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# Rational equations word problem: combined rates

CCSS Math: HSA.CED.A.2, HSA.REI.A.2

## Video transcript

Ian can rake a lawn and
bag the leaves in 5 hours. Kyandre can rake the same lawn
and bag the leaves in 3 hours. Working together,
how long would it take them to rake the
lawn and bag the leaves? So let's think about
it a little bit. It says Ian can rake a lawn and
bag of the leaves in 5 hours. So for Ian, I for Ian. For 1 lawn, he can
do 1 lawn in 5 hours. We could have also written
this as 5 hours per lawn. But we'll see that
writing it this way is more useful, because
it's actually a rate. Because this is the same
thing as 1/5 lawns per hour. Or 1/5 of a lawn per hour. That's the rate at
which Ian can rake a lawn, at 1/5 of
a lawn per hour. Now let's do the same
thing for Kyandre. And this actually probably
shouldn't be plural. 1/5 lawns, 1/5 of a lawn. So let me just
erase that S right there-- 1/5 of a lawn per hour. Now let's do the same
thing for Kyandre. Kyandre can rake the same lawn
and bag the leaves in 3 hours. So for Kyandre, Kyandre can,
for 1 lawn-- I'll assume it's a boy's name-- he
can do it in 3 hours. Or if we were to
write it as a rate, this is 1/3 of a lawn per hour. Now let's think about
what the combined rate is. So let's say if we
have Ian plus Kyandre. What's going to be
their combined rate? Well, they tell us that working
together, how long would it take them to rake
and bag the leaves? So let's let let's let t be
how long it will take them together. So that's how long they
would take together. And if we say t is how
long they take together, then we could say
that combined, they will do 1 lawn in
every t hours, if we're assuming t is in hours. For every t hours. Or as a rate,
their combined rate is going to be 1/t
of a lawn per hour. So that's their combined rate. So here, we have the rate of
Ian and the rate of Kyandre, and the combined rate. So the combined
rate's just going to be the sum of
each of their rates. If he can do 1/5
of a lawn per hour and he can do 1/3
of a lawn per hour, their combined rate is going
to be 1/5 of a lawn per hour plus 1/3 of an hour. Because in an hour he'll
do 1/5, and he'll do 1/3. So you'll add those two
together to figure out how much they can do in an hour. So their combined rate is
going to be 1/5 lawn per hour. And I won't write
the units here, just because it gets redundant. 1/5 lawn per hour I
could write over here. Plus 1/3 lawn per
hour for Kyandre. That's going to be
their total rate, which is 1/t lawns per hour. And now we just
have to solve for t. And we'll know the total number
of hours it will take them. So let's do that. So to do that, we just
have to add 1/5 plus 1/3. Well, we have a common
denominator of 15. So this is the same
thing as 3/15 plus 5/15 is going to be equal to 1/t. And then we have a
common denominator now. So 3 plus 5 is 8. So this is going to be 8/15. I'll go over here now. So now we have 3 plus 5 is 8. Over 15 is equal to 1 over t. If we want to solve
for t, we could take the reciprocal
of both sides. So if we flip the left
side, we get 15/8. And if we flip the right
side, we get t/1, or just t. So it'll take them 15/8 hours. Or if we want that
in kind of a way that we can think about it a
little bit better, 15/8-- so t is equal to 15 over 8 hours. And I should say 15
over 8 hours per lawn. This whole time here
we had lawns per hour. This was lawn per hour. And this was here as
well, lawns per hour. When we flip it, it
becomes hours per lawn. So that's exactly what we want. But 15/8 is the same
thing as 1 and 7/8 hours. And 7/8 of an hour-- we
can get our calculator out. If we have 60 minutes
in an hour, times 7/8, we get 52.5 minutes. So this is equal to our answer. Combined, it will take them 1
hour and 52.5 minutes per lawn. Or to do this lawn right over
here, the lawn in question. Hopefully, you
found that useful.