Difference between revisions of "2023 AMC 12B Problems/Problem 18"
(→Solution 2) |
m (→Solution 2) |
||
Line 28: | Line 28: | ||
==Solution 2== | ==Solution 2== | ||
− | We can use process of elimination by finding possible solutions to answer choices. Let <math>y_1</math> and <math>y_2 be the number of quizzes Yolanda took in the first and second semesters, respectively. Define < | + | We can use process of elimination by finding possible solutions to answer choices. Let <math>y_1</math> and <math>y_2</math> be the number of quizzes Yolanda took in the first and second semesters, respectively. Define <math>z_1</math> and <math>z_2</math> similarly for Zelda. |
− | Answer choice B is satisfied by < | + | Answer choice B is satisfied by <math>(y_1,y_2,z_1,z_2) = (289,1,1,289)</math>. |
− | Answer choice C and E are both satisfied by < | + | Answer choice C and E are both satisfied by <math>(y_1,y_2,z_1,z_2) = (17,17,17,17)</math>. |
− | Answer choice D is satisfied by < | + | Answer choice D is satisfied by <math>(y_1,y_2,z_1,z_2) = (7,5,5,7)</math>. |
− | Therefore the answer is < | + | Therefore the answer is <math>\boxed{(\text{A})}</math>. |
~cantalon | ~cantalon |
Revision as of 21:06, 15 November 2023
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 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 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
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.