Common denominator - calculator

About the calculator

This calculator can be used to convert fractions into fractions with the same common denominator. For the calculation of the common denominator either the product of the denominators or the least common multiple is used. Both the solution and the calculation path are always displayed.

Why convert to a common denominator?

To add or subtract fractions, they must be converted to a common denominator.

How to convert fractions to a common denominator?

To convert fractions to a common denominator, first find a common multiple and then expand the fractions so that they all have the common multiple as denominator.

Example:

The following 2 fractions are to be converted to common denominator:
2
3
und
1
4

A common multiple of 3 and 4 must be found. For example, this is 12.

Then
2
3
must be expanded by 4 and
1
4
by 3, so that both fractions have a 12 as denominator.
  • 2
    3
    =
    2 · 4
    3 · 4
    =
    8
    12
  • 1
    4
    =
    1 · 3
    4 · 3
    =
    3
    12

Find common multiple

To find a common multiple, there are several methods. Each has its advantages and disadvantages.

one denominator is multiple of the other denominators:

If the denominator of one fraction is multiple of the denominator of the other fraction (resp. the other fractions), then it can be used as the common multiple.

Example:

The following 2 fractions are to be converted to a common denominator:
3
4
and
5
12

12 is a multiple of 4. So
3
4
is expanded by 3 so that afterwards the denominator is 12:

3
4
=
3 · 3
4 · 3
=
9
12

Product of the denominators:

A simple method to find a common multiple is to multiply the denominators. This is easy to do in practice, but it often makes the numbers of the fractions larger than absolutely necessary.

Example:

The following 2 fractions are to be converted to a common denominator:
3
8
and
5
12

The product of the two denominators is 8 · 12 = 96. So
3
8
is expanded by 12 and
5
12
by 8.
  • 3
    8
    =
    3 · 12
    8 · 12
    =
    36
    96
  • 5
    12
    =
    5 · 8
    12 · 8
    =
    40
    96

However, a common denominator would also have been 24. In that case, the fractions would have been converted to
9
24
and
10
24
, which would usually simplify further calculations.

List multiples:

Another method is to calculate the multiples for both denominators until you find a number that is a multiple of both denominators. If you calculate the multiples in ascending order without omitting multiples, you will find the least common multiple.

Example:

The following 2 fractions are to be converted to a common denominator:
5
8
and
1
6
The first multiples of 8 are: 8, 16, 24, 32, ...
The first multiples of 12 are: 6, 12, 18, 24, 30, ...

The least common multiple is therefore 24.

  • 5
    8
    =
    5 · 3
    8 · 3
    =
    15
    24
  • 1
    6
    =
    1 · 4
    6 · 4
    =
    4
    24

If only a few multiples of the denominators have to be calculated until the least common multiple is found, then in practice it often makes sense to use this method. However, it may also happen that one would first have to calculate a great number of multiples until the least common multiple is reached. For example, 204 is the least common multiple of 4 and 102, and 4 has 51 multiples up to 204 (including 4 and 204).

Prime factorization:

The least common multiple of the denominators can also be calculated with the help of prime factorization. To do this, first, for the numbers in the denominators, the prime factorizations are performed and converted to canonical representation (combining equal prime factors to powers). Then, for each number that appears as a prime factor in at least one prime factorization, the power is marked for which the exponent is largest. At the end, the product of the marked powers or numbers is calculated.

Example:

The following 2 fractions are to be converted to a common denominator:
15
360
and
5
54

Decomposed into their prime factors, the following applies for 360 and 54:

360=2 · 2 · 2 · 3 · 3 · 5
 =23 · 32 · 5
54=2 · 3 · 3 · 3
 =2 · 33

The 2 occurs most often in the prime factorization of 360, the 3 occurs most often in the prime factorization of 54, and the 5 occurs only in the prime factorization of 360. So you calculate the least common multiple as follows:

kgV(360, 54)=23 · 33 · 5
 =8 · 27 · 5
 =1080

Then the fractions are expanded so that the denominators are 1080:

  • 15
    360
    =
    15 · 3
    360 · 3
    =
    45
    1080
  • 5
    54
    =
    5 · 20
    54 · 20
    =
    100
    1080

Euclidean algorithm:

For the greatest common divisor and the least common multiple, the following relation exists:

lcm(a, b)=
a∙b
gcd(a, b)

For the calculation of the GCD of 2 numbers the Euclidean algorithm can be used.

Example:

The following 2 fractions are to be converted to a common denominator:
10
21
and
5
15

First, the GCD of the two denominators is calculated with the help of the Euclidean algorithm:

21:15=1R6
15:6=2R3
6:3=2R0

result: GCD(21, 15) = 3

Then the lcm is calculated by dividing the product of the two denominators by the GCD of the two denominators:

lcm(15, 21)=
15∙21
GCD(15, 21)
 =
315
3
 =105

Then the fractions are expanded so that they have 105 as denominator:

  • 10
    21
    =
    10 · 5
    21 · 5
    =
    50
    105
  • 5
    15
    =
    5 · 7
    15 · 7
    =
    35
    105

good explanatory videos on YouTube:

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