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 "2017 AMC 10A Problems/Problem 6"

(→‎Solution 2 (Logic it out!))
(Added another video solution)
 
(22 intermediate revisions by 7 users not shown)
Line 3: Line 3:
 
Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which one of these statements necessarily follows logically?
 
Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which one of these statements necessarily follows logically?
  
−
<math>\textbf{(A)}\ </math>If Lewis did not receive an A, then he got all of the multiple choice questions wrong.<math>\\\qquad\textbf{(B)}\ </math>If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong.<math>\\\qquad\textbf{(C)}\ </math>If Lewis got at least one of the multiple choice questions wrong, then he did not receive an A.<math>\\\qquad\textbf{(D)}\ </math>If Lewis received an A, then he got all of the multiple choice questions right.<math>\\\qquad\textbf{(E)}\ </math>If Lewis received an A, then he got at least one of the multiple choice questions right.
+
<math>\textbf{(A)}\ \text{If Lewis did not receive an A, then he got all of the multiple choice questions wrong.}\\\textbf{(B)}\ \text{If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong.}\\\textbf{(C)}\ \text{If Lewis got at least one of the multiple choice questions wrong, then he did not receive an A. }\\\textbf{(D)}\ \text{If Lewis received an A, then he got all of the multiple choice questions right.}\\\textbf{(E)}\ \text{If Lewis received an A, then he got at least one of the multiple choice questions right.}</math>
  
−
==Solution==
+
==Solution 1 (Using the Contrapositive)==
 
Rewriting the given statement: "if someone got all the multiple choice questions right on the upcoming exam then he or she would receive an A on the exam."
 
Rewriting the given statement: "if someone got all the multiple choice questions right on the upcoming exam then he or she would receive an A on the exam."
−
If that someone is Lewis the statement becomes: "if Lewis got all the multiple choice questions right, then he got an A on the exam."
+
If that someone is Lewis the statement becomes: "if Lewis got all the multiple choice questions right, then he would receive an A on the exam."
−
The contrapositive: "If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong (did not get all of them right)" must also be true leaving B as the correct answer. B is also equivalent to the contrapositive of the original statement, which implies that it must be true, so the answer is <math>\boxed{\textbf{(B)}\text{ If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong.}}</math>.
+
 
 +
The contrapositive: "If Lewis did ''not'' receive an A, then he did ''not'' get all of them right"
 +
 
 +
The contrapositive (in other words): "If Lewis did not get an A, then he got at least one of the multiple choice questions wrong".
 +
 
 +
B is also equivalent to the contrapositive of the original statement, which implies that it must be true, so the answer is <math>\boxed{\textbf{(B)}\text{ If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong.}}</math>.
  
 
* Note that answer choice (B) is the contrapositive of the given statement. (That is, it has been negated as well as reversed.) We know that the contrapositive is always true if the given statement is true.
 
* Note that answer choice (B) is the contrapositive of the given statement. (That is, it has been negated as well as reversed.) We know that the contrapositive is always true if the given statement is true.
  
−
== Solution 2 (Logic it out!)==
+
~minor edits by BakedPotato66
  
−
<math>(A)</math> False. This can easily be identified as wrong, as Lewis might've only gotten <math>3</math> multiple-choice questions wrong, but still not received an A
+
== Solution 2 (Elimination)==
  
−
<math>(B)</math> This is true. If Lewis got all the multiple questions right, he would've gotten an A. If he didn't get an A, then he didn't get all of them right, since Lewis would have gotten an A if he got the multiple-choice section all right.
+
Note: An A is usually 90%-100% of the questions correct.
 +
 +
<math>\textbf{(A)}</math> False. If Lewis did not receive an A, that does not mean he got ''all'' the multiple-choice questions wrong. For example, he might get 19/20 or 18/20, which still accounts for an A.
  
−
<math>(C)</math> False. There might be another section, call it the writing section. Then Lewis still might get an A if the writing section can carry his grade.
+
<math>\textbf{(B)}</math> True. If Lewis did not receive an A, then he must have got at least one wrong. Otherwise, Lewis would have gotten an A.
  
