Let's order these improper fractions
and mixed numbers from least to greatest.
And there's two ways to do it.
You could put them all as mixed numbers,
or you could put them all as improper fractions.
I will write them all-- let's try it as mixed numbers first
and see what happens.
And then we could also try improper fractions.
So if we write them all as mixed numbers,
this expression right here, this number.
3 goes into 7 two times, and the remainder is 1.
So it's 2 and 1/3.
Then we have our 4 and 1/2, which
is already a mixed number.
Then this one over here-- 3 goes into 4 one time
with a remainder of 1, so this is 1 and 1/3.
And then you have 1 and 2/5, which
is already written as a mixed number.
And then finally, we have this quantity.
15 goes into 17 one time with a remainder of 2.
So this is 1 and 2/15.
So we already know the largest number is, just
looking at the whole number parts of these mixed numbers.
It's 4 and 1/2, so let me write it out here.
The second largest number is the 2 and 1/3.
And now the whole number part of all of these last three--
they're all the same, so we now have
to look at the fractional part.
And they all have different denominators.
So to simplify things, or so that we can compare apples
to apples, let's all give these the same denominator.
And the best common denominator will
be the least common multiple of 3, 5, and 15, which is 15.
So we can rewrite it.
So this is going to be 1 and something over 15.
To go from 3 to 15, you have to multiply the denominator by 5.
So let's also multiply the numerator by 5.
So this is 1 and 5/15.
Let's do this one.
This is 1 and something over 15.
We multiply the denominator by 3,
so we also have to multiply the numerator by 3.
So let's put a 6 there.
2 times 3 is 6.
And then finally, this one already
has 15 as the denominator.
This is 1 and 2/15.
So now this is easier to compare.
They all have 15 as the denominator
in the fractional part of the mixed number.
The smallest of these is 2/15, so I'm
going to write it out front.
That's the smallest.
Then the next smallest is 1 and 5/15,
which is this quantity right over here, which
is-- I could write it as 1 and 1/3.
And then the next smallest, or I guess the larger one
after that, is the 1 and 6/15, which
is the same thing as 1 and 2/5.
And we're done.
And we can then rewrite them in the way
that they were originally written.
So this one right over here in pink
was originally written as 17/15, so I'll write it as 17/15.
This 1 and 1/3 was originally written as 4/3.
So I can write it as 4/3.
The 1 and 2/5 was already written as 1 and 2/5.
And then the 2 and 1/3 was written as 7/3.
And the 4 and 1/2 was already 4 and 1/2.
Now, the other way to do it is to convert all of these
into improper fractions.
If we do it that way, the 7/3 is still a 7/3.
4 and 1/2-- to turn that into an improper fraction, 2 times 4
is-- well, one way to think about it--
4 is the same thing as 8/2.
8/2 plus 1/2 is 9/2.
Or a simple way, 2 times 4 is 8, plus 1 is 9.
9/2.
And then 4/3 is already written as an improper fraction.
1 and 2/5?
Well, 1 is the same thing as 5/5.
And 5/5 plus 2/5 is 7/5.
Or you could say 5 times 1 plus 2 is 7.
And then finally, we have the 17/15.
Now, in order to really compare apples to apples,
we have to have a common denominator here.
And the least common multiple of 3, 2, 3, 5 and 15--
it looks like, if we just had 3, 5's, and 15's here,
it would be 15.
But that's not divisible by 2, so we're
going to have to go with 30.
So we're going to have 30 as the common denominator for all
of these.
So 7/3-- to get the denominator from 3 to 30,
we multiplied by 10.
So we multiply the numerator by 10.
So we have 70/30.
Then we have 9/2, but now we're going
to make the denominator 30.
To go from 2 to 30, we had to multiply the denominator by 15.
So we're going to have to multiply the numerator by 15.
So 9 times 15 is 90, plus 45.
So that's 135.
Now we have 4/3.
We're going to rewrite the denominator as 30.
We're multiplying the denominator by 10,
so we want to multiply the numerator by 10.
So it's 40/30.
And then 7/5, we're going to write with a denominator of 30.
We're multiplying the denominator by 6,
so we have to multiply the numerator by 6.
7 times 6 is 42.
And then finally, 17/15.
If the denominator's 30, we're multiplying the denominator
by 2.
So we also have to multiply the numerator by 2.
17 times 2 is 34.
And now we can compare apples to apples.
All the denominators are the same,
so we can just compare the numerators.
The smallest numerator here is 34, so 34/30 is the smallest.
Then the next smallest-- it looks like 40/30.
And then the next smallest is 42/30.
And then the next smallest is 70/30.
And then the next smallest or the largest, now, is 135/30.
And you convert these back down to what they're originally
represented at.
You get this exact same ordering.