How do you know if a fraction is less than or greater?

Fractions contain a top number called the numerator and a bottom number called the denominator separated by a horizontal line that represents division. In a proper fraction, the numerator is smaller than the denominator and thus represents a part of a whole (the denominator). While it is easy to tell which integers are larger or smaller than each other based on their position on the number line, it can be harder to determine where fractions fall and whether one fraction is less than or greater than another fraction.

Compare fractions with the same denominator by determining the relationship between the numerators. For example, 3/5 is less than 4/5 because 3 is less than 4.

Compare fractions with different denominators by finding the least common denominators and converting the fractions to it so the numerators can be compared. Determine whether 8/15 is less than or equal to 4/5. Note that because 5 is a multiple of 15, the least common denominator is 15. Convert the fractions: 8/15 remains the same and 4/5 becomes 12/15. Write that 8/15 is less than 4/5 since the 8 is smaller than the 12.

Use a calculator to find the decimal forms of very large fractions or those that don't have a common denominator to compare the sizes. Determine whether 3/17 is less than or greater than 5/13. Perform the divisions: 3/17 = 0.177 (rounded) and 5/13 = 0.385 (rounded). Write that 3/17 is smaller than 5/13 because that decimal form is smaller than the other.

Comparing Fractions

Learning Objective(s)

·         Determine whether two fractions are equivalent.

·         Use > or < to compare fractions.

Introduction

You often need to know when one fraction is greater or less than another fraction. Since a fraction is a part of a whole, to find the greater fraction you need to find the fraction that contains more of the whole. If the two fractions simplify to fractions with a common denominator, you can then compare numerators. If the denominators are different, you can find a common denominator first and then compare the numerators.

Determining Equivalent Fractions

Two fractions are equivalent fractions when they represent the same part of a whole. Since equivalent fractions do not always have the same numerator and denominator, one way to determine if two fractions are equivalent is to find a common denominator and rewrite each fraction with that denominator. Once the two fractions have the same denominator, you can check to see if the numerators are equal. If they are equal, then the two fractions are equal as well.

One way to find a common denominator is to check to see if one denominator is a factor of the other denominator. If so, the greater denominator can be used as the common denominator.

Example
Problem	Are  and  equivalent fractions?
	Does ?	To solve this problem, find a common denominator for the two fractions. This will help you compare the two fractions. Since 6 is a factor of 18, you can write both fractions with 18 as the denominator.
		Start with the fraction . Multiply the denominator, 6, by 3 to get a new denominator of 18. Since you multiply the denominator by 3, you must also multiply the numerator by 3.
		The fraction  already has a denominator of 18, so you can leave it as is.
	does not equal	Compare the fractions. Now that both fractions have the same denominator, 18, you can compare numerators.
Answer	and  are not equivalent fractions.

When one denominator is not a factor of the other denominator, you can find a common denominator by multiplying the denominators together.

Example
Problem	Determine whether and  are equivalent fractions.
	6 • 10 = 60	Use 60 as a common denominator.
		Multiply the numerator and denominator of by 10 to get 60 in the denominator.
		Multiply numerator and denominator of  by 6.
		Now that the denominators are the same, compare the numerators.
Answer	Yes,  and  are equivalent fractions.	Since 30 is the value of the numerator for both fractions, the two fractions are equal.

Notice in the above example you can use 30 as the least common denominator since both 6 and 10 are factors of 30. Any common denominator will work.

In some cases you can simplify one or both of the fractions, which can result in a common denominator.

Example
Problem	Determine whether  and  are equivalent fractions.
		Simplify . Divide the numerator and denominator by the common factor 10.
		is still not in lowest terms, so divide the numerator and the denominator again, this time by the common factor 2.
		Compare the fractions. The numerators and denominators are the same.
Answer             Yes, and  are equivalent fractions.

Note: In the example above you could have used the common factor of 20 to simplify

Determining Equivalent Fractions

To determine whether or not two fractions are equivalent:

Step 1: Rewrite one or both of the fractions so that they have common denominators.

Step 2: Compare the numerators to see if they have the same value. If so, then the fractions are equivalent.

Which of the following fraction pairs are equivalent?

A)

B)

C)

D)

Comparing Fractions Using < and >

When given two or more fractions, it is often useful to know which fraction is greater than or less than the other. For example, if the discount in one store is

To determine which fraction is greater, you need to find a common denominator. You can then compare the fractions directly. Since 3 and 4 are both factors of 12, you will divide the whole into 12 parts, create equivalent fractions for

Now you see that

As long as the denominators are the same, the fraction with the greater numerator is the greater fraction, as it contains more parts of the whole. The fraction with the lesser numerator is the lesser fraction as it contains fewer parts of the whole.

Recall that the symbol < means “less than”, and the symbol > means “greater than”. These symbols are inequality symbols. So, the true statement 3 < 8 is read as “3 is less than 8” and the statement 5 > 3 is read as “5 is greater than 3”. One way to help you remember the distinction between the two symbols is to think that the smaller end of the symbol points to the lesser number.

As with comparing whole numbers, the inequality symbols are used to show when one fraction is “greater than” or “less than” another fraction.

Comparing Fractions

To compare two fractions:

Step 1: Compare denominators. If they are different, rewrite one or both fractions with a common denominator.

Step 2: Check the numerators. If the denominators are the same, then the fraction with the greater numerator is the greater fraction. The fraction with the lesser numerator is the lesser fraction. And, as noted above, if the numerators are equal, the fractions are equivalent.

Example
Problem	Use < or > to compare the two fractions  and .
	Is , or is ?	You cannot compare the fractions directly because they have different denominators. You need to find a common denominator for the two fractions.
		Since 5 is a factor of 20, you can use 20 as the common denominator.
		Multiply the numerator and denominator by 4 to create an equivalent fraction with a denominator of 20.
		Compare the two fractions.  is greater than .
Answer		If , then , since .

Summary

You can compare two fractions with like denominators by comparing their numerators. The fraction with the greater numerator is the greater fraction, as it contains more parts of the whole. The fraction with the lesser numerator is the lesser fraction as it contains fewer parts of the whole. If two fractions have the same denominator, then equal numerators indicate equivalent fractions.

Is 3 4 less or more than 2 3?

Which fraction is greater 4 5 or 5 6?

Is 2 3 or 7 12 bigger?