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## Modeling with rational functions

# Rational equations word problem: eliminating solutions

CCSS.Math:

## Video transcript

Two different hoses are being
used to fill a fish pond. Used together, the
two hoses take 12 minutes to fill the pond. If used alone, one hose is
able to fill the pond 10 minutes faster than the other. So 10 minutes faster than
the other hose. How long does each hose take
to fill the pond by itself? So let's think about
each of the hoses. We have a faster hose, and then
we have a slower hose. And let's say that the faster
hose fills the pond. Let's say it takes him
f minutes per pond. Now how long is it going to
take the slower hose? Well, the faster hose does
it in 10 minutes less. It's 10 minutes faster. So the slower hose is going
to take 10 minutes more. So the slower hose is going to
take f plus 10 minutes per fish pond or per pond. Now this is in minutes per pond,
but if we want to be able to add rates together, we
should really think about in terms of ponds per minute. So let's rewrite each
of these statements as ponds per minute. You could write this as f
minutes per 1 pond or f plus 10 minutes per 1 pond. And if you just take the inverse
of each of these statements, these ratios are
equivalent to saying 1 pond per f minutes. So it's really not saying
anything else. I'm just inverting the ratio. Or you could think of it as
1/f ponds per minute. Same logic right here. We could essentially rewrite
this ratio as 1 over f plus 10 ponds per minute. So now we have the rate
of the faster hose. We have the rate of
the slower hose. How many pounds per minute
for the faster hose? How many pounds per minute
for the slower hose? If we add these two rates,
we'll know the ponds per minute when they're
acting together. So if we have 1/f ponds per
minute plus 1 over f plus 10 ponds per minute. This is the faster hose, this
is the slower hose. This'll tell us how
many ponds for minute they can do together. Now, we know that information. They say the two hoses
take 12 minutes. So let me write that
over here. So combined. The combined take 12
minutes per pond. So what is their combined rate
in terms of ponds per minute? So you could view this as
12 minutes per one pond. You could take the inverse of
this or take the ratio in terms of ponds per minute
instead and you get 1/12 minutes. Sorry, 1/12 ponds per minute. In one minute combined, they'll
fill 1/12 of a pond. Which makes complete sense. Because it takes them
12 minutes to fill the whole thing. So in one minute they'll
only do 1/12 of it. So this is their combined rate
in ponds per minute. This is also their combined
rate in ponds per minute. So this is going to
be equal to 1/12. And now we just have to solve
for f and then f plus 10 is going to be how long it
takes the slower hose. So let's multiply. Let's see what we could do. We could multiply both sides of
this equation times f and times f plus 10. So let's do that. So I'm going to multiply both
sides of this equation times f and f plus 10 times both
sides of this equation. So f and f plus 10. Scroll down a little bit. Scroll to the left, so we
have some real estate. So let's distribute this
f times f plus 10. So if we multiply f times f plus
10 times 1/f, that f and that f will cancel out and we're
just going to be left with an f plus 10. That's when you multiply
that term times the f times f plus 10. Now, when you multiply this
term, when you multiply 1 over f plus 10 times f times f plus
10, this and this will cancel out and you're just
left with an f. So you have plus f is equal
to-- and you have over 12. Actually, well in a second, let
me multiply all sides of this equation by 12. I'll do that next in a second. So let's just say this is going
to be equal to 1/12 times f squared. f times f is f squared. Plus 10f. And now let's just multiply
both sides of this equation by 12. I could've done it in the last
step so that we don't have any fractions here. And so the left-hand side
we get 12 times f. We get 12f plus 120 plus 12f. The right-hand side, that and
that cancels out and you are left with f squared plus 10f. And now we have a quadratic. We just have to get it into
a form that we know how to manipulate or deal with. Before that, we can
simplify it. We have a 12f and a 12f. So this becomes 24f plus
120 is equal to f squared plus 10f. And then let's get all
of this stuff out of the left-hand side. They'll get all on the
right-hand side. So from both sides of this
equation, let's subtract 24f and a negative 120
or minus 120. So you have minus
24f minus 120. The left-hand side
just becomes 0. That was the whole point. Right-hand side is f squared. 10 minus 24f is negative 14f. Minus 120. Now we could factor this. Let's see, if you do 20 times--
yeah, that looks like it would work. So negative 20 and 6, when you
take their product, give you negative 120. And negative 20 plus
6 is negative 14. So we could factor this
right-hand side as 0 is equal to f minus 20 times f plus 6. When you multiply negative 20
times 6, you get negative 120. Negative 20 plus 6
is negative 14. And the only way that that's
going to be equal to 0 is if f minus 20 is equal to 0 or
f plus 6 is equal to 0. Add 20 to both sides
of this equation. You get f is equal to 20. Remember, f is how many minutes
does it take for the fast hose to fill the pond. And then, if you take
this one, you subtract 6 from both sides. You get f is equal
to negative 6. Now, when we're talking about
how many minutes does it take for the fast hose to fill the
pond, it doesn't make any sense to say that it
takes it negative 6 minutes to fill the pond. So we can't use this answer. We need a positive answer. So this is how many minutes it
takes the fast hose to fill the pond. f is equal to 20. So this right here, the
faster hose takes 20 minutes per pond. I'll write it here. 20 minutes per pond
is the fast hose. And then the slower hose, it
takes 10 minutes more. It's f plus 10. So it takes 30 minutes
per pond. And we're done. I don't want to confuse
you with this stuff. The faster hose takes
20 minutes. Slower hose takes 30
minutes per pond. If they were to do it together,
it would take 12 minutes, which is a little
bit more than half. If you had two faster hoses,
it would take 10 minutes. But this guy's a little bit
slower, so it's taking you a little bit more than
10 minutes. So it makes sense. It takes you 12 minutes when
they're working together.