For those who want an answer to this question specifically:
This year, Jim ears $1 and deposits x of that into his account, leaving him with (1-x) to spend. Next year, Jim earns an interest of x*b, leaving him with a total of x + x*b in his bank account, also to spend.
Thus, we want to find x such that:
(1 - x) = x + xb
1 = 2x + xb
1 = x(b+2)
x = 1/(b+2)
The key is not to get confused about the interest; don't make the mistake of thinking that he only has x*b to spend, remember that interest is added on to the principle.