Problem 24 · 1987 AJHSME
Stretch
Algebra & Patterns
score-constraintparity
A multiple choice examination consists of 20 questions. The scoring is +5 for each correct answer, −2 for each incorrect answer, and 0 for each unanswered question. John's score on the examination is 48. What is the maximum number of questions he could have answered correctly?
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Answer: D — 12.
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Hint 1 of 2
Write the score as 5c − 2w = 48 with c correct and w wrong. Before bounding anything, notice what parity 5c must have — that alone restricts c.
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Hint 2 of 2
Since 48 and 2w are both even, 5c must be even, which forces c to be even. So only even values of c are possible — test the largest ones downward.
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Approach: use parity to limit c, then check the largest candidates
- Let c = correct, w = wrong, so 5c − 2w = 48. The right side 48 and the 2w are both even, so 5c is even — meaning c itself must be even. Candidates: c = 14, 12, 10, …
- Try c = 14: 5(14) − 2w = 48 → 2w = 22 → w = 11, but then c + w = 25 > 20 questions. Too many. Try c = 12: 2w = 60 − 48 = 12 → w = 6, and c + w = 18 ≤ 20. This works.
- So the maximum is c = 12.
- Why this transfers: a parity check ('5c must be even ⇒ c even') instantly throws out odd choices like 9 and 11, so you only test a couple of values instead of all five.
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