RFID to detect when someone passes through a doorway? Locate users in a house?

Hi there. Im a newbie at all of this. For uni we're studying ubiquitous computing and have to build a simulation or a prototype using arduino and Pi's. For my project, I want to do an "intelligent" home. part of my project, i want to detect who's in each room in the house. Ive been looking for a few days how i might detect where multiple people are at any one time and the best solution i found was RFID. I thought i could put a reader on each side of the door way and when a user walks into the room with a ID in their pocked, the reader would pick it up and log their location, then when they walk out, the reader would pick it up again and log them out of the room and into the next one. the problem with RFID that i have noticed is that they only reach about 4 inches, which would mean the users would probably be able to walk in and out of rooms without being registered, unless i get a UHF reader with active tags, but this is out of the question...far too expensive. my budget for the whole project is around £100, which isn't a lot, I know, so i have to make it go as far as possible.

my question is, can you think of how i might be able to achieve logging the locations of different users in a house? either by using a cheaper, suitable RFID, or another way that I might not have thought of?