How Much Do I Owe You?

You know the scenario. A friend or family member says they’re hiring a villa for two weeks in the summer and you’re welcome to join them for as long as you can make it. Not just you, of course, but everyone else on their friends list. The place sleeps 12, if a few people are willing to bunk up, and it’d be so much fun. Just come for a couple of days, or the whole two weeks. The choice is yours.

Fun, undoubtedly, right up until the moment comes to split the bill. Not only did everyone not stay for the whole 14 nights, but consequently were not there for every one of the delicious meals cooked by the lovely people who staff the place, or for all the excursions, and they certainly weren’t all there to drink the champagne, beer and wine every night that was bought for everyone’s general enjoyment.

However, everyone was there for some of it, with the majority of people staying for maybe six or seven days, the rest grabbing a couple of days here and there when they could, and only a small few managing the full two weeks. So how do you divide up the costs of the villa rental, the cleaning, the general charges, the meals cooked, the day trips and all that booze?

Tricky. It would be all too easy at this point to accidentally trigger an argument or silent resentment that drives a wedge between some (soon to be former) friends, or even between the whole group. And who wants that? The whole idea was to have some time on holiday with close friends, away from petty everyday concerns and not to drag all that nonsense there along with your suitcases.

The perfect solution occurred to me on a recent trip with friends and family to a Riad in Marrakesh. At the end of the trip, there was a grand debate among the principle couples as to how to divide the whole bill equitably. We were all content to share every cost equally, regardless of haw many days we were all there but we all, of course, wanted to pay only in proportion to our time in the Riad and how much of the services we had used.

In a flash of inspiration drawn from medical science and applied statistics, I came up with what I shall call the Davenport Bill Splitting Method (DBSM). This relies on the simple but ingenious idea of calculating the number of person days that were spent at the villa; in other words, the number of times overall that a day/night was experienced by an individual staying there.

Let me give you an example.

Let’s say the villa was rented for a total of 14 days, and over that period 16 people stayed for various lengths of time. While that’s useful information, it doesn’t really capture how the villa was occupied over those 14 days, and certainly doesn’t help with us working out how to divide the entire cost between you.

So let’s look at it another way, by breaking down how many people were staying at the villa on each day:

• Day 1: 2 people
• Day 2: 3 people
• Day 3: 5 people
• Day 4: 6 people
• Day 5: 8 people
• Day 6: 10 people
• Day 7: 10 people
• Day 8: 12 people
• Day 9: 12 people
• Day 10: 9 people
• Day 11: 9 people
• Day 12: 6 people
• Day 13: 4 people
• Day 14: 4 people

If we add up the number of people who there were on each of the 14 days the villa was rented, that gives us a total of 100 person days at the villa, or 100 individual day/nights experienced by 16 people over 14 days.

Now, if we add up all the costs of villa hire, cleaning, cooking, excursions, booze, etc, etc, let’s say in our example it comes to £6000. Divided over the total number of person days that the villa was occupied that gives a cost per person day of £60.

So how much should you pay towards the total costs?

Well, that’s easy: you just count up the number of days that you were there and multiply it by £60. For example, if you stayed at the villa for two days, you owe £120. If you stayed for five days, you owe £300. And so on and so forth. And once everyone of the 16 guests has totalled the personal cost of their stay, it should add up to £6000.

Brilliant, eh!

So there you have it, folks: The DBSM, my gift to the world of mathematics and science, and to friend and family harmony the world over.

You're welcome.