Addition of Fractions | How To Add Like and Unlike Fractions

Addition of Fractions is the method of adding two fractions together to get a single fraction. A fraction represents the division of one number by another, written with a numerator (top number) and a denominator (bottom number) separated by a horizontal line.

Let’s learn the different kinds of addition in fractions, with the help of examples.

Addition of Fractions: Rules

Adding of fractions is a very fundamental concept in maths. A fraction represents a part of a whole, consisting of two parts: a numerator and a denominator.

Here are some rules to remember for adding fractions:

• First, we need to make sure the fractions have the same denominator. If not, we find a common denominator by multiplying or dividing both the numerator and the denominator by the same number.
• The numerators are added and the result is put over the common denominator.
• If possible, the fraction is simplified by dividing the numerator and denominator by a common factor.

There are two methods for adding fractions based on the denominators of the fractions, which are:

1. Adding Fractions with Same Denominators
2. Adding Fractions with Different Denominators

How to Add Fractions with Same Denominators

Fractions with the same denominator, are also called like fractions.

Let us suppose we have two fractions say, a/b and c/b. Here we see that the denominator of both fractions is the same.

We can easily add these fractions using the following formula,

a/b + c/b = (a+c)/b

Steps to Add Fractions with the Same Denominators:

Step 1: Find the denominators of the fractions you wish to add. In this case, the denominator will be the same for all fractions.

Step 2: Take the LCM of the denominator. The number itself is the LCM of the same number.

Step 3: Since the LCM and denominator is the same, multiply the numerators by 1.

Step 4: To find the solution, add the numerators specified in the fraction and use the same denominator and that will be your answer.

Let’s understand this with the help of an example.

Example: Add 2/5 and 3/5. These fractions have the same denominator.

Step 1: Here 5 and 5 are the denominator of the given fractions

Step 2: As both denominators are the same, the LCM will be 5 only.

Step 3: Since the LCM and Denominators are the same we have to multiply numerators and denominators by 1.

2/5 = (2 × 1)/(5 × 1) = 2/5

3/5 = (3 × 1)/(5 × 1) = 3/5

Step 4: Our numerators are 2 and 3 add it (2+3 = 5) and hence we got 5/5. Simplify it and the result will be 1.

How To Add Like Fractions Diagram

Let’s consider an example of 1/4 + 2/4, which is a like fraction. We will represent both fractions in the form of a circle as follows:

• The denominator is the same for both i.e., both circles are divided in equal parts which can be easily subtracted. So there is no need to modify the denominator because LCM will be the same as the denominator. Therefore, unlike denominators, the given picture will not be divided into further pieces.

• To reach the required result, all that is left to do is add the numerators while keeping the denominators constant. Additionally, see whether the final result can be attained by further simplifying the already acquired result.

As a result, our answer is (1+2)/4 = 3/4, as indicated in the image below.

How to Add Fractions with Different Denominators

Fractions with different denominators are also called unlike fractions.

Let us take two fractions say, a/b and c/d. They are unlike fractions as the denominator of both fractions is different.

We add unlike fractions using the following formula:

a/b + c/d = (a/b) × (d/d) (c/d) × (b/b) = ad/bd + cb/bd = (ad + bc)/bd

Steps to Add Fractions with Different Denominators:

Step 1: Find the opposite denominators of the fractions you wish to add .

Step 2: Determine the denominators’ least common multiples (LCM), or the lowest integer that can be divided equally by each denominator.

Step 3: To make the denominator equal, multiply the numerator and denominator by the LCM factor.

Step 4: Add the fractions’ numerators while maintaining the LCM as the denominator.

Step 5: If at all feasible, simplify the outcome.

Let’s try to understand this with the help of examples.

Example 1: Add 1/4 and 3/8. These fractions have different denominators, so we need to find a common denominator.

Step 1: Find the unlike denominators of the fractions you wish to add in step one.

The denominators, 4, and 8, are dissimilar.

Step 2: Determine the denominators’ least common multiple (LCM).

8 is the LCM of 4 and 8.

Step 3: To make the denominator equal, multiply the numerator and denominator by the LCM factor.

1/4 = (1 × 2)/(4 × 2) = 2/8

3/8 = (3 × 1)/(8 × 1) = 3/8

Step 4: Add the fractions’ numerators while maintaining the LCM as the denominator.

2/8 + 3/8 = 5/8

Step 5: If at all feasible, simplify the outcome. Since 5/8 cannot be made much simpler, that is the final solution.

Example 2: Add two numbers 3/4 and 1/5.

Step 1: Find the unlike denominators of the fractions you wish to add in step one.

The denominators, 4 and 5 are dissimilar.

Step 2: Determine the denominators’ least common multiple (LCM).

