Combining mixtures example

We're told a partially filled tank holds 30 liters of gasoline with an 18% concentration of ethanol. A fuel station is selling gasoline with a 25% concentration of ethanol. What volume in liters of the fuel station gasoline would we need to add to the tank to get gasoline with a 20% concentration of ethanol? Pause this video and see if you can figure this out.

All right, now let's work through this together. So let's first of all just remind ourselves how concentration relates to total volume to the volume of the ingredient. The way that we calculate concentration is that it is equal to the volume of the ingredient, which in this case is ethanol, over the total volume.

Now, this is already interesting because this first sentence tells us a lot. It tells us our concentration, it tells us our total volume. If we know two parts of this, in theory, we could figure out the third part.

Let's try that out. We know we're dealing with an 18% concentration. That's going to be equal to, they haven't told us our volume of ingredient, we just know that the ingredient is ethanol: volume of ethanol over the total volume. They have told us 30 liters, so if we multiply both sides by 30 liters, that's going to give us the volume of ethanol because those two cancel.

What we get is 18% of 30. Let's see, 18 times 3 is 54. So this is going to be 5.4 liters is equal to our volume of ethanol. Not only will this hopefully make a little bit clearer how these three relate, but this is also likely to be useful information for the rest of the problem.

But now let's go to where we're trying to get to. We're trying to find a volume in liters of the fuel station gasoline we would need in order to have this concentration. So let's set v equal to that, and we're trying to get a 20% concentration.

So what we could write is our 20% concentration is going to be equal to our new volume of ethanol. Actually, let me write that out. So it's going to be new volume of ethanol divided by our new total volume. Our new total volume, now, what's going to be our new total volume?

We're starting with 30 liters and then we're adding v liters to it. So our new total volume is going to be the 30 liters we started with plus the v liters that we're adding. And what's our new volume of ethanol? What's going to be the ethanol that we started with? The 30 liters times 18%, which is 5.4 liters, plus the volume of ethanol we're adding.

Well, to figure out the volume of ethanol we're adding, we just have to multiply the volume we're adding times the concentration of that volume. So it's going to be 25%, that's the concentration of the gas that we're adding, times v, plus 0.25v.

Now we have an equation to solve for v, and the best thing that I can think to do is let's start by multiplying both sides of this times 30 plus v. I'm also going to multiply this side times 30 plus v as well. These two characters cancel, and we are going to be left with, on the left-hand side, this over here—20% of that is going to be, if I distribute the 20%, 20% of 30 is 6, and then it's going to be, I'll write it as a decimal, 0.2v.

I'm just distributing the 20 over this expression here, and then that is going to be equal to, on the right-hand side, I just have the numerator here because the 30 plus v cancels with the 30 plus v. I have 5.4 plus 0.25v.

Now let's see, my v coefficient is larger on the right, so what I could do is try to subtract the 0.2v from both sides so I isolate the v's on the right. So let me do that: minus 0.2v, minus 0.2v. And then, actually, I'll just do one step at a time.

So that's going to get me on the left-hand side: 6 is equal to 5.4 plus, if I subtract here, this is 0.05v. Now I could subtract 5.4 from both sides, and what I'm going to get is—get is 6; or actually, 0.6, I have to be careful—is going to be equal to 0.00v.

Now, to solve for v, I can just divide both sides by 0.05. 0.05—that's going to get me, this is the same thing as 60 divided by 5. It gets me that v is equal to 12 liters, and we are done.

If you want, you can verify the new concentration when I add 12 liters of this concentration to the 30 liters of that concentration. You can verify that I now have a 20% concentration of ethanol.