Olympiad topic · P2–P6
Working backwards
Working-backwards questions appear in the Hua Xia Cup's P1 to P5 first rounds, qualifying rounds and finals.
Try one
★★ MediumA bus leaves the terminus.
At the first stop, half the passengers get off and then 4 get on.
At the second stop, half the passengers get off again and nobody gets on.
Now there are 10 passengers on board.
How many passengers were on the bus when it left the terminus?
- Hint 1: Work backwards: how many were on board before people got off at the second stop?
- Hint 2: Going back: before "half get off" there were twice as many; before "4 get on" there were 4 fewer.
- 1.Before the second stop: 10 × 2 = 20.
- 2.Before people got on at the first stop: 20 − 4 = 16.
- 3.Before people got off at the first stop: 16 × 2 = 32.
- ⚠️Common mistake: forgetting that undoing "4 get on" means taking 4 away.
Answer: 32 passengers
Four levels, step by step
- ★EasyFree2 question types · unlimited questionsOne or two steps.
- ★★MediumFree2 question types · unlimited questions"half, then a few more".
- ★★★HardFree2 question types · unlimited questionsA wrong calculator key, or three people taking turns.
- ★★★★Expert (competition level)VIP2 question types · unlimited questionsPeople doubling each other's amounts, worked back in a table.
Working backwards: what it is and how to solve it
You know the result after a chain of changes; work back to the number at the start.
For example: Mei gives half her cards to her sister, then 7 to a friend, and has 9 left. How many at first? Backwards: 9 + 7 = 16, 16 × 2 = 32.
The key idea: Start from the last step and undo each one: add becomes subtract, multiply becomes divide, in reverse order. A flow chart drawn from the end is clearest. In the usual method these are called inverse operations; an equation works too.
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