# Addition and Subtraction of fractions.

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## Addition 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 |

### Addition of fractions.

Definition.

To add two fractions, it is necessary:
- to reduce fraction to a least common denominator;
- add the numerators and put the answer over the same denominator;
- simplify the fraction;
- to convert improper fractions to mixed numbers if needed.

### 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 |

### Addition of mixed numbers

Definition.

To add two mixed numbers, it is necessary:
- to reduce fraction to a least common denominator;
- add the numerators and put the answer over the same denominator;
- simplify the fraction;
- if this fraction is improper then convert fraction to a mixed number;
- add the integer portions of the two mixed numbers;
- if adding the fractional parts created a mixed number then add its integer portion to the sum.

### 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:
- to reduce fraction to a least common denominator;
- subtract the numerators and put the answer over the same denominator;
- simplify the fraction.

### 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:
- to reduce fraction to a least common denominator;
- make the first numerator larger than the second if it is not;
- subtract the second numerator from the first;
- place that difference over the common denominator;
- Subtract the integer portions of the two mixed numbers;
- simplify the fraction.

### 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 |

**Fraction**

**Forms of fractions (vulgar fraction, proper fraction, improper fractions, mixed numbers, decimals)**

**The basic property of fraction**Simplifying fractions Least common denominator of fractions Converting improper fractions (composed fractions) to mixed numbers Converting mixed numbers to improper fractions (composed fractions) Addition and Subtraction of fractions Multiplication of fractions Division of fractions Comparing fractions Convert Decimals to Common Fractions

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