20 is the LCM of 4 and 5.

Step 3: To make the denominator equal, multiply the numerator and denominator by the LCM factor.

3/4 = (3 × 5)/(4 × 5) = 15/20

1/5 = (1 × 4)/(5 × 4) = 4/20

Step 4: Add the fractions’ numerators while maintaining the LCM as the denominator.

15/20 + 4/20 = 19/20

Step 5: If at all feasible, simplify the outcome.Since 19/20 cannot be made much simpler, that is the final solution.

How To Add Unlike Fractions Diagram

Let’s understand the addition of two unlike fractions, 2/3 + 1/4 with the help of following diagram :

To add 2/3 and 1/4 which can be done by taking the LCM of the denominators, which may be expressed by dividing the circle into equal pieces as follows:

Now that each division of the two circles is the same in size, we can add them as a single unit.

Therefore, 8/12 + 3/12 = 11/12 i.e., 11 parts out of 12 equal parts.

Learn more about, Adding Fractions with Unlike Denominators

How to Add Fractions with Whole Numbers

To add fractions with whole numbers, first we convert the whole number into a fraction. Take the denominator of the whole number as one. Then add the fraction as you would when adding fractions with different denominators.

Steps to add fractions with whole numbers:

Step 1: Convert the whole number into an Improper Fraction.

• Multiply the whole number by the denominator of the given fraction.
• The result becomes the new numerator, and the denominator will be the same as given fraction.

Step 2: Add the improper fraction and the given fraction just like adding with same denominators.

Step 3: Simplify the resulting fraction, if possible.

Example: Suppose we want to add 2/3 and 1. To add a fraction and a whole number, simply express the whole number as a fraction with the same denominator as the other fraction.

Solution :

Step 1: Convert the whole number to an improper fraction: Multiply 1 by the denominator of the given fraction which is 3. So,1 can be written by 3/3 .

Step 2: Now , we have to add 2/3 + 3/3 which is equal to 5/3 .(just like adding fractions with same denominators)

Step 3: We can write 5/3 in mixed fraction 1(²/₃).

How To Add Mixed Fractions

We add mixed fractions by simply changing the mixed fractions into improper fractions and then adding them accordingly.

A mixed fraction is written as, a(b/c)

a(b/c) = (ac + b)/c

How to Add Mixed Fractions Step-By-Step:

Step 1: Convert the mixed fractions into improper fractions.

• Multiply the whole number with the denominator and add the numerator, the result will become our numerator and the denominator will be the same as given fractions.
• a(b/c) will be written as ((a × c) +b) / c.

Step 2: Determine the denominators’ least common multiples (LCM), or the lowest integer that can be divided equally by each denominator.

Step 3: To make the denominator equal, multiply the numerator and denominator by the LCM factor.

Step 4: Add the fractions’ numerators while maintaining the LCM as the denominator.

Step 5: Simplify the result, if necessary.

Here is an example to illustrate how to add mixed fractions.

Example: Consider two numbers 2(3/4) and 1(1/2). Add the given mixed fractions.

Step 1: Convert the mixed fractions into improper fractions.

2(3/4) = (2 × 4 + 3) / 4 = 11/4

1(1/2) = (1 × 2 + 1) / 2 = 3/2

Step 2: Find a common denominator.

The denominators are 4 and 2, so the LCM of 4 and 2 is 4.

Step 3: To make the denominator equal, multiply the numerator and denominator by the LCM factor.

11/4 remains the same.

3/2 = (3 × 2) / (2 × 2) = 6/4

Step 4: Add the fractions’ numerators while maintaining the LCM as the denominator.

11/4 + 6/4 = 17/4

Step 5: Simplify

The fraction 17/4 cannot be simplified further since 17 and 4 do not have a common factor other than 1.

So, the sum of 2(3/4) and 1(1/2) is 17/4 or you can write it in mixed fraction format as 4(1/4).

How to Add Fractions with Variables

Here are the steps to Add Fractions with Variables :

Step 1: Find the fractions variables that need to be added.

Imagine that we have two fractions that contain variables, such as (a/b) and (c/d).

Step 2: Find a common denominator.

We require a common denominator in order to add fractions. Find the denominators’ least common multiple (LCM), which in this example would be bd.

Step 3: Modify the fractions such that they have a common denominator. To do this, multiply each fraction’s numerator and denominator by the amounts required to arrive at the common denominator.

The result of multiplying the numerator and denominator of the fraction (a/b) by d is (ad/bd).

The fraction (bc/bd) is created by multiplying the numerator and denominator of (c/d) by b.

Step 4: Add the revised fractions .Add the numerators of the modified fractions together, keeping the common denominator constant.

(bc/bd) + (ad/bd) = (ad + bc)/bd.

