Compare or order fractions - Calculator

About the calculator

With this calculator you can either compare 2 fractions or order 2 or more fractions. Both the solution and the calculation path are always displayed.

To enter a negative fraction, "mixed numbers" must be selected and the minus sign must be entered in the field for the integer part. You can also enter an integer by entering it in the input field for the integer part and leaving the fields for the numerator and denominator empty.

How to compare 2 fractions?

positive fractions with a common denominator:

If 2 positive fractions share the same denominator, they can be compared simply by comparing the numerators of both fractions. If the numerator of one of the two fractions is greater than the numerator of the other fraction, then the fraction itself is greater. And if the numerators of both fractions are equal, then the fractions are equal.

Examples:

  • 5
    12
    <
    7
    12
  • 7
    10
    >
    3
    10
  • 7
    12
    =
    7
    12

negative fractions with a common denominator:

If both fractions with the same denominator are negative, the opposite is true. Then the fraction whose numerator is smaller is greater.
If one fraction is positive and one is negative, then the positive fraction is always greater than the negative one.

Examples:

  • −
    5
    12
    > −
    7
    12
  • −
    7
    10
    < −
    3
    10
  • −
    2
    15
    <
    3
    15

Fractions with different denominators:

If the denominators of the two fractions differ, they must first be converted to a common denominator. To do this, a common multiple must be found and then the fractions are expanded so that they both have the same denominator.

If one denominator is a multiple of the other denominator, then it should be used as the common multiple. Then the fraction with the smaller denominator must be expanded so that afterwards both fractions have the larger denominator as denominator.

Example:

The following two fractions are to be compared with each other:
2
6
and
5
18

18 is a multiple of 6 and therefore
2
6
should be expanded so that the denominator is 18, afterwards.

2
6
=
2 · 3
6 · 3
=
6
18

Since
6
18
is greater than
5
18
ist,
2
6
is also greater than
5
18
.

If the denominator of one of the two fractions is not a multiple of the denominator of the other fraction, then it makes sense to expand both fractions by the denominator of the other fraction. The common multiple is then the product of the two denominators.

Example:

The following two fractions are to be compared:
5
7
and
2
3

So you expand the first fraction by 3 and the second by 7.

  • 5
    7
    =
    5 · 3
    7 · 3
    =
    15
    21
  • 2
    3
    =
    2 · 7
    3 · 7
    =
    14
    21

Since
15
21
is greater than
14
21
,
5
7
is greater than
2
3
.

Compare fractions, mixed numbers and integers

A fraction can also be compared with a mixed numer or an integer and a mixed number can also be compared with another mixed number or an integer. In the following, it is assumed that the denominators of fractions and mixed numbers are always greater than the numerators. If this is not the case, then the fractions or mixed numbers must first be converted so that this is the case.

If you want to compare a fraction, mixed number or integer with another fraction, mixed number or integer, you first check whether the signs are different. If one fraction (or one number) is positive and the other is negative, then the positive fraction (or the positive number) is always greater than the negative one.

Examples:

  • −5
    2
    5
    < 2
    7
    8
  • 2
    3
    > −5
  • −2
    1
    2
    < 1
    2
    5

If the sign is the same, then you look at the integer part. For proper fractions this is 0 and for integers this is the number itself. If they differ, then the fraction (or number) for which the integer part is smaller in combination with the sign is smaller.

Examples:

  • 5
    2
    7
    > 2
    5
    7
  • −2
    3
    4
    < −1
    5
    7
  • 3 < 4
    2
    5
  • 1
    1
    4
    >
    5
    8

If both the sign and the integer part are the same, then look at the fractional part. For integers, you can imagine a fraction with a 0 in the numerator. If none of the fractions has a 0 in the numerator, the two fractions must be converted to a common denominator. Then compare the numerators. Again, the signs must be taken into account.

Example:

The following two mixed numbers are to be compared: 2
2
7
and 2
3
10

Both mixed numbers are positive and both have a 2 as an integer part. The fractions of the two mixed numbers must therefore be compared. To do this, they must first be converted to a common denominator.

  • 2
    2
    7
    = 2 +
    2 · 10
    7 · 10
    = 2
    20
    70
  • 2
    3
    10
    = 2 +
    3 · 7
    10 · 7
    = 2
    21
    70

Next, the two numerators are compared. 20 is smaller than 21. So 2
20
70
is smaller than 2
21
70
.

From this it follows: 2
2
7
< 2
3
10


Example with negative sign:

The following two mixed numbers are to be compared: −1
2
3
and −1
7
9

Both mixed numbers are negative and have a 1 as integer part. The two fractional parts must therefore be compared again. 3 is a divisor of 9. It therefore makes sense to expand the fractional part of −1
2
3
by 3.
  • −1
    2
    3
    = −(1 +
    2 · 3
    3 · 3
    ) = −1
    6
    9

The numerator of −1
6
9
is smaller than the numerator of −1
7
9
. However, since both mixed numbers are negative, −1
6
9
is not smaller, but −1
7
9
.

