How to split the rent if two roommates live there from the beginning and a third one joins in the middle of the month? I've been thinking over it and I can't figure it out.

Consider the rent of the house is $X$. Now there are two roommates from the beginning and a third one joins in the middle of the month. Now how is the rent split?

Solution 1:
half month = $X/2$
1 share in first half $= X/2/2=X/4$
1 share in second half $= X/2/3=X/6$
Total from each person $= (X/4+X/6) + (X/4+X/6) + X/6$ which is
                   $5X/12 +5X/12 + X/6$
Solution 2:
Person A and Person B stayed = 30 days.
Person C stayed = 15 days.
Total days $= 30+ 30 + 15 = 75$
Therefore,  each share would be   $= 2X/5 + 2X/5 + X/5$
Both solutions seems to be correct, yet yields different shares. Can someone tell me the difference between the two and which old would be correct ??
 A: The first solution assumes that the paid rent should be proportional to both the time spent and the area consumed. The second solution assumes that only the time should be considered. I suppose the first one is more realistic as the utility one person draws from the house can be assumed to be bigger if more personal space is available (whether this is the case to full extent - for example there is less social interaction possible in an empty house - is a different question). For some aspects the second method seems to be more appropriate, for exacmple if the rent is not fixed per month (for the house as a whole) but there are consumption dependent parts in it (e.g. water and electricity - three people may be assumed to consume more, for simplicity assumed in a proportional way, but that is also not absolutely true).
If we assum ethat the contract with the landlord fixed a rent that is the same every month independent of the number of inhabitants, then only method one is valid - just imagine the landlord would drop by every day to cash in $X/30$.
A: In your first method, you are splitting the rent such that there is the same cost per day, regardless of how many people are in the apartment each day.
In the second method, you are splitting the rent such that there is the same cost per day per person.
Neither is really more 'correct' than the other, as they are measuring different things.