Step 5: Simplify, if necessary.

How to Add Fractions with Co-Prime Denominators

Co-Prime Denominators are the denominators with no common factors other than 1.

To add fractions with co-prime denominators we follow the steps added below,

Let’s suppose we have to add a/b + c/d where, b and d are co-prime denominator.

Step 1: Multiply the numerator of first fraction(a) with denominator of second fraction(d) and numerator of second fraction(c) with denominator of first fraction(b) and they both are added.

Step 2: Now both the denominators are multiplied (bd)

Step 3: Now both numbers from step 1 and step 2 are divided {(ad + bc)/(bd)} to get the required fractions.

This can be explained by the following example :

= (2/7) + (5/6)

= {(2×6 + 5×7)/(7×6)}

= (12 + 35)/(42)

= 47/42

Gist of Adding Fractions

To add fractions, you can follow these three steps:

1. Make sure the denominators (bottom numbers) are the same.
2. Add the numerators (top numbers), and put that answer over the denominator.
3. Simplify the fraction if possible.

Addition of Fractions Solved Examples

Here are some solved examples on addition of fractions:

Example 1: Add 2/3 and 1/3

Solution:

3 is the denominator in the both given fractions. (Like Fractions)

First Fractional Number: 2/3

Second Fractional Number: 1/3

So, 2/3 + 1/3

= (2+1) / 3

= 3/3

= 1

Example 2: Add 1/4 and 2/3

Solution:

First Fraction: 1/4

Second Fraction: 2/3

LCM of 4 & 3 is 12.

First Fractional Number: (1×3)/(4×3) = 3/12

Second Fractional Number: (2×4)/(3×4) = 8/12

So, 3/12 + 8/12 = (3+8) / 12

= 11/12

Example 3: Add 3/2 and 1

Solution:

1 is represented as, (1 × 1) / (2 × 2) = 2/2

Now,

= 3/2 + 1

= 3/2 + 2/2

= (3+2)/2

= 5/2

5/2 in mixed fraction is 2(1/2)

Example 4: Add 1(2/5) and 2(1/5)

Solution:

Change mixed fraction to improper fraction

1(2/5) = ((1 × 5)+2)/5 = 7/5

2(1/5) = ((2 × 5)+1)/5 = 11/5

7/5 + 11/5

= 18/5

= 3(3/5)

Addition of Fractions Worksheet

Solving the worksheet below will help you improve your understanding of Adding Fractions.

1. Add 13/6 and 7/6.

2. Kabir buys a cake and ate 1/5 of it and gives the 2/3 of the remaining to Somesh, then find the amount of cake left with Kabir.

3. Add 43/35 and 9/7.

4. Add 3(4/7) and 4(3/7).

5. Add 9(3/5) and 11(3/7).

6. Ram spent 1/3 of his money on rent, and did 1/2 on food and other needs. What fraction of his salary does he spend altogether?

Subtraction of Fractions

In mathematics, addition and subtraction are known to be comparable operations. While subtracting involves taking a number away from another, addition involves adding two or more numbers. As a result, the same rule applies to addition and subtraction of fractions.

When the fractions have identical denominators, subtracting them involves directly subtracting the numerators while maintaining the common denominator.

However, when the fractions have different denominators, the initial step is to rationalise them before proceeding with the subtraction.

How To Add Fractions- FAQs

1. How to Add Fractions with Same Denominator?

To add fractions with same denominators, simply add the numerators together while keeping the common denominator unchanged. The sum will have the same denominator.

2. How to Add Fractions with Different Denominators?

To add fractions with different denominators, we first take the LCM of the denominator and then add then accordingly.

3. What are Fractions?

Fractions are the rational numbers that are represented as parts of a number. They are written as a numerator divided by a denominator. For example, the fraction 1/2​ represents one part out of two, or 50%.

4. What are Like Fractions?

Adding fractions with like denominators is the process of combining two or more fractions that have the same number in their denominators.

5. How to Add Improper Fractions?

Improper fractions are the fractions in which the numerator of the fraction is greater than denominator of the fraction and Improper Fractions can be easily added by the rules of addition of fraction as added above.

6. What are the rules to Add and Subtract Fractions?

To add or subtract fractions, make sure they have a common denominator. If they don’t, find the LCM of the denominators and convert each fraction to an equivalent fraction with this common denominator. Once the fractions have the same denominator, simply add or subtract the numerators while keeping the denominator the same. Then simplify the resulting fraction, if possible.

7. How to do addition of 3 Fractions altogether?

3 fractions with same denominator(Like Fractions) can be easily added by taking the denominator of the fraction common and adding the numerator of the fraction directly. But if all the 3 fractions are unlike fractions then before adding them we first take the LCM of the denominator fractions and then add them accordingly.