Working backwards is a good strategy to use when you're told how a situation ends up, and you're supposed to figure out how it began. For example:

Q: Jim bought a bunch of grapes. He ate one half of the grapes. Then 
   his friend Gary ate one fourth of the remianing grapes. There were 6 
   grapes left. How many grapes were in the bunch to begin with?

A: Jim ends up with 6 grapes:

     oooooo

   Just before that, Gary ate 1/4 of the number he had.  If we divide the
   6 grapes into pairs, 

     oo oo oo

   we see that if we add another pair, then Gary ate two:

      oo oo oo oo
               \/__________ Gary ate these

   Can you take the next step? 


Here's an example of a problem that can be worked either forwards (using algebra) or backwards (using arithmetic). It's interesting to note which solution turns out to be simpler!

Q:  Alice, Bonita, and Carmen have just finished playing three rounds 
    of a game. In each round there was only one loser. Alice lost the 
    first round. Bonita lost the second. Carmen lost the third. After 
    each round the loser was required to double the chips of each of 
    the other by giving away some of her own chips. After three rounds 
    each of the girls had 8 chips. How many chips did they have at the 
    start?

A1: Let's say Alice, Bonita, and Carmen start out with A, B, and C chips 
    respectively. Now we can follow through the game and see what they 
    end up with.

    First round: Alice loses, and gives Bonita and Carmen B and C chips 
    respectively, in order to double what each of them has by giving them 
    as much as they already have:

      Alice:  A - B - C
      Bonita: B + B = 2B
      Carmen: C + C = 2C

    Second round: Bonita loses, and gives Alice (A - B - C) chips and 
    Carmen 2C chips:

      Alice:  (A - B - C) + (A - B - C) = 2A - 2B - 2C
      Bonita: 2B - (A - B - C) - 2C = -A + 3B - C
      Carmen: 2C + 2C = 4C

    Now do the same with the third round, and you'll have three equations.
    Since you know the resulting expressions are all equal to 8, solve 
    those equations, and you'll know what they started with.

A2: You know what happens at each step, and you know how things ended up. 
    So let's just work backwards.  At the end:

      A has 8
      B has 8
      C has 8

    Before the third round, when A and B were doubled and C lost what A 
    and B gained:

      A had 8 / 2 = 4
      B had 8 / 2 = 4
      C had 8 + 4 + 4 = 16

    Before the second round, when A and C were doubled and B lost what A 
    and C gained:

      A had 4 / 2 = 2
      C had 16 / 2 = 8
      B had 4 + 2 + 8 = 14

    Go back one more round and you're done!