \frac{1 \ job}{14 \ hours} \cdot t = 1 \ job\]
multiply both sides by 14 hours/1 job and simplify to get
\[ t = 14 \ hours\]
(Which we knew... but this shows how to do the math)

Answer 6

with two people, each with their own rate
Matt's rate 1 job / 14 hours
Arianna's rate 1 job/ 11 hours
then we can add their rates, and multiply by the time ("t" because we don't know the number)
\[ \left(\frac{1\ job}{14\ hours}+ \frac{1\ job}{11\ hours}\right)t= 1\ job\]

Answer 7

You can add the two fractions by using a common denominator of 11*14= 154
can you add
\[ \frac{1}{14}+ \frac{1}{11} \] ?

Answer 8

2/25

Answer 9

So it's take them 5 hours?

Answer 10

oh, I think I see what you did.

It's more complicated

Answer 11

you need the bottoms of both fractions to be the same number (154 in this case)
to get 154 in the first fraction we multiply the bottom by 11, and to keep things from changing we also multiply its top by 11, like this:
\[ \frac{1}{14} \cdot \frac{11}{11 } \]
(the idea is 11/11 is 1 so it does not change the value, just how it "looks")
anyway, multiply top times top, and bottom times top.
can you do that ?

Answer 12

***anyway, multiply top times top, and bottom times bottom can you do that ?

Answer 13

so 1 times 14 and 11 times 11?

Answer 14

1 * 11 for the tops
and 14*11 for the bottom

Answer 15

top times top
bottom times bottom

Answer 16

Oh ops. so 11/154

Answer 17

ok, now we do the 2nd fraction
1/11
we want its bottom to be 154
what do we multiply top and bottom by ? any idea ?

Answer 18

multiple it by 14

Answer 19

yes, we multiply the 2nd fraction by 14/14
what do we get ?

Answer 20

14/154?

Answer 21

\[ \frac{1}{11} \cdot \frac{14}{14}= \frac{14}{154}\]

Answer 22

so we now know
\[ \frac{1}{14}+\frac{1}{11} \]
is the same as adding
\[ \frac{11}{154}+\frac{14}{154} \]
when adding fractions with the same bottom, we add their tops
and keep the same bottom

Answer 23

you should get
\[ \frac{11}{154}+\frac{14}{154}= \frac{11+14}{154} = \frac{25}{154} \]

Answer 24

Yes, I got that but now I don't know what to do after that.

Answer 25

Do I divide?

Answer 26

and your problem is
\[ \left( \frac{1}{14}+\frac{1}{11} \right)t = 1 \\ \frac{25}{154}t = 1 \]

Answer 27

now multiply both sides by 154/25
and simplify

Answer 28

so I multiply 25/154 by 154/25?

Answer 29

yes, on the left side. and to keep things equal, you multiply the right side by 154/25
like this
\[ \frac{154}{25} \cdot \frac{25}{154} t = 1 \cdot \frac{154}{25}\]

Answer 30

as you know, when multiplying fractions, you multiply top times top
and bottom times bottom
on the left side of the equation we do
\[ \frac{154 \cdot 25}{25 \cdot 154} \]
when multiplying you can change the order.
for example, change the order up top:
\[ \frac{25 \cdot 154}{25 \cdot 154} \]

and we can also write that as
\[ \frac{25}{25}\cdot \frac{154}{154} \]
notice if you mutiply top times top and bottom times bottom we get back what we started with... this shows it is the same thing.

but anything divided by itself is 1
25/25 is 1
and 154/154 is 1
and 1*1 is 1

that is the long way to show that the left side turns into
1*t or just t
\[ t = \frac{154}{25} \]
you might want to write the right side as a decimal
t= 6.16 hours

Answer 31

Thank you so much!! You were really helpful :)