Converting units: US volume

How many cups are in 3 and 1/2 gallons? So before even addressing this question, let's just think about how large a cup is. Actually, I'll give you a little bit of overview of how many cups there are in a pint, how many pints in a quart, and how many quarts in a gallon. Let me just draw a cube here, and let's imagine that this is a gallon. The most common time we see a gallon is when you see a gallon of milk. So let's say that that whole thing is a gallon. You can imagine if it had a handle, it would be kind of a big gallon of milk. Now, there are 4 quarts per gallon. Let me write this over here. There are 4 quarts per gallon. So if I were to draw the quarts here, I could divide this gallon into 4 quarts, and then each of these sections would be a quart. So you would have 4 quarts. So this right here that I've just drawn in blue would be exactly 1 quart. And obviously, there's 4 of them in this entire gallon. Now, you can divide the quarts into pints. You have 2 pints per quart. So this quart that I drew here, I can divide it into 2, like that, and this little section that I'm highlighting in magenta is a pint. That is a pint right over there. And then finally, there are 2 cups per pint. So this pint right here, I can divide it into 2, and each of these will be a cup. So this section right here will be a cup. Now, we can go straight and figure out exactly how many cups there are per gallon. Actually, that might be an interesting way to think about it. If you have 4 quarts-- let's multiply it right here. So you have 4 quarts per gallon times 2 pints per quart. What does this give you? This gives you 4 times 2 is equal to 8. And then the quarts cancel out, and you have 8 pints per gallon. And that makes complete sense because we had 4 quarts in this gallon, and then each of those quarts have 2 pints in them. So 4 times 2. So 8 pints per gallon. And then we can multiply that times 2 cups per pint. So I could just copy and paste this right here. Actually, I should've cut and paste. Let me select it again. I want to do that so I get that real estate back. So edit, cut, edit, paste. There you go. So now you multiply this times 2 cups per pint. And the reason why this will work is because you have pints in the numerator. It cancels out with the pints in the denominator. And you will be left with-- I'll go back to the yellow-- 8 times 2 is 16. In the numerator, we have cups per gallon. Now, we just figured how many cups there are per gallon. That makes sense. This section right here is exactly 1/16 of this entire cube, this entire gallon. But we haven't even answered our question. We want to figure out how many cups there are in 3 and 1/2 gallons. So let's write it over here. So we're concerned with 3 and 1/2 gallons. I don't like working with mixed numbers. I like to turn them into improper fractions. 3 and 1/2 is the same thing as 2 times 3 is 6, plus 1 is 7. This is the same thing as 7/2. If you divided 7 by 2, you would get 3 with a remainder of 1, or this would be 3 and 1/2, so this is the exact same thing. So we want to know how many cups are in 7/2 gallons. So what we want to do is end up with cups, and we want the gallons to cancel out. So we have gallons in the numerator right here. It's definitely not in the denominator. And so we want to divide by gallons. And then we're going to have a numerator. We have cups in the numerator. And how many cups are there per gallon? Well, we just figured that out. There are 16 cups per gallon. When you multiply these two quantities, the gallons will cancel out, and you'll just be left with cups, and that's what we wanted. So it's going to be 7/2 times 16. So this is going to be 7 times 16/2 cups. You could divide 16 by 2 to get 8. 2 divided by 2 is 1. So it just becomes 7 times 8 divided by 1, or just 7 times 8, which is 56. So this is equal to 56 cups. And this should make sense. This should be a much larger number because cups are a much smaller unit. So if you have 3 and 1/2 gallons, you will have many, many, many more cups in that 3 and 1/2 gallons, so this makes sense.