Math Lesson 3.4.3 - Addition of improper fractions and/or mixed numbers

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Welcome to our Math lesson on Addition of improper fractions and/or mixed numbers, this is the third lesson of our suite of math lessons covering the topic of Operations with Fractions, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Addition of improper fractions and/or mixed numbers

So far, we have considered only proper fractions to explain the addition of fractions but we can follow, in the same way, for the addition of improper fractions as well. So, there is nothing new to add here. As for mixed numbers, we can complete the addition in two ways: Adding the whole parts and fractional parts separately and then, writing the sum as a mixed number by making the proper arrangements, and Writing both mixed numbers as improper fractions and then applying the aforementioned rules for addition of fractions. At the end, you can convert back the result in a mixed number. If the operations are made correctly, you must obtain the same result through both methods. Let's consider a few examples to clarify this point.

Example 3

Complete the following operations:

13/4 + 7/6
= 2 1/3 + 7/4
= 5 3/8 + 2 1/6

Express all results as mixed numbers.

Solution 3

Let's use both the above methods in each of these examples, so that everything will be cleared at the end. Since LCM (4 and 6) = 12, we have

13/4 + 7/6 = (13 × 3)/(4 × 3) + (7 × 2)/(6 × 2)
= 39/12 + 14/12
= 53/12

When written as a mixed number, this result becomes

53/12 = 53 ÷ 12 = 4 (5) = 4 5/12

The second method consists of writing all fractions as mixed numbers first and then completing the additions. We have

13/4 = 13 ÷ 4 = 3 (1) = 3 1/4
7/6 = 7 ÷ 6 = 1 (1) = 1 1/6

Hence,

13/4 + 7/6 = 3 1/4 + 1 1/6
= (3 + 1)(1/4 + 1/6)
= (43/12 + 2/12)
= 4 5/12

As you see, the results calculated through two different methods are the same. This time, there is only one conversion needed in each case. We have

2 1/3 = (2 × 3 + 1)/3 = 7/3

Thus, since LCM (3 and 4) is 12, we have

2 1/3 + 7/4 = 7/3 + 7/4
= 28/12 + 21/12
= 49/12
= 49 ÷ 12
= 4 (1)
= 4 1/12

The second method consists of writing all fractions as mixed numbers first and then completing the additions. We have

7/4 = 7 ÷ 4 = 1 (3) = 1 3/4

Thus, since LCM (3 and 4) = 12, we have

2 1/3 + 7/4 = 2 1/3 + 1 3/4
= (2 + 1)(1/3 + 3/4)
= 3(4/12 + 9/12)
= 3 13/12
= (3 + 1) 1/12
= 4 1/12

Here, we have made use of the fact that

13/12 = (12 + 1)/12 = 12/12 + 1/12 = 1 + 1/12 = 1 1/12

Again, the results obtained through the two different methods fit. When using the first method, we have to convert both mixed numbers to improper fractions. Thus,

5 3/8 = (5 × 8 + 3)/8 = 43/8

and

2 1/6 = (2 × 6 + 1)/6 = 13/6

Thus, since LCM (8 and 6) = 24, we obtain

5 3/8 + 2 1/6 = 43/8 + 13/6
= (43 × 3)/(8 × 3) + (13 × 4)/(6 × 4)
= 129/24 + 52/24
= 181/24

When written as a mixed number, this result becomes

181/24 = 181 ÷ 24 = 7 (13) = 7 13/24

The second method consists of writing all fractions as mixed numbers first (we have them written as mixed numbers already) and then completing the additions. We have

5 3/8 + 2 1/6 = (5 + 2)(3/8 + 1/6)
= 7((3 × 3)/(8 × 3) + (1 × 4)/(6 × 4))
= 7(9/24 + 4/24)
= 7 13/24

Again, you can see that we obtained the same result through both methods.

More Operations with Fractions Lessons and Learning Resources

Fractions Learning Material
Tutorial ID	Math Tutorial Title	Revision Notes	Revision Questions
3.4	Operations with Fractions
Lesson ID	Math Lesson Title	Lesson	Video Lesson
3.4.1	Addition of Fractions
3.4.2	Addition of fractions with different denominators
3.4.3	Addition of improper fractions and/or mixed numbers
3.4.4	Subtraction of Fractions
3.4.5	Multiplication of Fractions
3.4.6	Division of Fractions
3.4.7	Power of Fractions
3.4.8	Application of PEMDAS Rule in Operations with Fractions
3.4.9	Applications of Operations with Fractions in Practice

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Continuing learning fractions - read our next math tutorial: Converting Fractions to Decimals and Vice-versa

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