# How to split money fairly after a vacation

After a week of fun and relax with friends, splurging money without second thoughts, it is time to make sums and make sure everybody paid what is fair. How to do this?

First, everybody should sum up all money they spent for the group, excluding personal expenses. Assume there are four people: Alice spent 887 euros, Bob spent 736, Charlie 609 and David 48. The fairest way of splitting money is making sure that after the transfers have been made everybody spent the exactly average, in this case 570 euros.

Spent | Gives to others in total | Spent after transfers | |
---|---|---|---|

Alice | 887 | -317 | 887-317=570 |

Bob | 736 | -166 | 736-166=570 |

Charlie | 609 | -39 | 609-39=570 |

David | 48 | 522 | 48+522=570 |

This means that Alice should receive 317 euros from the other three, Bob 166, Charlie 38 and David, who did not spend much during the vacation, should give 522 euros in total. By summing the money spent and the money received from the others, everybody will have spent exactly the average of 570 euros.

But how much of the 522 euros that David gives should go to Alice, how much to Bob, and how much to Charlie? And, even though in aggregate Bob should receive 166 euros from the others, he should certainly give some to Alice who spent more than him during the vacation. How to deal with this?

Consider Alice, who spent 887 euros for the group of four friends; dividing fairly, Alice spent 887/4=221.75 euros for each member, including herself. This means that everybody else should give Alice 221.75 euros to repay for her expenses. Similarly, Bob spent 736/4=184 euros for each member, which means that he should receive that amount from the others. Now, since Bob owes Alice 221.75 euros and Alice owes Bob 184 euros, they can settle this by having Bob give Alice 221.75-184=37.75 euros. If we put into a table how much each person should receive from the others, we have:

Spent | Receives from each friend | |
---|---|---|

Alice | 887 | 887/4=221.75 |

Bob | 736 | 736/4=184 |

Charlie | 609 | 609/4=152.25 |

David | 48 | 48/4=12 |

We can now find exactly how much everybody should give to each other friend by taking the difference of the quantities above, as we did for Alice and Bob. If we do this, we get the following table, with positive amounts denoting money given and negatives denoting money received:

↓ gives to → | Alice | Bob | Charlie | David | Total Given |
---|---|---|---|---|---|

Alice | 0 | -37.75 | -69.50 | -209.75 | -317 |

Bob | 37.75 | 0 | -31.75 | -172 | -166 |

Charlie | 69.50 | 31.75 | 0 | 140.25 | -39 |

David | 209.75 | 172 | 140.25 | 0 | 522 |

Total Received | 317 | 166 | 39 | -522 |

Summing the numbers in each row results in the numbers we saw at the beginning, i.e. how much each person should give to or receive from each other. In addition, we know how much each person owes to each other person by looking at the relevant cell in the table.

The math is simple, but here’s a widget to do this easily for your own vacation with up to six friends.

#### Total spending by each friend:

#### Money transfers:

↓ to → | Friend A | Friend B | Friend C | Friend D | Friend E | Friend F | Gives |
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Friend A | |||||||

Friend B | |||||||

Friend C | |||||||

Friend D | |||||||

Friend E | |||||||

Friend F | |||||||

Receives |