Dividing fractions, i.e. dividing fractions, is similar to multiplying fractions. However, one fraction is multiplied by the reciprocal of the other fraction, which we will see below. After explaining the rules for dividing simple fractions, we will then show how to divide mixed fractions. The calculator for dividing fractions allows you to perform any calculations. All steps of the division together with the intelligent shortening of the entered fractions are comprehensively derived in the calculator.
The general page on fractions provides you with a lot of basic information on fractions and the transformation of fractions. Would you like to know how to perform the other arithmetic operations on fractions? Then visit our guides on the topics Multiplying fractions, Adding fractions or Subtracting fractions.
Contents on the topic "Dividing fractions"
How to divide fractions?
Fractions are divided by multiplying one fraction by the reciprocal of the other fraction. Fractions are then multiplied by multiplying all the numerators above the fraction bars and also all the denominators below the fraction bars.
|Example: Division of fractions|
|12 ÷ 34 = 12 × 43 = 1 × 42 × 3 = 46|
In the example, the reciprocal fraction was formed first, i.e. the reciprocal of the right-hand fraction with numerator 3 and denominator 4. The numerator and denominator were therefore swapped so that the left-hand fraction is now multiplied by the right-hand reciprocal fraction. Unlike the addition of fractions or the subtraction of fractions, the new numerators and also the new denominators can now be multiplied together.
In the following, we show step by step with examples first how to cleverly shorten fractions before division in order to be able to continue calculating easily afterwards with numbers that are as small as possible. Then we divide whole numbers with fractions, divide mixed fractions and finally present you with a video on dividing fractions.
How do you truncate fractions before multiplication?
Early shortening, i.e. shortening the fractions before dividing the left fraction with the right fraction, subsequently avoids complicated calculations with large numbers. On the one hand, the individual fractions involved in the division can be shortened if necessary. In addition, when dividing fractions, you can also shorten them "crosswise", i.e. you can shorten the numerator of one fraction with the denominator of the reciprocal fraction or the denominator of one fraction with the numerator of the reciprocal fraction, as we will show in the following examples. By the way, you can find more about shortening on our overview page on fractions.
Shorten individual fractions before dividing
The following example shows the advantage of shortening the fractions involved in the division before dividing.
|Example 1: Shortening individual fractions before division|
420 ÷ 721 = 420 × 217 = 4 × 2120 × 7 = 84140 = 35
shorten both fractions beforehand
420 ÷ 721 = 15 ÷ 13 = 15 × 31 = 1 × 35 × 1 = 35
As you can easily see, we have saved ourselves a lot of work by shortening the two fractions before the division (left fraction shortened by 5 and right fraction shortened by 7). While the first calculation can only be solved by pocket calculator for some, the second division is much easier to calculate by shortening beforehand.
Cross-cut fractions before dividing
The following example shows the advantage of being able to cross-cut when dividing fractions, i.e. to cut the numerator of one fraction with the denominator of the then reciprocal fraction to be multiplied and vice versa.
|Example 2: Cross-cut before division|
421 ÷ 207 = 421 × 720 = 4 × 721 × 20 = 28420 = 115
Cross shorten beforehand.
We start as before:
. 421 ÷ 207 = 421 × 720 = 4 × 721 × 20
Now shorten left numerator and right denominator with 5
4 × 721 × 20 = 1 × 721 × 5
Now shorten right numerator and left denominator with 7
1 × 721 × 5 = 1 × 13 × 5 = 115
Here, too, the benefit of shortening beforehand becomes clear. Instead of making the numerator and denominator untruncated very large by multiplying them by the reciprocal fraction following the division and then awkwardly truncating these large numerators and denominators again at the end of the calculation, it makes a lot of sense to carry out the truncation before multiplying the fraction and the reciprocal fraction. Not only can you shorten the individual fractions, but, as we have seen, you can also intelligently shorten them crosswise after forming the reciprocal fraction.
How to divide whole numbers by fractions?
When we want to divide whole numbers by a fraction, we take advantage of the fact that whole numbers can easily be converted into a fraction: Every whole number can be represented as a "unit". The whole number 4 can thus be represented by the fraction 4 ones, as we can see in the following example.
|Example: Multiply whole number by fraction|
|4 ÷ 32 = 4 × 23 = 41 × 23 = 4 × 21 × 3 = 83|
As described at the beginning, the integer 4 was converted into a fraction and then the division of this fraction with the other fraction of the task was carried out.
How to divide mixed fractions?
Mixed fractions or mixed numbers are composed of a whole number and an ordinary fraction that are added together even though there is no plus sign between them. To divide mixed fractions, first convert the whole number for each mixed fraction into the corresponding fraction so that the resulting fraction can then be divided with the other fraction of the task.
|Example: Division of mixed fractions|
|214 ÷ 13 = 94 ÷ 13 = 94 × 31 = 9 × 34 × 1 = 274 = 634|
The integer part of the mixed fraction, i.e. the two, was converted here into 8 quarters and added to the corresponding one quarter. The mixed fraction was thus converted into a non-genuine fraction. Fractions are called improper if the numerator is greater than the denominator.
Conversion of mixed fractions into improper fractions
A mixed fraction or a mixed number is converted into a non-genuine fraction by multiplying the integer part by the denominator and then adding the numerator to it. The denominator remains unchanged.
Example of conversion
The mixed fraction from the above example is thus converted into a non-genuine fraction as follows.
The integer 2 is multiplied by the denominator 4 and added to the previous numerator 1.
Dividing the two fractions
Now the two fractions of the example can be divided.
Video on dividing fractions
Finally, a video on dividing fractions by Math Antic. The first example of dividing fractions is followed by a somewhat more difficult example. Later Math Antics explains the division of mixed fractions.
More online calculators
As source for the information in the 'Fractions' category, we have used in particular: