2023 AMC 12B Problems/Problem 18
Contents
Problem
Last academic year Yolanda and Zelda took different courses that did not necessarily administer the same number of quizzes during each of the two semesters. Yolanda's average on all the quizzes she took during the first semester was points higher than Zelda's average on all the quizzes she took during the first semester. Yolanda's average on all the quizzes she took during the second semester was points higher than her average for the first semester and was again points higher than Zelda's average on all the quizzes Zelda took during her second semester. Which one of the following statements cannot possibly be true?
Yolanda's quiz average for the academic year was points higher than Zelda's.
Zelda's quiz average for the academic year was higher than Yolanda's.
Yolanda's quiz average for the academic year was points higher than Zelda's.
Zelda's quiz average for the academic year equaled Yolanda's.
If Zelda had scored points higher on each quiz she took, then she would have had the same average for the academic year as Yolanda.
Solution 1
Denote by the average of person with initial in semester Thus, , , .
Denote by the average of person with initial in the full year. Thus, can be any number in and can be any number in .
Therefore, the impossible solution is
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Solution 2
We can use process of elimination by finding possible solutions to answer choices. Let and be the number of quizzes Yolanda took in the first and second semesters, respectively. Define and similarly for Zelda.
Answer choice B is satisfied by .
Answer choice C and E are both satisfied by .
Answer choice D is satisfied by .
Therefore the answer is
~cantalon
Solution 3 (Algebra)
Let Yolanda's average for semester be , the number of quizzes Yolanda took in semester be , Zelda's average for semester be , the number of quizzes Zelda took in semester be , Yolanda's average for semester be , the number of quizzes Yolanda took in semester be , Zelda's average for semester be , the number of quizzes Zelda took in semester be , Yolanda's average for the entire year be , Zelda's average for the entire year be .
From the problem we know that
at most is , is at least , meaning that is at most .
Therefore,
Hence, is not possible.
Solution 4
Denote and as the quiz averages of Yolanda in the st and nd semesters, respectively. Similarly, denote and as the quiz averages of Zelda in the st and nd semesters.
We have , so . We also know that , implying .
The average quiz scores for both students must lie between the averages of each semester, i.e and Since , , and , we have and . Therefore the maximum difference between the two yearly averages is
Therefore, is not possible.
-Benedict T (countmath1)
Video Solution 1 by OmegaLearn
Video Solution 2
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
See Also
2023 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 17 |
Followed by Problem 19 |
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 AMC 12 Problems and Solutions |
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