Multiplying whole numbers and fractions
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Let's think about how or what 1/2 times 5 represents. So one way to think about it is that this could be five 1/2's added together. So you could view this as 1/2 plus 1/2 plus 1/2 plus 1/2 plus 1/2, which is the same thing as 1 plus 1 plus 1 plus 1 plus 1, over 2, which is equal to 5/2. The other way to think about this is that you start with 5 things. So let's say, that's 1 thing. Let me copy and paste that so they all look the same. So then let me paste it. So that's 2 things. That's 3 things. That's 4 things. And that's 5 things. So the other way to think about it is you start with 5 things, and you take 1/2 of them. So what would be 1/2 of this? Well, let's see. You have 5 things, so you would get-- 5 divided by 2 would be 2 and 1/2. So you would get this far. Let me make it like this. So you would get this one. You would get this one. And you would get this one. Now, is this the same thing as 5/2? Well, what happens if we divide each of these wholes into halves? So let's do that. So if we just multiplied-- so we just divide each of these into 2. So instead of having 5 wholes, we now have 10 halves. How many of those halves have we filled in? Well, we have filled in 1, 2, 3, 4, 5. So this is also equal to 5/2. So far we just did it thinking about what multiplication actually means. But if you said, well, how did I compute this? Well, the way you could think about it, and multiplying fractions is actually straightforward from that point of view, is as long as you can express both of them as fractions, and 5 we already know is the same thing as 5 ones, so this we can just multiply times 5/1. So now that I've expressed both of them as fractions, I can just multiply the numerator. So 1 times 5 over 2 times 1. And what's that going to be equal to? Well, 1 times 5 is 5. 2 times 1 is 2. So once again, we get 5/2.