Math Lesson 3.4.2 - Addition of fractions with different denominators

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Welcome to our Math lesson on Addition of fractions with different denominators, this is the second 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 fractions with different denominators

In this case, the items to be added do not represent the same kind of division. Therefore, the first thing to do is to convert the fractions involved in the operation to have the same denominator, as the denominator represents the method of division. This is because it would not be right to add items of different sizes. For example, if we want to add 3/5 and 1/6, first we must find the LCM of 5 and 6 which is 30, then write the two fractions with a denominator of 30 by making the appropriate arrangements in the numerators (i.e. to write the equivalent fractions to the original ones but with denominator 30) and eventually, add the new fractions like we did earlier. We have:

3/5 + 1/6 = (3 × 6)/(5 × 6) + (1 × 5)/(6 × 5)
= 18/30 + 5/30
= 23/30

This operation is made clear through the figure below. $Math Tutorials: Operations with Fractions Example$

Example 2

A painter was able to paint 5/12 of a house in the first day and 3/8 of the house in the second day. What part of the house did he paint in two days?

Solution 2

This exercise can be solved in two ways: (1) directly multiplying the denominators and finding a common denominator but which probably is not the least of them (LCM). In this case, further simplifications are needed after calculating the sum, and (2) finding the LCM of 12 and 8 first and then writing both fractions with the same denominator determined by the value of LCM. In this case, further simplifications are rarely required. Let's solve this exercise in both ways: We have:

Part of house painted in two days = 5/12 + 3/8
= (5 × 8)/(12 × 8) + (3 × 12)/(8 × 12)
= 40/96 + 36/96
= 76/96 of the house

As explained above, now we have to simplify both fractions until their respective numerators and denominators become relatively prime. Thus, we have

Part of house paitned in two days = 76/96
= (76 ÷ 4)/(96 ÷ 4)
= 19/24 of the house

We can find the LCM (12 and 8) first. We have LCM (12 and 8) = 24. Hence, we obtain directly the denominator of the final fraction, so now we have a shorter way ahead. Thus,

Part of house painted in two days = 5/12 + 3/8
= (5 × 2)/(12 × 2) + (3 × 3)/(8 × 3)
= 10/24 + 9/24
= 19/24 of the house

As you see, the result is the same in both cases.

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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Fractions Video tutorial: Operations with Fractions. Watch or listen to the Operations with Fractions video tutorial, a useful way to help you revise when travelling to and from school/college
Fractions Revision Notes: Operations with Fractions. Print the notes so you can revise the key points covered in the math tutorial for Operations with Fractions
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Check your calculations for Fractions questions with our excellent Fractions calculators which contain full equations and calculations clearly displayed line by line. See the Fractions Calculators by iCalculator™ below.
Continuing learning fractions - read our next math tutorial: Converting Fractions to Decimals and Vice-versa

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