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Difference between revisions of "2022 AMC 10B Problems/Problem 11"
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Converting everything to conditional statements (if-then form), the given statement becomes <cmath>M\implies\neg B.</cmath>
 
Converting everything to conditional statements (if-then form), the given statement becomes <cmath>M\implies\neg B.</cmath>
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The answer choices become
+
Its contrapositive is <math>B\implies\neg M,</math> which is <math>\boxed{\textbf{(B)}}.</math>
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<math>\textbf{(A) }</math> All schools smaller than Euclid HS sold fewer T-shirts than Euclid HS.
 
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<math>\textbf{(B) }</math> No school that sold more T-shirts than Euclid HS is bigger than Euclid HS.
 
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<math>\textbf{(C) }</math> All schools bigger than Euclid HS sold fewer T-shirts than Euclid HS.
 
−
 
 
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<math>\textbf{(D) }</math> All schools that sold fewer T-shirts than Euclid HS are smaller than Euclid HS.
 
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<math>\textbf{(E) }</math> All schools smaller than Euclid HS sold more T-shirts than Euclid HS.
 
  
 
~MRENTHUSIASM
 
~MRENTHUSIASM

Revision as of 18:50, 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 \[M\implies\neg B.\] Its contrapositive is $B\implies\neg M,$ which is $\boxed{\textbf{(B)}}.$

~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

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