Addition and Subtraction of Fractions (With Pictures)

A fraction represents a part of a whole. As you perform arithmetic operations with whole numbers, similarly you can perform these operations with fractions. But the process of adding and subtracting fractions is a bit different from that of whole numbers.

In this article, you will understand how to add fractions and how to subtract fractions.

Let’s first understand some related basic terms.

Terms Related With Addition & Subtraction of Fractions

The terms used in the article are

Proper Fraction

A fraction in which the numerator value is always less than the denominator value is known as a proper fraction. For example, $\frac {2}{3}$, $\frac {4}{7}$, $\frac{21}{73}$, and $\frac{1}{4}$ are proper fractions. The value of a proper fraction is always less than $1$.

Improper Fraction

A fraction in which the numerator value is always greater than the denominator value is known as an improper fraction. For example, $\frac {3}{2}$, $\frac{5}{4}$, $\frac {37}{19}, and $\frac {14}{5} are improper fractions. The value of an improper fraction is always greater than $1$.

Mixed Fraction

A fraction that is formed by combining a whole number and a fraction. For example, in $3 \frac {1}{4}$, $3$ is a whole part and $\frac {1}{4}$ is a fractional part. All mixed fractions can also be written as an improper fraction and vice-versa.

Like Fractions

A group of two or more fractions having the same denominators is called like fractions. 

For example consider a group of fractions $\frac {1}{2}$, $\frac {3}{4}$, $\frac {1}{4}$, $\frac {3}{2}$, $\frac {4}{3}$, $\frac {1}{3}$, $\frac {5}{2}$,

$\frac {1}{2}$, $\frac {3}{2}$, and $\frac {5}{2}$ are like fractions. Similarly, $\frac {3}{4}$ and $\frac {1}{4}$ are like fractions. And also, $\frac {4}{3}$ and $\frac {1}{3}$ are like fractions.

Unlike Fractions

A group of two or more fractions having different denominators is called like fractions.

For example consider a group of fractions $\frac {1}{2}$, $\frac {3}{4}$, $\frac {1}{4}$, $\frac {3}{2}$, $\frac {4}{3}$, $\frac {1}{3}$, $\frac {5}{2}$.

$\frac {1}{2}$ and $\frac {3}{4}$ are unlike fractions. Similarly, $\frac {1}{2}$, and $\frac {4}{3}$ are unlike fractions. And also $\frac {5}{2}$ and $\frac {4}{3}$ are unlike fractions.

How to Add and Subtract Fractions?

The addition and subtraction of fractions are performed in a similar way and use the same steps to perform the operation. The first step in performing addition or subtraction of fractions is to check whether the fractions involved in the operation are like fractions or unlike fractions.

If the fractions are like fractions, the numerators of the fractions are added or subtracted and the denominator of the result is the same as that of the fractions used in addition or subtraction.

If the fractions are unlike fractions, first of all, the fractions are converted to like fractions and then the resultant fractions are added or subtracted.

Adding and Subtracting Like Fractions

The process for adding and subtracting fractions with like denominators is quite simple because we just need to add or subtract the numerators keeping the denominator the same.

Adding Like Fractions

Let us add the fractions $\frac {3}{7}$ and $\frac {2}{7}$ using rectangular models. In this case, both the fractions have the same denominators. These fractions are called like fractions. The following figure represents both the fractions in the same model.

$\frac {3}{7}$ indicates that $3$ out of $7$ parts are shaded blue.

$\frac {2}{7}$ indicates that $2$ out of $7$ parts are shaded green.

To get the sum of $\frac {3}{7}$ and $\frac {2}{7}$, count the total number of colored boxes (blue and green both). There are $5$ colored boxes out of a total of $7$.

Therefore, $\frac {3}{7} + \frac {2}{7} = \frac {5}{7}$ 

Now, let us add the fractions with like denominators in numerical terms. In our case, we need to add $\frac {3}{7} + \frac {2}{7}$. The following steps will be used to add these two fractions (or any two fractions in general).

Step 1: Add the numerators of the given fractions. Here, the numerators are $3$ and $2$, so it will be $3 + 2 = 5$

Step 2: Keep the same denominator in the result. Here, the denominator of both the fractions is $7$. So, the denominator in the resultant fraction will be $7$.

Step 3: Therefore, the sum of $\frac {3}{7} + \frac {2}{7} = \frac {3 + 2}{7} = \frac {5}{7}$

Subtracting Like Fractions

Let us subtract the fractions 3/7 and 2/7 using the same rectangular models. We will represent 3/7 by shading 3 out of 7 parts and 2/7 by shading 2 out of 7.

