Like Fraction

Two or more fractions that have the same denominator are known as like fractions. In like fractions, the bottom numbers (i.e., denominators) are identical. The addition and subtraction of like fractions is simple: we add or subtract the numerators and keep the same denominator. 1/4, 3/4, 5/4, and 7/4 are a few examples of like fractions with denominators of 4. Here we are going to learn about the two different categories of fractions. 

Fractions are classified into two categories according to their denominators. Here is a table that explains the difference between like fractions and unlike fractions.

The three properties of like fractions are: 

• Commutative property of like fractions 
   
• Associative property of like fractions    
   
• Distributive property of like fractions 

The commutative property of like fractions states that the order of arranging the fractions in multiplication won’t affect the result. The order of the fractions in the addition and multiplication will not affect the result.

Commutative property of addition and subtraction: If p/q and c/q are two like fractions, then the addition is:

(p/q) + (c/q) = (c/q) + (p/q)
 
For example, if we add 2/7 and 3/7, then

  2/7 + 3/7 = 5/7 

  3/7 + 2/7 = 5/7
 
The addition of two like fractions is commutative. 

Next, if we can subtract two like fractions, such as p/q and c/q, then

(p/q) -  (c/q) ≠  (c/q) - (p/q)

Subtraction of like fractions is not commutative. 

For example, if we subtract 3/7 - 4/7, then, 

3/7 - 4/7 = -1/7

Then, 

4/7 - 3/7 = 1/7 

Hence, 3/7 - 4/7 ≠ 4/7 - 3/7 

Commutative property of multiplication and division: If we multiply two like fractions, the order of the fractions will not change the final product. 

(p/q) × (c/q) = (c/q) × (p/q)

For instance, 2/3 × 4/3 

= 8/9 

Then, 

4/3 × 2/3 = 8/9

Hence, 2/3 × 4/3 = 4/3 × 2/3 

The multiplication of like fractions is commutative. 

The commutative property is not applicable to division. It can be expressed as, 

 If (p/q) ÷ (c/q) ≠  (c/q) ÷ (p/q)

For example, check if 4/2 ÷ 8/2 = 8/2 ÷ 4/2 

So, first let’s find the value of 4/2 ÷ 8/2,  

4/2 ÷ 8/2 = 4/2 × 2/8 

    = 4/8 = 1/2

4/8 is simplified into 1/2 

Then finding the value of 8/2 ÷ 4/2,  

8/2 ÷ 4/2 = 8/2 × 2/4 

     = 16/8 

16/8 is simplified into 2/1. 

So, 4/2 ÷ 8/2 ≠ 8/2 ÷ 4/2 

According to this property, the grouping of fractions in addition and multiplication does not change the result. 

Associative property of addition: The addition of three like fractions, such as p/q, c/q, and d/q, will be like:

p/q + (c/q + d/q) = (p/q + c/q) + d/q

For example, 2/4 + (5/4 + 3/4) = 2/4 + 8/4 = 10/4
Then, 

(2/4 + 5/4) + 3/4 = 7/4 + 3/4 = 10/4 

So, 2/4 + (5/4 + 3/4) = (2/4 + 5/4) + 3/4.

Associative property of subtraction: The subtraction of like fractions does not follow the associative property. If p/q, c/q, and d/q are the three fractions, then, 

p/q - (c/q - d/q) ≠ (p/q - c/q) - d/q

For instance, 7/4 - (5/4 - 3/4) = 7/4 -  2/4 = 5/4 

Then, 

(7/4 - 5/4) - 3/4 = 2/4 - 3/4 = -1/4 

So, 7/4 - (5/4 - 3/4)  ≠ (7/4 - 5/4) - 3/4 

Associative property of multiplication: Like fractions’ multiplication is associative. If p/q, c/q, and d/q are the three fractions, then,
 
p/q × (c/q × d/q) = (p/q × c/q) × d/q

For example, 2/4 × (5/4 × 3/4) = 2/4 × 15/4 = 30/4 
Then, 
(2/4 × 5/4) × 3/4 = 10/4 × 3/4 = 30/4 

