### 2.3 More complex additions and subtractions

Complex sums with more than two fractions can be carried out, but it is advisable to break these down into smaller units:

13 | - | 2 | + | 20 | - | 9 |

8 | 7 | 3 | 5 |

has the common denominator 840! It is easier to rearrange into two additions and then do a subtraction with the results - this also sorts out the mixture of signs in the expression:

( | 13 | - | 2 | ) | + | ( | 20 | - | 9 | ) |

8 | 7 | 3 | 5 |

Notice that the first set of brackets contains the sum of the positive fractions, whilst the second set is the sum of the negative fractions in the original expression. These can be evaluated:

[(13 × 3) + (20 × 8)] | - | [(2 × 5) + (9 × 7)] | = | 199 | - | 73 |

24 | 35 | 24 | 35 |

[(199 × 35) - (78 × 24)] | = | 5093 | = 5 | 173 |

(24 × 35) | 840 | 840 |

Brackets were used to break the expression into separate units and to address ambiguities that could have arisen from the mixture of signs in the expression. Note that the expressions moved within brackets may change sign, for instance:

4 | - | 5 | - | 3 |

5 | 7 | 4 |

can also be written as:

4 | - | ( | 5 | + | 3 | ) |

5 | 7 | 4 |

but is not the same as:

4 | - | ( | 5 | - | 3 | ) |

5 | 7 | 4 |