In this video we’re gonna be comparing fractions that either have the same numerator or the same denominator. We’ll be using less than, greater than, and equal signs to express the differences. And we’ll also be reminding ourselves that we can only sensibly compare fractions when they’re referring to the same whole. For example, half a mouse is not equal to half of an elephant.
First though let’s quickly remind ourselves what numerators and denominators are. A numerator is what we call “the upstairs number” on a fraction and a denominator is what we call “the downstairs number” on a fraction.
Now let’s get a scrumptious looking chocolate cake. That’s a whole cake. And if we split it into one piece and eat one of those pieces, then we’ve eaten the whole cake on our own. Yum, but ouch! So one represents the whole cake. And we’ve eaten one-oneth or one over one — one out of one — of that cake.
Now let’s take another cake exactly like the last cake. But this time we’re gonna split it into two equal pieces so that we can share the cake between two people. Each person would get a half, one over two, a half of the cake. We take another similar cake and split it into four pieces to share between four people. So each person gets one out of the four pieces; that’s one-quarter of the cake. Now we got a third cake and we split that into six equal pieces. And each person getting a piece of that cake would have one out of the six pieces; that’s a sixth of a cake.
So if we share a cake between two people, each person gets a bigger share of the cake than they would do if we shared it between four people. And if we share a cake between four people, each person gets a bigger share of the cake than they would if they would have shared it between six people.
So we can use this symbol here to represent greater than or larger than. So a half is greater than a quarter. And a quarter is greater than a sixth. Now that sign might look a bit confusing to start off with. But what you gotta remember is it’s got a great big end and it’s got a small end. And the big end goes against the big number and the small end goes against the small number. Likewise over here a quarter is bigger, so the big end of the sign goes to the quarter. A sixth is smaller, so the small end of the sign goes against the sixth.
Now there’s another version of that sign as well called the less than sign. So we know a quarter is less than a half, so the small end of the sign goes against the smaller number. The big end of the sign goes against the bigger number. A sixth is less than a quarter, so the small end of the sign goes against the smaller number and the larger end goes against the larger number.
Now some of you might be thinking whoa, hold on a second! You’re saying that a half is bigger than a quarter, but two is smaller than four. But what you gotta remember that a denominator is telling us how many people were sharing the cake between. And if we share a cake between more people, we’re getting a smaller amount each. So if we’ve got one piece of a cake that we shared between four people, that’s gonna be smaller than we would’ve got if we would’ve shared that- if we have got that one piece of a cake shared between two people. Remember this quarter is smaller than this half of the same cake.
So to summarize that, with fractions if the numerators are equal, then the fraction with the larger denominator represents the smaller proportion. We can see these have all got one as their numerator. A sixth represents a smaller proportion than a half. It’s got a bigger denominator than half, but it represents a smaller piece of the cake we’re sharing it out between more people.
Now let’s compare some more pieces cut from identical cakes. This time we’ve cut them into halves, quarters, and eighths. And using what we just learned, we can see that because they’ve all got one as their numerator, as the denominator increases the pieces of cake get smaller. So a half is bigger than a quarter and a quarter is bigger than an eighth. But what if I took more than one piece from some of the cakes? If I took two of the quarters or four of the eighths, then I’d have an equal amount of cake in each case. And obviously we use the equal sign to represent when we’ve got equal bits of cake. So a half is equal to two- quarters or two-quarters is equal to four-eighths. In fact in this case we can do these all three ways. And we can also say that a half is equal to four-eighths.
Now in all the examples that we’ve looked at so far in this video, the cakes we’ve been cutting up were the same size, so we can compare fractions directly. But it’s important to remember that you have to be careful when comparing fractions. If there’re fractions of different wholes, then you can’t easily compare them. So a quarter of a small cake is not equal to a quarter of a large cake.
Okay back to two cakes which are the same size. And we’ve cut these equal sized cakes into eight equal sized pieces. And the cake on the left, we’ve selected one of those eighths and the cake on the right, we’ve selected three of the eighths. So now we got two fractions in which the denominators are the same. So given that the cakes are the same size and we’ve cut them up into the same number of pieces, eight, then the more pieces of that cake that we have, the more cake we’ve got overall.
So three-eighths is gonna be bigger than one-eighth. So we can draw the sign in with the big end of the sign against the three-eighths and the small end of the sign in against the one-eighth. Or we could’ve written the fraction the other way round and written the sign the other way around. So three-eighths is greater than one-eighth.
So for fractions with the same denominator, then a higher numerator means a larger proportion. If we’ve got more pieces of the same size cake, then we’ve got a larger proportion of that same size cake.
So moving away from cake for a moment, we can also make comparisons on the number line. So if we have a half and compare that to a third, we can see that a half is bigger than a third. Because if I just draw a little dotted line coming down here, this bottom red arrow is slightly smaller than the top red arrow. And that ties in with what we’ve just learned. If we’ve got the same numerator, then the bigger denominator will be the smaller fraction. The bigger denominator means we’re sharing the cake out between more people. So each person is gonna get a smaller piece of cake. And don’t forget we could also write that round the other way: a third is less than a half.
Now let’s think about a quarter. Again if we compare a third and a quarter, the arrow representing a quarter is shorter than the arrow representing a third. So a quarter is less than a third or a third is greater than a quarter.
And again because we’ve got the same numerator in each case, the larger the denominator the smaller the proportion that represents. Remember we’re sharing that cake between more people, so everyone’s gonna get a smaller piece.
Okay it’s time to test yourself on what we’ve learned. I want you to write those symbols: less than or greater than or equals to to complete these statements. Now we’re assuming that these are fractions of the same overall sized whole. So just pause the video now and then check your answers in a moment.
In number one then, we’ve got the same numerator. So the bigger denominator means that we’re sharing that out between more people. So it’s gonna be the smaller fraction, so a third is greater than a ninth. The small end of the sign points to the small fraction. With number two, we’ve also got the same numerator, but hold on! We’ve also got the same denominator. So three-fifths is the same as three-fifths in this case. So we put our equal sign; those two fractions are the same. For number three, we’ve got the same numerator again. And the larger denominator means we’re sharing that out between more people. That’s gonna be the smaller fraction. So the small end of the sign goes against that and the large end of the sign goes against the other fraction.
For number four, we’ve got the same denominator. So all these little pieces are the same size. Now for the first fraction, we’ve got one of those fifths of a cake and in the second fraction we’ve got three of the fifths of a cake, so we’ve got more in the second fraction. So the big end of the sign goes against the bigger fraction. The small end of the sign goes against the smaller fraction. In number five, we’ve also got the same denominator. So again we’re looking for which is the biggest numerator; that’s gonna be the biggest fraction cause we’re gonna have more of those sevenths of cake. So it’s gonna go this way around: four- sevenths is more than only two-sevenths. And the last question here number six, we don’t have the same numerator and we don’t have the same denominator. So what we gotta remember is our equivalent fractions thing. So look — two — if I multiply two by two, I get four. And if I multiply three by two, I get six. So we’ve multiplied the numerator by two and the denominator by two, the same number in each case, so that means that they’re equivalent fractions. So in fact these are equal.