Add or subtract fractions calculator

Result:
Calculation path:

With this calculator 2 fractions or mixed numbers can be added or subtracted. The result is given including the calculation path.

Convert to like fractions

To add or subtract 2 fractions, they must have the same denominator. The common denominator is (usually) a common multiple of both denominators. For example, if the denominator of the first fraction is 2 and the denominator of the second fraction is 3, then 6, 12, 18, 24,... are possible common denominators. Often, common denominators can be found realtively intuitively or through a little "trial and error". But sometimes it can be a little more difficult.

one denominator is multiple of the other denominator:

If one denominator can be divided by the other denominator, then it makes sense to choose this denominator as a common multiple. If, for example, you want to add
1
2
and
5
8
, then you should choose 8 as the common denominator.

least common multiple:

One could also form (in ascending order) new multiples of the two denominators until one multiple occurs for both denominators.

For example, if there is a 5 in one denominator and a 7 in the other, then you list multiples of 5 and of 7:

Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, ...
Multiples of 7: 7, 14, 21, 28, 35, ...

The 35 occurs in both lists and since there is no smaller number that occurs in both lists, it is the least common multiple. So it makes sense to expand the two fractions so that there is a 35 in the denominator of both.

Product of the two denominators:

An easy way to find a common denominator is to simply multiply the two denominators. For example, if one denominator is 10 and the other is 12, then you might choose 10∙12 = 120 as the common denominator. This makes it possible to find a common denominator very quickly, but this method has the disadvantage that the numbers in the fractions often become very large.

Adding fractions

If you want to add 2 fractions, then they must first be converted to fractions with the same denominator. Then add the two numerators and write the result in the numerator of the resulting fraction and write the common denominator in the denominator. If you want, you can still simplify the fraction afterwards.

Example:

Task:
2
3
+
1
5

15 is chosen as the common denominator:
2
3
+
1
5
=
10
15
+
3
15

The numerators are added and the common denominator is written in the denominator:
10
15
+
3
15
=
10 + 3
15
=
13
15

13
15
is in lowest terms.

Subtracting fractions

The subtraction of 2 fractions works equivalent to the addition of 2 fractions. The two fractions must first be converted to fractions with the same denominator. Then the numerator of the second fraction is subtracted from the numerator of the first fraction and the result is written to the numerator of the resulting fraction. And in the denominator of the resulting fraction, the common denominator is written. After that, the fraction can should be reduced if possible.

Example:

Task:
3
4
−
5
7

28 is chosen as the common denominator:
3
4
−
5
7
=
21
28
−
20
28

Perform subtraction:
21
28
−
20
28
=
21 − 20
28
=
1
28

Adding mixed numbers:

If you want to add 2 mixed numbers, there are 2 possibilities. Either you convert the mixed number into a improper fraction before adding them, or you add the two mixed numbers without converting them into an improper fraction.

Addition with conversion to improper fraction:

The mixed number is first converted to a improper fraction. To do this, multiply the number that precedes the fraction by the denominator and add it to the numerator. Then the fractions are converted so that they have the same denominator. At the end the fraction is converted back to a mixed number and if you want you can simplify the fraction.

Example:

Task: 2
2
3
+1
1
5

Conversion to fractions:
2
2
3
+1
1
5
=
2∙3 + 2
3
+
1∙5 + 1
5
=
8
3
+
6
5

The fractions are expanded so that they have 15 as common denominator:
8
3
+
6
5
=
40
15
+
18
15

Perform addition:
40
15
+
18
15
=
40 + 18
15
=
58
15

Convert to mixed number:
58
15
=3
13
15

The fraction is in lowest terms.

Addition without conversion to a improper fraction:

Between the integer part and the fraction, the plus symbol, which is normally omitted, is inserted again. For example, one converts the number 2
1
3
into 2 +
1
3
. After that, the integers are added and after that the fractions are converted so that they share a common denominator. If the fraction is an improper fraction after adding (numerator is greater in amount than denominator), then the fraction is converted to a mixed number and added to the sum of the integer parts.

Example:

Task: 2
2
3
+1
4
5

Write plus signs between the integer parts and the fractions:
2
2
3
+1
4
5
= 2 +
2
3
+ 1 +
4
5

Add integer parts:
2 +
2
3
+ 1 +
4
5
= 3 +
2
3
+
4
5

Convert to fractions with same denominator and perform addition:
3 +
2
3
+
4
5
= 3 +
10
15
+
12
15
= 3 +
22
15

Convert improper fraction to mixed number and add to sum of integer parts:
3 +
22
15
= 3 + 1
7
15
= 4
7
15

subtract mixed numbers

Subtraction with conversion to improper fraction:

Subtraction of mixed numbers with conversion to improper fraction works equivalently to addition of mixed numbers with conversion to improper fraction. The mixed numbers are converted to improper fractions. Then the fractions are converted to fractions with common denominator and subtracted. The result is converted back to a mixed number.

Example:

Task: 3
1
3
- 1
3
5

Conversion to fractions:
3
1
3
- 1
3
5
=
10
3
-
8
5

Converting fractions to fractions with common denominators and performing subtraction:
10
3
-
8
5
=
50
15
-
24
15
=
50
15
-
24
15
=
50 - 24
15
=
26
15

Convert to mixed number:
26
15
= 1
11
15

Subtraction without conversion to improper fraction:

Subtraction of mixed numbers without conversion to improper fraction works very similar to addition of mixed numbers without conversion to improper fraction. Again, plus symbols are written between the integer parts and the fractions. Then the fractions are converted to fractions with a common denominator and for the integers the subtraction is performed. If the first fraction (the minuend) is greater than or equal to the second fraction (subtrahend), then subtraction can simply be performed for the two fractions. If the second fraction (the subtrahend) is larger, then before subtracting the fractions, the integer must first be decreased (typically by 1) and the value by which the integer was decreased is added to the first fraction. If afterwards the second fraction is no longer larger than the first fraction, the subtraction can be performed for the two fractions.

Example:

Task: 3
1
3
- 1
3
5

Insert plus symbols and convert fractions to fractions with equal denominators :
3
1
3
- 1
3
5
= 3 +
1
3
- (1 +
3
5
) = 3 +
1
3
- 1 -
3
5
= 3 +
5
15
- 1 -
9
15

Subtract integer parts:
3 +
5
15
- 1 -
9
15
= 2 +
5
15
-
9
15

5
15
is less than
9
15
. Therefore, the 2 must be decreased by 1 and the 1 must be added to the first fraction:
2 +
5
15
-
9
15
= 1 +
15
15
+
5
15
-
9
15
= 1 +
20
15
-
9
15

Now that the first fraction is greater than the second fraction, subtraction can be performed for the fractions:
1 +
20
15
-
9
15
= 1 +
20 - 9
15
= 1 +
11
15
= 1
11
15

good explanatory videos on YouTube:

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