# How to compare fractions

Sometimes we need to compare two fractions to discover which is larger or smaller.

There are two main ways to compare fractions: using decimals, or using the same denominator.

## The Decimal Method of Comparing Fractions

Convert each fraction to decimals, and then compare the decimals.

• Convert each fraction to a decimal.
• We can use a calculator (3÷8 and 5÷12), or the method on Converting Fractions to Decimals.
• Anyway, these are the answers I get:

3 8 = 0.375, and 5 12 = 0.4166…

So 5 12 is bigger.

### The Same Denominator Method

1. The denominator is the bottom number in a fraction.
2. It shows how many equal parts the item is divided into
3. When two fractions have the same denominator they are easy to compare: is less than 4 9 5 9

But when the denominators are not the same we need to make them the same (using Equivalent Fractions).

Look at this:

• When we multiply 8 × 3 we get 24,
• and when we multiply 12 × 2 we also get 24,

so let's try that (important: what we do to the bottom we must also do to the top):

 × 3 3 = 9 8 24 × 3
and
 × 2 5 = 10 12 24 × 2

We can now see that 9 24 is smaller than 10 24 (because 9 is smaller than 10).

so 5 12 is the larger fraction. is less than 3 8 5 12

### Making the Denominators the Same

There are two main methods to make the denominator the same:

• Common Denominator Method, or the
• Least Common Denominator Method

They both work, use which one you prefer!

Using the Common Denominator method we multiply each fraction by the denominator of the other:

 × 15 5 = 75 6 90 × 15
and
 × 6 11 = 66 15 90 × 6

We can see that 75 90 is the larger fraction (because 75 is more than 66)

so 5 6 is the larger fraction. is more than 5 6 11 15

## Using the Number Line to Compare Fractions

In previous posts, we’ve learned how to place different numbers on the number line. Today we’re going to learn how to represent fractions on a number line. This is very useful when we need to compare them.

### Fractions with the same denominator

As we know, it’s easy to compare fractions with the same denominator. In these cases, the fraction with the larger numerator is the bigger fraction. For example, eight-thirds is bigger than three thirds. ### Fractions with the same numerator

Something that seems a bit more complicated, but only on the surface, is comparing fractions with the same numerator. In this situation, the fraction with the smaller denominator is always the bigger fraction. For example, fifteen-thirds is bigger than fifteen fourths. ### Using the number line for fractions with different numerators and denominators

The comparison becomes more complicated when the fractions we’re comparing don’t have the same numerator or denominator. Which fraction is greater: eleven thirds or thirteen fourths?

This is where the number line is particularly useful. We can use it to give us a visual representation of the fractions, which we can then compare without having to do any mathematical calculations.

Although the number of subdivisions of each unit will be different on the number line for each fraction, it’s important to make sure the unit is the same size on each one.

Then we can compare the fractions visually.  Once we’ve placed the two fractions with different numerators and denominators on the number line, we can tell which of the two is the greater simply by looking. Without having to make any calculations or operate with fractions, we can see that thirteen-fourths is less than eleven thirds. The first fraction is placed close to the three, while the second fraction is closer to the four.

• It’s easy, right?
• Learn and Practice How to Find Sums of Fractions
• Learning How to Subtract Fractions
• How to Find a Sum of Fractions
• Learn and Practice How to Multiply Fractions
• Learn How to Subtract Fractions

## Comparing Fractions with Different Denominators • As we've begun our study of fractions, we've learned how to identify equivalent fractions.
• We used the equal fractions property to create equivalent fractions.
• We will also use this same property to help us compare fractions with different denominators.

If the numerator and denominator of a fraction are multiplied (or divided) by the same nonzero number, then the resulting fraction is equivalent to the original fraction.

Remember how we multiplied the numerator and denominator by the same number in order to create equivalent fractions?

We will use the same process to compare fractions. Take a look….

### Example 1 –  Are these two fractions equal? If you are confused, please take a look at the video lesson below that will fully explain Example 1.

That example was pretty easy since 8 is a factor of 24.

Now let's take a look at another example that is a little harder. In this example we will use our knowledge of least common multiple.

