Addition and Subtraction of Fractions

While adding and subtracting fractions, we need to check whether the fractions have the same denominators or different denominators and then the calculation starts. Let us learn more about the addition and subtraction of fractions in this article.

Addition and subtraction of fractions is done using similar rules in which the denominators are checked before the addition or subtraction starts. After the denominators are checked, we can add or subtract the given fractions accordingly. The denominators are checked in the following way.

• If the denominators of the given fractions are the same, we add or subtract only the numerators and we retain the denominator.
• If the denominators are different, we convert the fractions to like fractions so that the denominators become the same, and then we add or subtract, whatever is required.

Let us learn about these in the following sections.

The process for adding and subtracting fractions with like denominators is quite simple because we just need to work with the numerators.

Adding Fractions with Like Denominators

Let us add the fractions 1/5 and 2/5 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.

• 1/5 indicates that 1 out of 5 parts are shaded yellow.
• 2/5 indicates that 2 out of 5 parts are shaded blue.

Out of the 5 parts, 3 parts are shaded. In the fractional form, this can be represented as 3/5.

Now, let us add the fractions with like denominators in numerical terms. In this case, we need to add 1/5 + 2/5. Let us use the following steps to understand the addition.

• Step 1: Add the numerators of the given fractions. Here, the numerators are 1 and 2, so it will be 1 + 2 = 3
• Step 2: Retain the same denominator. Here, the denominator is 5.
• Step 3: Therefore, the sum of 1/5 + 2/5 = (1 + 2)/5 = 3/5

It should be noted that we use the same method for subtracting fractions.

Subtracting Fractions with Like Denominators

Let us subtract the fractions 2/5 and 1/5 using rectangular models. We will represent 2/5 in this model by shading 2 out of 5 parts. We will further shade out 1 part from our shaded parts of the model which would represent removing 1/5.

We are now left with 1 part in the shaded parts of the model.

Now, let us subtract the fractions with like denominators in numerical terms. In this case, we need to subtract 2/5 - 1/5. Let us understand the procedure using the following steps.

• Step 1: We will subtract the numerators of the given fractions. Here, the numerators are 2 and 1, so it will be 2 - 1 = 1
• Step 2: Retain the same denominator. Here, the denominator is 5.
• Step 3: Therefore, the difference of 2/5 - 1/5 = (2 - 1)/5 = 1/5

Example: Add 1/5 + 1/3

Solution: For adding unlike fractions we need to use the following steps

• Step 1: Find the Least Common Multiple (LCM) of the denominators. Here, the LCM of 5 and 3 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. Here, it will be \(\frac {1}{5}\)×\(\frac {3}{3}\)=\(\frac {3}{15}\)
• Step 3: Similarly, an equivalent fraction of 1/3 with denominator 15 is \(\frac {1}{3}\)×\(\frac {5}{5}\)=\(\frac {5}{15}\)
• Step 4: Now, that we have converted the given fractions to like fractions we can add the numerators and retain the same denominator. This will be 3/15 + 5/15 = 8/15

Subtracting Fractions with Unlike Denominators

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.

Example: Subtract 5/6 - 1/3

Solution: For subtracting unlike fractions we need to use the following steps.

• Step 1: Find the Least Common Multiple (LCM) of the denominators. Here, the LCM of 6 and 3 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. Here, it will be \(\frac {5}{6}\)×\(\frac {1}{1}\)=\(\frac {5}{6}\)
• Step 3: Similarly, an equivalent fraction of 1/3 with denominator 6 is \(\frac {1}{3}\)×\(\frac {2}{2}\)=\(\frac {2}{6}\)
• Step 4: Now, that we have converted the given fractions to like fractions we can subtract the numerators and retain the same denominator. This will be 5/6 - 2/6 = 3/6. This can be further reduced to 1/2

Adding and subtracting mixed fractions is done by converting the mixed fractions to improper fractions and then the addition or subtraction is done as per the requirement. Let us understand these with the help of the following examples.

Example: Add the mixed fractions: \(2\dfrac{1}{4}\) + \(1\dfrac{3}{4}\)

Solution: First let us convert the mixed fractions to improper fractions.

• Step 1: Convert the given mixed fractions to improper fractions. So, \(2\dfrac{1}{4}\) will become 9/4; and \(1\dfrac{3}{4}\) will become 7/4
• Step 2: Add the fractions by adding the numerators because the denominators are the same. This will be 9/4 + 7/4= 16/4.
• Step 3: Reduce the fraction, if required. This will become, 16/4 = 4. Therefore, \(2\dfrac{1}{4}\) + \(1\dfrac{3}{4}\) = 4.

Now, let us understand the subtraction of mixed fractions using the same method.

Example: Subtract the mixed fractions: \(5\dfrac{1}{3}\) - \(2\dfrac{1}{3}\)

Solution: First let us convert the mixed fractions to improper fractions.

