Let x = 0.36363636363636....

Let's multiply x by 100:

100x = 36.3636363636363636....

Now let's subtract x from this:

100x - x = 36.3636363636363636... - 0.36363636363636... = 36

99x = 36

x = 36/99 = 4/11

So 1.363636363636... = 1 + 4/11 = 15/11

Thus p = 15 and q = 11.

Thus p + q = 26.

Note: Since 15/11 = 30/22, p + q could also be 52. Your answer is not unique.

-Dan