Math

The number of stamps Joe: Alex: Sam = 3:8:5

Let the ratio be 3x : 8x : 3x

Sam keep 1/3 of his stamps the rest of the stamps given to

Joe and Alex the rest if 4/5 of 3x that is 4x

in ratio 3:5 that is 1.5x : 2.5x

Alex had 96 stamps more than Joe

Alex: 2.5x + 8x

Joe: 1.5x + 3x

2.5x + 8x - 3x - 1.5x = 96

6x = 96

x = 16

Joe: 1.5 * 16 + 3 * 16 = 72 stamps

So Joe had 72 stamps in the end

Let's set Joe has 3x stamps initially. So

3x + 5x * 4/5 * 3/8 + 96 = 8x + 5x * 4/5 * 5/8

3x + 3/2x + 96 = 8x + 5/2x

6x + 96

x = 16

So 3 * 16 + 3/2 * 16 = 72

Setting Joe has 3x, Alex has 8x, Sean has 5x,

3x + (1 - 1/5) * 5x * 3/8 = 8x + (1 - 1/5) * 5x * 5/8 - 96

x = 16

∴ 3x + 16 + (1 - 1/5) * 5 * 16 * 3/8 = 72

∴ Joe have 72 stamps in the end.

In this case, me look up the apply math. 

Middle School method
Sean kept 1/5 of his stamps. So the rest is

5 - 5 * 1/5 = 4.

And the rest was given to Joe and Alex in

ratio of 3:5. So Joe got 4 * \( {3 \over 5 + 3}\) = 3/2; Alex

got 4 * \( {5 \over 3 + 5}\) = 5/2.

So the new ratio of Joe and Alex is (3 + 3/2):

(8 + 5/2) = 4 1/2 : 10 1/2.

Alex had 96 stamps more than Joe. So 96 / (10 1/2 - 4 1/2) = 96 / 6 = 16 stamps. (The "I" in ratio represent 16 stamps)

So Joe had 4 1/2 * 16 = 72 stamps.