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## Algebra (all content)

### Unit 13: Lesson 11

Modeling with rational functions- Analyzing structure word problem: pet store (1 of 2)
- Analyzing structure word problem: pet store (2 of 2)
- Combining mixtures example
- Rational equations word problem: combined rates
- Rational equations word problem: combined rates (example 2)
- Mixtures and combined rates word problems
- Rational equations word problem: eliminating solutions
- Reasoning about unknown variables
- Reasoning about unknown variables: divisibility
- Structure in rational expression

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# Rational equations word problem: combined rates (example 2)

CCSS.Math:

Sal solves a word problem about the combined deck-staining rates of Anya and Bill, by creating a rational equation that models the situation. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- Why must the rate problem always be set up in deck/hr to get the correct answer.

i.e. Why can't you put 8hrs/deck=A hrs/deck + B hrs/deck

If you solve this way using 2A=B you get an incorrect answer but I don't understand why, this was how I tried to solve it originally.

I'm just confused about how to pick where the A and B will go (on the numerator or denominator) and how to set up the problem.

Please help!(46 votes)- CasualJames, I'm with you, but why is the answer wrong? In real life seems logical: you can take 8 hours to run a mile or you can run a mile in 8 hours. But mathematically it makes no sense. That is: 1/2 + 1/2 = 1 yet 2+2=4 so you obviously cannot shift the denominators as you will. But the question remains: Is there a bullet proof method to decide which one is the right denominator or is it mere intuition?(18 votes)

- why do i have to take the inverse?(5 votes)
- because if we didnt inverse it, it would be like this

8 = A + B

B = 2A

8= A + 2A

8= 3A

A = 8/3

A = 2.66 (and repeating)

B = 5.33(and repeating)

now think about it.

logically how would two people working together take more time than working indivisually?(5 votes)

- Why does the following not produce correct results?

a+b=8

a+2a=8

3a=8

a=8/3=2.66

b=16/3=5.33(2 votes)- Anya and Bill are working TOGETHER. So they need to finish at the same time. Because Anya could working faster, she could do more than bill does. So why should Anya only use 2.66 hours to finish and just sit there waiting for Bill to finish his part?

Hope that helps! :)(2 votes)

- At4:06, why does Sal decide to multiply "1/8 = 1/A +1/2A" by 8A? I am wondering why does it answer the question correctly? I ask because I added "1/A + 1/2A" to come up with "1/8 = 1/3A." Then I inversed it to get "8 = 3A" which is "A = 8/3" or "A = 2 2/3 hours per deck (or 2 hours and 40 minutes). While I came up with 2 hours and 40 minutes for Anya, Sal came up with 12 hours for her. Somebody please help me understand why is my calculation incorrect? Thank you.(1 vote)
- you can't just add denominators (a+2a), you have to make them equal (multiply 1/a times 2 to make it equal to second fraction):

1/a + 1/2a

2*1/2*a + 1/2a

2/2a + 1/2a = 3/2a

more info here: http://www.khanacademy.org/math/arithmetic/fractions/v/adding-and-subtracting-fractions(4 votes)

- If two people work on a task at the same rate, what's the reasoning behind, the task being done in half the amount of time?

Is it something calculated mathematically, or is there some logical reasoning?

I understand that, it will be quicker, but what's the reasoning behind it being half the amount of time, than the times of the individuals?(2 votes)- It is mathematical. If 2 people are working together on one task, and they work at the same rate of speed, then each person completes 1/2 the work needed to complete the job and the whole job gets finished because 1/2+1/2 = 1 (the job is done).(3 votes)

- There are no exercises, what do I now?(1 vote)
- If KA has no exercises for this topic, you can search the internet for other sites that do.(4 votes)

- Working together, it takes two different sized hoses

35

minutes to fill a small swimming pool. If it takes

55

minutes for the larger hose to fill the swimming pool by itself, how long will it take the smaller hose to fill the pool on its own?(1 vote)- Because the hoses are different we need to consider how much the pool is filled per minute.

For the larger hose 1/55 of the pool will be filled in a minute. The smaller hose would then be 1/x per minute. Now we can add both hoses together (1/x + 1/55) and set it equal to how much of the pool is filled when both are working (1/35)

1/x +1/55 = 1/35 and solve

1/x=1/35-1/55

1/x=4/385

4x=385

x=96.25

So it would take 96.25 minutes to fill the pool with just the smaller hose.(3 votes)

**Another way of looking into this problem**

Divide porch into 3 Parts. At the end of 8 hours Anaya would paint 2/3 of porch and Bill would Paint 1/3 of porch. (because bill is twice as slow as Anaya OR Anaya is twice as fast as Bill).**Conclusion 1**: Anaya takes 8 hrs to paint 2/3 rd of a porch**Conclusion 2**: Bill takes 8 hrs to paint 1/3 rd of porch.

