1987 AJHSME Problems/Problem 24
Problem
A multiple choice examination consists of questions. The scoring is for each correct answer, for each incorrect answer, and for each unanswered question. John's score on the examination is . What is the maximum number of questions he could have answered correctly?
Solution
Solution 1
Let be the number of questions correct, be the number of questions wrong, and be the number of questions left blank. We are given that
Adding equation to double equation , we get
Since we want to maximize the value of , we try to find the largest multiple of less than . This is , so let . Then we have
Finally, we have . We want , so the answer is , or .
Solution 2
If John answered 16 questions correctly, then he answered at most 4 questions incorrectly, giving him at least points. Therefore, John did not answer 16 questions correctly. If he answered 12 questions correctly and 6 questions incorrectly (leaving 2 questions unanswered), then he scored points. As all other options are less than 12, we conclude that 12 is the most questions John could have answered correctly, and the answer is .
See Also
1987 AJHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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