Here we introduce you to subtracting fractions, that is, subtracting one fraction from another. We start with the simple subtraction of fractions with the same denominator. Afterwards, you will get all the information on subtracting fractions with different denominators and on subtracting mixed fractions. The article is concluded by a video on subtracting fractions. With the calculator for subtracting fractions, you can perform any calculations here.
Fractions are subtracted by first making them equal and then subtracting the numerators, i.e., subtracting them from each other. Each fraction is thus first expanded so that all the fractions to be subtracted have the same denominator. The numerators of the fractions with the same denominator are then subtracted, while the common denominator remains the same.
In the following, we will show step by step, using examples, first the subtraction of fractions with the same denominator, then the subtraction of fractions with different denominators, and finally the subtraction of mixed fractions.
How Do You Subtract Like Fractions (Same Denominator)?
If the fractions to be subtracted already have the same name (they all have the same denominator), you can simply subtract the numerators of the fractions from each other. The common denominator remains the same. In this way, you finally get the difference between the fractions.
Example: Subtraction of fractions with the same denominator
.
24
−
14
=
2 − 14
=
14
In this example, both fractions have the same denominator, i.e., both have the same number below the fraction bar. Therefore, they have the same name. To subtract the two fractions, only the two numbers, i.e., the two numerators above the fraction bar, have to be subtracted from each other.
Fractions are unequally named exactly when the respective numbers below the fraction bar, i.e., the two denominators of the fractions to be subtracted, are different. For subtraction, unlike fractions must first be made like fractions, just as in the addition of fractions. If they then have the same name, i.e., the same denominator, only the numerators above the fraction must be subtracted from each other, while the common denominator remains unchanged.
Example: Subtraction of unlike fractions
13
−
14
=
412
−
312
=
4 − 312
=
112
The two fractions to be subtracted from each other here initially have the different denominators 3 and 4. For subtraction, they must first be made equal. To do this, both fractions are transformed so that they have the same denominator, i.e., a common denominator. The fractions are always transformed in such a way that their value, i.e., the fraction number, does not change. There are basically several ways of transforming fractions, which are described on the homepage under the topic Fractions.
Make fractions have the same denominator
Two fractions can be made equal by expanding one fraction with the denominator of the other. So you multiply both the numerator and the denominator of one fraction by the denominator of the other fraction.
Expand
Extending a fraction is a transformation in which the value of the fraction, i.e., the fraction number, is not changed. This is because the represented fraction is only divided into smaller sections, i.e., the fraction or the division is refined.
Fractions are expanded by multiplying both the numerator and the denominator by the same number.
Making the same denominator using the example
We can thus make the two fractions from the above example have the same denominator as follows.
The left fraction is expanded with the denominator 4 of the right fraction. Expanding by 4 means multiplying the numerator and denominator of the left-hand fraction by 4.
13
=
1 × 43 × 4
=
412
The right-hand fraction is expanded with the denominator 3 of the left-hand fraction. Expanding by 3 means multiplying the numerator and denominator of the right-hand fraction by 3.
14
=
1 × 34 × 3
=
312
Now the two fractions with the same denominator can be subtracted, as shown in the example below:
412
−
312
=
4 − 312
=
112
Note
The described “like fraction” is based on expanding the two fractions so that the two different denominators are finally multiplied together. However, this often leads to the fact that the values of the expanded fractions can become very large, which makes the subsequent calculations more time-consuming. To make them equal, the smallest common denominator (main denominator) of the fractions should therefore be determined. The main denominator is the lowest common multiple (kgV) of the denominators and is therefore often smaller than the multiplication of the two denominators. You can read more about the lowest common denominator at fraction-calculator.
Mixed fractions are composed of an integer and an ordinary fraction. They are also called mixed numbers. To subtract mixed fractions, the whole number of each fraction is first converted into the corresponding fraction so that the two fractions can then be subtracted from each other. For this purpose, as with every fraction subtraction, they must be made equal if necessary so that the numerators can be subtracted while the denominator remains the same.
Example: Subtraction of Mixed Fractions
223
−
213
=
83
−
73
=
13
The integer part of the two mixed fractions, i.e., the 2 in each case, was converted here into six-thirds each and added to its associated fraction. The two mixed fractions were thus converted into improper fractions. Fractions are called improper if the numerator is greater than the denominator.
Conversion of mixed fractions into improper fractions
You convert a mixed fraction or mixed number into an improper fraction by multiplying the integer part by the denominator and then adding the numerator to it. Meanwhile, the denominator remains unchanged.
Conversion using the example
The two mixed fractions from the above example are thus converted into improper fractions as follows.
The left mixed number is transformed as follows: The integer 2 is multiplied by the denominator 3 and added to the previous numerator 2.
223
=
2 × 3 + 23
=
83
The right mixed number is transformed as follows:
The integer 2 is multiplied by the denominator 3 and added to the previous numerator 1.
213
=
2 × 3 + 13
=
73
Subtraction of the two fractions
Since the two transformed fractions already have the same denominator, they can now be subtracted from each other.
Here is a video on subtracting fractions by Math Antics. After an introduction, the video first explains how to subtract fractions with the same denominator. Math Antics explains how to expand fractions in order to obtain the common denominator for subtracting two fractions. Later, shortening for the common denominator is described. Finally, the subtraction of mixed fractions is described.
