WEBVTT

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>> Welcome to the Cypress
College math review

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on Applications of
Rational Equations.

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Objective one: Work
Rate Problems.

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If it takes me five
hours to paint a room,

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then I can do one-fifth
of that job in one hour.

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So the part of the
job that I can do

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in one hour is one over five.

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The part of the job that I can
do in one hour is always one

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over the time it takes
me to do the whole job.

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Another way to say that is

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that my rate is one-fifth
of the job per hour.

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A park fountain has
two sprinklers,

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which are used to
fill a fountain.

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One sprinkler can
fill the fountain

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in 30 minutes while the second
sprinkler can fill the fountain

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in 20 minutes.

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How long will it take
to fill the fountain

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with both sprinklers operating.

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So let's x be the time it
would take both sprinklers

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to fill the fountain if they
were both working together.

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The set up for these work
rate problems is similar

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for each problem.

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The part of the job done by the
first sprinkler in one minute

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or its rate plus the part of the
job done by the second sprinkler

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in one minute or its
rate is equal to the part

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of the job done together in one
minute or the rate together.

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As we saw the formula is one
over the time it would take

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that individual to do the
job all by themselves.

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So we have one over
time plus one

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over time equals one over time.

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It took 30 minutes for the
first sprinkler to do the job

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by itself, so one over
30 would be their rate.

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It took 20 minutes for
the second sprinkler,

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so one over 20 would
be their rate.

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And then one over x because
x is the time it would take

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if the sprinklers
were working together.

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So the rate working together
is one over x or the part

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of the job done together
in one minute is one

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over x. This is the equation
that models this problem.

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Pause the video if
you want to go

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through the steps
for solving it.

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And here's the answer.

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It takes Sylvester
and Tom four hours

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to paint a room working
together.

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Sylvester is a very
slow painter.

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It takes him twice as long
to paint as it does Tom.

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How long would it take each

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of them individually
to paint the room.

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So Sylvester takes twice
as long as Tom does.

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So if we let x be the
time it would take Tom

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to paint the room, then 2x
would be the time it would take

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Sylvester to paint the room.

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Similar to the last problem,
the part of the job done

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by Sylvester in one hour, plus
the part of the job done by Tom

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in one hour equals the part

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of the job done together
in one hour.

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The formula once
again one over time.

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The time it would
take Sylvester is 2x

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if he was doing the whole job by
himself, so we're going to put

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that in the first denominator.

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The time it would take Tom to
paint the room by himself is x,

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so we're going to put that
in the second denominator.

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And then working together
it takes them four hours

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to paint the room.

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So we're going to put that
in the final denominator.

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This is the equation
that models this problem.

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Pause the video if you want
to go through the steps

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to solve it, and then
here's the answer.

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It takes two hours to
empty a certain tank

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if it started out
completely full.

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It takes three hours
to fill that same tank.

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If both the fill pipe and the
drain are accidently left open,

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how long would it take
to empty a full tank?

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So we'll let x be the time it
would take to empty a full tank

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if both the fill pipe
and the drain are open.

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So that's the time
that it takes for both

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of them working together.

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This time we have two
different systems working

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against each other.

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So what is our goal?

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Our goal is to empty the tank.

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So the drain is doing the job.

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The fill pipe is
working against us,

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working against us
means subtract.

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So the part done by the drain
in one hour minus the part done

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by the fill pipe in one
hour equals the part

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of the job done together in one
hour working against each other.

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The formula once
again, one over time.

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And it would take the drain
two hours to empty a full tank.

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So that's the time it would
take it working by itself.

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So we put that in the first
denominator since we're working

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with the drain in
the first term.

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The second term deals
with the fill pipe,

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once again working against us.

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It takes them three hours
to fill the entire tank,

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so one over three
in the second term.

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And then x is the
time it would take

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if they were both
working together,

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so one over x is our final term.

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This equation models
the problem.

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Pause the video if
you want to go

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through the steps to solve it.

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And here's the answer.

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Pause the video and
try these problems.

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Objective two: Distance,
Rate, Time Problems.

