Question

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 $3$ 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 $18$ points higher than her average for the first semester and was again $3$ 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 $22$ 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 $3$ points higher than Zelda's.

Zelda's quiz average for the academic year equaled Yolanda's.

If Zelda had scored $3$ points higher on each quiz she took, then she would have had the same average for the academic year as Yolanda.

Solution:

Set Zelda's first-semester average to $0.$ Then Yolanda's first semester is $3,$ her second semester is $3+18=21,$ and Zelda's second semester is $21-3=18.$ Each person's yearly average is a weighted average of their two semester averages, so Yolanda's year average lies between $3$ and $21,$ and Zelda's lies between $0$ and $18.$ The largest possible gap Yolanda $-$ Zelda is therefore at most $21-0=21,$ so it can never be $22.$ All the other statements are achievable for suitable quiz counts.

Thus, the correct answer is A.

2023 AMC 12B Problem 18

Solution:

Problem 18 in Other Years