Problem 19 · 2022 AMC 8
Medium
Arithmetic & Operations
careful-counting
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Answer: C — 4 students.
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Hint 1 of 2
Median isn't about the whole class — with 20 scores it's just the average of the 10th and 11th from the top. So the question really only cares about the scores sitting right at those two middle spots.
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Hint 2 of 2
To make that middle average 85, you need the 11th-highest score to reach 85. Count how many already sit at ≥ 85, and the gap up to 11 is how many you must push up.
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Approach: the median only depends on the 11th-from-top score — push that to 85
- Insight: with 20 students the median is the average of the 10th and 11th scores, so forget the rest of the class — the median becomes 85 exactly when the 11th-highest score climbs to 85 (then the 10th, already ≥ that, makes both middles 85).
- From the dot plot, 7 students already score ≥ 85 (the 85, 90, 95, 100 columns). Adding 5 points turns an 80 into an 85.
- To have 11 students at ≥ 85 (so the 11th-highest is 85), raise 11 − 7 = 4 of the 80-scorers. Fewer than 4 leaves the 11th spot below 85.
- You'll see this again: a median is a positional statistic — it depends only on the middle item(s), not on how high or low the extremes are. To move a median, move scores across that middle position.
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