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### Raising the Antifreeze Ratio

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Date: 08/07/99 at 11:32:45
From: Steven Robles
Subject: Antifreeze Ratio

I can't understand this problem:

A 5-gallon radiator containing a mixture of water and antifreeze was
supposed to contain a 50% antifreeze solution. When tested, it was
found to have only 40% antifreeze. How much must be drained out and
replaced with pure antifreeze so that the radiator will then contain
the desired 50% antifreeze solution?

Thank you.
```

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Date: 08/20/99 at 19:48:12
From: Doctor Jesse
Subject: Re: Antifreeze Ratio

Hey - this *is* a hard one! Good for you, Steven, for not giving up on
it.

Like most word problems, the hard part is in setting up the equation.
Let's look at what we know:

A 5-gallon radiator containing a mixture of water and antifreeze
was supposed to contain a 50% antifreeze solution.

Well, that means that the tank is supposed to contain 2.5 gallons of
antifreeze, and 2.5 gallons of water.

When tested, it was found to have only 40% antifreeze.

Hmm... That means the tank had (5gal * 0.4) = 2 gallons of antifreeze
and (5gal * 0.6) = 3 gallons of water in it.

How much must be drained out and replaced with pure antifreeze so
that the radiator will then contain the desired 50% antifreeze
solution?

If you could take a half-gallon of water out of the tank and replace
it with a half-gallon of antifreeze, you'd be done... But you can't do
that. When you pour out the tank, whatever you pour out will be 40%
antifreeze and 60% water, which is what makes this problem tricky. But
think of it this way:

We have 2 gallons of antifreeze in the tank now, and we need to get it
so that there are 2.5 gallons of antifreeze instead. If we let x be
the amount of pure antifreeze we need to add to the tank, then we can
set up an equation:

2 + x = 2.5

But wait - for every gallon of pure antifreeze we add to the tank, we
first have to take out an equal quantity of 40% antifreeze... so,
subtracting that part gives us the equation:

2 - 0.4x + x = 2.5

Solve for x, and you will know how much 40% antifreeze had to be
poured out, and how much pure antifreeze added, to get to the 50%
mixture. I bet you can take it from here.

You also might want to check your answer when you are done. You can do
this by going to the auto parts store and buying 2.5 gallons of pure
antifreeze... just kidding! You can check it by plugging x into the
equation you just solved, and making sure that the answer on the left
side really does come out to be 2.5 gallons.

Good luck, Steven!

- Doctor Jesse, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Ratio and Proportion

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