Math question

[aerocontrols]

My (recently former) roomate and I have a disagreement on bills.

There are three of us, and one of us moved out this month after 2 weeks. He says his share of the bills should be calculated like this:

2 weeks for him + 4 weeks for me + 4 weeks for third roomate = 10 weeks total.

His share 2/10 = 20%

My share 4/10 = 40%

Third roomie's share 4/10 = 40%

Working independently, I came up with the following:

1/3 of the bills per roomate per month.

Since he was here half of a month, he pays 1/2 of his normal 1/3 = 1/6 = 2/12 = 16.66%

I pay my 1/3 + 1/4 of old roomie's 1/3 = 5/12 = 41.6666%

Third roomate pays his 1/3 + 1/4 of old roomie's 1/3 = 5/12 = 41.6666%

Obviously, I want him to be correct, and I think that his is, (It certainly seems simpler) but I still want to know why I'm wrong. This reminds me, in fact, of several puzzles (what happened to the missing dollar tip puzzle, for instance) but I can't see what's wrong with my method, if in fact my method is the wrong one.

MattJ

[Fade]

Yours seems more correct, unless this happened to be the month of february.

Those few extra days would begin to tip the bills in favour of the two of you paying more, and him paying less.

[Walter Wayne]

I don't think one method can be said to be correct.

His method involves looking at the number of person-months spent there and dividing it accordingly. It makes sense for variable costs (utilities) to be calculated this way as costs would go down during the period only 2 people are there.

Example if each of you used 1 litre of water/month (you should shower more often) you would use 2.5 litres of water with a ratio of (1:2:2)

Your method makes slightly more sense for fix costs. During the first half of the month everybody has one third of the apartment, while in the second half of the month you and the remaining room mate each have a half of the appartment.

So
mover out = 0.5 month * 0.33 appartment = 16.7%
hangers on = 0.5 month*0.33 appt + 0.5 month*0.5 appt = 41.7%

Walt

[BobK]

Use a cost per day figure.

Using May = 31 days

Total cost per month / 31 = Cost per day

Cost per day / 3 = Individual Cost per day

Individual Cost per day x Days residence = Amount owed

Or use percentage per day

Total percentage for 1 day = 100% / 31 days = 3.2258%

3.2258 / 3 = 1.075% per individual per day

14 days x 1.075% = 15.05% for two weeks

Total cost per month x 15.05% = Amount owed

Either formula will give the same result within a couple cents due to rounding.

If your former roomie wants to pay 20% I wouldn't quibble with them.

[Goryus]

His is quite unfair to him. More accurately, he owes one third of the rent on half the payment, assuming he was there exactly half the month. The two of you owe half the payment each on the remaining half, plus one third the payment on the initial half.

The flaw in his logic is the assumption that all of the weeks are equally weighted. Since the cost is less for the first half (it's split among three of you) and more for the second half (where it's split among only two), they are not.

That makes your figure of 16+2/3 the more correct of the two (note that since 14 days (2 weeks) is technically a little less than half the month, BobK's figure is even more accurate).

[bjornart]

Originally posted by Walter Wayne
I don't think one method can be said to be correct.

His method involves looking at the number of person-months spent there and dividing it accordingly. It makes sense for variable costs (utilities) to be calculated this way as costs would go down during the period only 2 people are there.

Example if each of you used 1 litre of water/month (you should shower more often) you would use 2.5 litres of water with a ratio of (1:2:2)

Your method makes slightly more sense for fix costs. During the first half of the month everybody has one third of the apartment, while in the second half of the month you and the remaining room mate each have a half of the appartment.

So
mover out = 0.5 month * 0.33 appartment = 16.7%
hangers on = 0.5 month*0.33 appt + 0.5 month*0.5 appt = 41.7%

Walt

What he said. Except I'd have made him pay for the whole month, or more. Even without a contract or agreement (written or oral) I'd say that giving notice before moving out is customary. :D

[aerocontrols]

Originally posted by bjornart


What he said. Except I'd have made him pay for the whole month, or more. Even without a contract or agreement (written or oral) I'd say that giving notice before moving out is customary. :D

He gave plenty of notice. :)

Thanks, everybody else. I should note that the 'two weeks' was an estimate, really. The three of us moved, but he only moved for the beginning of the month, except his stuff was there longer than he was, etc...

MattJ

[Jesse]

Originally posted by Fade
two of you paying more, and him paying less. You're being silly.

[Michael Redman]

Originally posted by aerocontrols
My (recently former) roomate and I . . .Does that mean he moved back in with you? :confused:

[aerocontrols]

Originally posted by Michael Redman
Does that mean he moved back in with you? :confused:

probably maybe not

[rwguinn]

Originally posted by Walter Wayne
I don't think one method can be said to be correct.


Walt
I beg your pardon? I gues it depends on what the definition of correct is?
there are more than 4 weeks in any month-with the exception of February. Therein lies the error.

[Walter Wayne]

Originally posted by rwguinn

I beg your pardon? I gues it depends on what the definition of correct is?
there are more than 4 weeks in any month-with the exception of February. Therein lies the error. Actually it was obvious that both were assuming half a month. A good approximation to two weeks. The calculations on one look at how many person-months of occupation took place, however the other took it to account that renters have more space when there are less people.

Walt

[peptoabysmal]

What month has ten weeks? You are trying to figure out how much percent of four weeks you each stayed and apply that to the 100% of the rent that needs to be paid. I think doing it your roommate's way definately makes the percentage work out to 100% but does not accurately reflect what percentage of four weeks you each owe for. I think that his way makes each of you pay your percentage of 10 weeks.

You could do it like this:

(MaxPercent / NumPersons) / (TotalTime/TimeStayed)

The percentage of total time stayed by each is:
(100 / 3) / (4/2) = 16.67

(100 / 3) / (4/4) = 33.33

Add those percentages:
33.33 + 33.33 + 16.67 = 83.33

Not all three stayed the total amount of time so:
Subtract total time stayed from the 100% of rent that needs to be payed:
100 - 83.33 = 16.67%

The difference divided by three to share the cost of the third not staying the full four weeks is:
16.67 / 3 = 5.56%

The two who stayed four weeks pay (each):
33.33 + 5.56 = 38.89%

The one staying two weeks pays:
16.67 +5.56 = 22.23%

(38.89 * 2) + 22.23 = 100.01%

The accuracy is a bit off due to rounding errors, but you could have two of you pay 38.89% and the third who stayed two weeks pay 22.22% and it will total 100%.