In the above diagram when $\frac {2}{7}$ is taken out from $\frac {3}{7}$ ($\frac {2}{7}$ is subtracted from $\frac {3}{7}$), we are left with $\frac {1}{7}$.

So, $\frac {3}{7} – \frac {2}{7} = \frac {1}{7}$.

Now, let us subtract the fractions with like denominators in numerical terms. In our case, we need to perform $\frac {3}{7} – \frac {2}{7}$. The following steps will be used to subtract these two fractions (or any two fractions in general).

Step 1: We will subtract the numerators of the given fractions. Here, the numerators are $3$ and $2$, so it will be $3 – 2 = 1$.

Step 2: Keep the same denominator in the result. Here, the denominator of both the fractions is $7$. So, the denominator in the resultant fraction will be $7$.

Step 3: Therefore, the sum of $\frac {3}{7} – \frac {2}{7} = \frac {3 – 2}{7} = \frac {1}{7}$.

Adding and Subtracting Unlike Fractions

To add or subtract unlike fractions, first of all, the denominators are made equal and converted to like fractions. After that, we use the steps of addition or subtraction of like fractions.

Adding Unlike Fractions

Let us subtract the fractions $\frac {1}{2}$ and $\frac {1}{3}$ using the same rectangular models. We will represent $\frac {1}{2}$ by shading $1$ out of $2$ parts and $\frac {1}{3}$ by shading $1$ out of $3$. 

$\frac {1}{2}$ indicates that $1$ out of $2$ part is shaded blue.

$\frac {1}{3}$ indicates that $1$ out of $3$ part is shaded green.

We’ll represent these fractions with common denominators, i.e., like fractions.

And, now let’s add these fractions.

So, $\frac {1}{2} + \frac {1}{3} = \frac {5}{6}$ 

Now, let us add the fractions with like denominators in numerical terms. 

Consider the following example to understand the process.

Add $\frac {2}{3} + \frac {1}{5}$

The following steps will be used to add these two fractions (or any two fractions in general).

Step 1: Find the Least Common Multiple (LCM) of the denominators. Here, the LCM of $3$ and $5$ is $15$.

Step 2: Convert the given fractions to like fractions by writing the equivalent fractions for the respective fractions such that their denominators remain the same. 

In this case, it will be $\frac {2}{3} \times \frac {5}{5} = \frac {10}{15}$ and $\frac {1}{5} \times \frac {3}{3} = \frac {3}{15}$  

Step 3: Now, that we have converted the given fractions to like fractions we can add the numerators and keep the same denominator. 

$\frac {10}{15} + \frac {3}{15} = \frac {13}{15}$

Subtracting Unlike Fractions

Let us subtract the fractions $\frac {1}{4}$ and $\frac {1}{2}$ using the same rectangular models. We will represent $\frac {1}{4}$ by shading $1$ out of $4$ parts and $\frac {1}{2}$ by shading $1$ out of $2$. 

$\frac {1}{2}$ indicates that $1$ out of $2$ parts are shaded blue.

$\frac {1}{4}$ indicates that $1$ out of $4$ parts is shaded blue.

$\frac {1}{2}$ indicates that $1$ out of $2$ parts is shaded green.

Now, we’ll represent these fractions with common denominators, i.e., like fractions.

And, now let’s subtract these fractions.

Therefore, $\frac {1}{2} – \frac {1}{4} = \frac {1}{4}$ 

For subtracting unlike fractions, we follow the same steps as we did for the addition of unlike fractions. 

Let us understand this with the help of an example.

Subtract $\frac {1}{2}$ from $\frac {5}{6}$.

The following steps will be used to add these two fractions (or any two fractions in general).

Step 1: Find the Least Common Multiple (LCM) of the denominators. Here, the LCM of $2$ and $6$ is $6$.

Step 2: Convert the given fractions to like fractions by writing the equivalent fractions for the respective fractions such that their denominators remain the same. 

In this case, it will be $\frac {1}{2} \times \frac {3}{3} = \frac {3}{6}$ and $\frac {5}{6} \times \frac {1}{1} = \frac {5}{6}$  

Step 3: Now, that we have converted the given fractions to like fractions we can add the numerators and keep the same denominator. 

So, $\frac {5}{6} – \frac {3}{6} = \frac {2}{6}$.

And hence $\frac {5}{6} – \frac {1}{2} = \frac {2}{6}$.

