Equivalence between mixed fractions and improper fractions

Perhaps, prior to addressing the correct way in which a given amount can be expressed in both mixed and improper fraction form, it is appropriate to review a number of concepts, which will allow to understand in their mathematical context, each of these expressions, as well as the equivalence between them.

Fundamental definitions

In this sense, it will then also be relevant to delimit this theoretical review to three specific notions: Fractions, Improper Fractions and Mixed Fractions, as this will allow to be aware of the nature of the expressions on the basis of which equivalence is established:


In this way, one can begin by saying that mathematics has defined fractions as a type of expression, by which it realizes fractional numbers, that is, that serve to express non-whole or non-exact amounts. Likewise, this discipline has pointed out that fractions are composed of two elements:

  • Numerator: First, you will find the numerator, which occupies the top position of the expression, referring to the amount of the entire fraction to which the fraction refers.
  • Denominator: for its part, the denominator will occupy the bottom of the fraction, indicating in how many parts the whole is divided.

Improper fractions

Thus, it will be of great help to repair in the definition of the Improper Fractions, these expressions that are basically characterized by having a numerator greater than the denominator. Consequently, it could then be said that improper fractions account for non-whole quantities where the number of parts that have been taken is greater than the number of parts in which the whole is divided. Examples of such fractions would be as follows:

Mixed fractions

Finally, it will be noted that mixed fractions are a type of expression of fractional numbers, characterized by being composed of an integer – which counts an integer or exact – and a fraction, which on the other hand indicates an unexact quantity. Such expressions tend to correspond more commonly to everyday use than to the mathematician itself, where it is more customary to use improper fractions. Here are some examples:

Equivalence mixed fractions and improper fractions

Although they are made up of different elements, mixed and improper fractions can be used to account for the same amount, hence it is claimed that these expressions are equivalent. However, to express this equivalence it will be necessary to carry out a series of operations, which allow to convert a mixed fraction into an improper and vice versa. Here are each of the cases:

Turning a mixed fraction into an improper fraction

If you were facing a mixed fraction, that is, an expression composed of an integer and a fraction, and you wanted to convert it to a mixed fraction, then you should follow these steps:

  • First, the whole number must be multiplied by the denominator value, and summed with the fraction numerator, in order to obtain the numerator of the inappropriate fraction.
  • The same denominator of the fraction, which had the mixed fraction, is supported.

This conversion operation could be expressed mathematically as follows:

Turning an improper fraction into a mixed fraction

On the other hand, if the conversion to be made was that of an improper fraction at a mixed fraction, the following steps should be taken:

  • The numerator will be divided by the denominator.
  • In this way, the obtained quotient will be taken as the whole number of the mixed fraction.
  • For its part, the rest of this division will be held as the numerator of the fraction.
  • The original denominator of the improper fraction is assumed as the denominator.

This conversion operation can be expressed mathematically as follows:

Graphic example

However, it may be that the best way to conclude an explanation of how you can account for the same amount by using indifferently unsuitable and mixed fractions is through a graphic example, which allows you to visualize what each expression, and how an equivalence can really be conceived between the two, beyond the mathematical fact, as can be seen below:

Assuming that two rectangles are held, and these have been divided into some parts, and in turn some of these have been taken, then the following expressions will be taken:

Reviewing them will show how both expression 6/4 expresses how each of the all have been divided into four parts, of which six have been taken, while the mixed fraction will also express how the all have been divided into four parts, taking all of them, and only two parts of the other, to express equally that six parts of the two have been taken, divided into four. Therefore, they are equivalent expressions.

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Equivalence between mixed fractions and improper fractions
Source: Education  
September 26, 2019

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