Comparing Fractions 2 - Fraction, Arithmatic, Mathematics Video Lecture - JEE | Best Video for JEE

Use less than, greater than or
equal to compare the two
fractions 21/28 and 6/9.
Now, your impulse might just
be to look at these numbers
and say, oh, look, these are
larger numbers than those.
This must be bigger.
You cannot do that.
And we'll talk about in future
videos on why that is, or I
encourage you to think about why
you cannot do that, but to
do it properly, at least in my
mind, the best way is to get
these both in simplified forms
and to just think about what
they represent.
And in the future when we
learn to convert them to
decimals, that's actually the
best way to do it, or actually
for me, the easiest way.
But let's try to do what
we can so far.
So 21/28, we want to put this
in simplified form.
21 is the same thing
as 3 times 7.
28 is the same thing
as 4 times 7.
They're both divisible by 7.
Now, if you divide the numerator
and the denominator
by 7, 3 times 7 divided by
7 is just going to be 3.
4 times 7 divided by 7 is
just going to be 4.
So this guy right here is 3/4,
is equivalent to 3/4.
Now, let's think about
what 6/9 is.
Since these numbers are smaller,
this is actually a
good chance to visualize it.
So let's visualize if we have
ninths, let me draw the 9
ninths first. So if I were to
split it into thirds, I just
want to make it so
it's reasonably--
oh, that's not right.
Maybe there and there, and then
divide each of these into
thirds, so just like that, a
little bit more space to that
one, and then we have it
like that, and then
this one into thirds.
Let me draw it like that.
So here we have 9 pieces of a
candy bar, whatever you want
to think about it, or 9/9.
6/9 literally means one, two,
three, four, five six.
6/9 references 6 of the 9
potential pieces, 6 of these
one-ninth of a candy bar piece,
however you want to
think about it.
Now, 6 is the same thing as 2
times 3, so you can view it as
two groups of 3.
Let me do it this way.
You can view it as
two groups of 3.
So this is one group of 3.
This is one group of 3 right
there, and that is another
group of 3.
These are two groups of 3.
Now, what is 9 equivalent to?
9 is equivalent to 3 times
3, or three groups of 3.
So 9 is equivalent to that's
one group of 3, that is two
groups of 3, and now this
is three groups of 3.
So we could redraw this as--
so if we just draw just the
groups of 3, it would
look like this.
And these two groups of 3 that
the 6 represents, if we just
draw the groups of three, it
would be that guy and that the
guy right over there.
So how many thirds do we have,
if you consider each of those
groups of three as a third?
Well, we have 2/3.
This is equivalent to 2/3.
But doing the process that we're
used to doing, we could
imagine 6 is 2 times 3.
9 is 3 times 3.
If you divide 2 times
3 by 3, you get 2.
If you divide 3 times 3
by 3, you get the 3.
So 6/9 is the same thing as 2/3,
two of the thirds, of the
one-thirds.
That's what 6/9 is
equivalent to.
Now, let's think about
what's larger.
2/3, which is the same
thing as 6/9, or 3/4?
I didn't do 21/28 because
it would've
taken forever to draw.
So what does 3/4 look like?
What is 3/4?
So if we were to take this
exact same square-- so
let me redraw it.
So if we were to take that same
exact candy bar, so these
should be the same size, instead
of cutting it up into
three pieces, if we were to cut
it up into four pieces,
what do we get?
So let's see, that's cutting
up into two pieces, and now
let me cut it up into
four pieces.
So now this one has fourths,
this one has thirds.
3 in the denominator here-- so
let me make it very clear.
This represents 2/3.
Now, this is going to represent,
we have four
pieces, and if we're coloring in
three of them, so if we're
talking about 3/4, that's one--
let me draw it like
this, actually.
That is one, two-- let
me scroll up a
little bit-- three.
This is three of the fourths.
And the way I drew it, it is not
completely obvious to you.
Actually, the way I drew it
because-- let me draw it a
little bit neater than this.
Let me draw it a little bit
neater and it'll become a
little bit more obvious.
I didn't draw the fourths.
Here, drawing neatly
actually matters.
So let me draw it like
that and then split
this group-- nope.
I want to make it even, so I
want to make it like this.
OK, so if those are the fourths,
if we're talking
about three of the fourths,
we're talking about one, two,
three fourths, and you can see
that 3/4 is larger than 2/3.
2/3 gets us to this far of the
candy bar, while 3/4 is a
little bit more of
the candy bar.
2/3 is about that far, 3/4
is even more, so 3/4
is larger than 2/3.
And we'll show you in the
future how to do it
numerically.
You can use a calculator, if you
like, or you could do it
by hand, but you can see from
this visually that 3/4 is
greater than 2/3.
Now, they want to compare
these two fractions.
So 21/28 is the same
thing as 3/4.
So 21/28 is going to be greater
than 2/3, which is the
same thing as 6/9.
And so we are done.