### 2.1 Adding fractions

Adding fractions together is simplest if the denominator is identical. For instance:

5 | + | 8 | = | (5 + 8) | = | 13 | = 4 | 1 |

3 | 3 | 3 | 3 | 3 |

Note that the denominator remains unchanged in this process. What we did was to add five objects, each with a value of one third, to eight objects with the same individual value.

What happens if the denominators in the two fractions are not identical? One or both fractions have to be transformed so that they have a common (shared) denominator. In this example, we have expressed the mixed number used in the example above as two separate fractions, by writing the whole number as a fraction whose denominator is 1:

5 | 1 | = | 5 | + | 1 |

4 | 1 | 4 |

Here, we can convert both denominators to 4:

5 | + | 1 | = | (5 × 4) | + | 1 | = | 20 | + | 1 | = | (20 + 1) | = | 21 |

1 | 4 | (1 × 4) | 4 | 4 | 4 | 4 | 4 |

Note that both numerator and denominator are multiplied by the same amount. So long as you carry out exactly the same multiplication to top and bottom, the value of a fraction remains unchanged:

15 | = | 30 | = | (7 × 30) |

3 | 6 | (7 × 6) |

In the example, the denominator and numerator have both been increased. If it is possible to decrease the denominator, the fraction is said to have been simplified:

210 | = | (210 ÷ 14) | = | 15 | = | 5 | = 5 |

42 | (42 ÷ 14) | 3 | 1 |

If when adding two fractions there is no simple way to make the denominators equal (as is the case when one is an exact multiple of the other), the denominators are multiplied together to make a new common denominator before the fractions are added:

12 | + | 11 | = | (12 × 7) | + | (11 × 6) | = | (84 + 66) | = | 150 |

6 | 7 | (6 × 7) | (7 × 6) | 42 | 42 |

The improper fraction in the answer can be converted to a mixed number, and it may then be possible to simplify the fraction part:

150 | = 3 | 24 | = 3 | (24 ÷ 6) | = 3 | 4 |

42 | 42 | (42 ÷ 6) | 7 |