Adding Mixed Fractions

Adding mixed fractions means finding the sum of mixed fractions. A mixed fraction is a combination of a whole number and a proper fraction. Combining two or more mixed fractions is known as adding mixed fractions. Let us learn more about adding mixed fractions in this article.

Adding mixed numbers with like denominators means adding those mixed fractions that have the same denominator. For example, \(2\dfrac{2}{3}\), \(1\dfrac{1}{3}\) are mixed fractions with like denominators. These mixed fractions can be added using the usual rules of addition of fractions. However, we need to note a few facts about mixed fractions that would help us solve these questions easily. Here is a list of a few points that need to be kept in mind while adding mixed fractions in a better way:

• A mixed fraction \(a\dfrac{b}{c}\) can also be written as a + (b/c)
• To convert a mixed number to an improper fraction, the whole number is multiplied with the denominator of the proper fraction and the result is added to the numerator of the proper fraction by retaining the denominator. For example, to convert \(1\dfrac{4}{7}\) to an improper fraction, we multiply 1 and 7, i.e, 1 × 7 = 7 and the result is added to 4, i.e., 7 + 4 = 11. Thus, the mixed fraction gets converted to an improper fraction and is written as 11/7.
• To convert an improper fraction to a mixed number we need to divide the numerator of the improper fraction by its denominator. The quotient becomes the whole number part, the remainder becomes the numerator of the proper fraction and the denominator remains as it is. For example, to convert 13/6 to a mixed number, we first divide 13 by 6 and get the quotient as 2 and the remainder as 1. Thus, when 13/6 is converted to a mixed fraction it becomes \(2\dfrac{1}{6}\)

Let us take an example to understand the steps for adding mixed fractions with like denominators.

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

This can be solved using 2 methods.

Method 1

• Step 1: The whole numbers of both the fractions are added first, i.e., 2 + 1 = 3
• Step 2: The fractional parts of both the numbers are added now, i.e., (2/3) + (1/3) = 3/3
• Step 3: The result obtained in step 2 should be converted to its simplest form, if needed, i.e., 3/3 = 1
• Step 4: Now we add the results of step 1 and step 3, i.e., 3 + 1 = 4

Hence, the result is \(2\dfrac{2}{3}\) + \(1\dfrac{1}{3}\) = 4

Method 2

Now, let us solve this question using the second method which is the basic method of addition of fractions.

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

Solution: Let us convert the mixed fractions to improper fractions.

• Step 1: Convert both the mixed fractions to improper fractions. Therefore, \(2\dfrac{2}{3}\) will become 8/3; and \(1\dfrac{1}{3}\) will become 4/3
• Step 2: Add the fractions by adding the numerators because the denominators are the same. This will be 8/3 + 4/3 = 12/3.
• Step 3: Reduce the fraction, if required. This will become, 12/3 = 4. Therefore, \(2\dfrac{2}{3}\) + \(1\dfrac{1}{3}\) = 4.

Mixed fractions with unlike denominators are a group of those mixed fractions that do not have the same denominator. Let us learn how to add mixed fractions with unlike denominators using an example with the help of the following steps.

Example: Add the mixed fractions with unlike denominators: \(3\dfrac{1}{4}\) + \(6\dfrac{1}{2}\)

Solution:

• Step 1: We will convert the given fractions to improper fractions first, i.e., \(3\dfrac{1}{4}\) = 13/4 and \(6\dfrac{1}{2}\) = 13/2
• Step 2: Now, the fractions can be written as: (13/4) + (13/2)
• Step 3: The denominators are different, so we have to find the Least Common Multiple (LCM) of the denominators, i.e., LCM of 2 and 4 = 4.
• Step 4: With the help of the LCM, we will write their respective equivalent fractions so that they become like fractions. The first fraction already has a denominator as 4, so it will remain the same, i. e., 13/4. But the second fraction will change to 26/4
• Step 5: Now, we have both the fractions with the same denominators, that is, they have been converted to like fractions. So, we can add them, i.e., (13/4) + (26/4) = (13 + 26)/ 4 = 39/4
• Step 6: This improper fraction (39/4) can be changed to a mixed fraction as, 39/4 = \(9\dfrac{3}{4}\)

Hence, the result of \(3\dfrac{1}{4}\) + \(6\dfrac{1}{2}\) = \(9\dfrac{3}{4}\)

Another way for adding mixed fractions with unlike denominators is to first add the whole number parts of the given fractions and then add the proper fractions. For example, \(3\dfrac{1}{4}\) + \(6\dfrac{1}{2}\) = (3 + 6) + (1/4 + 1/2). It can be solved as follows.

= 9 + (1/4 + 2/4) (as the LCM of 2 and 4 is 4)

= 9 + 3/4

= \(9\dfrac{3}{4}\)

Therefore, any of the above two methods can be used to add mixed fractions.

