Mike has 4 times as many stamps as Andrew. If Mike gives Andrew 8 stamps, he will have twice as many stamps as Andrew. How many stamps does each boy have?

Mike has 4 times as many stamps as Andrew

Let Mike = M and Andrew = A, eq.1 will be M = 4A

However, if Mike gives Andrew 8 stamps, then the number of stamps Mike has is now twice the number of stamps Andrew has. This will give us another equation. 

eq. 2 will be M-8 = 2 (A+8) 

The expression on the left tells us that Mike gave away eight stamps, while the expression on the right tells us that whatever this new number is is equal to twice the total number of Andrew's stamps after receiving eight from Mike.
Simplifying this further gives us:

M-8 = 2A + 16
M = 2A + 16 + 8 = 2A + 24

Use 2A + 24 and substitute this for M in eq. 1. This gives us:

2A + 24 = 4A
24 = 4A - 2A
24 = 2A
12 = A, and therefore M = 48 (because M=4A)

To begin with, Andrew had 12 stamps and Mike had 48 stamps. 
If only M-8 = 2A is used, this will only meet the condition given in situation 1 (M=4A) but not situation 2, wherein after giving away 8 stamps, Mike's stamps is twice as many as Andrew's. You can check it. :)