How to solve a counterfeit coin problom?

First problem:
You have 8 coins.1 is lighter and is fake.Weight using balance weighing scale at most twice.Find the fake coin.
Answer:Split 8 coins into 3 groups.Group A(3 coins),B(3 coins) and C(2 coins).
1. weight A against B
a)If equal, Fake coin is in C.
-balance the 2 coins in C and lighter coin is counterfeit.
b)If A or B is lighter,take 2 random coin from lighter group and weigh.
-balance,third coin is fake
-if lighter,then it is fake