The addition of mixed fractions and proper fractions involves the same procedure except for a few changes. Let us understand this using the following examples.

Case 1: Mixed fraction and the proper fraction having the same denominator.

Example: Add the mixed fraction and the proper fraction \(3\dfrac{2}{5}\) + 1/5

Note that, \(3\dfrac{2}{5}\) = 3 + (2/5). Therefore,

\(3\dfrac{2}{5}\) + (1/5) = 3 + (2/5) + (1/5)

= 3 + (3/5)

= \(3\dfrac{3}{5}\)

Therefore, \(3\dfrac{2}{5}\) + (1/5) = \(3\dfrac{3}{5}\)

Case 2: Mixed fraction and the proper fraction having different denominators.

Example: Add the mixed fraction and the proper fraction \(5\dfrac{1}{2}\) + 2/3

\(5\dfrac{1}{2}\) + 2/3

= (11/2) + (2/3) [We have converted \(5\dfrac{1}{2}\) to an improper fraction, 11/2]

= [(11 × 3) / (2 × 3)] + [(2 × 2) / (3 × 2)] [Since the LCM of 2 and 3 is 6]

= (33/6) + (4/6)

= 37/6

= \(6\dfrac{1}{6}\)

Therefore, \(5\dfrac{1}{2}\) + 2/3 = \(6\dfrac{1}{6}\).

How to Add Mixed Numbers?

Mixed fractions can be added in different ways. If the mixed fractions have like denominators then the whole number part and the fractional part can be added separately and combined to get the result. For mixed fractions with unlike denominators, they are first converted into improper fractions. After this we need to make the denominators the same, so we find their LCM, convert them into respective equivalent fractions and then add the numerators.

How to Add Mixed Fractions with Whole Numbers?

To add mixed fractions with whole numbers, we add the whole number part of the mixed fraction with the given whole number and finally combine it with the fractional part to get the result.
For example, \(3\dfrac{3}{5}\) + 4
= 3 + (3/5) + 4
= (3 + 4) + (3/5)
= 7 + (3/5)
= \(7\dfrac{3}{5}\)

This can also be solved by converting the mixed number to an improper fraction and then the fractions can be added using the usual method of addition of fractions.

What are the Steps in Adding Fractions and Mixed Fractions?

The steps to add fractions and mixed fractions can be understood with the help of the following example. For example, let us add \(5\dfrac{4}{7}\) + (1/7)

• Step 1: Convert the mixed fraction to an improper fraction. Here, \(5\dfrac{4}{7}\) will become (39/7)
• Step 2: Now, check if the denominators are the same or not. Here both the denominators are the same. (39/7) + (1/7)
• Step 3: If yes, add the numerators of both the fractions and write down the result over the common denominator. So, (39/7) + (1/7) = 40/7
• Step 4: If the denominators are not the same, then find out the LCM of the denominators to make them equal and follow step 3. This step is not needed because the denominators are the same.
• Step 5: The previous step gives the result in the form of an improper fraction. Convert it to a mixed fraction. So, 40/7 = \(5\dfrac{5}{7}\)

How to Add Mixed Fractions with Proper Fractions?

Mixed fractions can be added with proper fractions easily. We just need to convert the mixed fractions into improper fractions and then add them using the same rules. For example, let us add \(2\dfrac{2}{5}\) + 3/5 using the following steps:

• We will convert the mixed fraction to an improper fraction. So, \(2\dfrac{2}{5}\) will become (12/5)
• Now, we will check if the same denominators are the same. If the denominators are the same, their numerators can simply be added. If they are not the same, then we find their LCM, convert them to equivalent fractions and then add them. In this case, the denominators are the same, so their numerators can be added. Here, 12/5 + 3/5 = 15/5. This sum can then be simplified to its lowest form. So, 15/5 = 3.

What are the Steps of Adding Mixed Fractions with Same Denominators?

Addition of mixed fractions with the same denominator can be easily done by combining the whole numbers separately and the fractional parts separately. Then, they are added and combined to get the final answer.

For example, let us add \(6\dfrac{1}{6}\) + \(2\dfrac{4}{6}\)
= (6 + 2) + (1/6) + (4/6)
= 8 + (5/6)
= \(8\dfrac{5}{6}\)

How to Add Mixed Fractions with Different Denominators?

The addition of mixed fractions with different denominators is done by first converting the mixed fractions to improper fractions. Then, we find their LCM, convert them into equivalent fractions and then add the numerators. Finally, the sum is converted back to a mixed fraction.

For example, let us add \(4\dfrac{5}{8}\) + \(3\dfrac{1}{2}\)
= (37/8) + (7/2)
= (37/8) + (28/8)
= 65/8
= \(8\dfrac{1}{8}\)

How to Add and Subtract Mixed Fractions?

The addition and subtraction of mixed fractions is done in a similar way. The mixed fractions are converted to improper fractions and then added or subtracted as per the usual rules.