Difference between revisions of "2022 AMC 10B Problems/Problem 11"

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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 not bigger than Euclid HS, and <math>\neg M</math> 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  
+
Converting everything to conditional statements (if-then form), the given statement becomes <cmath>M\implies\neg B.</cmath>
 +
The answer choices become
 +
 
 +
<math>\textbf{(A) }</math> All schools smaller than Euclid HS sold fewer T-shirts than Euclid HS.
 +
 
 +
<math>\textbf{(B) }</math> No school that sold more T-shirts than Euclid HS is bigger than Euclid HS.
 +
 
 +
<math>\textbf{(C) }</math> All schools bigger than Euclid HS sold fewer T-shirts than Euclid HS.
 +
 
 +
<math>\textbf{(D) }</math> All schools that sold fewer T-shirts than Euclid HS are smaller than Euclid HS.
 +
 
 +
<math>\textbf{(E) }</math> All schools smaller than Euclid HS sold more T-shirts than Euclid HS.
  
 
~MRENTHUSIASM
 
~MRENTHUSIASM

Revision as of 18:46, 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 \[M\implies\neg B.\] The answer choices become

$\textbf{(A) }$ 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.

~MRENTHUSIASM

See Also

2022 AMC 10B (ProblemsAnswer KeyResources)
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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