Problem 22 · 2001 AMC 8
Stretch
Counting & Probability
careful-counting
On a twenty-question test, each correct answer is worth 5 points, each unanswered question is worth 1 point, and each incorrect answer is worth 0 points. Which of the following scores is NOT possible?
Show answer
Answer: E — 97.
Show hints
Hint 1 of 2
Scores bunch up near the top in a jagged way — so test the high choices first by asking "what's the most you can score without getting everything right?"
Still stuck? Show hint 2 →
Hint 2 of 2
Turning one correct (5 pts) into one unanswered (1 pt) drops your score by 4, which creates a gap just below the perfect 100.
Show solution
Approach: find the gap just below the maximum
- The ceiling is all 20 correct: 20 × 5 = 100. To score less you must give up a correct answer, and the gentlest downgrade is correct (5) → unanswered (1), a loss of 4. So the very next attainable score is 100 − 4 = 96 (= 19 right + 1 unanswered).
- That leaves 97, 98, 99 stranded with no way to reach them. Among the choices, 97 is the impossible one.
- Quick check that the rest work: 95 = 19 right (the last as wrong); 90 = 18 right; 91 = 18 right + 1 unanswered; 92 = 18 right + 2 unanswered. The lesson: scoring systems with unequal point values leave "holes," and the biggest holes sit right under the maximum.
Mark:
· log in to save