Help setting up a linear equation from a word problem.

Oh yes, you all will be my algebra tutors.

My word problem is this:

A radiator holds 20L of 20% antifreeze solution. For this particular model of car it should be 40%. How much do you need to drain and refill with 100% solution to make it 40%?

I know that the amount that needs to be refilled is my variable, x. What I can't figure out is how to set up the equation. Anyone want to help me understand this?

Re: Help setting up a linear equation from a word problem.

It's been a while since I've done, well, math, but I think this makes sense. I thought about it in terms of volume of antifreeze in the tank.

The antifreeze concentration is the amount of antifreeze per volume. So at 20%, there is one liter of antifreeze per five liters of water. In a 20L tank, the amount of antifreeze is 20 (liters in the tank) * 0.2 (@ 20% concentration). We want 20 liters of antifreeze at 40%. So, the amount of antifreeze is 20 (liters in the tank) * 0.4 (at 40% concentration).

I thought of x as 100% antifreeze.

So:

20(0.2) - 0.2x + x = 20(0.4)

20 is the total volume of the tank. 0.2 is the concentration of the antifreeze. We're going to remove some of that 20% antifreeze, and we have to do it in terms of x, which is 100% antifreeze. So we remove 0.2x.

Simplifying, we get 4 - 0.2x + x = 8, then 4 + 0.8x = 8, then 0.8x = 4, then x = 5.

Re: Help setting up a linear equation from a word problem.

It may be simpler to think about it as (20 - x)(0.2) + x = 20(0.4). (20 - x) represents what is in the tank after you remove something (the -x part), but before you add anything. By definition, that is at 20% concentration, so it's (20 - x)(0.2).