Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2022 AMC 10B Problems/Problem 11"
Line 15: Line 15:
 
Let <math>B</math> denote a school that is bigger than Euclid HS, and <math>M</math> denote a school that sold more T-shirts than Euclid HS.
 
Let <math>B</math> denote a school that is bigger than Euclid HS, and <math>M</math> denote a school that sold more T-shirts than Euclid HS.
  
−
It follows that <math>\neg B</math> denotes a school that is not bigger than Euclid HS, and <math>\neg M</math> denotes a school that did not sell more T-shirts than Euclid HS.
+
It follows that <math>\neg B</math> denotes a school that is <b>not bigger than</b> Euclid HS, and <math>\neg M</math> denotes a school that <b>did not sell more</b> T-shirts than Euclid HS.
  
 
Converting everything to conditional statements (if-then form), the given statement becomes <cmath>B\implies\neg M.</cmath>
 
Converting everything to conditional statements (if-then form), the given statement becomes <cmath>B\implies\neg M.</cmath>
 
Its contrapositive is <math>M\implies\neg B,</math> which is <math>\boxed{\textbf{(B)}}.</math>
 
Its contrapositive is <math>M\implies\neg B,</math> which is <math>\boxed{\textbf{(B)}}.</math>
 +
 +
Note that "not bigger than" does not mean "smaller than", and "not selling more" does not mean "selling less". There is an equality case. Therefore, none of the other answer choices is equivalent to <math>B\implies\neg M.</math>
  
 
~MRENTHUSIASM
 
~MRENTHUSIASM

Revision as of 20:41, 17 November 2022

Problem

All the high schools in a large school district are involved in a fundraiser selling T-shirts. Which of the choices below is logically equivalent to the statement "No school bigger than Euclid HS sold more T-shirts than Euclid HS"?

$\textbf{(A) }$ All schools smaller than Euclid HS sold fewer T-shirts than Euclid HS.

$\textbf{(B) }$ No school that sold more T-shirts than Euclid HS is bigger than Euclid HS.

$\textbf{(C) }$ All schools bigger than Euclid HS sold fewer T-shirts than Euclid HS.

$\textbf{(D) }$ All schools that sold fewer T-shirts than Euclid HS are smaller than Euclid HS.

$\textbf{(E) }$ All schools smaller than Euclid HS sold more T-shirts than Euclid HS.

Solution

Let $B$ denote a school that is bigger than Euclid HS, and $M$ denote a school that sold more T-shirts than Euclid HS.

It follows that $\neg B$ denotes a school that is not bigger than Euclid HS, and $\neg M$ denotes a school that did not sell more T-shirts than Euclid HS.

Converting everything to conditional statements (if-then form), the given statement becomes \[B\implies\neg M.\] Its contrapositive is $M\implies\neg B,$ which is $\boxed{\textbf{(B)}}.$

Note that "not bigger than" does not mean "smaller than", and "not selling more" does not mean "selling less". There is an equality case. Therefore, none of the other answer choices is equivalent to $B\implies\neg M.$

~MRENTHUSIASM

See Also

2022 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 10 		Followed by
Problem 12
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 10 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2022_AMC_10B_Problems/Problem_11&oldid=182079"