The set of rational numbers is the one whose elements can be represented by fractions, which, in turn, are divisions between whole numbers. In this way, adding two fractions is the same as adding the results of two divisions. That's why adding or subtracting fractions is the most difficult basic math operation to perform.
The addition and subtraction of fractions can be divided into two cases: the first for fractions that have equal denominators and the second for those who have different denominators. We have divided this last, more complicated one into four steps to help students organize their thinking.
First case: Fractions with equal denominators
To add or subtract fractions that have equal denominators, do the following: Add (or subtract) the numerators and keep the denominator of fractions as the denominator of the result. Note the example below:
4 + 3 = 4 + 3 = 7
2 2 2 2
Second case: Fractions with different denominators
To add (or subtract) fractions with different denominators, it is necessary to replace them with others that have the same denominators, but that are equivalent to the first ones. To find these
equivalent fractions, follow the instructions below. For a better understanding of the reader, we will use the example below to illustrate an addition/subtraction of fractions through the proposed step by step.2 + 10 – 2
4 12 50
Step One: Finding a Common Denominator
To find the common denominator, do the least common multiple of the denominators of all fractions involved in the numerical expression. From this MMC, it is possible to find all the equivalent fractions needed to perform the operation in question.
Example: How fractions have different denominators, it is not possible to add or subtract them directly. The MMC among its denominators will be:
4, 12, 50| 2
2, 6, 25| 2
1, 3, 25| 3
1, 1, 25| 5
1, 1, 5| 5
1, 1, 1| 300
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The number 300 will be the denominator of the equivalent fractions, so we can write:
2 + 10 – 2 =+–
4 12 50 300 300 300
Second step: Finding the first numerator
To find the first numerator, use the first fraction of the original sum. Divide the MMC found by the denominator of the first fraction and multiply the result by its numerator. The number obtained will be the numerator of the first equivalent fraction.
Example: (300:4)·2 = 75·2 = 150. So just put the numerator of the first fraction in its place. Watch:
2 + 10 – 2 = 150 +–
4 12 50 300 300 300
Step Three: Find the rest of the numerators
Repeat the above procedure for each fraction present in the operation. At the end, you will have found all the equivalent fractions.
Example: Now performing the same procedure for the last two fractions, we will find the results (300:12)·10 = 25·10 = 250 and (300:50)·2 = 6·2 = 12.
2 + 10 – 2 = 150+250– 12
4 12 50 300 300 300
Fourth step: First case
After finding all the equivalent fractions, they will have the same denominators and their addition or subtraction can be done exactly as in the first case – of fractions that have the same denominators. In the example used, the result of the first sum of fractions is equivalent to the result of the second, therefore:
2 + 10 – 2 = 150+250– 12 = 150 + 250 – 12 = 400 – 12 = 388
4 12 50 300 300 300 300 300 300
In this way, we can write the following:
2 + 10 – 2 = 388
4 12 50 300
By Luiz Paulo Moreira
Graduated in Mathematics
Would you like to reference this text in a school or academic work? Look:
SILVA, Luiz Paulo Moreira. "Fraction addition and subtraction"; Brazil School. Available in: https://brasilescola.uol.com.br/matematica/adicao-subtracao-fracao.htm. Accessed on June 28, 2021.
