Addition and Subtraction of Fractions

A fraction represents a part of a whole. For instance, the fraction 4/7 means 4 out of 7 of a whole. Fractions are used in our daily lives, from cooking to measuring distances and dividing objects. A fraction consists of a numerator and a denominator. A fraction can be written as:  

Fraction: Numerator / Denominator  
 

Fractions mainly are of three types - 

• Proper
   
• Improper
   
• Mixed. 
  
   

Let’s see how they differ from one another.
 

• In proper fractions, the numerator is always less in value than the denominator. Examples: 2/4, 1/3, and 5/6.
   
• If the numerator is greater than or equal to the denominator, then it is called an improper fraction. Examples include 2/1, 3/3, and 9/4.
   
• If we see a combination of a fraction and a whole number, we can classify that as a mixed fraction. e.g., 3 2/4, 5 1/4, and 8 6/8.  

There are two methods to perform addition and subtraction of fractions: 

• If the denominators are the same, add or subtract numerators only and retain the denominator.

• If the denominators differ, we can convert unlike fractions to like fractions by making the denominators equal. Then add or subtract only numerators and leave the denominator as it is.  
  
   

Adding and Subtracting Fractions with Like Denominators

Adding and subtracting fractions with the same denominators is an easy operation. If the denominators of fractions are the same, we refer to them as like fractions. With like denominators, we only add or subtract the numerators and then keep the denominators constant. Take a look at the examples below to understand the addition and subtraction of like fractions.  

For instance, add 2/7 and 4/7

Here, both the fractions have the same denominators.  

So, only add the numerators: 2 + 4 = 6

Thus, 2/7 + 4/7 = 6/7

Subtracting fractions with like denominators:

Subtract 5/7 and 3/7.

First, we need to verify that the fractions have the same denominator. Here, 7 is the denominator of both fractions. 

So, subtract the numerators: 5 - 3 = 2

Keep the denominators the same.

 Thus, 5/7 - 3/7 = 2/7

Adding and Subtracting Fractions With Unlike Denominators

When adding and subtracting fractions with different denominators, we must convert the unlike fractions into like fractions. First, we change the different denominators to a common denominator. Adding or subtracting the numerators, while keeping the denominator the same, are as follows -  

For instance, add 1/3 and 3/4

First, we must change the unlike denominators to a common denominator. For that, we need to find the least common multiple of 3 and 4. 

Multiples of 3 include 3, 6, 9, 12, 15, 18, ...
Multiples of 4 include 4, 8, 12, 16, 20, …

Here, the smallest common multiple that appears in both lists is 12. Hence, 12 is the LCD of 3 and 4. 

Next, we can convert the fractions to get a denominator of 12.

Convert 1/3: (1 × 4) / (3 × 4) = 4/12
Convert 3/4: (3 × 3) / (4 × 3) = 9/12

Now the denominators are the same. Next, add the numerators together.

    4/12 + 9/12 = (4 + 9) / 12 = 13/12

13/12 is an improper fraction, so convert it into a mixed fraction.

Divide 13 by 12: 13 ÷ 12
Quotient = 1
Remainder = 1

Fraction can be expressed as a mixed fraction in the following way: Q(R/D)

Here, Q is the quotient, R is the remainder, and D is the denominator. 

 So 13/12 can be written as 1 1/12.

Subtracting fractions with unlike denominators

The rules to add unlike fractions can also be applied to subtract unlike fractions. For a better understanding, take a look at this example. 

Subtract 4/5 - 1/4

Find the LCD of 5 and 4. 

20 is the least common multiple of both 5 and 4. So, we can convert the unlike fractions into like fractions. 

Convert 4/5: (4 × 4) / (5 × 4) = 16/20
Convert 1/4: (1 × 5) / (4 × 5)  = 5/20

Next, subtract the numerators.

    16/20 - 5/20 = (16 - 5) / 20 = 11/20

Thus, 16/20 - 5/20 = 11/20

Adding and Subtracting Fractions With Whole Numbers

Addition and subtraction can also be performed between fractions and whole numbers. Read the steps mentioned below to know how it can be done.

Step 1: The first step is to convert the whole number into a fraction by adding 1 as the denominator. E.g., if we need to add 2/3 and 3, then 3 should be written as 3/1. Please note that writing 3 as 3/1 does not change its value.

Step 2: Unlike fractions must be converted to like fractions. This can be done by finding the LCD.