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Humberto and Wakesha
went for a run.

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Humberto ran 6.5
miles in the same time

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that it took Wakesha
to run five miles.

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Wakesha runs 1.5 miles an hour
slower than Humberto does.

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How fast do each of them run?

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What are we given here?

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Humberto runs 6.5 miles.

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So that's a distance.

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We have his distance.

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Wakesha ran five miles.

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That's her distance.

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So we have the two distances.

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Wakesha ran 1.5 miles an hour
slower than Humberto did.

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So that's a relationship
between their rates.

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So we don't know their rates,

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but we know a relationship
between their rates.

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For any problem where you're
given both of the distances,

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the best way to set the problem

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up is using a relationship
of the times.

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Similarly, for any problem
where you're given both times,

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the best way to set the problem

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up is using a relationship
of the distances.

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But that's not our
case in this problem.

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Distance equals rate times time.

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Solve for time.

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Time would be distance
over rate.

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So we're supposed to set this
up in terms of relationship

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of the times because we're
given both distances.

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What do we know about the times?

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Humberto ran 6.5 miles in
the same time, ah, same time.

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So the time it took Humberto
equals the time it took Wakesha.

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Now we put in the formula
for time, distance over rate.

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So we have the distance
for Humberto, which is 6.5,

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and the distance for
Wakesha, which is five.

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Now we have to figure out what
to put in the denominator.

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We have to figure
out their rates.

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Well it says that Wakesha
runs 1.5 miles an hour slower

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than Humberto.

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We don't know either
of their rates,

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but since Wakesha's is
written in terms of Humberto's,

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let's let x be the rate
that Humberto ran at,

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which would mean that x
minus 1.5 would be the rate

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that Wakesha ran at.

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Put those in the denominators
in the appropriate fractions.

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This is the equation
that models this problem.

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Pause the video if you want
to go through the steps

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to solve it, and
here's the answer.

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Danika flew her Beechcraft
Baron to Santa Fe, New Mexico,

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a distance of 2,700 miles.

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She had a tailwind
for the entire flight.

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On her return trip, after
traveling for the same amount

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of time, she noticed that she
had only gone 2,160 miles.

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She had a headwind for
the entire flight back.

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The wind was blowing
at 25 miles an hour.

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What was the airspeed
of her plane?

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So what are we given?

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Twenty-seven hundred miles.

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That's a distance.

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Two thousand one
hundred sixty miles.

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That's a distance.

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So we're given both distances.

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For any problem where you
are given both distances,

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the best way to set the problem

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up is using a relationship
of the times.

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So from our formula, distance
equals rate times time,

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we solve for time and we
get distance over rate.

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Now we need a relationship
of the times.

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What do we know about the times.

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On her return trip
after traveling

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for the same amount of time.

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So the time on the way
there where she was flying

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with the wind equals
the time on the way back

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where she was flying
against the wind.

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So time equals distance
over rate.

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So we have distance over rate
on the way there equals distance

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over the rate on the way back.

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The distances were given,
so 2700 for the first one

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on the way there,
2160 on the way back.

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Now we have to figure
out the rates.

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Recall that flying with
the tailwind is the same

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as flying with the wind.

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Flying with the headwind is the
same as flying against the wind.

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The airspeed of the plane is the
speed that the plane would fly

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at if it was calm air.

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So our unknown is that airspeed.

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So let's let r be the
airspeed of the plane.

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Then r plus 25 would be the
speed of the plane flying

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with the wind since the wind
was blowing at 25 miles an hour.

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R minus 25 would be the speed

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of the plane flying
against the wind.

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So we put in our plus 25
for our first denominator

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since we're going with
the wind and our minus 25

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for our second denominator since
we were going against the wind.

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And this equation
models that problem.

00:10:53.096 --> 00:10:55.016 A:middle
Pause the video if you want
to go through the steps

00:10:55.016 --> 00:10:56.976 A:middle
to solve it, and then
here's our answer.

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Tatiana and Nikolai
went for a run

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from their house to the park.

