Sum of mixed fractions

Perhaps the best thing, before moving forward in an explanation of the correct way in which the Sum of Mixed Fractions should be performed, is to review several definitions, which will allow us to understand this operation, within its indicated context.

Fundamental definitions

In this regard, it may also be prudent to focus this review on four specific notions: Fractions, Improper Fractions, Integers, and Mixed Fractions, as these are the elements, involved in the sum of mixed fractions. Here are each of these concepts:


In this way, it will begin to say that fractions have been generally explained by Mathematics as a type of expression, by which it realizes fractional numbers, that is, of those amounts that are not accurate or not whole. Likewise, this discipline has pointed out that fractions will always be composed of two elements, each of which have been explained as follows:

  • Numerator: First, the Numerator is designated as the numeric element of the fraction, which occupies its top. As for their task, the various sources point out that the Numerator complies with indicating which or how many parts of the whole have been taken, or wished to be represented.
  • Denominator: The Denominator will be the element at the bottom of the fraction. Your mission will be to point out in how many parts the whole has been divided.

Improper fractions

With regard to the Inappropriate Fractions, these have been described by the different sources as that mathematical expression, composed of numerator and denominator, used to represent fractional numbers, and which is characterized by always having a numerator that is of greater value than the denominator that accompanies it.


On the other hand, Whole Numbers have been designated as those numerical elements that are used to express whole or exact amounts. These numbers will in turn make up the numerical set Z, and will consist of all positive integers, their negative reverses and zero, so they will be considered useful for accounting for exact amounts, absence or lack of specific quantities and even of the total absence of quantity.

Mixed fractions

Finally, it will also be relevant to cast lights on the definition of Mixed Fractions, which have been described as those mathematical expressions, used to account for fractional or non-exact amounts, and which are mainly characterized by be composed of an integer and a fraction of its own, that is, a fraction where always and without exception the numerator is greater than the denominator.

Sum of mixed fractions

With these definitions in mind, it may certainly be much easier to understand each of the steps, elements and procedures involved in the sum of mixed fractions, an operation that basically consists of combining the values of each of the expressions, in order to obtain the total of these expressions.

However, taking into account the mixed nature of these mathematical expressions, the different authors warn of the need to add the elements involved according to their nature, that is, that in a sum of mixed fractions will be careful to add by one side the whole numbers, and on the other the fractional or fractional numbers, in order to result in a mixed fraction.

However, Mathematics also warns two types of cases in terms of the sum of mixed fractions, and they will basically be based on whether the product fraction of the sum has turned out to be a fraction of its own or an improper fraction, in which case it will be necessary to perform a additional procedure, where it is taken to a mixed fraction, where the fraction is its own – as must happen in any mixed expression – then adding this expression to the obtained subtotal. Here are each of these possible cases:

Sum of mixed fractions (when fractions are their own)

In the event that the sum of mixed fractions yields a result where the sum of the fractions that make up this mixed expression is its own, that is, it has a numerator less than the denominator, it is sufficient to add on one side, the whole numbers, and the fractions , on the other, getting the desired result, as seen below:

Then the integers will be added first:

3 + 4= 7

Second, the fractions will be added up, for which the following procedure will be used:

Having the two totals, each shall be expressed as part of the resulting mixed fraction:

Sum of mixed fractions (when some fraction is improper)

However, it may happen that when adding up the fractions of the mixed expression, an improper fraction is obtained, that is, with a numerator much larger than the denominator, which considering that the definition of mixed fraction implies the combination of an integer and a fraction of its own is not correct, so then this inappropriate fraction obtained, must be converted into a mixed fraction, to add it equally to the total of the whole numbers, as seen below:

In this case, you will start by getting the total of the integers:

8 + 5= 13

The sum of the fractions, which are part of the mixed fractions, should then be carried out:

Doing so results in an improper fraction, so you should look to convert this expression into a mixed fraction:

Obtained this mixed fraction, it will then be added to the total of the whole numbers:

Second method

However, the different mathematical sources state that another method of performing the sum between two or more mixed fractions will be one that includes converting each of the expressions involved into improper fractions, and then adding them up. Once the total between these fractions has been achieved, the result will be converted back to a mixed fraction.


Sum of mixed fractions
Source: Education  
September 30, 2019

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