>> Welcome to the Cypress College math review
on Applications of Rational Equations.
Objective one: Work Rate Problems.
If it takes me five hours to paint a room,
then I can do one-fifth of that job in one hour.
So the part of the job that I can do
in one hour is one over five.
The part of the job that I can do in one hour is always one
over the time it takes me to do the whole job.
Another way to say that is
that my rate is one-fifth of the job per hour.
A park fountain has two sprinklers,
which are used to fill a fountain.
One sprinkler can fill the fountain
in 30 minutes while the second sprinkler can fill the fountain
in 20 minutes.
How long will it take to fill the fountain
with both sprinklers operating.
So let's x be the time it would take both sprinklers
to fill the fountain if they were both working together.
The set up for these work rate problems is similar
for each problem.
The part of the job done by the first sprinkler in one minute
or its rate plus the part of the job done by the second sprinkler
in one minute or its rate is equal to the part
of the job done together in one minute or the rate together.
As we saw the formula is one over the time it would take
that individual to do the job all by themselves.
So we have one over time plus one
over time equals one over time.
It took 30 minutes for the first sprinkler to do the job
by itself, so one over 30 would be their rate.
It took 20 minutes for the second sprinkler,
so one over 20 would be their rate.
And then one over x because x is the time it would take
if the sprinklers were working together.
So the rate working together is one over x or the part
of the job done together in one minute is one
over x. This is the equation that models this problem.
Pause the video if you want to go
through the steps for solving it.
And here's the answer.
It takes Sylvester and Tom four hours
to paint a room working together.
Sylvester is a very slow painter.
It takes him twice as long to paint as it does Tom.
How long would it take each
of them individually to paint the room.
So Sylvester takes twice as long as Tom does.
So if we let x be the time it would take Tom
to paint the room, then 2x would be the time it would take
Sylvester to paint the room.
Similar to the last problem, the part of the job done
by Sylvester in one hour, plus the part of the job done by Tom
in one hour equals the part
of the job done together in one hour.
The formula once again one over time.
The time it would take Sylvester is 2x
if he was doing the whole job by himself, so we're going to put
that in the first denominator.
The time it would take Tom to paint the room by himself is x,
so we're going to put that in the second denominator.
And then working together it takes them four hours
to paint the room.
So we're going to put that in the final denominator.
This is the equation that models this problem.
Pause the video if you want to go through the steps
to solve it, and then here's the answer.
It takes two hours to empty a certain tank
if it started out completely full.
It takes three hours to fill that same tank.
If both the fill pipe and the drain are accidently left open,
how long would it take to empty a full tank?
So we'll let x be the time it would take to empty a full tank
if both the fill pipe and the drain are open.
So that's the time that it takes for both
of them working together.
This time we have two different systems working
against each other.
So what is our goal?
Our goal is to empty the tank.
So the drain is doing the job.
The fill pipe is working against us,
working against us means subtract.
So the part done by the drain in one hour minus the part done
by the fill pipe in one hour equals the part
of the job done together in one hour working against each other.
The formula once again, one over time.
And it would take the drain two hours to empty a full tank.
So that's the time it would take it working by itself.
So we put that in the first denominator since we're working
with the drain in the first term.
The second term deals with the fill pipe,
once again working against us.
It takes them three hours to fill the entire tank,
so one over three in the second term.
And then x is the time it would take
if they were both working together,
so one over x is our final term.
This equation models the problem.
Pause the video if you want to go
through the steps to solve it.
And here's the answer.
Pause the video and try these problems.
Objective two: Distance, Rate, Time Problems.
Humberto and Wakesha went for a run.
Humberto ran 6.5 miles in the same time
that it took Wakesha to run five miles.
Wakesha runs 1.5 miles an hour slower than Humberto does.
How fast do each of them run?
What are we given here?
Humberto runs 6.5 miles.
So that's a distance.
We have his distance.
Wakesha ran five miles.
That's her distance.
So we have the two distances.
Wakesha ran 1.5 miles an hour slower than Humberto did.
So that's a relationship between their rates.
So we don't know their rates,
but we know a relationship between their rates.
For any problem where you're given both of the distances,
the best way to set the problem
up is using a relationship of the times.
Similarly, for any problem where you're given both times,
the best way to set the problem
up is using a relationship of the distances.
But that's not our case in this problem.
Distance equals rate times time.
Solve for time.
Time would be distance over rate.
So we're supposed to set this up in terms of relationship
of the times because we're given both distances.
What do we know about the times?
