Word problems with multiple units
Your car gets 25 miles per gallon, and you want to go on a 400-mile road trip. Right now, gas costs $3 per gallon. How much will the gas for your road trip cost? So let's see. They tell us they're going on a 400-mile road trip. 400 miles. So the first thing that I'd want to think about is, well, how many gallons am I using? And then once I know how many gallons I'm using, I know it's $3 per gallon, so I can multiply the number of gallons by $3. So to figure out the number of gallons, would I want to multiply 400 miles-- would I want to multiply that times the miles per gallon, which is 25, or would I want to multiply by the gallons per mile? Well, if I multiply by the gallons per mile, and I multiply that times 400, then I would get the number of gallons. So let's just think about that. I want to multiply that times-- and I'll write the units first-- the gallons per mile. And what are the gallons per mile? Well, we have 25 miles per gallon. We have 25 miles for every 1 gallon. Or you could say we have 1 gallon for every 25 miles. So I really just took the reciprocal of 25 miles per gallon and made it 1/25 gallons per mile. Now what do we get when we multiply these two things? The whole purpose was to figure out how many gallons we're going to use. Well, we see that our miles cancel out. Miles cancel out with miles. And then I have 400 times 1/25 gallons, which is the same thing as 400 divided by 25 gallons. So this is equal to 400/25 gallons, which is the same thing as 400 divided by 25 is equal to 16 gallons. Now it's always important to do a reality check here, not just to try to blindly cancel out units. Does this actually makes sense that 16 is a much lower number than 400? Well, sure it does. And actually, if you have any experience filling up a car, you would sense that, OK, well, that's about the size. On around 16 gallons, a car tends to go 300 or 400 miles if it gets pretty good fuel mileage. So that just makes sense from experience. And it also make sense based on how it's stated. You get 25 miles per gallon. So you're going to need fewer gallons than you're going to need miles. So this all seems to make sense so far. But we haven't answered their question. They want to know, how much will my trip cost? Right now, we've just figured out how much fuel we're going to use. So then we could take our 16 gallons. And to figure out the dollar cost, are we going to multiply it by dollars per gallon or gallons per dollar? Well, if we're thinking just about unit conversion, we want to multiply times the dollars per gallon. So I could write it like this. I could write it like dollars per gallon. Actually, let me just write out the word "dollar." Dollars per gallon. The units will cancel out. And it also makes sense. Whatever number of dollars per gallon, I multiply it times the number of gallons, and that's going to tell me how much it's going to cost. This happens at the fuel pump every day. Hey, it's $3 per gallon. I'm going to fill up 16 gallons. Hey, 3 times 16. So let's do that. So it's $3 per gallon. We see the gallons cancel out. And we are left with 16 times $3, which is the same thing as $48. And we are done.