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
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
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.