Comparing Fractions

Fractions are a part of a whole. Sometimes we have a whole divided into different number of parts. For example, three pizzas can be divided into 8 or 10 or 12 parts. In these cases, it might get difficult to understand how to compare fractions if we try to compare among all the pizzas. We need to compare fractions to find out which fraction is larger and which fraction is smaller. We use two methods to compare such fractions - Decimal method and Same Denominator method. However, before we learn what is comparing fractions, we need to know what a fraction is.

What Is A Fraction?

As mentioned before, fractions are a part of a whole. So, if you divide a whole cake into a number of equal parts, we call each part a fraction of the whole cake. It is represented by a/b where a is the numerator and b is the denominator. The numerator is the number of equal parts that are used while the denominator is the total number of equal parts in which the whole is divided. Suppose you have a pizza divided into 8 equal slices and you eat 1 slice, you can say you have ⅛ of the pizza and ⅞ is left.

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What Are Like And Unlike Fractions?

To understand how comparing fractions work, it is essential to first understand what like and unlike fractions are. Like fractions are those that have the same denominator. For example ⅖, ⅕, ⅗ is a group of like fractions. On the other hand, comparing unlike fractions is basically comparing fractions with different denominators. For example, ½, 4/7, ⅔, ⅗ are a group of unlike fractions. 

Comparing Fractions Using Decimal Method

We can compare fractions by converting fractions into decimals. Suppose we have different fractions - ½, ⅔, ¾ and ⅘ which we want to compare. In the decimal method, we convert each of them into a decimal.
½ = 0.5
⅔ = 0.66
¾ = 0.75
⅘ = 0.80
Here we can easily compare and say which one is bigger and which one is smaller.

Comparing Fractions Examples

Here are a few Comparing Fractions Questions:

EXAMPLE 1:
Which of the following fractions is larger: 3/7 or 5/9?

Solution:
3/7 = 0.42

5/9 = 0.56

Since 0.56 is greater than 0.42, we can conclude that 5/9 is greater than 3/7.

Comparing Fractions Using The Same Denominator Method

Comparing fractions is easier if the denominators of different fractions are the same. Hence, if there is a group of like fractions, they can be compared easily. For example, when you have to compare two fractions 21/50 and 37/50, you have to only compare the numerators. In this case, both fractions have the same denominator 50 and 37 is greater than 21. Hence, we can conclude that 37/50 is greater than 21/50. However, this method can only be used when the denominator of all the fractions in the group are the same.

Example:

Which of the following fractions is smaller: 90/90 or 80/90

Solution:
90/90 = 1

80/90 = 0.89

Since 1 is greater than 0.89, we can conclude that 80/90 is smaller than 90/90.

The above method can be used only when you are comparing fractions using the same denominator.

What happens if the denominators are different? What will you do when you are comparing fractions with unlike denominators and you do not want to convert them into decimals? Well, you can convert the unlike fractions into like fractions by multiplying them with a common multiple and converting them to the same denominators. Here is an example:

Example:

Which of the following fractions is larger: 7/9 or 4/7

Solution:
We convert 7/9 and 4/7 into the same denominators by finding out the LCM of the denominators. The LCM in this case is 63. So we multiply both 7 and 9 by 7 and get 49/63.

\[\frac{7 × 7}{9 × 7}\] = \[\frac{49}{63}\]

 Similarly, we multiply 4 and 7 with 9 and get 36/63. 

\[\frac{4 × 9}{7 × 9}\] = \[\frac{36}{63}\]

As numerator 49 is higher than numerator 36, 7/9 is higher than 4/7.

FAQ (Frequently Asked Questions)

1. Can we compare fractions with the whole like 1?

Yes. If there is a fraction which you have to compare with one, you can treat one as any other fraction. For example, if you have to compare 3/4 with 1, you can do it using the decimal method or the same denominator method.
3/4 = 0.75 while
1 = 1.00
In this case, 1 is greater than 0.75 which is 3/4 . For the same denominator function, you can take 1 as 4/4 and then compare it with 3/4. Thus, numerator 4 is higher than numerator 3 which means, whole number 1 is higher than ¾.

2. How can we solve a question that has a mixed fraction?

In case of a mixed fraction, you have to first convert it to improper fractions (fractions with numerators lower than the denominator). Once this is done, you have to check if the denominator is the same. If not, you have to convert the denominators into the same number and then compare the fractions. Below is an example:

You have 5 ⅓ chocolate pieces in your hand and your friend has 5 ⅖ chocolate pieces in her hand. Out of the two, who has more chocolate?

Solution:

We have to first convert the mixed fractions to simple fractions.
5 ⅓ = (3 x 5) + 1 =  16/3
5 ⅖ = (5 x 5) + 2 =27/5

Now we have to convert these fractions into like fractions by making the denominators the same. For this, we find the LCM to be 15 and multiply numerator 16 and denominator 3 with 5 which is 80/15
Similarly, we multiply numerator 27 and denominator 5 by 3 which is 81/15
Since 81 is greater than 80, 81/15 is greater than 80/15. So 27/15 is greater than 16/3, which means that your friend has more chocolate than you.