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Tatiana ran at eight miles
an hour and Nikolai ran

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at six miles and hour.

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Nikolai arrived one-half
hour after Tatiana did.

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How far is it from
their house to the park?

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So what are we given?

00:11:23.786 --> 00:11:25.926 A:middle
We're given their
individual rates.

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Tatiana eight miles an hour,
Nikolai six miles an hour.

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We're not given the distances.

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We're given a relationship
of the times,

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but we're not given the
times for each of them.

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If the distance is the same for
both parts of a problem though,

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let that be your variable and
once again set it up in terms

00:11:45.286 --> 00:11:46.876 A:middle
of relationship of the times.

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Since distance equals rate
times time, time equals distance

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over rate, we're going
to let d be the distance

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from their house to the park.

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What do we know about
their times?

00:11:58.486 --> 00:12:00.416 A:middle
What's the relationship
that we're looking for here?

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Nikolai arrived one-half
hour after Tatiana did.

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So the time it took Nikolai
equals the time it took Tatiana

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plus one-half hour,
one-half hour longer.

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So we put in the formula,
distance over rate

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for each of the unknown times.

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And then we have our setup.

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The distance from the house to
the park is our variable, d,

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so we'll but that in
for the distances.

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The rate for Nikolai
was six miles an hour.

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The rate for Tatiana
was eight miles an hour,

00:12:38.996 --> 00:12:41.506 A:middle
and we have our equation
that models this problem.

00:12:42.686 --> 00:12:44.836 A:middle
Pause the video if you want
to go through the steps

00:12:45.366 --> 00:12:47.926 A:middle
to solve it, and then
here's our answer.

00:12:53.696 --> 00:12:54.976 A:middle
Pause the video and
try these problems.

00:13:06.046 --> 00:13:08.026 A:middle
Objective three:
Proportion problems.

00:13:09.016 --> 00:13:12.106 A:middle
If six out of 15 homes
in a community have wells

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for their water supply,
how many homes have wells

00:13:15.676 --> 00:13:18.176 A:middle
in a community of 18,000 homes?

00:13:18.796 --> 00:13:21.886 A:middle
So we'll let x be the
number of homes with wells

00:13:21.886 --> 00:13:23.056 A:middle
in the entire community.

00:13:23.966 --> 00:13:26.886 A:middle
We're going to set this up as
a proportion when we see this

00:13:26.986 --> 00:13:29.386 A:middle
if six out of 15 homes.

00:13:30.776 --> 00:13:33.456 A:middle
So we're going to put the
number with wells on top

00:13:34.146 --> 00:13:36.236 A:middle
and the total number
of homes on the bottom.

00:13:36.236 --> 00:13:37.516 A:middle
You need to be consistent.

00:13:37.966 --> 00:13:39.536 A:middle
So we set up our proportion.

00:13:40.446 --> 00:13:43.836 A:middle
Six with wells out of 15 total.

00:13:44.756 --> 00:13:48.726 A:middle
We don't know the number
of houses with wells

00:13:48.726 --> 00:13:51.756 A:middle
in the entire community, so
we're going to put x for that.

00:13:52.456 --> 00:13:53.956 A:middle
And then the total
number of homes

00:13:53.956 --> 00:13:56.036 A:middle
in the community was 18,000.

00:13:57.386 --> 00:13:59.526 A:middle
This equation models
that problem.

00:14:00.766 --> 00:14:02.716 A:middle
Pause the video if you want
to go through the steps

00:14:02.756 --> 00:14:04.976 A:middle
to solve it, and then
here's our answer.

00:14:12.486 --> 00:14:18.576 A:middle
Today's exchange rate for the
Australian dollar is 1.26744

00:14:18.576 --> 00:14:21.396 A:middle
Australian dollars
equals one U.S. dollar.

00:14:22.346 --> 00:14:25.026 A:middle
Porter has 324 Australian
dollars.

00:14:25.956 --> 00:14:28.046 A:middle
How many U.S. dollars
will he get in trade?

