I have killed many many many rats over the years. When I was in college I lived in a rat infested house and killed over 30 in one fall.
Rats are highly intelligent. You will almost always catch the babies and young ones first, followed by the females. The last rat you catch with typically be the king rat, the alpha male. I have observed this at 3 seperate residences. The king rat will typically be the most difficult to catch.
Snap style traps can be very effective but rats learn quickly that they are dangerous. Observing rats, I have seen the large ones hold back and watch while smaller rats are caught/killed first. Always place the traps along fences or walls, with the longest side of the trap touching the wall. Rats typically travel along fences or edges, and placing the trap in the same direction of increases the effectiveness of the trap. If you know where the rats nest is, face the trap to the snap points towards the nest. Rats leaving at night to eat will be headed face first into the traps. Once your catch rate drops off and rats get wise to snap style traps, you need to appeal to their intelligence. Place bait on the traps and dont set them for a few days. Let the rats get conditioned to the traps as a food source, then try just setting one or two out of a set.
Another good method is the trash can. An adult rat can jump about 24-30 inches. Using a 160L trash can put a piece of dowel or broomstick through the top with a cardboard paper towel roll or soup can in the middle. Place the bait on the soup can. THe rat will walk out on the dowel try and grab the bait and fall into the garbage can(which it cant jump out of). Check this type of trap frequently. If theres a rat in the pail either fill it water, or finish it off with a shovel.
These methods will get most of the rats, but to kill the king rat the most effective(and time consuming) method is spotlighting. First determine where the rat is travelling, then get a lawnchair and six pack of lucky. Prepare a .177 calibre pellet gun by duct taping an LED flashlight to the barrel. Figure out where the rat is hanging out, set food out and wait. Large rats will almost always start moving right at dusk. When you see it come out, shine the flashlight in its eyes and it will freeze momentarily. In Victoria I caught killed a very large rat this way. Over 30cm long, with no tail and scars all over its face that I presume were from snap traps.
Happy hunting!