Problem 7 · 2025 AMC 8
Easy
Counting & Probability
complementary-counting
On the most recent exam in Prof. Xochi's class,
- 5 students earned a score of at least 95%,
- 13 students earned a score of at least 90%,
- 27 students earned a score of at least 85%, and
- 50 students earned a score of at least 80%.
How many students earned a score of at least 80% and less than 90%?
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Answer: D — 37 students.
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Hint 1 of 2
These groups aren't separate piles — "at least 80%" already contains everyone who scored at least 90%. They're nested, like measuring cups inside each other.
Still stuck? Show hint 2 →
Hint 2 of 2
So to get just the 80–90% band, take the big group and subtract the part of it you don't want. Which two of the four counts do you need? (The 85% and 95% lines are decoys.)
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Approach: subtract the inner group from the outer (don't add the bands)
- The four counts are nested, not separate: the 13 who scored ≥ 90% sit inside the 50 who scored ≥ 80%. So you subtract, you don't add.
- Students in [80%, 90%) = (those ≥ 80%) − (those ≥ 90%) = 50 − 13 = 37. The 85% and 95% counts are just there to distract.
- Why this transfers: with "at least" cutoffs, the count for a band is the difference of two cumulative counts — spot the nesting and subtract instead of trying to build each band from scratch.
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