### Example 2 – Which fraction is greater? Take a look at this example on video!

You must always make sure that the fractions have the same denominator.

Let's take a look at one more example for comparing fractions with different denominators. This time we'll compare more than two fractions.

### Example 3 – Ordering from least to greatest Hopefully you now feel comfortable comparing and ordering proper fractions.

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## 8 Ways to Compare Fractions in the Late Elementary Grades

In this post, I want to share a collection of methods that you can use with late elementary students, in Grades 3 to 5, to compare fractions. Plus, download a FREE set of problems in which students are asked to use and to think about different strategies for comparing fractions. When I was younger, I only learned one way to compare fractions. It wasn’t until I was tutoring math as a community college student that I started to explore the different ways of representing and comparing fractions. There have been a few times as a teacher when I have had “Aha!” moments that have given me even deeper insights into fractions.

Students’ ability to justify their choice of strategy for comparing fractions is part of Standard for Mathematical Practice 3, “construct viable arguments and critique the reasoning of others.”

### Eight Ways To Compare Fractions In Grades 3–5

Below I've outlined eight ways to compare fractions. Once you develop these different strategies for comparing fractions with your students, check their reasoning skills using the download for this post.

My printable Explaining Your Reasoning to Compare Fractions Activity asks students to compare fractions using given strategies, by choosing between strategies, and by creating a problem of their own.  ### 1. Equivalent Denominators

This is the easiest situation in which to compare fractions. If two fractions have equivalent denominators, then compare the numerators to determine which faction is greater. Students at the earliest stages of learning about fractions should be able to do this.

## COMPARING FRACTIONS

 Example 1. 23 is to 58 as 2 × 8 is to 3 × 5 as 16 is to 15.
 16 and 15 are the numerators we would get if we expressed 23 and 58

with the common denominator 24. as

 And since 16 is larger than 15, we would know that 23 is larger than 58 .
 Example 2.   Which is larger, 47 or 59 ?
• as
• 36  is to  35.
• 36 is larger than 35.  Therefore,
• Note:  We must begin multiplying with the numerator on the left:
• 4 × 9.
 Example 3. 14 is to 12 as which whole numbers?
cross-multiplying,
2. as
3. 2  is to  4.
4. That is,

Example 4.   What ratio has 2½ to 3?

 Answer.   First, express 2½ as the improper fraction 52 .  Then, treat the

whole number 3 as a numerator, and cross-multiply:

2½ is five sixths of 3.

 Equivalently, since  3 = 62 (Lesson 21, Question 2), then
 52 is to 62 as  5 is to 6.

In general:

To express the ratio of a fraction to a whole number, multiply the whole number by the denominator.

 67 is to 3  as  6 is to 21.

For an application of this, see Lesson 26.

 Example 5.   On a map, 34 of an inch represents 60 miles.  How many

miles does 2 inches represent?

Solution.  Proportionally,

 34 of an inch  is to  2 inches  as  60 miles  is to  ? miles.

Therefore:

3 is to 8  as  60 miles  is to  ? miles.

• Since  20 × 3 = 60,  then 20 × 8 = 160 miles.
• The theorem of the same multiple.
• Or, inversely:

8 is to 3  as  ? miles is to 60 miles.

1. Now,
2. 8 is two and two thirds times 3.
3. (Lesson 18, Example 5.)  Therefore, the missing term will be
 Two and two thirds times 60 = Two times 60 + two thirds of 60 (Lesson 16) = 120 + 40 = 160 miles.

More than or less than ½

## Comparing fractions

This is a more difficult task the two comparisons above. Depending on your child’s level, this maybe a step you skip and then come back to later.

These fractions are compared by changing the denominators to a common number. This can be done by multiplying the top and bottom of the fraction by the same number since this will give a fraction with an equivalent value. Do this as shown below to both fractions to get a common denominator and then compare them.

So, in the example above, now that the denominators are equal, the fraction with the greatest numerator is the largest.

Some students may respond to questions about which fraction is bigger by saying it depends on what size the whole is and, depending on the wording of the question, they may actually be correct! Encourage them to think in terms which is the larger or smaller share and be precise with your questions. e.g. “which is the largest fraction?” not just “which is the largest?”