We can easily translate this algebraically. Let M = Mike's original # of stamps, and J - Jean's original # of stamps.

Mike has twice as many stamps as Jean has -->
\(M=2J\)

After he gives Jean 6 stamps, he still has 8 more stamps than Jean does -->
\(M-6=(J+6)+8\)
*NB: remember that in an EXCHANGE problem, when he gives her 6 stamps, he loses 6 stamps (M-6) but she also gains 6 stamps (J+6).

Now simplify the 2nd equation:
\(M-6=(J+6)+8\)
\(M-6=J+14\)
\(J=M-20\)

Now substitute this in for J in the 1st equation to solve for M:
\(M=2J\)
\(M=2(M-20)\)
\(M=2M-40)\)
\(M=40\)

The answer is E.

This type of 2-variable word problem is know as an EXCHANGE problem, and it can be tricky. For more on exchange problems and translating other tricky word problems, here is a video lesson:
https://www.youtube.com/watch?v=5Kg6mjN ... ex=11&t=0s
(It's listed for the GRE, but the exam same types of tricky word problems appear on the GMAT and GRE).