From this it follows: −1
2
3
> −1
7
9

Order fractions

with the same sign:

To order fractions that all have the same sign, all fractions must first be converted to a common denominator. They can then be ordered by comparing the numerators of the fractions. A simple way to find a common denominator is to calculate the product of all denominators that are not a multiple of another denominator. Alternatively, you could also determine the least common multiple by calculating further multiples of the denominators until you have found a multiple of all denominators.

Example:

The following 4 fractions are to be ordered:
4
6
,
5
8
,
2
4
,
3
5

8 is a multiple of 4 and therefore every multiple of 8 is also a multiple of 4. Therefore, only a common multiple of 5, 6 and 8 needs to be found. If you multiply the 3 numbers, you get 240. To convert the 4 fractions to a common denominator, you can expand them so that 240 becomes the denominator of all 4 fractions.

  • 4
    6
    =
    4 · 40
    6 · 40
    =
    160
    240
  • 5
    8
    =
    5 · 30
    8 · 30
    =
    150
    240
  • 2
    4
    =
    2 · 60
    4 · 60
    =
    120
    240
  • 3
    5
    =
    3 · 48
    5 · 48
    =
    144
    240

If you order these fractions, you get:

120
240
<
144
240
<
150
240
<
160
240

The following therefore applies:

2
4
<
3
5
<
5
8
<
4
6

with different signs:

If the fractions you want to order include both positive and negative fractions, first order them according to their sign. Then the negative fractions are converted to a common denominator and the positive fractions are converted to a common denominator. The common denominator of the positive fractions and the common denominator of the negative fractions may differ. Then order the negative fractions first and after that the positive fractions according to their numerators.

Example:

The following 4 fractions are to be ordered: −
4
6
,
5
8
,
2
4
, −
3
5

First, the fractions are ordered according to their sign:

−
4
6
, −
3
5
,
5
8
,
2
4

Next, the negative fractions and after that the positive fractions are converted to a common denominator. For the negative fractions, 30 is a suitable common denominator and 8 for the positive ones.

  • −
    4
    6
    = −
    4 · 5
    6 · 5
    = −
    20
    30
  • −
    3
    5
    = −
    3 · 6
    5 · 6
    = −
    18
    30
  • 2
    4
    =
    2 · 2
    4 · 2
    =
    4
    8

Fractions after expanding:

−
20
30
, −
18
30
,
5
8
,
4
8

If you order these fractions, you get:

−
20
30
< −
18
30
<
4
8
<
5
8

The following therefore applies:

−
4
6
< −
3
5
<
2
4
<
5
8

Ordering fractions, mixed numbers and integers

In the following, it is again assumed that the denominator of fractions and mixed numbers is greater than the numerator and that decimal numbers have been converted into fractions.

In principle, ordering mixed numbers or a combination of fractions, mixed numbers and/or integers works very similarly to ordering positive and negative fractions. First, they are ordered according to the signs and integer parts. Then fractions and mixed numbers with the same sign and integer parts are converted to a common denominator. Then the fractions and mixed numbers with the same signs and integer parts are ordered according to the numerators.

Example:

The following 10 fractions, mixed numbers and integers are to be ordered:

id12345678910
fraction/
number		−2
3
4
−22
5
8
−2
5
8
22
4
7
−2
7
10
1
5
6
−3
1
4
3
4

First, the values are ordered according to the signs and the integer parts. In addition, the 2 is smaller than the positive mixed numbers with a 2 as integer part and the −2 is larger than the negative mixed numbers with a 2 as integer part.

id91472108536
fraction/
number		−3
1
4
−2
3
4
−2
5
8
−2
7
10
−2
3
4
1
5
6
22
5
8
2
4
7

The values in the fields with a green background are now ordered correctly. The values in the fields with a red background may not yet be ordered correctly. −3
1
4
,
3
4
und 1
5
6
are correctly sorted because there are no other mixed numbers with "−3" or "1" in front of the fraction part and because there is no other positive proper fraction. −2 is correctly ordered because it is always greater than mixed numbers with "−2" in front of the fraction and 2 is always smaller than positive mixed numbers with a 2 as the integer part.

Now the mixed numbers with "−2" or "2" in front of the fraction must be ordered. To do this, they must first be converted to a common denominator. A common denominator of 4, 8 and 10 is 40 and a common denominator of 7 and 8 is 56.

id91472108536
fraction/
number		−3
1
4
−2
30
40
−2
25
40
−2
28
40
−2
3
4
1
5
6
22
35
56
2
32
56

The mixed numbers with the fractions of the same name are next ordered by their numerators.

id91742108563
fraction/
number		−3
1
4
−2
30
40
−2
28
40
−2
25
40
−2
3
4
1
5
6
22
32
56
2
35
56

If you wish, you can replace the expanded fractions by their original form.

id91742108563
fraction/
number		−3
1
4
−2
3
4
−2
7
10
−2
5
8
−2
3
4
1
5
6
22
4
7
2
5
8

good explanatory videos on YouTube:

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