• Step 1: Convert the given mixed fractions to improper fractions. So, \(5\dfrac{1}{3}\) will become 16/3; and \(2\dfrac{1}{3}\) will become 7/3
• Step 2: Subtract the fractions by subtracting the numerators because the denominators are the same. This will be 16/3 - 7/3 = 9/3
• Step 3: Reduce the fraction, if required. This will become, 9/3 = 3. Therefore, \(5\dfrac{1}{3}\) - \(2\dfrac{1}{3}\) = 3

Adding and subtracting fractions with whole numbers can be done using the following method. Let us understand this using an example.

Example: Add 7/4 + 5

Solution: Let us add 7/4 + 5 using the following steps.

• Step 1: Write the whole number in the form of a fraction. In this case the whole number is 5 which can be written as 5/1. So, now we need to add 7/4 + 5/1
• Step 2: Now, find the LCM of the denominators and convert the given fractions to like fractions. Here the LCM of 4 and 1 is 4. And after converting them to like fractions we get, (7 × 1)/(4 × 1) + (5 × 4)/(1 × 4) = 7/4 + 20/4
• Step 3: Add the numerators while the denominator remains the same. Here, 7/4 + 20/4 = 27/4 = \(6\dfrac{3}{4}\)

Now, let us understand the subtraction of a fraction from a whole number with the help of the following example.

Example: Subtract 6 - 3/5

Solution: Let us subtract 6 - 3/5 using the following steps.

• Step 1: Write the whole number in the form of a fraction. In this case the whole number is 6 which can be written as 6/1. So, now we need to subtract 6/1 - 3/5
• Step 2: Now, find the LCM of the denominators and convert the given fractions to like fractions. Here the LCM of 1 and 5 is 5. And after converting them to like fractions we get, (6 × 5)/(1 × 5) - (3 × 1)/(5 × 1) = 30/5 - 3/5
• Step 3: Subtract the numerators while the denominator remains the same. Here, 30/5 - 3/5 = 27/5 = \(5\dfrac{2}{5}\)

Important Notes on Adding and Subtracting Fractions

• For adding and subtracting like fractions, we can directly work with the numerators while the denominators remain the same.
• For adding and subtracting unlike fractions, never add or subtract the numerators and denominators directly. Convert them to like fractions and then add or subtract.

☛ Related Topics

• Adding Fractions
• Subtraction of Fractions
• Multiplying Fractions
• Division of Fractions
• Adding Fractions with Unlike Denominators
• Subtracting Fractions with Unlike Denominators
• Like Fractions Calculator
• Fractions Calculator

How to Add and Subtract Fractions?

For adding and subtracting fractions, we first need to check the denominators. If the denominators are the same, we simply add or subtract the numerators and retain the same denominator. In the case of unlike fractions, when the denominators are not the same, we convert the unlike fractions to like fractions by finding the LCM of the denominators. This helps in writing their respective equivalent fractions and then they are added or subtracted, as required.

How to Add and Subtract Fractions with Different Denominators?

In order to add and subtract fractions with different denominators, we need to convert the fractions to like fractions so that the denominators become the same. Once the denominators are the same, we can add or subtract the numerators. In order to convert the given fractions to like fractions, we need to find the LCM of the denominators and then write their respective equivalent fractions. The equivalent fractions with the same denominators can then be added or subtracted, as the case may be.

How to Add and Subtract Fractions with Whole Numbers?

For adding and subtracting fractions with whole numbers we use the following method.

• Write the whole number in the form of a fraction by writing 1 as its denominator. For example, if we need to add 8/7 + 5, we will write the whole number in the form of a fraction. In this case the whole number is 5 which can be written as 5/1. So, now we need to add 8/7 + 5/1. We will find the LCM of the denominators and convert the given fractions to like fractions. Here the LCM of 7 and 1 is 7. And after converting them to like fractions we get, (8 × 1)/(7 × 1) + (5 × 7)/(1 × 7) = 8/7 + 35/7 = 43/7 = \(6\dfrac{1}{7}\)
• The same method will be used for subtraction, for example, if we need to subtract 7 - 2/5, we will write the whole number 7 as 7/1 and then subtract. This will make it 7/1 - 2/5. We will find the LCM of the denominators and convert the given fractions to like fractions. Here the LCM of 5 and 1 is 5. And after converting them to like fractions we get, (7 × 5)/(1 × 5) - (2 × 1)/(5 × 1) = 35/5 - 2/5 = 33/5 = \(6\dfrac{3}{5}\)

How to Add and Subtract Fractions with Mixed Numbers?

To add and subtract fractions with mixed numbers, we convert the mixed numbers to improper fractions. Now, if they are like fractions, we can simply add or subtract the numerators and retain the same denominator. For adding or subtracting unlike fractions, we convert them to like fractions. We find the LCM of the denominators and convert the addends to their equivalent fractions and add them in the same way as we add like fractions.

What are the Rules for Adding and Subtracting Fractions?

The basic rules for adding and subtracting fractions are given below:

• We need to check if the denominators of the fractions are same or different.
• If the denominators are the same, we can simply add or subtract the numerators.
• If the denominators are not the same, we need to convert them to like fractions and then we add or subtract.