−
<math>(D)</math> False. He might've screwed up the multi-choice section but got extra credit to still level his grade up.
+
<math>\textbf{(C)}</math> False. Again, Lewis can get 19/20 or 18/20, which is still an A.  
  
−
<math>(E)</math> False. Lewis could've gotten 0 multi-choice correct but still receive an A.
+
<math>\textbf{(D)}</math> False. The above situation can happen.
 +
 +
<math>\textbf{(E)}</math> False. Lewis can get 17/20 or less but it is not an A.
  
 +
Therefore, our answer is <math>\boxed{\textbf{(B)}}.</math>
  
−
writing section??????? lol.
+
~minor edits by Terribleteeth
 +
 
 +
~edits by BakedPotato66
 +
 
 +
<!-- edits by MrThinker-->
  
 
==Video Solution==
 
==Video Solution==
 
https://youtu.be/pxg7CroAt20
 
https://youtu.be/pxg7CroAt20
 +
(TheBeautyofMath)
  
 +
==Video Solution 2==
 
https://youtu.be/7j5JigR0pbs
 
https://youtu.be/7j5JigR0pbs
  
 
~savannahsolver
 
~savannahsolver
 +
 +
==Video Solution 3==
 +
https://www.youtube.com/watch?v=iHlOCfubaq8
 +
 +
~Math4All999
  
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2017|ab=A|num-b=5|num-a=7}}
 
{{AMC10 box|year=2017|ab=A|num-b=5|num-a=7}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 09:46, 14 September 2024

Problem

Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which one of these statements necessarily follows logically?

$\textbf{(A)}\ \text{If Lewis did not receive an A, then he got all of the multiple choice questions wrong.}\\\textbf{(B)}\ \text{If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong.}\\\textbf{(C)}\ \text{If Lewis got at least one of the multiple choice questions wrong, then he did not receive an A. }\\\textbf{(D)}\ \text{If Lewis received an A, then he got all of the multiple choice questions right.}\\\textbf{(E)}\ \text{If Lewis received an A, then he got at least one of the multiple choice questions right.}$

Solution 1 (Using the Contrapositive)

Rewriting the given statement: "if someone got all the multiple choice questions right on the upcoming exam then he or she would receive an A on the exam." If that someone is Lewis the statement becomes: "if Lewis got all the multiple choice questions right, then he would receive an A on the exam."

The contrapositive: "If Lewis did not receive an A, then he did not get all of them right"

The contrapositive (in other words): "If Lewis did not get an A, then he got at least one of the multiple choice questions wrong".

B is also equivalent to the contrapositive of the original statement, which implies that it must be true, so the answer is $\boxed{\textbf{(B)}\text{ If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong.}}$.

  • Note that answer choice (B) is the contrapositive of the given statement. (That is, it has been negated as well as reversed.) We know that the contrapositive is always true if the given statement is true.

~minor edits by BakedPotato66

Solution 2 (Elimination)

Note: An A is usually 90%-100% of the questions correct.

$\textbf{(A)}$ False. If Lewis did not receive an A, that does not mean he got all the multiple-choice questions wrong. For example, he might get 19/20 or 18/20, which still accounts for an A.

$\textbf{(B)}$ True. If Lewis did not receive an A, then he must have got at least one wrong. Otherwise, Lewis would have gotten an A.

$\textbf{(C)}$ False. Again, Lewis can get 19/20 or 18/20, which is still an A.

$\textbf{(D)}$ False. The above situation can happen.

$\textbf{(E)}$ False. Lewis can get 17/20 or less but it is not an A.

Therefore, our answer is $\boxed{\textbf{(B)}}.$

~minor edits by Terribleteeth

~edits by BakedPotato66


Video Solution

https://youtu.be/pxg7CroAt20 (TheBeautyofMath)

Video Solution 2

https://youtu.be/7j5JigR0pbs

~savannahsolver

Video Solution 3

https://www.youtube.com/watch?v=iHlOCfubaq8

~Math4All999

See Also

2017 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 5 		Followed by
Problem 7
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=2017_AMC_10A_Problems/Problem_6&oldid=227173"