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## Modeling with rational functions

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

CCSS.Math:

## 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.