More on equivalent fractions

What I want to do in this video is really digest the idea that if we have some fraction, as long as we multiply the numerator and the denominator of the fraction by the same number, then we're going to have an equivalent fraction. So let's think about that. Let's say we multiply the denominator here by 2. I'm claiming that as long as we multiply the numerator by 2, we are going to get an equivalent fraction. So here, the denominator was 6. So here, our denominator will be 12. If our numerator here is 4, well, we've got to multiply by 2 again, multiply our numerator by 2, to get 8. So I'm claiming that 8/12 is the same fraction as 4/6. And to visualize that, let me redraw this whole. But instead of having 6 equal sections, we now have 12 equal sections. So each of the six we can turn into 2. That's essentially what multiplying by 2 does. We now have twice as many equal sections. Now that we have twice as many equal sections-- literally one, two, three, four, five, six, seven, eight, nine, 10, 11 12-- how many of them are actually shaded in yellow? Well, one, two three, four five, six, seven, eight-- 8/12. And there's no magic here. If we have twice as many sections, we're going to have to shade in twice as many of them in order to have the same fraction of the whole. And it goes the other way, too. This isn't just true with multiplication. It's also true that if we divide the numerator and the denominator by the same quantity, we are going to have an equivalent fraction. So that's another way of saying, well, what happens if I were to divide by 2? So if I were to divide by 2-- so let me divide by 2-- I'm going to have 1/2 the number of equal sections. Or I will only have three equal sections. And I'm claiming if I do the same thing in the numerator, that this is going to represent the same fraction. So 4 divided by 2 is 2. So I'm claiming that 2/3 is the same fraction as 4/6 is the same fraction as 8/12. Well, let's visualize that. So here, this is 6 equal sections. But now, we're going to have only three equal sections. So we can merge some of these equal sections. So we can merge these two right over here. And we can merge these two right over here. And then, we can merge these two right over here. So our whole is still the same whole. But now, we only have three equal sections. And two of them are actually shaded in. So these are all equivalent fractions. So the big takeaway here is start with a fraction. If you multiply the numerator and the denominator by the same quantity, you're going to have an equivalent fraction. If you divide the numerator and the denominator by the same quantity, you're also going to have an equivalent fraction. So with that in our brains, let's tackle a little bit of an equivalent fractions problem. Let's think about-- if someone says, OK, I have 5/25, and I want to write that as some value, let's call that value t, over 100, what would t be? Well, we can see in the denominator to go from 25 to 100, you had to multiply by 4. So if you want an equivalent fraction, you have to multiply the numerator by 4 as well. So t will need to be equal to 20. So t is equal to 20. 5/25 is the same thing as 20/100. But what if someone says, well, 5/25 is equivalent to blank, let's say question mark, over 5? Well now what would you do? Actually, let's do it the other way-- is equal to 1 over question mark. Well, you could say, look, to get our numerator from 5 to 1, we have to divide by 5. We have to divide by 5 to go from 5 to 1. And so similarly, we have to divide the denominator by 5. So if you divide the denominator by 5, 25 divided by 5 is going to get you 5. So these are all equivalent fractions. 1/5 is equivalent to 5/25, which is equal to 20/100.