This is making my head hurt.
September 13, 2011 7:01 AM   Subscribe

Attention mathletes! Help me with some real-life algebra.

I have a vacation house for three nights and a total cost of $2100. 16 people are coming. Some people will be there for all three nights, some for only two. I need a formula to figure out how much each person should pay, with the 2-night people paying less than the 3-night people.

I realize it would be helpful to know how many people are coming how many nights. At this point, two people are coming for 2 nights and fourteen for 3 nights, but I have a feeling it will change closer to the date of the vacation and be more like four or six people for 2 nights and twelve or ten people for 3 nights.

There's gotta be a simple, 8th-grade algebra equation for figuring this out that will make me fell stupid for not seeing it sooner. Please, make me feel stupid!
posted by kidsleepy to Education (17 answers total) 2 users marked this as a favorite
 
Yes, you need a new unit of measure-- people-nights. For each person, count the number of nights they'll be there, then sum across people. Divide the total cost by that sum. That's how much each person should pay per night.
posted by supercres at 7:05 AM on September 13, 2011 [6 favorites]


I think...

X = rate for one night
Y = number of people coming for two nights
Z = number of people coming for three nights

2100 = X(2Y + 3Z)

Solve for X. (Y and Z will be given.)

People staying for 2 nights pay 2X, people staying for 3 nights pay 3X.
posted by oracle bone at 7:05 AM on September 13, 2011 [1 favorite]


Its best to think that the place costs $700 a night.

For each night, divide 700 by the number of people that are staying there. That means nights with lots of people will be less expensive per person, but that makes sense to me.
posted by vacapinta at 7:05 AM on September 13, 2011 [6 favorites]


If you really want it to be completely equal per night, add up the total number of nights stayed; so in your base estimate, it would be 14*3+2*2=46, then divide 2100 by that number, which comes to about $45.65. Then each person would pay number of nights they're staying times $45.65.

If people are prepaying, then you should just go by the number of nights they say they're coming now, because if anyone switches to two nights it will throw off what everyone else needs to pay. Alternately, you can just charge everyone for three nights, which will encourage them to stay longer!
posted by kaibutsu at 7:06 AM on September 13, 2011


Dividie by total number of person-nights, then each person pays that times the number of nights.

E.g. 4 * 2 + 3 * 12 = 44 person-nights. $2100 / 44 = $47.72 USD per person-night. Staying two nights? Pay $95.45 :) Three? $143.18.

Right?
posted by krilli at 7:07 AM on September 13, 2011


You want to determine cost per person-night. So,

( 2 (people) x 2 (nights) ) + ( 14 (people) x 3 (nights) ) = 46 person-nights.

$2100 / 46 person-nights = $45.65

Multiply that figure by the number of nights each guest is staying, and you get

2 x 45.65 = $91.30
3 x 45.65 = $136.95

It's a little weird to have that many digits of precision, so I'd charge your two-night guests $91, and your three-night guests $137.
posted by Mayor West at 7:08 AM on September 13, 2011


2*A + 3*B = Total number of nights (where A is the number of people staying 2 nights, and B is the number of people staying 3 nights - right now the total number of nights is 46).

$2100/total number of nights = cost per night = $45.65

So people who stay two nights should pay $91.30 and people who stay three nights should pay $136.96.

As A and B change, so will the rest of it.

On preview - what everyone else said.
posted by muddgirl at 7:08 AM on September 13, 2011


The problem with the method everyone else is giving you is that, in the limiting case, where only one person is staying 3 nights - everyone else is subsidizing that night too.
posted by vacapinta at 7:08 AM on September 13, 2011 [1 favorite]


On second thought, vacapinta is right. Fewer people there on a given night will have "more" of the house each. Treat each night like a separate commodity and divide the cost by the people using it. Additionally, that way, you won't get thrown off if people stay for more/less (except that if people back out from staying a given night, the other people staying that night will have to pay more unexpectedly.)
posted by supercres at 7:09 AM on September 13, 2011 [1 favorite]


There are two ways I can think of to do this, depending on what you feel is fairest.

Method #1: The person-nights method advocated by supercres and (implicitly) by oracle bone.

Method #2: The cost of each night is split evenly among the people who are there that night, advocated by vacapinta. So divide $700 among the people who stay three nights, divide $1400 among everyone, and add to get the total cost.

These two methods will not, in general, give the same results, so you'll have to go with your gut on which one seems "fairer".
posted by Johnny Assay at 7:09 AM on September 13, 2011


For the current set-up, that means that the people who are only staying two nights pay $87.50 and the people staying three nights pay $137.50.
posted by muddgirl at 7:12 AM on September 13, 2011


How well do you know all these people? It's not terrible to charge each person 60$/night and then refund them once you know the exact number of nights everyone stayed. That makes it much easier on you, I think, too.

Incidentally, using the person-night method, if 6 people stay 2 nights and 10 stay 3, the charge is 55$/night.
posted by jeather at 7:16 AM on September 13, 2011


I'm seconding vacapinta's reccommendation, I think. The way I started doing it is by deciding the number of people staying each night (x, y & z), and then finding 700/x, 700/y and 700/z. A person staying for the first and second night pays $(700/x + 700/y), a person staying for the second and third night pays $(700/y + 700/z) and a person staying all three nights pays $(700/x + 700/y +700/z).

So, for example if 16 people stay for the first two nights, and 10 for all three, the cost would be:

Night 1 = 700/16 = 43.75
Night 2 =700/16 = 43.75
Night 3 = 700/10 = 70.00

So the two-nighters pay 43.75+43.75 = $87.50
While the three-nighters pay 43.75+43.75+70.00= $157.50

The only wrinkle is that it depends on knowing ahead of time how many people are staying each night, and it's possible people are asking for the cost before making up their minds. In which case, I think you're justified in using an estimate or lump sum or something.
posted by missix at 9:40 AM on September 13, 2011


Just so you know, this would be really easy to think about if you stored the information in a spreadsheet program like Excel or Numbers.
posted by oceanjesse at 9:53 AM on September 13, 2011


Also something to consider in terms of fairness: Did everyone agree to a 3 night thing, then some backed down to 2? If so, making people pay person/nights, which is slightly unfair to the 2nighters because of the "more house" thing mentioned above, would become more fair, since the 3nighters now have to pay more than they had originally thought, so the person/night cost analysis might be more fair since those who backed down are now helping to subsidize(slightly) the 3nighters.
posted by Grither at 10:40 AM on September 13, 2011


Response by poster: Everyone agreed to three nights, but it turns out some people can't get out of work and are stuck with two nights. Everyone should figure out their schedule well in advance of the vacation, but it's not until New Year's Eve so it's still some months away.

We're all good friends so the money doesn't have to be on an equal ratio (the 2-nighters don't necessarily have to pay only two-thirds of what the 3-nighters pay). The 2-nighters are already on the losing side because they don't get to party with the rest of us, so the discount is just to make them feel a little better.

The face that there's no simple formula makes me feel better (I'm not stupid!) but also worse (dang, I'm still going to have to figure it out). Thanks for all the ideas though, and if you've got more, keep 'em coming!
posted by kidsleepy at 11:02 AM on September 13, 2011


This will make you insane. Normalize it to $50/person/night, and throw the tiny bit extra to the booze fund or other entertainment supplies for the weekend.
posted by judith at 2:07 PM on September 13, 2011 [2 favorites]


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