Intro to multiplying 2 fractions

Let's think about what it means to multiply 2 over 3, or 2/3, times 4/5. In a previous video, we've already seen how we can actually compute this. This is going to be equal to-- in the numerator, we just multiply the numerators. So it's going to be 2 times 4. And in the denominator, we just multiply the denominator. So it's going to be 3 times 5. And so the numerator is going to be 8, and the denominator is going to be 15. And this is about as simple as we can make it. 8 and 15 don't have any factors common to each other, than 1, so this is what it is. It's 8/15. But how, why does that actually makes sense? And to think about it, we'll think of two ways of visualizing it. So let's draw 2/3. I'll draw it relatively big. So I'm going to draw 2/3, and I'm going to take 4/5 of it. So 2/3, and I'm going to make it pretty big. Just like this. So this is 1/3. And then this would be 2/3. Which I could do a little bit better job making those equal, or at least closer to looking equal. So there you go. I have thirds. Let me do it one more time. So here I have drawn thirds. 2/3 represents 2 of them. It represents 2 of them. One way to think about this is 2/3 times 4/5 is 4/5 of this 2/3. So how do we divide this 2/3 into fifths? Well, what if we divided each of these sections into 5. So let's do that. So let's divide each into 5. 1, 2, 3, 4, 5. 1, 2, 3, 4, 5. And I could even divide this into 5 if I want. 1, 2, 3, 4, 5. And we want to take 4/5 of this section here. So how many fifths do we have here? We have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. And we've got to be careful. These really aren't fifths. These are actually 15ths, because the whole is this thing over here. So I should really say how many 15ths do we have? And that's where we get this number from. But you see if 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15. Where did that come from? I had 3, I had thirds. And then I took each of those thirds, and I split them into fifths. So then I have five times as many sections. 3 times 5 is 15. But now we want 4/5 of this right over here. This is 10/15 right over here. Notice it's the same thing as 2/3. Now if we want to take 4/5 of that, if you have 10 of something, that's going to be 8 of them. So we're going to take 8 of them. So 1, 2, 3, 4, 5, 6, 7, 8. We took 8 of the 15, so that is 8/15. You could have thought about it the other way around. You could have started with fifths. So let me draw it that way. So let me draw a whole. So this is a whole. Let me cut it into five equal pieces, or as close as I can draw five equal pieces. 1, 2, 3, 4, 5. 4/5, we're going to shade in 4 of them. 4 of the 5 equal pieces. 3, 4. And now we want to take 2/3 of that. Well, how can we do that? Well, let's split each of these 5 into 3 pieces. So now we have essentially 15ths again. So 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15. We want to take 2/3 of this yellow area. We're not taking 2/3 of the whole section. We're taking 2/3 of the 4/5. So how many 15ths do we have here? We have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. So if you have 12 of something, and you want to take 2/3 of that, you're going to be taking 8 of it. So you're going to be taking 1, 2, 3, 4, 5, 6, 7, 8 or 8 of the fifteenths now. So either way, you get to the same result. One way, you're thinking of taking 4/5 of 2/3. Another way you could think of it as you're taking 2/3 of 4/5.