Fractions <−> decimal numbers converter

Click on the digit from which the repeating part should start.

Result:
Calculation path:

This calculator can be used to convert decimal numbers to fractions and mixed numbers (or vice versa). Periodic decimal numbers can also be entered. In addition to the result, a calculation path is also provided.

Convert decimal number to fraction

A decimal number can be converted to a fraction very easily. The number is written into the numerator, leaving out the decimal point. If there is only a 0 before the decimal point, this is also not written down. In the denominator you write a 1 followed by as many zeros as the decimal number has decimal places. If you want, you can simplify the fraction afterwards.

Examples:

0.3 has one decimal place. So 0.3 can be converted into
3
10
.

1.25 has 2 decimal places. So 1.25 can be converted into
125
100
. If you want you can simplify the fraction by 25 and get
5
4
.

25.221 has 3 decimal places. So 25.221 can be converted into
25221
1000
.

Convert decimal number to mixed number

If a decimal number is to be converted into a mixed number, one proceeds very similarly to the conversion into a fraction. This time, however, only the decimal places are written in the numerator and the digits before the decimal point are written before the fraction. In the denominator again a 1 is written followed by as many zeros as the decimal number has decimal places.

Examples:

1.25 = 1
25
100
= 1
1
4

25.221 = 25
221
1000

3.5 = 3
5
10
= 3
1
2

Convert periodic decimal number to fraction or mixed number

First you have to look at whether the periodic part starts directly at the first decimal place or behind it. If it starts directly at the first decimal place, this is the easy case. If it starts at a later decimal place, a little more effort is needed.

Periodic part starts at first decimal place:

Let us first consider the case where the period bar starts directly at the first decimal place and where there is just a 0 in front of the decimal point. Then the decimal places are written into the numerator and as many 9s are appended to the denominator as the number has decimal places below the bar.

Examples:

0.12 has 2 decimal places below the bar for the periodic part. So 0.12 is converted into
12
99
. This can still be reduced by dividing the numerator and denominator by 3 and you get
4
33
.

0.3 has one decimal place below the bar for the periodic part. So the number is converted into
3
9
. The fraction can be reduced by dividing the numerator and denominator by 3 and you get
1
3
.

0.125 has 3 decimal places below the bar for the periodic part so it is converted into
125
999
.


Now the case where the bar starts directly at the first decimal place, but the number has at least one digit before the decimal point not equal to 0. First convert the number into a mixed number by first converting the decimal places into a fraction as described above and then writing the digits before the decimal point as an integer in front of the fraction. For example, one converts 15.125 into 15
125
999
or 5.3 into 5
3
9
= 5
1
3
.

If you want to represent the number as a fraction rather than a mixed number, you still have to convert the mixed number to a improper fraction by converting the integer part to a fraction that has the same denominator as the fraction and then add the two fractions.

Example:

5.3 was converted to the mixed number 5
1
3
. Now the 5 has to be converted into a fraction with denominator 3 and then the two fractions are added.

5
1
3
=
5 ∙ 3
3
+
1
3
=
15
3
+
1
3
=
16
3

Periodic part starts after first decimal place:

Now we come to the case that the vinculum starts behind the first decimal place. In this case, the comma must be shifted to the right until it is directly in front of the digit above which the vinculum begins. This is done by multiplying the decimal number by a power of 10 (10, 100, 1000, 10000, ...). For example, if the number 1.221 is to be converted into a fraction, then multiply the number by 100 to get 122.1. This periodic decimal number is converted to a fraction as described above. At the end, you still have to divide the fraction by the number it was multiplied by at the beginning to move the decimal point.

Example:

1.221 is to be converted into a fraction.

First, multiply by 100 to shift the decimal point so that it is directly in front of the number above which the vinculum begins. This results in 122,1.

Now the 122.1 is converted to a mixed number and then to a fraction::

122.1 = 122
1
9
=
122 ∙ 9
9
+
1
9
=
1099
9

Now the fraction still has to be divided by 100. This is achieved by multiplying the denominator by 100:

1099
9∙100
=
1099
900

So it applies: 1.221 =
1099
900

This can still be converted to a mixed number:
1099
900
=
900
900
+
199
900
= 1
199
900

Convert fraction or mixed number to decimal number

To convert a fraction into a decimal number, the numerator must be divided by the denominator. If a mixed number is to be converted to a decimal number, then the result of the division is added to the integer part that precedes the fraction.

Examples:

4
3
25
is to be converted into a decimal number. First, the 3 is divided by the 25 and then the 4 is added to the result:
4
3
25
= 4 + 3 : 25 = 4 + 0.12 = 4.12

good explanatory videos on YouTube:

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