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Volume of a rectangular prism: fractional dimensions

Video transcript
Let's see if we can calculate the volume of this rectangular prism, or I guess you think this thing is the shape of a brick or a fish tank right over here. And what's interesting is now that the dimensions are actually fractions, we have a width. Yeah, we could call this the width. The width here is 3/5 of a unit. The length here is 1 and 1/6 units, and the height here is 3/7 of a unit. So I encourage you to pause this video and try to figure out the volume of this figure on your own before we work through it together. So there's a couple of ways to think about it. One way to think about it is you're trying to pack unit cubes in here, and one way to think about how many unit cubes could fit in here is to think about the area of this base right over here. So sometimes you'll see volume is equal to the area of the base times the height. This right over here is the height, and let me make it clear. This is the area of the base. Area of the base times the height. Well, what's the area of the base? Well, the area of the base is the same thing as the length times the width, so you might see it written like that. You might see it written as area of base is going to be your length times your width. Length times width is the same thing as your area of the base, so that's that right over there. And of course, you still have to multiply times the height. Or another way of thinking about it, you're going to multiply your length times your width times your height. You're going to multiply the three dimensions of this thing to figure out how many unit cubes could fit into it, to figure out the volume. So let's calculate it. The volume here is going to be-- what's our length? Our length is 1 and 1/6 units. Now, when I multiply fractions as I'm about to do, I don't like to multiply mixed numbers. I like to write them as improper fractions, so let me convert 1 and 1/6 to an improper fraction. So 1 is the same thing as 6/6. Plus 1 is 7/6. So this is going to be 7/6-- that's my length-- times 3/5-- that's my width-- times the height, which is 3/7. And we know when we multiply fractions, we can multiply the numerators, so it's going to be 7 times 3 times 3. And the denominator, we can just multiply the denominators. So it's going to be 6 times 5 times 7. Now, we could just multiply these out, but just to try to get an answer that has as simplified as I can make it, let me-- we see we have a 7 in the numerator and a 7 in the denominator, so let's divide the numerator and the denominator by 7. And what that does is that becomes 1, and those become 1. We also see what the numerator and the denominator has 3. They're both divisible by 3. We see a 3 up here. We see of 3 over here. So let's divide both the numerator and the denominator by 3. So we divide by 3. Divide by 3. 3 divided by 3 is 1. 6 divided by 3 is going to be equal to 2. So in our numerator, what are we left with? This is going to be equal to what we were just left with, that green 3. It's going to be equal to 3 over 2 times 5. 2 times 5 is 10. 2 times 5 right over here. So it's going-- the volume over here is 3/10 units cubed, or we could fit 3/10 of a unit cube inside of this brick, or this fish tank, or whatever you want to call it.