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# Addition and Subtraction of fractions.

### Addition of fractions with equal denominators.

Definition.
To add two fractions with equal denominators, it is necessary to add their numerators, and to leave without modifications a denominator:
 a + b = a + b c c c

### Examples of addition of fractions with the same denominators:

Example 1.
Find the sum of two fractions with equal denominator:
 1 + 2 = 1 + 2 = 3 5 5 5 5

Example 2.
Find the sum of two fractions with equal denominator:
 3 + 2 = 3 + 2 = 5 7 7 7 7

Definition.
To add two fractions, it is necessary:

### Examples of addition of fractions

Example 3.
Find the sum of two fractions:
 1 + 1 = 1·2 + 1 = 2 + 1 = 2 + 1 = 3 = 3 = 1 3 6 3·2 6 6 6 6 6 3·2 2

Example 4.
Find the sum of two fractions:
 29 + 44 = 29·3 + 44·2 = 87 + 88 = 87 + 88 = 30 45 30·3 45·2 90 90 90

 = 175 = 35·5 = 35 = 18 + 17 = 1 17 90 18·5 18 18 18

Definition.
To add two mixed numbers, it is necessary:

### Examples of addition of mixed numbers

Example 5.
Find the sum of two mixed numbers:
 2 + 1 1 = 2·2 + 1 1·3 = 4 + 1 3 = 1 + 4 + 3 = 3 2 3·2 2·3 6 6 6

 = 1 + 7 = 1 + 6 + 1 = 1 + 1 1 = 2 1 6 6 6 6

Example 6.
Find the sum of two mixed numbers:
 1 5 + 2 3 = 1 5·4 + 2 3·3 = 1 20 + 2 9 = 3 + 20 + 9 = 6 8 6·4 8·3 24 24 24

 = 3 + 29 = 3 + 24 + 5 = 3 + 1 5 = 4 5 24 24 24 24

## Subtraction of fractions

### Subtraction of fractions with equal denominators.

Definition.
To receive a difference of fractions with equal denominatorsTo receive a difference of fractions with equal denominators, it is necessary to subtract their numerators , and to leave without modifications a denominator:
 a - b = a - b c c c

### Examples of subtraction of fractions with the same denominators:

Example 7.
Find the difference between two fractions with the same denominators:
 3 - 1 = 3 - 1 = 2 5 5 5 5

Example 8.
Find the difference between two fractions with the same denominators:
 8 - 5 = 8 - 5 = 3 41 41 41 41

### Subtraction of fractions.

Definition.
To subtract two fractions, it is necessary:

### Examples of subtraction of fractions

Example 9.
Find the difference between two fractions:
 5 - 1 = 5 - 1·3 = 5 - 3 = 5 - 3 = 2 = 2 = 1 6 2 6 2·3 6 6 6 6 2·3 3

Example 10.
Find the difference between two fractions:
 3 - 1 = 3·3 - 1·5 = 9 - 5 = 9 - 5 = 4 = 2·2 = 2 10 6 10·3 6·5 30 30 30 30 15·2 15

### Subtraction of mixed numbers

Definition.
To subtract two mixed numbers, it is necessary:

### Examples of subtraction of mixed numbers

Example 11.
Find the difference between two mixed numbers:
 2 1 - 1 1 = 2 1·3 - 1 1·2 = (2 - 1) + 3 - 2 = 2 3 2·3 3·2 6 6

 = 1 + 3 -2 = 1 + 1 = 1 1 6 6 6

Example 12.
Find the difference between two mixed numbers:
 3 1 - 1 3 = 3 1·4 - 1 3·3 = 3 4 - 1 9 = 6 8 6·4 8·3 24 24

 = 2 24 + 4 - 1 9 = 1 + 28 - 9 = 1 + 19 = 1 19 24 24 24 24 24

Example 13.
Find the difference between two mixed numbers:
 1 1 - 3 2 = 1 1 - 3 2·2 = 1 1 - 3 4 = (1-3) + 1 - 4 = 6 3 6 3·2 6 6 6

 = -2 - 3 = -2 - 3 = -2 - 1 = -2 1 6 2·3 2 2