Now it boils down to*ratios and proportions*problem

Time taken by Anaya to Paint 1 Porch can be given by

2/3 = 8 then 1 = X

2/3 = 8

3/2 *2/3 = 8 * (3/2) ............................... (Multiply both side by 3/2)

1 = 12

Similarly Time taken by Bill to Paint 1 Porch can

1/3 = 8

1/3 * 3 = 8 * 3....................(Multiply both side by 3)

1 = 24

Hence Anaya takes 12 hrs to paint 1 Porch and Bill takes 24 hrs to Paint 1 Porch.

Hope that helps !(2 votes)- How can we prove that both of them together take 8 hours to paint?(1 vote)
- As for any equation, you can check your results by plugging them back into your equation. In the problem in the video, you can check the results found by doing? 1/12 + 1/24 = 1/8

See if the 2 sides become equal. If they do, you have the correct results.(2 votes)

- why multiply both sides of the equation with 8A

I think we can figure out the answer without multiply 8A.

1/8= 1/A + 1/2A

1/8= 3/2A

2A=24

A=12

I'm confusing that how did 8A come from.(1 vote)- Your method does work. You are working with the fractions and then using properties of proportions to finish.

Sal's method is good if you don't want to do a lot of math with fractions. The 8A is the LCM for all the denominators in the equation. If you multiply the equation by 8A, the denominators cancel out and you have an equation with no fractions. Some people find this easier to do than your approach. So, it's good to have options.(2 votes)

## Video transcript

Working together Anya and
Bill stained a large porch deck in 8 hours. Last year Anya stained
the deck by herself. The year before Bill painted it
by himself, but took twice as long as Anya did. How long did Anya and Bill take
when each was painting the deck alone? So let's define some
variables here. Let's define A. Let's define A as number
of hours for Anya to paint a deck. And so we could say that
Anya paints, or has A hours for one deck. Or we can invert this and
say that she can do 1/A decks per hour. Now, let's do another variable
for Bill just like that. Let's define B as the
number of hours for Bill to paint a deck. And so for Bill, he can paint--
it takes him B hours per deck, per one deck. We could put it over
1 there if we want. We don't have to. And this is the same thing as
saying that he can do 1 deck per every B hours. Or another way to think
of it, is he can do 1/B deck per hour. Now, when they work together,
Anya and Bill stained a large porch deck in 8 hours. So let's write this down. Anya plus Bill-- I'll
do it in orange. Anya and Bill. This information right here. Anya and Bill stained a large
porch deck in 8 hours. So we could say 8
hours per deck. Or 8 hours per 1 deck, which is
the same thing as saying 1 deck per 8 hours. And this is going to be
the combination of each of their rates. So this 1 deck of per 8 hours,
this is going to be equal to Anya's rate, 1/A decks per
hour, plus Bill's rate. Plus-- do that same color. Plus 1/B decks per hour. So we have one equation
set up. Let me scroll down
a little bit. And I won't write the
units any more. We have 1/8 is equal
to 1/A plus 1/B. Now, we have two unknowns, so
we need another equation if we're going to solve these. Let's see. It tells us, it tell this right
here the year before Bill painted it by himself,
but took twice as long as Anya did. So the number of hours it takes
Bill to paint the deck is twice as long as the number
of hours it takes Anya to paint a deck. So B is equal to 2A. The number of hours Bill takes
is twice as the number of hours Anya takes per deck. So B is equal to 2A. So we can rewrite this equation
as-- I'll stick to the colors for now. We could say 1/8 is equal
to 1 over Anya. Instead of writing 1 over Bill
we would write, so plus 1 over Bill is 2 times Anya. The number of hours Bill takes
is two times the number of hours Anya takes. So 2 times Anya. And now we have one equation
and one unknown. And we can solve for A. And the easiest way to solve for
A right here is if we just multiply both sides of
the equation by 2A. So let's multiply both
sides of this by 2A. Multiply the left side by 2A. Actually, let's multiply both
sides by 8A, so we get rid of this 1/8 as well. And then multiply the right-hand
side by 8A as well. The left-hand side, 8A divided
by 8 is just A. A is equal to-- 1/A times 8A
is just going to be 8. 1/2A times 8A. 8A divided by 2A is 4. So A, the number of hours it
takes Anya to paint a deck-- and I made it lowercase, which
I shouldn't have. Well, these should all be capital A's. This is A is equal to
8A over A is 8. 8A over 2A is 4. So the number of hours it takes
Anya to paint a deck, or A, is 12 hours. Now what are they asking
over here? They're asking, how long did
Anya and Bill take when each was painting alone? So we figured out Anya. It takes her 8 hours. And then we know that it takes
Bill twice as long as Anya. Bill is two times A. So bill is going to
be 2 times 12. So Bill is equal to 2 times
Anya, which is equal to 2 times 12 hours. Which is equal to 24 hours. So when they're alone it would
take Anya 12 hours, it would take Bill 24 hours. When they do it combined,
it takes 8 hours. Which makes sense. Because if Bill took 12 hours
by himself, combined they would take 6 hours, they would
take half as long. But Bill isn't that efficient. He's not as efficient as
Anya, so it takes them a little bit longer. Takes them 8 hours. But it makes sense that combined
they're going to take less time than individually.