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Comparing fractions with > and < symbols
When you write a fraction, there are actually words for the top number and the bottom number. And the words are a lot more fancy than the word "top number" and "bottom number." What mathematicians typically use is the word "numerator" for the top number and "denominator" for the bottom number. And what I want to do now that we know that the top number is the numerator of the fraction and the bottom number is the denominator, I want to compare pairs of fractions that have either the same denominator or the same numerator. So let's look at this first pair. I want to compare 4/7 to 3/7. And I have two wholes right over here. They're the same hole, and I've divided them into sevenths. I've divided them into 7 equal chunks. And I want to see what's larger, 4/7 or 3/7. So what I can do is, I can fill in 4/7. Let me select 4 out of the 7. So, that's 1, 2, 3, 4. And the fact that trying to even get to 4/7 I had to have 3/7 first gives you good clue that 4/7 is probably larger, or it is larger. But now let's color in 3/7, just so we can compare. So 1, 2, 3/7. And so it's pretty clear that on the left-hand side, we are shading in more of the whole than on the right-hand side. So, 4/7 represents a larger fraction, more of the whole than 3/7. And the way that we can state that comparison mathematically is with the greater than symbol. We can write 4/7 is greater than 3/7. Now, the greater than and less than symbols can sometimes be confusing. This is greater than. This is less than. And the way that I remember it is that the greater than symbol, either symbol, the small pointy side is always on the side of the smaller number, and the big open side is always on the side of the larger number. So here, big open side is opening towards the 4/7, small pointy side opening to the 3/7. 4/7 is greater than 3/7. Now, what about 3/7 and 3/4? So, here I have different denominators, but I have the same numerator. And so I encourage you to pause this video and draw maybe little boxes like this and try to judge for yourself which of these is a larger fraction, represents a larger number. Well, let's color them in. So, let's think about 3/7 first. And we actually already drew it here, but I can do it really fast right over here. So that is 3/7. I've colored in 3 of the 7 equal groups. And what would 3/4 be? Well, that's 1/3, 2/4, and 3/4. And so it's pretty clear that 3/4 represents a larger fraction of the whole, that 3/4 is larger, or that 3/7 is smaller. So we could write that 3/7 is less than 3/4. So, notice, same exact numerator. When I divided it-- because this fraction symbol could also be viewed as division. When I have it as more equal groups, so it's a fraction of more equal groups, so 3 out of 7 versus 3 out of 4, this is a smaller number, which which makes sense. Now, let's compare these two. We have the same denominator, different numerators-- 3/4 versus 2/4. Well, 3/4 we've already looked at. We can just shade in 3 of these. So 3 of these fourths. So that's 3/4 right over there. And then 2/4, well, we're going to only have 2 of the fourths, 1, 2. So 2/4 is clearly the smaller number. 3/4 is the greater number. So, once again, we could write 3/4 is greater than 2/4. And then finally, I encourage you to pause the video. Try to come up with whether 2/4 or 3/6 is a larger number. Well, let's color it in again. We've already seen 2/4. We just have to color in 2 of our fourths. So let's just color in 2 of our fourths right over there. And then 3/6, we've split our whole into 6 equal sections-- 1, 2, 3, 4, 5, 6. We need to color in 3 of them. And as you see, we're coloring in the exact same amount of the whole. These two fractions are equivalent. These are equivalent fractions. 2/4 is equal to 3/6. And as you see here, they're both filling in half of the whole. If we were to just draw the whole and split it only-- let me do this in a different color. If we have our whole, and we were to split it only into two sections, we are shading in exactly 1 out of the 2 sections. So you could say that 2/4 is equal to 3/6, and they both equal 1/2. So 1/2 is equal to 2/4 is equal to 3/6.