00:14:29.186 --> 00:14:32.356 A:middle
So we'll let x be the number of
U.S. dollars that Porter gets

00:14:32.356 --> 00:14:35.346 A:middle
in trade and this problem
sets up as a proportion

00:14:35.786 --> 00:14:38.426 A:middle
because the relationships
are the same

00:14:38.426 --> 00:14:41.256 A:middle
between the Australian
dollars and the U.S. dollars

00:14:41.256 --> 00:14:43.516 A:middle
that he will get and
the exchange rate.

00:14:45.116 --> 00:14:47.706 A:middle
We'll choose to put the
Australian dollars on the top.

00:14:47.706 --> 00:14:49.836 A:middle
It doesn't matter, but
you have to be consistent.

00:14:50.066 --> 00:14:52.076 A:middle
And we'll put the U.S.
dollars on the bottom.

00:14:53.946 --> 00:14:59.176 A:middle
So the exchange rate is 1.26744
Australian dollars, so that goes

00:14:59.176 --> 00:15:01.256 A:middle
on the top where I
have Australian dollars

00:15:01.556 --> 00:15:02.946 A:middle
and one U.S. dollar.

00:15:04.046 --> 00:15:07.756 A:middle
He has 324 Australian
dollars so that's going to be

00:15:07.756 --> 00:15:11.346 A:middle
in the numerator, and x is
the number of U.S. dollars.

00:15:11.716 --> 00:15:13.726 A:middle
This is the equation
that models that problem.

00:15:15.436 --> 00:15:17.426 A:middle
Pause the video if you want
to go through the steps

00:15:17.456 --> 00:15:19.936 A:middle
to solve it, and
here's our answer.

00:15:26.576 --> 00:15:28.696 A:middle
The ratio of employees
at a small company

00:15:28.696 --> 00:15:30.796 A:middle
who have their paychecks
directly deposited

00:15:30.866 --> 00:15:35.246 A:middle
into their bank accounts to
those who do not is nine to two.

00:15:35.586 --> 00:15:39.456 A:middle
If the number of people who
have direct deposit is 14 more

00:15:39.456 --> 00:15:40.676 A:middle
than the number who do not,

00:15:41.326 --> 00:15:43.796 A:middle
how many employees do
not have direct deposit?

00:15:44.506 --> 00:15:46.546 A:middle
When we see that
relationship nine to two,

00:15:46.546 --> 00:15:49.526 A:middle
we know we can set this
problem up as a proportion.

00:15:50.906 --> 00:15:52.916 A:middle
What are the things
that we do not know.

00:15:53.576 --> 00:15:56.296 A:middle
We do not know how many
people have direct deposit,

00:15:56.496 --> 00:15:58.106 A:middle
and we do not know
the number of people

00:15:58.106 --> 00:15:59.596 A:middle
that do not have direct deposit.

00:16:00.426 --> 00:16:01.866 A:middle
It says that the
number of people

00:16:01.866 --> 00:16:05.766 A:middle
who have direct deposit is 14
more than those who do not,

00:16:06.626 --> 00:16:09.096 A:middle
so we should let x be
the number of employees

00:16:09.096 --> 00:16:14.616 A:middle
that do not have direct deposit
and the 14 more, x plus 14,

00:16:14.946 --> 00:16:17.996 A:middle
would be the number of employees
that do have direct deposit.

00:16:18.386 --> 00:16:20.146 A:middle
You get to choose
what to put on top.

00:16:20.146 --> 00:16:22.956 A:middle
I'll put those that have direct
deposits on top and those

00:16:22.956 --> 00:16:23.916 A:middle
that don't on the bottom.

00:16:24.176 --> 00:16:27.106 A:middle
So nine had direct
deposits and two did not.

00:16:28.316 --> 00:16:31.426 A:middle
In our variables,
x plus 14 was those

00:16:31.426 --> 00:16:35.486 A:middle
that do have direct deposits,
and we put that on top.

00:16:35.486 --> 00:16:38.936 A:middle
And then x was the number that
do not have direct deposits.

00:16:39.636 --> 00:16:41.466 A:middle
This equation models
our problem.

