# The Roommates' Dilemma

2 roommates - Arjun and Bishan - rent a house with two rooms for a month. One room is 700 sqft in size and the other 300 sqft.

Let's say that the rent is Rs. 100 for 1 month for both rooms combined. The roomates decide that Arjun would stay in the larger room for half the month after which they would swap and Bishan would stay in the 700 sqft room. After half the month, Arjun decides that he does not want to move to the smaller room and asks Bishan if they can come to a compromise.

Bishan proposes the following: He says that he will continue to let Arjun stay in the 700 sqft room for the remaining half of the month. In exchange for allowing this, Bishan wants Arjun to pay Rs. 70 of the total Rs.100 because Arjun's room is 70% of the apartment (when they are due the bill at the end of the month).

Arjun agrees that rent must be payed proportional to the area of the room. However, he claims that as per the original agreement, they were both entitled to stay in each room for half a month each. So, Arjun claims, Bishan only deserved the larger room for half the month, and so, Arjun only needs to compensate Bishan for the 2nd half of the month. Thus, says Arjun, that he will pay 70% of the rent of 2nd half of the month - i.e., 70% of Rs 50 (2nd half's rent) which is Rs.35.

As per the method proposed by Bishan, the total rent payable by Arjun is Rs.70 (ie., 70% of Rs.100).

As per the method proposed by Arjun, the total rent payable by himself is Rs.60 (ie., Rs.25 for the 1st half of the month and Rs. 35 for the next half)

What should they do?

• The first computation seems to make more sense - I can't see the puzzley part of it. – Arnaud Mortier Aug 17 at 21:02
• @ArnaudMortier You're probably correct. Do you have any suggestions for more appropriate forums where I can post this. Any would be appreciated. – rahs Aug 17 at 21:40
• @rahs in US this would be a legal contract issue. If they signed a contract they should abide by it. – DEEM Aug 17 at 23:50

What should they do?
Not be roommates, if Arjun is going to this kind of twisted "logic" to get one over on Bishan.

But here's a way to look at this —
Clearly the larger room has a benefit, and one which should cost more, as they themselves conclude in arriving at their compromise. It would make sense for the person in the larger room to pay half the rent, but then pay the person in the smaller room an additional Rs.20 per month in exchange for the benefit of having the bigger room. If they don't swap rooms at the half month as they originally agreed, this would make the rent paid per month the 70/30 split they've compromised on.

If they traded off mid-month as per the original plan, they would each "pay" the other $$\frac{\text{Rs.}20}2$$ for their respective half-months in the larger room - which of course cancels out - and they end up each just paying half the rent.

Now that the switch hasn't happened, Arjun should not get to enjoy the benefit of the larger room for that first half month without paying anything for it. By the above thinking, that half-month is worth Rs.10, which is what Arjun should pay Bashun for the first half of the month, in addition to half (Rs.25) of the half month's rent (Rs.50) ... for a total of Rs.35 for the first half of the month. For the second half of the month, Arjun pays 70% of the remaining Rs.50 as they've just agreed, or another Rs.35. (Equivalently, Arjun pays half the rent for half the month = Rs.25, and pays Bashun Rs.10 as compensation for the other half-month of having the larger room. Either way, the math works out the same.)

Arjun owes Rs.70 for the month, and shouldn't try getting out of it.

The only puzzle here is why you chose to post this here. If this is an actual dispute you or someone you know is having, this is not the ideal site to ask this question on. If it's more along the line of "Both arguments seem to use logic and math to arrive at an answer, so why are the two answers different?" then perhaps my explanation of why Bishan's number is the appropriate one will help answer that.

• Thanks for your answer. This question did originally stem from a possible real-life scenario, but then turned into a math exercise for myself. I guess there are better forums this can be posted on - I'm just not aware which. Thanks again – rahs Aug 18 at 4:50