If we call the Roommates A, J, C, and W, then the math is as follows:
Deposit needed paid back to A from C and W at 75 each. Next month's rent would be paid by C, so J and W needed to pay her 300 each. A needed to pay 10 days rent to the other three, which is 75 total, but then internet for the next month gets split and 22.29 is owed to J by C and W.
The additional parameter: Try doing this with the least amount of checks written only by J and W.
I finally got it down to a W->J check, J->C check, and J->A check. But honestly, that's the first real life puzzle I've had of that degree in a long time.