Humberto ran 6.5 miles in the same time, ah, same time.
So the time it took Humberto equals the time it took Wakesha.
Now we put in the formula for time, distance over rate.
So we have the distance for Humberto, which is 6.5,
and the distance for Wakesha, which is five.
Now we have to figure out what to put in the denominator.
We have to figure out their rates.
Well it says that Wakesha runs 1.5 miles an hour slower
than Humberto.
We don't know either of their rates,
but since Wakesha's is written in terms of Humberto's,
let's let x be the rate that Humberto ran at,
which would mean that x minus 1.5 would be the rate
that Wakesha ran at.
Put those in the denominators in the appropriate fractions.
This is the equation that models this problem.
Pause the video if you want to go through the steps
to solve it, and here's the answer.
Danika flew her Beechcraft Baron to Santa Fe, New Mexico,
a distance of 2,700 miles.
She had a tailwind for the entire flight.
On her return trip, after traveling for the same amount
of time, she noticed that she had only gone 2,160 miles.
She had a headwind for the entire flight back.
The wind was blowing at 25 miles an hour.
What was the airspeed of her plane?
So what are we given?
Twenty-seven hundred miles.
That's a distance.
Two thousand one hundred sixty miles.
That's a distance.
So we're given both distances.
For any problem where you are given both distances,
the best way to set the problem
up is using a relationship of the times.
So from our formula, distance equals rate times time,
we solve for time and we get distance over rate.
Now we need a relationship of the times.
What do we know about the times.
On her return trip after traveling
for the same amount of time.
So the time on the way there where she was flying
with the wind equals the time on the way back
where she was flying against the wind.
So time equals distance over rate.
So we have distance over rate on the way there equals distance
over the rate on the way back.
The distances were given, so 2700 for the first one
on the way there, 2160 on the way back.
Now we have to figure out the rates.
Recall that flying with the tailwind is the same
as flying with the wind.
Flying with the headwind is the same as flying against the wind.
The airspeed of the plane is the speed that the plane would fly
at if it was calm air.
So our unknown is that airspeed.
So let's let r be the airspeed of the plane.
Then r plus 25 would be the speed of the plane flying
with the wind since the wind was blowing at 25 miles an hour.
R minus 25 would be the speed
of the plane flying against the wind.
So we put in our plus 25 for our first denominator
since we're going with the wind and our minus 25
for our second denominator since we were going against the wind.
And this equation models that problem.
Pause the video if you want to go through the steps
to solve it, and then here's our answer.
Tatiana and Nikolai went for a run
from their house to the park.
Tatiana ran at eight miles an hour and Nikolai ran
at six miles and hour.
Nikolai arrived one-half hour after Tatiana did.
How far is it from their house to the park?
So what are we given?
We're given their individual rates.
Tatiana eight miles an hour, Nikolai six miles an hour.
We're not given the distances.
We're given a relationship of the times,
but we're not given the times for each of them.
If the distance is the same for both parts of a problem though,
let that be your variable and once again set it up in terms
of relationship of the times.
Since distance equals rate times time, time equals distance
over rate, we're going to let d be the distance
from their house to the park.
What do we know about their times?
What's the relationship that we're looking for here?
Nikolai arrived one-half hour after Tatiana did.
So the time it took Nikolai equals the time it took Tatiana
plus one-half hour, one-half hour longer.
So we put in the formula, distance over rate
for each of the unknown times.
And then we have our setup.
The distance from the house to the park is our variable, d,
so we'll but that in for the distances.
The rate for Nikolai was six miles an hour.
The rate for Tatiana was eight miles an hour,
and we have our equation that models this problem.
Pause the video if you want to go through the steps
to solve it, and then here's our answer.
Pause the video and try these problems.
Objective three: Proportion problems.
If six out of 15 homes in a community have wells
for their water supply, how many homes have wells
in a community of 18,000 homes?
So we'll let x be the number of homes with wells
in the entire community.
We're going to set this up as a proportion when we see this
if six out of 15 homes.
So we're going to put the number with wells on top
and the total number of homes on the bottom.
You need to be consistent.
So we set up our proportion.
Six with wells out of 15 total.
We don't know the number of houses with wells
in the entire community, so we're going to put x for that.
And then the total number of homes
in the community was 18,000.
This equation models that problem.
Pause the video if you want to go through the steps
to solve it, and then here's our answer.
Today's exchange rate for the Australian dollar is 1.26744
Australian dollars equals one U.S. dollar.
Porter has 324 Australian dollars.
