Modeling with rational expressions
- [Instructor] 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 is, in this case, it is ethanol, over the total volume, over 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. And so 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 multiplied both sides by 30 liters, that's going to give us the volume of ethanol, 'cause those two cancel. And 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 it 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 gonna 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? Well it's going to be the ethanol that we started with, the 30 liters times 18%, which is 5.4 liters, 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 gonna be 25%, that's the concentration of the gas that we're adding, times V, plus 0.25v. And 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 plus, 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. And 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, so 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 6 is, or actually 0.6, I have to be careful, is going to be equal to 0.05v. And 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. And 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.