Therefore, 2/4 × (5/4 × 3/4) = (2/4 × 5/4) × 3/4 

Associative property of division: The division of like fractions is not associative. If p/q, c/q, and d/q are the three fractions, then, 

 p/q ÷ (c/q ÷ d/q) ≠ (p/q ÷ c/q) ÷ d/q
 

For instance, 2/3 ÷ (4/3 ÷ 1/3) = 2/3 ÷ 4 = 1/6 
Then, 

(2/3 ÷4/3) ÷ 1/3, first we find the value of 2/3 ÷4/3

2/3 × 3/4 = 2/4 = 1/2

1/2 ÷ 1/3 = ½ ×3/1 = 3/2

So, 2/3 ÷ (4/3 ÷ 1/3) ≠ (2/3 ÷4/3) ÷ 1/3

This property states that multiplication distributes over addition or subtraction. That is, multiplying a number by a sum or difference is the same as multiplying it by each term individually and then adding or subtracting. 

Distributive property of multiplication over addition: If p/q, c/q, and d/q are the three fractions, then,

 p/q × (c/q + d/q) = (p/q × c/q) + (p/q × d/q) 

For example, 1/4 × (2/4 + 1/4) = (1/4 × 2/4) + (1/4 × 1/4)

1 × 2 / 4 × 4 + 1 × 1 / 4 × 4

= 2/16 + 1/16

1/4 × (2/4 + 1/4)  = 3/16

So, multiplication is distributive in nature over addition. 

Distributive property of multiplication over subtraction:

Multiplication is distributive over subtraction. If 
p/q × (c/q - d/q) = (p/q × c/q) - (p/q × d/q) 

For instance, 1/5 × (2/5 - 1/5) = (1/5 × 2/5) - (1/5 × 1/5)

      1/5 × 2/5 = 2/25 

       1/5 × 1/5 = 1/25

Subtract 2/25 - 1/25 = 1/25 

1/5 × (2/5 - 1/5) =2/25 - 1/25 

Addition, subtraction, multiplication, and division are the four arithmetic operations that can be performed on like fractions: 

Addition of like fractions: By adding two fractions together, we find the sum in a single fraction. In the case of like fractions, we only add the numerators and keep the denominator the same. 

For example, 3/6 + 4/6 

   = 3 + 4 = 7

The sum is 7/6. 

Subtraction of like fractions: We find the difference between two quantities by subtracting one fraction from the other. 

For example, 4/2 - 1/2 

Subtracting the numerators as the denominators are the same:  4 -1 = 3 = 3/2. 

Thus, the difference is 3/2. 

Multiplication of like fractions: To find the product of two fractions, we multiply the numerators and the denominators separately. 

For example, 5/3 × 4/3 

      = 5 × 4 = 20 

      = 3 × 3 = 9

Thus, 5/3 × 4/3 = 20/9 

Division of like fractions: When we divide two like fractions, the second fraction will become the reciprocal. Then the first fraction will be multiplied by the reciprocal. Here the denominators are the same, therefore, they cancel out after taking reciprocal. 

For example, 8/5 ÷ 3/5 

     = 8/5 × 5/3 

Here, the denominator 5 will cancel out, and the fraction is 8/3.  

Thus, 8/5 ÷ 3/5 = 8/3 

In our daily lives, fractions are used to measure and share objects and to perform financial calculations. Some real-world applications of like fractions are:  

• Cooking and baking: When we cook or bake, we use like fractions to calculate the required ingredients for the recipe. For example, if we bake a cake, and we add 1/4 cup of flour, and we need to double the measurement of flour. So we can add the like fractions to get the correct measurement. 
  1/4 + 1/4 = 2/4 = 1/2 
  
  This means we need to add 1/2 cup of flour to the recipe. 

• Sharing and distribution: To ensure accurate distribution while sharing, we use like fractions. For instance, if four students need to share 1/5 of a pizza each, they can calculate how much they get together. 
  
  1/5 + 1/5 +1/5 + 1/5 = 4/5

• Time management: Students can easily calculate their study times using like fractions, it helps them to schedule their timings fruitfully. For example, if a student studies for 1/3 of an hour on Mathematics and 2/3 of an hour on Science, the total study time is: 
  
  1/3 + 2/3 = 3/3 = 1 hour.