Problem 8 · 2019 AMC 8
Medium
Fractions, Decimals & Percents
percent-multiplier
Gilda has a bag of marbles. She gives 20% of them to her friend Pedro. Then Gilda gives 10% of what is left to another friend, Ebony. Finally, Gilda gives 25% of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?
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Answer: E — 54%.
Show hints
Hint 1 of 2
Each "gives away X%" is the same as "keeps (100−X)%." Don't track what leaves — track the fraction that stays, because those just multiply.
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Hint 2 of 2
Keeps in order: 80%, then 90% of that, then 75% of that. Multiply 0.8 × 0.9 × 0.75 — no need to ever pick an actual number of marbles.
Show solution
Approach: multiply the surviving fractions
- Flip each gift into what's kept: after Pedro she keeps 80%, after Ebony she keeps 90% of that, after Jimmy she keeps 75% of what's left.
- Percentages of percentages just multiply: 0.8 × 0.9 × 0.75 = 0.54 = 54%.
- Why this transfers: for a chain of successive percentage changes, convert each to its multiplier (a 20% loss = ×0.8) and multiply them — you never need the starting amount, and order doesn't matter.
Another way — pretend she started with 100:
- Start with 100 marbles. Pedro takes 20 → 80 left. Ebony takes 10% of 80 = 8 → 72 left. Jimmy takes 25% of 72 = 18 → 54 left.
- 54 out of 100 = 54% — same result, and picking 100 keeps every step a whole number.
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