Difference between revisions of "1987 AJHSME Problems/Problem 24"
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==Problem== | ==Problem== | ||
− | A multiple choice examination consists of <math>20</math> questions. The scoring is <math>+5</math> for each correct answer, <math>-2</math> for each incorrect answer, and <math>0</math> for each unanswered question. John's score on the examination is <math>48</math>. What is the maximum number of questions he could have answered correctly? | + | <!-- don't remove the following tag, for PoTW on the Wiki front page--><onlyinclude>A multiple choice examination consists of <math>20</math> questions. The scoring is <math>+5</math> for each correct answer, <math>-2</math> for each incorrect answer, and <math>0</math> for each unanswered question. John's score on the examination is <math>48</math>. What is the maximum number of questions he could have answered correctly?<!-- don't remove the following tag, for PoTW on the Wiki front page--></onlyinclude> |
<math>\text{(A)}\ 9 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 11 \qquad \text{(D)}\ 12 \qquad \text{(E)}\ 16</math> | <math>\text{(A)}\ 9 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 11 \qquad \text{(D)}\ 12 \qquad \text{(E)}\ 16</math> | ||
==Solution== | ==Solution== | ||
+ | ===Solution 1=== | ||
Let <math>c</math> be the number of questions correct, <math>w</math> be the number of questions wrong, and <math>b</math> be the number of questions left blank. We are given that | Let <math>c</math> be the number of questions correct, <math>w</math> be the number of questions wrong, and <math>b</math> be the number of questions left blank. We are given that | ||
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==See Also== | ==See Also== | ||
+ | [[2010 AMC 10B Problems/Problem 15|2010 AMC10B Problem 15]] | ||
{{AJHSME box|year=1987|num-b=23|num-a=25}} | {{AJHSME box|year=1987|num-b=23|num-a=25}} | ||
[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 08:56, 11 June 2024
Contents
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 .
See Also
1987 AJHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.