Problem 1 · 2009 AMC 8
Easy
Algebra & Patterns
work-backward
Bridget bought a bag of apples at the grocery store. She gave half of the apples to Ann. Then she gave Cassie 3 apples, keeping 4 apples for herself. How many apples did Bridget buy?
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Answer: E — 14 apples.
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Hint 1 of 2
The story runs forward (give away, give away, keep), but you know the END, not the start. So play the movie in reverse.
Still stuck? Show hint 2 →
Hint 2 of 2
To undo a story, reverse the order AND flip each action: gave away → add back, took half → double.
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Approach: work backward, inverting each step
- Start from what's known: 4 apples kept. The last thing before that was giving Cassie 3, so undo it — add them back: 4 + 3 = 7. That 7 is what she had right after giving Ann half.
- Giving away half means 7 is the OTHER half, so the original was double: 2 × 7 = 14.
- Why this transfers: whenever a problem hands you the finish and asks for the start, work-backward turns it into pure arithmetic — just remember to invert each operation (subtract becomes add, halve becomes double).
Another way — algebra (one equation):
- Let she bought n. After giving Ann half she has n/2; after Cassie's 3 she has n/2 − 3 = 4.
- So n/2 = 7 and n = 14.
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