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:
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
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!