00:16:42.536 --> 00:16:44.506 A:middle
Pause the video if you want
to go through the steps

00:16:44.506 --> 00:16:46.896 A:middle
to solve it, and
here's our answer.

00:16:50.136 --> 00:16:52.976 A:middle
This problem could have been
worded a little differently.

00:16:54.106 --> 00:16:57.086 A:middle
What if instead of saying
what it did earlier,

00:16:57.086 --> 00:16:59.976 A:middle
it said the ratio of
employees at a small company

00:16:59.976 --> 00:17:02.456 A:middle
who have their paychecks
directly deposited

00:17:02.456 --> 00:17:08.666 A:middle
into their bank accounts to the
total employees is nine to 11.

00:17:09.406 --> 00:17:11.386 A:middle
To the total employees.

00:17:11.746 --> 00:17:13.946 A:middle
So now we're going to have
a relationship of those

00:17:13.996 --> 00:17:18.496 A:middle
that have direct deposits to the
total employees in the company.

00:17:19.956 --> 00:17:21.816 A:middle
So that's nine to 11.

00:17:23.256 --> 00:17:26.646 A:middle
We set up our problem exactly
the same way as far as the x

00:17:26.646 --> 00:17:29.536 A:middle
and the x plus 14,
x plus 14 is those

00:17:29.536 --> 00:17:31.176 A:middle
that do have direct deposits.

00:17:32.186 --> 00:17:34.516 A:middle
What do we put in the
denominator over here

00:17:34.516 --> 00:17:36.906 A:middle
on the right, the total
number of employees?

00:17:37.416 --> 00:17:41.126 A:middle
There were x employees that
do not have direct deposits

00:17:41.726 --> 00:17:44.036 A:middle
and there were x
plus 14 employees

00:17:44.146 --> 00:17:46.576 A:middle
that do have direct deposits.

00:17:46.636 --> 00:17:48.096 A:middle
So the total employees

00:17:48.096 --> 00:17:50.576 A:middle
in the company would
be the sum of those.

00:17:50.576 --> 00:17:53.086 A:middle
So you'd add x and x plus 14.

00:17:55.386 --> 00:17:58.146 A:middle
This equation models
the new problem,

00:17:58.726 --> 00:18:01.426 A:middle
and the solution is exactly
the same as we got before.

00:18:07.566 --> 00:18:11.896 A:middle
Javier can run 300
yards in 34 seconds.

00:18:13.036 --> 00:18:18.116 A:middle
At that rate, how far
can he run in 52 seconds?

00:18:19.816 --> 00:18:21.346 A:middle
A lot of people when
they look at this,

00:18:21.396 --> 00:18:22.896 A:middle
to start with they're
going to think, oh,

00:18:22.896 --> 00:18:24.756 A:middle
this is a distance,
rate, time problem.

00:18:25.766 --> 00:18:29.776 A:middle
Well it is, but it also can be
done with a simple proportion,

00:18:30.006 --> 00:18:31.306 A:middle
and that's what I'm
going to do here.

00:18:32.286 --> 00:18:34.006 A:middle
When we see at that rate,

00:18:34.796 --> 00:18:36.746 A:middle
we know that it can be
done as a proportion.

00:18:37.966 --> 00:18:42.396 A:middle
So let's x be the distance that
Javier can run in 52 seconds.

00:18:42.766 --> 00:18:45.266 A:middle
I'm going to put yards on top
and seconds on the bottom.

00:18:46.626 --> 00:18:54.676 A:middle
So 300 yards in 34 seconds,
unknown x yards in 52 seconds.

00:18:55.736 --> 00:18:57.846 A:middle
This is the models our problem.

00:18:58.946 --> 00:19:01.076 A:middle
Pause the video if you want
to go through the steps

00:19:01.116 --> 00:19:02.976 A:middle
to solve it, and
here's our answer.

00:19:11.366 --> 00:19:13.796 A:middle
Alberto uses three-quarters
cups of sugar

00:19:14.216 --> 00:19:15.726 A:middle
in his banana muffin recipe.