Adding and Subtracting Mixed Fractions

Adding and subtracting mixed fractions is done by converting the mixed fractions to improper fractions and then the addition or subtraction is done according to the rules of whole numbers and fractions the requirement.

Adding Mixed Fractions

To understand the addition of mixed fractions, let’s understand these with the help of the following example.

Add $2\frac {1}{4}$ and $3\frac {1}{4}$ 

$2\frac {1}{4}$ indicates that $2$ whole parts ($1 + 1$) ($1$ whole = $4$ parts) are shaded blue.

$3\frac {1}{4}$ indicates that $3$ whole parts ($1 + 1 + 1$) ($1$ whole = $4$ parts) are shaded green.

Now, adding these two fractions, we get

Therefore, $2\frac {1}{4} + 3\frac {1}{4} = 5\frac {1}{2}$. 

For adding mixed fractions, we follow the steps of addition of whole numbers and fractions 

Let us understand this with the help of an example.

Add $4\frac {1}{4}$ and $5\frac {1}{2}$. 

The following steps will be used to add these two fractions (or any two fractions in general).

Step 1: Add the whole parts $4 + 5 = 9$

Step 2: Add the fractions $\frac {1}{4}+ \frac {1}{2} = \frac {3}{4}$ 

Step 3: Write the whole part and fraction together

Step 4: $4\frac {1}{4} + 5\frac {1}{2} = 9\frac {3}{4}$.

Subtracting Mixed Fractions

To understand the subtraction of mixed fractions, let’s understand these with the help of the following example.

Subtract $9\frac {1}{2}$  and $3\frac {1}{4}$. 

The following steps will be used to add these two fractions (or any two fractions in general).

Step 1: Subtract the whole parts $9 – 3 = 6$

Step 2: Subtract the fractions $\frac {1}{2} – \frac {1}{4} = \frac {1}{4}$  

Step 3: Write the whole part and fraction together

Therefore, $9\frac {1}{2} – 3\frac {1}{4} = 6\frac {1}{4}$.

Key Takeaways

• Adding or Subtracting Like Fractions
  • Add or subtract the numerators
  • Keep the denominators the same
  • The numerator of the resultant fraction is the sum or the difference and the denominator is that of the original fraction
• Adding or Subtracting Unlike Fractions
  • Calculate the LCM of the denominators
  • Convert fractions to equivalent fractions with LCM as the denominator
  • Add or subtract the numerators
  • The numerator of the resultant fraction is the sum or the difference and the denominator is LCM
• Adding or subtracting Mixed Fractions
  • Add or subtract the whole parts
  • Add or subtract the fractional parts
  • Write the whole and fraction parts together

Practice Problems

1. Add te following
   1. $\frac{2}{4} + \frac{1}{4}$ 
   2. $\frac{3}{11} + \frac{5}{11}$
   3. $\frac{6}{23} + \frac{12}{23}$
   4. $\frac{5}{47} + \frac{9}{47}$
   5. $\frac{5}{6} + \frac{1}{3}$
   6. $\frac{2}{9} + \frac{5}{18}$
   7. $\frac{3}{5} + \frac{5}{7}$
   8. $\frac{6}{11} + \frac{7}{9}$
   9. $2/13 + 4/9$
   10. $12\frac{5}{9} +7 \frac{5}{9}$
   11. $6\frac{3}{4} + 2\frac{3}{4} $
   12. $15\frac{3}{4} + 6\frac{1}{2}$
   13. $24\frac{3}{8} + 11\frac{1}{4}$ 
2. Subtract the following
   1. $\frac{12}{23} – \frac{11}{23}$
   2. $\frac{15}{43} – \frac{8}{43}$
   3. $\frac{16}{21} – \frac{9}{21}$
   4. $\frac{5}{9} – \frac{2}{9}$
   5. $\frac{11}{15} – \frac{1}{5}$ 
   6. $\frac{2}{3} – \frac{6}{7}$
   7. $\frac{3}{8} – \frac{1}{16}$
   8. $\frac{5}{9} – \frac{4}{27}$
   9. $67 \frac{2}{3} – 45 \frac{1}{3}$
   10. $34 \frac{5}{6} – 12 \frac{1}{6}$
   11. $52 \frac{2}{3} – 28 \frac{1}{6}$
   12. $21 \frac{4}{5} – 18 \frac{1}{2}$
   13. $18 \frac{6}{7} – 3 \frac{10}{21}$

Suggested Reading

• Multiplication and Division of Integers
• Addition and Subtraction of Integers
• How To Solve Radical Equations?