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# Comparing improper fractions and mixed numbers

Worked examples comparing improper fractions and mixed numbers. Created by Sal Khan.

Video transcript

I've got pairs of mixed
numbers and improper fractions, and I want to think about
which of the two is larger. So 1 and 7/8, 39/10. So you could do
this in your head. You could say 10
goes into 39, I'll even write it out, 10 goes
into 39 3 times, 3 times 10. And you want to find the
largest number of times 10 goes into this
without going over. So you couldn't
write a 4 over here, because then that would be 40. That would go over 39. 3 times 10 is 30. And then you have
a remainder of 9. So you could rewrite this
expression right over here. Instead of 39/10 you could
write it as 30/10 plus 9/10. And 30/10 is just 3. So this is equal to 3 and 9/10. And you could do
that in your head. You could say 10 goes into 39 3
times, and the remainder is 9. You have your 9/10. And that's essentially just
doing this in your head. So now we can compare,
and we can literally just look at the whole number parts. This is 1 and
something, 1 and 7/8, and we're comparing it to
essentially 3 and 9/10. 3 and 9/10 is clearly
a larger number. We have a 3 out
here instead of a 1, so we will write less than. And the way I always remember
it is, the opening always faces the larger number. And the point is small. It always points to
the smaller number. Now let's do this next one. 4 and 7/8 versus 49/9. So let's convert this
to a mixed number. 9 goes into 49 5 times,
and 5 times 9 is 45. So the remainder
is going to be 4. The remainder is 4,
so this is 5 and 4/9. Once again, we
can literally just look at the whole number parts. 5 is clearly larger than 4,
so once again, less than. Point facing the
smaller number, opening facing the larger number. Now 2 and 1/2 versus 11/10. 10 goes into 11 only 1 time. And if you care about
the remainder, it's 1. So it's 1 and 1/10. Which is clearly
smaller than 2 and 1/2. You just look at the
whole number parts. 2 is clearly larger than 1. So we want our opening of our
less than or greater than sign to face the larger number. So we would write it like this. And this is greater than, so 2
and 1/2 is greater than 11/10. The little point facing
the smaller number. 5 and 4/9 versus 40/7. 7 goes into 40, so
let me rewrite this, 7 goes into 40 5 times. And then you're going to
have a remainder of 5, because 7 times 5 is 35. You have a remainder
of 5 to get to 40. So it's 5 and 5/7. And if that looks like I'm
doing some type of voodoo, just remember, I'm really
just breaking it up. I'm just really
saying that 40/7 is the same thing as 35 plus 5/7. The largest multiple of 7
that is less than this number. And this is the same
thing as 35/7 plus 5/7. And then this, 35/7 5. And 5/7 is just 5/7 there. This one is interesting because
we have the same whole number out front on our mixed numbers. 5 versus 5. So now we actually do
have to pay attention to the fractional part
of our mixed number. We essentially have
to compare 4/9 to 5/7. And there's a couple
of ways to do this. You could get them to
have the same denominator. That's probably the
easiest way to do it. So you could rewrite-- so what's
the least common multiple of 9 and 7? They share no factors, so
really the least common multiple is going to be their product. So if we want to rewrite
4/9 we would write 63 in the denominator,
that's 9 times 7. If we multiply the
denominator by 7 we also have to multiply
the numerator by 7. So that will be 28. Now 5/7, we're going to
make the denominator 63. We're multiplying the
denominator times 9. Then we have to multiply the
numerator times 9 as well. 5 times 9 is 45. So here it's easy to see. 45/63 is clearly
larger than 28/63. And so we could write this. And because the whole number
of parts are the same, and 5/7 is the same
thing as 45/63, and 4/9 is the same
thing as 28/63, we can write that 5 and
4/9 is less than 40/7. Another way that you could have
thought about 4/9 versus 5/7 is you could have said, well,
how does 4/9 compare to 4/7? We have the same numerator. The denominator here is larger
than the denominator here. But when you have a
number in the denominator, the larger it is, the
smaller the fraction. The smaller the absolute
value of the fraction. So this right over here is
a smaller quantity than 4/7. And 4/7 is clearly a
smaller quantity than 5/7. So 4/9 is clearly
smaller than 5/7, so we would have
gotten the same result.