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Comparing and Ordering Fractions

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Comparing Like Fractions

When comparing two fractions with same denominator, the larger fraction is the one with the greater numerator.

Let's illustrate through an example.

Example 1

 
Imagine you take
1
6
of a birthday cake and your sister takes
3
6
. Who has more cake?

Cake 1

 
We need to compare
1
6
and
3
6
.

The denominator of both the fractions is same i.e. 6.

To compare fractions when the denominators of two fractions are the same, the one with the greater numerator is larger.

Since 3 is greater than 1,
1
6
<
3
6
.

So, your sister has more cake than you.

Comparing Unlike Fractions

To compare fractions with different denominators, we must convert the fractions to equivalent fractions with a common denominator and then look for numerators.

 

Example 2

 
Compare
3
6
and
2
3
.

Convert these fractions to equivalent fractions with a common denominator in order to compare them.

Follow these steps:

List out the multiples.

Multiples of 6 = 6, 12, 18, 24, ......

Multiples of 3 = 3, 6, 9, 12, ......

The smallest multiple the two denominators have in common is 6.

The least common denominator (LCD) of the fractions is the least common multiple of their denominators.

Now, convert these fractions to equivalent fractions with a denominator 6. As we can notice the first fraction already has the denominator 6.

So, convert the other fraction with denominator 6.

 
2 x 2
3 x 2
=
4
6
 
Now, we have
3
6
and
4
6
with the same denominator.

Let's look for numerators to compare.

We know, 4 is greater than 3.

So,
3
6
<
4
6
 
We can rewrite the answer as
1
2
<
2
3
.

Ordering Fractions

If the denominator is the same, look at the numerators, and put the fractions in order.

If the denominator is different, we need to convert our fractions to equivalent fractions of the same denominator and then compare them to put in order.

 

Example 3

 
Order these fractions from least to greatest:
1
10
,
12
10
,
7
10
,
9
10
 

Here the denominator of all the fractions is same so we look at the numerators.

1 is the smallest number, followed by 7, 9, and 12.

1 < 7 < 9 < 12

Arrange the fractions from least to greatest:

 
1
10
<
7
10
<
9
10
<
12
10
 
 

Example 4

 
Order these fractions from least to greatest:
3
4
,
5
6
,
1
2
,
2
3
 

These fractions have unlike fractions. We will use the least common denominator (LCD) to write these fractions as equivalents fractions with same denominators, and then compare.

The least common denominator of
3
4
,
5
6
,
1
2
,
2
3
is 12.
 
3 x 3
4 x 3
=
9
12
 
 
5 x 2
6 x 2
=
10
12
 
 
1 x 6
2 x 6
=
6
12
 
 
2 x 4
3 x 4
=
8
12
 

We know,

6 < 8 < 9 < 10

So,

 
6
12
<
8
12
<
9
12
<
10
12
 

Switch fractions back to their original form:

 
1
2
<
2
3
<
3
4
<
5
6
Summary
  • When comparing two fractions with same denominator, the larger fraction is the one with the greater numerator.
  • To compare fractions with different denominator, we must convert the fractions to equivalent fractions with a common denominator and then look for numerators.
  • If the denominator is the same, look at the numerators, and put the fractions in order.
  • If the denominator is different, we need to convert our fractions to equivalent fractions of the same denominator and then compare them to put in order.

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