Comparing Fractions

 

Learning Objective(s)

·         Determine whether two fractions are equivalent.

·         Use > or < to compare fractions.

 

 

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.

 

 

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.

 

 

 

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

 

 

 

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.

 

 

 

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

 

 

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)   Show/Hide Answer A) Incorrect. Although the same numbers, 5 and 7, are used in each fraction, the numerators and denominators are not equal, so the fractions cannot be equivalent. The correct answer is .   B) Incorrect. 30 is divisible by 10, and 12 is divisible by 6. However, they do not share a common multiple: 6 · 2 = 12, and 10 · 3 = 30. This means the fractions are not equivalent. The correct answer is .   C) Correct. Take the fraction  and multiply both the numerator and denominator by 4. You are left with the fraction . This means that the two fractions are equivalent.   D) Incorrect. The numerators of the two fractions are the same, but the denominators are different. This means the fractions are not equivalent. The correct answer is .

 

 

 

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  off the original price and the discount in another store is  off the original price, which store is offering a better deal? To answer this question, and others like it, you can compare fractions.

 

 

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  and , and then compare.

 

 

Now you see that  contains 4 parts of 12, and  contains 3 parts of 12. So,  is greater than .

 

 

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.

 

 

 

 

 

Which of the following is a true statement?   A)   B)   C)   D)   Show/Hide Answer A) Incorrect.  is equivalent to , and since 25 > 24, , which means . The correct answer is .   B) Incorrect. Both fractions simplify to , so one isn’t greater than or less than the other one. They are equivalent. The correct answer is .   C) Incorrect. Finding a common denominator, you can compare  to , and see that , which means . The correct answer is .   D) Correct. Simplifying , you get the equivalent fraction . Since you still don’t have a common denominator, write as an equivalent fraction with a denominator of 8: . You find that , so  as well.

 

 

 

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.