How many U.S. dollars will he get in trade?
So we'll let x be the number of U.S. dollars that Porter gets
in trade and this problem sets up as a proportion
because the relationships are the same
between the Australian dollars and the U.S. dollars
that he will get and the exchange rate.
We'll choose to put the Australian dollars on the top.
It doesn't matter, but you have to be consistent.
And we'll put the U.S. dollars on the bottom.
So the exchange rate is 1.26744 Australian dollars, so that goes
on the top where I have Australian dollars
and one U.S. dollar.
He has 324 Australian dollars so that's going to be
in the numerator, and x is the number of U.S. dollars.
This is the equation that models that problem.
Pause the video if you want to go through the steps
to solve it, and here's our answer.
The ratio of employees at a small company
who have their paychecks directly deposited
into their bank accounts to those who do not is nine to two.
If the number of people who have direct deposit is 14 more
than the number who do not,
how many employees do not have direct deposit?
When we see that relationship nine to two,
we know we can set this problem up as a proportion.
What are the things that we do not know.
We do not know how many people have direct deposit,
and we do not know the number of people
that do not have direct deposit.
It says that the number of people
who have direct deposit is 14 more than those who do not,
so we should let x be the number of employees
that do not have direct deposit and the 14 more, x plus 14,
would be the number of employees that do have direct deposit.
You get to choose what to put on top.
I'll put those that have direct deposits on top and those
that don't on the bottom.
So nine had direct deposits and two did not.
In our variables, x plus 14 was those
that do have direct deposits, and we put that on top.
And then x was the number that do not have direct deposits.
This equation models our problem.
Pause the video if you want to go through the steps
to solve it, and here's our answer.
This problem could have been worded a little differently.
What if instead of saying what it did earlier,
it said the ratio of employees at a small company
who have their paychecks directly deposited
into their bank accounts to the total employees is nine to 11.
To the total employees.
So now we're going to have a relationship of those
that have direct deposits to the total employees in the company.
So that's nine to 11.
We set up our problem exactly the same way as far as the x
and the x plus 14, x plus 14 is those
that do have direct deposits.
What do we put in the denominator over here
on the right, the total number of employees?
There were x employees that do not have direct deposits
and there were x plus 14 employees
that do have direct deposits.
So the total employees
in the company would be the sum of those.
So you'd add x and x plus 14.
This equation models the new problem,
and the solution is exactly the same as we got before.
Javier can run 300 yards in 34 seconds.
At that rate, how far can he run in 52 seconds?
A lot of people when they look at this,
to start with they're going to think, oh,
this is a distance, rate, time problem.
Well it is, but it also can be done with a simple proportion,
and that's what I'm going to do here.
When we see at that rate,
we know that it can be done as a proportion.
So let's x be the distance that Javier can run in 52 seconds.
I'm going to put yards on top and seconds on the bottom.
So 300 yards in 34 seconds, unknown x yards in 52 seconds.
This is the models our problem.
Pause the video if you want to go through the steps
to solve it, and here's our answer.
Alberto uses three-quarters cups of sugar
in his banana muffin recipe.
The recipe makes 12 muffins.
Alberto is baking for his daughter's school bake sale.
He wants to make 52 muffins.
How much sugar will he need?
So we're going to let x be the amount of sugar that he needs
to make the 52 muffins.
We can set this up as a proportion.
I chose to put the cups of sugar on top.
So we have three-quarters cups of sugar.
I'll write that as a decimal, .75, to 12 muffins.
We don't know the amount of sugar to make 52 muffins
so we'll but that in as x and 52 is the number of muffins,
and we have our proportion.
This equation models the problem.
Pause the video if you want to go through the steps
to solve it, and here's our answer.
Pause the video and try these problems.
Objective four.
Number problems.
The numerator of a certain fraction is one less
than the denominator.
If three is added to both the numerator and the denominator,
the result is equal to seven-eighths.
What was the original fraction?
So to figure out the original fraction,
we need to know the numerator and we need
to know the denominator.
It says that the numerator is one less than the denominator.
So we should let d be the denominator of the fraction
and one less than d, so d minus one is the numerator
of the fraction.
So our original fraction was d minus one over d. D minus one
for the numerator and d for the denominator.
In the second sentence, it says that three is added
to both the numerator and the denominator.
So we take and add three to d minus one and we add three
to d. The result is equal to seven-eighths.
So equals seven-eighths.
This is the equation that models that problem.
Pause the video if you want to go through the steps
to solve it, and here's our answer.
Pause the video and try this problem.