Addition and Subtraction of Fractions: Formula, Types

In Addition and Subtraction of Fractions, numerator is added and the answer is put above the same denominator. If the denominators are different, convert them into like fractions by taking the LCM. A fraction indicates the number of elements that make up a whole. Fractions are categorized as Proper Fraction, Improper Fraction, Mixed Fraction, Like Fraction, Unlike Fraction, Equivalent Fraction. Fraction whose numerator is less than the denominator is called Proper Fraction while the fraction whose numerator is greater than the denominator is called improper fraction. The combination of a fraction with a whole number is a mixed fraction. 

Also read: Ratio to Percentage

Key Terms: Fraction, Proper Fraction, Mixed Fraction, Improper Fraction, Addition of Fraction, Subtraction of Fraction, Like Fractions, Unlike Fraction

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What is Fraction?

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In arithmetic, a fraction is a number stated as a quotient, where the numerator is divided by the denominator. Both are integers in a simple fraction. There is a fraction in the numerator or denominator of a complex fraction. If the numerator is less than its denominator, the fraction is a Proper fraction. It's called an improper fraction if the numerator is bigger, and it can also be expressed as a mixed number.

Numerator and Denominator

Addition and Subtraction of Fractions

One can perform Addition and Subtraction of Fractions by categorizing the fraction into three categories. Categorizing the fraction in the following category will help in easily finding the result. Following are the categories:

• Add or subtract fraction with the same denominator
• Add or subtract fraction with different denominator
• Add or subtract mixed number fraction

Also Read: Rules of Adding and Subtracting Integers

Adding and Subtracting Fractions with Like Denominators

Like Denominators are the ones having the same denominators and are also called Like Fractions. For example, 3/5, 6/5, 1/5, etc.

Adding Fractions with Like Denominators

By simply taking the common denominator as the same and adding the numerator we can get the addition of fractions with like denominators.

For example: add 1/9+5/9+8/9

As we can see the denominator of the fraction is the same, we can consider the denominator to be common for all the numerators, like

= (1+5+8)/9

= 14/9

Hence, the basic formula to add the Fractions with Like Denominators can be written as:

Numerator1(Nr1)/Denominator (Dr) + Numerator2(Nr2)/Denominator(Dr) +……. +Numeratorn(Nrn)/ Denominator(Dr)

(Nr1+Nr2+…..+Nrn) /Dr

Subtracting Fractions with Like Denominators

To subtract fractions with a denominator, take the denominator as common as it is the same in all the fractions and just subtract the numerator.

For example: Subtract: 9/8-3/8-1/8

As we can see the denominator of the fraction is the same, we can consider the denominator to be common for all the numerators, like

= (9-3-1)/8

= 5/8

Hence, the basic formula to subtract the Fractions with Like Denominators can be written as

Numerator1(Nr1)/Denominator (Dr) - Numerator2(Nr2)/Denominator(Dr) - ……. - Numeratorn(Nrn)/ Denominator(Dr)

(Nr1-Nr2+…..-Nrn) /Dr

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Adding and Subtracting Fraction with Unlike Denominator

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Unlike Denominators are those denominators whose value is different in every fraction and hence also called Unlike Fractions. For example, 5/9, 8/7, 1/5, etc.

Adding Fractions with Unlike Denominators

To add Unlike denominators, we will have to take the LCM of the denominator and convert the fractions to equivalent fractions with the denominators as the LCM and finally add the fractions.

let us take an example for more clarity,

Adding 1/7 with 5/3

Take LCM of (7,3) = 21

(1/7) *(3/3) = 3/21 and (5/3) *(7/7) = 35/21

As the denominator has become the same in both the fractions (i.e., like fraction) we can follow the same steps as Adding Fractions with Like Denominators

i.e., (3+35)/21 = 38/21

Subtracting Fractions with Unlike Denominators

To subtract Unlike denominators, follow the same steps, i.e., we will have to take the LCM of the denominator and convert the fractions to equivalent fractions with the denominators as the LCM and finally subtract the fractions.

let us take an example for more clarity,

subtract 5/9, with 3/7

Take LCM of (9,7) = 63

= (5/9)*(7/7) – (3/7)*(9/9)

As the denominator has become the same in both the fractions (i.e., like fraction) we can follow the same steps as Subtracting Fractions with Like Denominators

= (45-27)/63

= 18/63 or 2/7

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Adding and Subtracting of Mixed Fraction

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A mixed fraction is an improper fraction that is represented as the sum of a whole number and a proper fraction.

The denominator can be the same or different along with Mixed fractions. 

Addition and Subtraction of Mixed Fractions with the Same Denominator

There are two methods to solve the Mixed fraction and are discussed below.

Method I

Follow the steps to convert the mixed fraction into a simple fraction.

