Problem 12 · 2010 AMC 8
Medium
Fractions, Decimals & Percents
fix-the-invariant
Of the 500 balls in a large bag, 80% are red and the rest are blue. How many of the red balls must be removed from the bag so that 75% of the remaining balls are red?
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Answer: D — 100 red balls.
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Hint 1 of 2
Only red balls leave the bag, so the number of blue balls never moves. Track that fixed quantity instead of chasing the changing reds.
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Hint 2 of 2
When one quantity stays constant through a change, anchor on it. ‘75% red’ means ‘25% blue,’ and you already know exactly how many blue there are.
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Approach: anchor on the unchanging blue count
- Start: 80% of 500 = 400 red, leaving 100 blue. Removing reds can't touch those 100 blue.
- At the end red is 75%, so blue is the other 25%. Those 25% are still exactly 100 balls, so the new total = 100 / 0.25 = 400 balls.
- We dropped from 500 to 400, all reds: 100 red balls removed.
- Why this transfers: in any ‘remove/add until the percentage changes’ problem, find the quantity that doesn't change and let it carry the new total. Chasing the moving part directly is the slow road.
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