Hey, this is Presh Talwalkar.
Alice and Bob can complete a job in two hours.
Alice and Charlie can complete the same job in three hours.
Bob and Charlie can complete the same job in four hours.
How long will the job take if Alice, Bob, and Charlie work together?
Assume each person works at a constant rate whether working alone or working with others.
This problem has been asked to students in US colleges.
To the professor's surprise,
many of the students set up the wrong equations and could not solve this problem.
Can you figure it out?
Give this problem a try and when you're ready keep watching the video for the solution.
Before I get to the solution, let me go over a common mistake in how students get to the wrong answer.
They read the first sentence, that Alice and Bob can complete a job in two hours,
and translate the names and the numbers into an equation.
They say this must mean that A + B = 2.
They look at the second sentence, that Alice and Charlie can complete the job in three hours,
and they similarly convert it to A + C = 3.
The third condition, that Bob and Charlie can complete the job in four hours,
gets converted to the equation B + C = 4.
The question of how long it will take for all three of them working together
gets translated into the question of "what is A + B + C = ?"
So to solve this system of equations...
They want to solve for A + B + C, so they can add up all the equations together.
We end up getting two terms of A, two terms of B, and two terms of C,
to equal 2 + 3 + 4.
If we group the factors [summands], we get 2A + 2B + 2C = 9.
We then divide by two and that gets us to
A + B + C = 9/2 = 4.5 (four and a half).
So evidently this will be the answer that many students get.
They would say that it takes 4.5 hours for all three of them, when working together.
But let's think about: does this answer make any sense?
We know that Alice and Bob take two hours,
Alice and Charlie take three hours and Bob and Charlie take four hours,
but somehow when all three are working together they take four and a half hours?
This makes no sense. When three people work together it should take less time then when only two people work together!
But 4.5 hours is more time, so this answer must be wrong.
Not only were the equations set up incorrectly,
but any student who submits this answer
is not thinking about whether the answer makes any logical sense.
So how do we solve this problem?
We need to set up the equations in the correct method. We know that Alice and Bob can complete a job in two hours,
so how do we translate this into an equation?
Well, if they complete the job in two hours, that means the percentage of the job that Alice does in two hours
plus the percentage of the job that Bob does in two hours equals 100%, or that equals 1.
Now since they work at a constant rate,
we can say the amount of the job that Alice does in two hours is
2 times the amount of the job that she does in one hour.
And the same thing goes for Bob.
So we now have a natural choice for our variables
we can say the percentage of the job that Alice does in one hour will be the variable A, and the
percentage of the job that Bob does in one hour will be the variable B.
This leads to the equation that 2A + 2B = 1.
And that's how we can translate this we can group this out to be 2 times (2x) the quantity A + B = 1.
So we can now translate the second sentence. We have Alice and Charlie completing the job in three hours.
This will translate into 3 times the quantity A + C = 1,
where C is the percentage of the job that Charlie completes in one hour.
We also have that Bob and Charlie can complete the job in four hours,
so that would mean 4 times the quantity B + C = 1.
Now we want to figure out... What would happen if they all three work together?
So we are needing to solve for...
the time (t) x (A + B + C) = 1.
We need to solve for this variable "t".
So how do we do that?
Well, we can similarly add up all the equations
But we have different quantities of each of these variables, so in order to get the same number of each variable,
we're going to do a little trick.
We're going to multiply each equation so that there's a leading coefficient of 12,
which is the lowest common multiple of 2, 3 and 4, so
the first equation will multiply by six.
This will get 12 times the quantity A + B to be equal to 6.
The second equation will multiply by four and the third equation will multiply by three.
We can now add up all of these equations
We'll end up with 12A (2 times),
12B (2 times), and 12C (2 times), and this will be equal to 6 + 4 + 3.
We can factor out the 24 of each of these variables,
and that will be equal to 6 + 4 + 3, which equals 13.
We now divide by 13...
and we end up with 24 divided by 13 times the quantity A + B + C = 1, and
THAT is what we wanted to figure out.
So we go back to our setup and we can see that we get to the answer of 24 over 13.
So the job will take 24 over 13 hours or about 1 hour and 51 minutes.
And this is a sensible answer because it takes less time than any pair working together.
Did you figure this out?
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