Step 3: Just add the numerators and retain the denominator. 

Let us take a look at some examples to have a better understanding. 
 

Add 3/5 + 4

Convert 4 into a fraction, 4/1

Now we can find the least common denominator (LCD) of 5 and 1. 

5 is the LCD of both denominators. 

Next, we can convert the fractions to have a common denominator of 5. 

 3/5 have already 5 as the denominator. 

Convert 4/1: (4 × 5) / (1 × 5) = 20/5

Add the numerators together. 

3/5 + 20/5 = (3 + 20) / 5 = 23/5

23/5 is an improper fraction. So convert it to a mixed fraction. 

Divide 23 by 5
Quotient = 4
Remainder = 3

Thus, the mixed fraction will be: 
23/5 = 4 3/5
 

Next, we can understand the subtraction of fractions with whole numbers. 

​​​​​​​
Subtract 5 - 3/5

Convert 5 into a fraction as 5/1

Next, find the LCD of 1 and 5. 

5 is the least common denominator of both denominators. 

Convert 5/1 to have a 5 as the common denominator. 

5/1 = (5 × 5) / (1 × 5)  = 25/5

Now we can subtract the fractions.

25/5 - 3/5 = (25 - 3) / 5 = 22/5

22/5 is an improper fraction, so we can convert it into a mixed fraction if needed. 

22 ÷ 5 
Quotient = 4
Remainder = 2

 22/5 = 4 2/5

Thus, 5 - 3/5 = 4 2/5

 Adding and Subtracting Mixed Fractions

A whole number and a proper fraction makes a mixed fraction. Adding and subtracting mixed fractions involves converting the mixed fractions into improper fractions and performing the operations accordingly. For example, add 2 2/5 + 1 4/5

For converting mixed fractions into improper fractions, the formula is 

 Improper fraction = [(Whole number × Denominator) + Numerator] / Denominator

Convert 2 2/5: [(2 × 5) + 2] / 5 = (10 + 2) / 5 = 12/5
Convert 1 4/5: [(1 × 5) + 4] / 5 = 9/5

Next, we can add the improper fractions. 

12/5 + 9/5 = (12 + 9) / 5 = 21/5

To convert 21/5 into a mixed fraction, divide the numerator by the denominator. 

21 ÷ 5
Quotient = 4
Remainder = 1

So, 21/5 = 5 1/5

Thus, 2 2/5 + 1 4/5 = 5 1/5

Next, we can subtract the mixed fractions: 5 1/4 - 1 3/4

First, we can convert the mixed fractions into improper fractions.

Convert 5 1/4: [(5 × 4) + 1] / 4 = (20 + 1) / 4 = 21/4 
Convert 1 3/4: [(1 × 4) + 3] / 4 = (4 + 3) / 4 = 7/4 

Next, subtract the improper fractions.

  21/4 - 7/4 = (21 - 7) / 4 = 14/4

Simplifying: 14/4 = 7/2 

7/2 is an improper fraction; we can convert it into a mixed fraction. 

7 ÷ 2 
Quotient = 3
Remainder = 1

7/2 = 3 1/2

Thus, 5 1/4 - 1 3/4 = 3 1/2

Learning how to add and subtract fractions will help us solve various real-life calculations more easily. Here are some real-world applications of addition and subtraction of fractions: 

• While we cook or bake, we use fractions to measure ingredients accurately. For example, to bake a cake, 1/3 cup of flour and 2/4 a cup of sugar are needed. To calculate the total amount of ingredients used, we add:
   1/4 + 2/4 = 3/4
  
   
• Addition and subtraction of fractions help manage time and schedule shifts. For instance, if an employee in a company works 513 hours in the morning and 223 hours in the evening, the total working time is:
  5 1/3 + 2 2/3 = 16/3 + 8/3 = 24/3 = 8
  So, the total working time is 8 hours. 
   
• In construction and engineering, addition and subtraction of fractions help builders to measure construction materials. For example, if an engineer splits a 4 2/3 sq.ft. room into two rooms, each measuring 1 1/4 sq.ft., the remaining area is 2 1/6 sq ft. 
  
   
• Sports coaches and athletes can track the distances they cover using fractions. For example, if a runner completes 1/2 mile in the morning and 1/5 mile in the evening, the total distance covered is:   1/2 + 1/5 = 5/10 + 2/10 = 7/10