• Convert the improper fractions from the mixed fractions.
• Check whether the denominators are equal.
• Now, take the denominator as common and add the numerators.
• And lastly, simplify the answer.

Example: 5 ¼ + 6 ¾

Solution: 5 ¼ can be written as 21/4 and 6 ¾ can be written as 27/4

i.e., (21/4) + (27+4)

= (21+27)/4

= 48/4 or 12

Method II

Alternatively, the following steps are needed to be followed to add the mixed fractions.

• Add the whole number and the fractions separately
• Combine the whole number and fractions
• Now Improper fraction is to be converted to mixed fraction.

For example: 5 ¼ + 6 (5/4) 

Solution: (5+6) (1/4 + 5/4)

= 11 (6/4)

= 11 (3/2)

= 25/2

Subtraction of Mixed Fractions with the Same Denominator

There are two methods to solve the Mixed fraction and are discussed below.

Method I

Follow the steps to convert the mixed fraction into a simple fraction.

• Convert the improper fractions from the mixed fractions.
• Check whether the denominators are equal.
• Now, take the denominator as common and subtract the numerators.
• And lastly, simplify the answer.

Example: 8 (7/4) - 6 (3/4)

Solution: 8(7/4) can be written as 39/4 and 6 ¾ can be written as 27/4

i.e., (39/4) - (27+4)

= (39-27)/4

= 12/4 or 3/1

Method II

Alternatively, the following steps are needed to be followed to subtract the mixed fractions.

• subtract the whole number and the fractions separately
• Combine the whole number and fractions
• Now Improper fraction is to be converted to mixed fraction.

For example: 9(7/8) – 5(3/8)

Solution: (9-5) (7/8-3/8)

= 4(4/8)

= 4(1/2)

= 9/2

Also Read: Mathematics Preparation Tips

Addition and Subtraction of Mixed Fractions with the Same Denominator

There are two methods to solve the Mixed fraction and are discussed below.

Method I

Follow the steps to convert the mixed fraction into a simple fraction.

• Take the LCM of the denominators
• Convert the fraction into an equivalent fraction with the help of LCM
• Convert the improper fractions from the mixed fractions.
• Check whether the denominators are equal.
• Now, take the denominator as common and add the numerators.
• And lastly, simplify the answer.

Example: 5 (2/3) + 6 ¾

Solution: 5 (2/3) can be written as 17/3 and 6 ¾ can be written as 27/4

LCM of 3 & 4 is 12

17/3 * 4/4 and 27/4 * 3/3

= 68/12 + 51/12

=119/12

Method II

Alternatively, the following steps are needed to be followed to add the mixed fractions.

• Take the LCM of the denominators
• Convert the fraction into an equivalent fraction with the help of LCM
• Add the whole number and the fractions separately
• Combine the whole number and fractions
• Now Improper fraction is to be converted to mixed fraction.

For example: 2 (3/8) + 8 (5/4) 

Solution: LCM of 8 and 4 is 8

Hence, (2+8) (3/8*1/1 + 5/4*2/2)

= 10 (3+(10/8))

= 10(13/8)

= 93/8

Subtraction of Mixed Fractions with the Different Denominator

There are two methods to solve the Mixed fraction and are discussed below.

Method I

Follow the steps to convert the mixed fraction into a simple fraction.

• Take the LCM of the denominators
• Convert the fraction into an equivalent fraction with the help of LCM
• Convert the improper fractions from the mixed fractions.
• Check whether the denominators are equal.
• Now, take the denominator as common and add the numerators.
• And lastly, simplify the answer.

Example: 5 (2/3) - 6 ¾

Solution: 5 (2/3) can be written as 17/3 and 6 ¾ can be written as 27/4

LCM of 3 & 4 is 12

17/3 * 4/4 and 27/4 * 3/3

= 68/12 - 51/12

=17/12

Method II

Alternatively, the following steps are needed to be followed to add the mixed fractions.

• Take the LCM of the denominators
• Convert the fraction into an equivalent fraction with the help of LCM
• Subtract the whole number and the fractions separately
• Combine the whole number and fractions
• Now, Improper fraction has to be converted to mixed fraction.

For example: 7 (7/8) - 2 (3/4) 

Solution: LCM of 8 and 4 is 8

Hence, (7-2) (7/8*1/1 + 3/4*2/2)

= 5(7-(6/8))

= 5(1/8)

= 41/8

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Things to Remember

• Like fraction is the one having the same denominator and numerator can be different.
• Unlike fraction has different denominators in each fraction.
• Mixed fractions are of type a(b/c).
• LCM has to be taken in the case of mixed fractions.
• If the numerator is less than the denominator the fraction is said to be a proper fraction.
• An improper fraction has a numerator greater than the denominator.

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