00:19:16.066 --> 00:19:17.756 A:middle
The recipe makes 12 muffins.

00:19:18.226 --> 00:19:20.636 A:middle
Alberto is baking for his
daughter's school bake sale.

00:19:21.176 --> 00:19:23.136 A:middle
He wants to make 52 muffins.

00:19:23.626 --> 00:19:24.846 A:middle
How much sugar will he need?

00:19:25.626 --> 00:19:28.176 A:middle
So we're going to let x be the
amount of sugar that he needs

00:19:28.176 --> 00:19:29.716 A:middle
to make the 52 muffins.

00:19:30.206 --> 00:19:31.906 A:middle
We can set this up
as a proportion.

00:19:32.636 --> 00:19:34.566 A:middle
I chose to put the
cups of sugar on top.

00:19:34.886 --> 00:19:37.396 A:middle
So we have three-quarters
cups of sugar.

00:19:37.626 --> 00:19:41.816 A:middle
I'll write that as a
decimal, .75, to 12 muffins.

00:19:42.316 --> 00:19:45.596 A:middle
We don't know the amount
of sugar to make 52 muffins

00:19:45.596 --> 00:19:49.646 A:middle
so we'll but that in as x and
52 is the number of muffins,

00:19:50.016 --> 00:19:51.486 A:middle
and we have our proportion.

00:19:53.206 --> 00:19:54.976 A:middle
This equation models
the problem.

00:19:55.696 --> 00:19:57.606 A:middle
Pause the video if you want
to go through the steps

00:19:57.606 --> 00:19:58.976 A:middle
to solve it, and
here's our answer.

00:20:06.056 --> 00:20:06.976 A:middle
Pause the video and
try these problems.

00:20:18.386 --> 00:20:19.456 A:middle
Objective four.

00:20:19.816 --> 00:20:20.746 A:middle
Number problems.

00:20:21.206 --> 00:20:24.096 A:middle
The numerator of a certain
fraction is one less

00:20:24.096 --> 00:20:25.006 A:middle
than the denominator.

00:20:25.806 --> 00:20:28.716 A:middle
If three is added to both the
numerator and the denominator,

00:20:29.086 --> 00:20:30.976 A:middle
the result is equal
to seven-eighths.

00:20:31.276 --> 00:20:33.216 A:middle
What was the original fraction?

00:20:33.336 --> 00:20:36.786 A:middle
So to figure out the
original fraction,

00:20:36.786 --> 00:20:38.756 A:middle
we need to know the
numerator and we need

00:20:38.756 --> 00:20:39.756 A:middle
to know the denominator.

00:20:40.826 --> 00:20:44.826 A:middle
It says that the numerator is
one less than the denominator.

00:20:45.226 --> 00:20:48.516 A:middle
So we should let d be the
denominator of the fraction

00:20:48.906 --> 00:20:53.646 A:middle
and one less than d, so d
minus one is the numerator

00:20:53.646 --> 00:20:54.316 A:middle
of the fraction.

00:20:54.706 --> 00:20:59.406 A:middle
So our original fraction was d
minus one over d. D minus one

00:20:59.406 --> 00:21:01.696 A:middle
for the numerator and
d for the denominator.

00:21:02.366 --> 00:21:06.176 A:middle
In the second sentence, it
says that three is added

00:21:06.176 --> 00:21:08.346 A:middle
to both the numerator
and the denominator.

00:21:08.936 --> 00:21:12.956 A:middle
So we take and add three to
d minus one and we add three

00:21:12.956 --> 00:21:17.576 A:middle
to d. The result is
equal to seven-eighths.

00:21:17.576 --> 00:21:19.486 A:middle
So equals seven-eighths.

00:21:20.736 --> 00:21:23.126 A:middle
This is the equation
that models that problem.

00:21:24.036 --> 00:21:26.736 A:middle
Pause the video if you want
to go through the steps

00:21:26.736 --> 00:21:29.586 A:middle
to solve it, and
here's our answer.

00:21:37.346 --> 00:21:38.936 A:middle
Pause the video and
try this problem.