Difference between revisions of "2021 AMC 10B Problems/Problem 17"

Revision as of 04:03, 8 April 2021

Problem

Ravon, Oscar, Aditi, Tyrone, and Kim play a card game. Each person is given $2$ cards out of a set of $10$ cards numbered $1,2,3, \dots,10.$ The score of a player is the sum of the numbers of their cards. The scores of the players are as follows: Ravon-- $11,$ Oscar-- $4,$ Aditi-- $7,$ Tyrone-- $16,$ Kim-- $17.$ Which of the following statements is true?

$\textbf{(A) }\text{Ravon was given card 3.}$

$\textbf{(B) }\text{Aditi was given card 3.}$

$\textbf{(C) }\text{Ravon was given card 4.}$

$\textbf{(D) }\text{Aditi was given card 4.}$

$\textbf{(E) }\text{Tyrone was given card 7.}$

Solution 1

Oscar must be given 3 and 1, so we rule out $\textbf{(A) \ }$ and $\textbf{(B) \ }$ . If Tyrone had card 7, then he would also have card 9, and then Kim must have 10 and 7 so we rule out $\textbf{(E) \ }$ . If Aditi was given card 4, then she would have card 3, which Oscar already had. So the answer is $\boxed{ \textbf{(C) }\text{Ravon was given card 4.}}$

~smarty101 and smartypantsno_3

Solution 2

Oscar must be given 3 and 1. Aditi cannot be given 3 or 1, so she must have 2 and 5. Similarly, Ravon cannot be given 1, 2, 3, or 5, so he must have 4 and 7, and the answer is $\boxed{ \textbf{(C) }\text{Ravon was given card 4.}}$ .

-SmileKat32

Solution 3 (Comprehensive but Unnecessary)

By observations, we consider the scores from lowest to highest. We make the following logical deduction:

$\begin{align*} \text{Oscar's score is 4.} &\Longrightarrow \text{Oscar is given cards 1 and 3.} \\ &\Longrightarrow \text{Aditi is given cards 2 and 5.} \\ &\Longrightarrow \text{Ravon is given cards 4 and 7.} \hspace{13mm} (*) \\ &\Longrightarrow \text{Tyrone is given cards 6 and 10.} \\ &\Longrightarrow \text{Kim is given cards 8 and 9.} \end{align*}$

Therefore, the answer is $\boxed{\textbf{(C) }\text{Ravon was given card 4.}}$

Of course, if we read the answer choices sooner, then we can stop at $(*)$ and pick $\textbf{(C)}.$

~MRENTHUSIASM

Video Solution by OmegaLearn (Using logical deduction)

https://youtu.be/zO0EuKPXuT0

~ pi_is_3.14

Video Solution by TheBeautyofMath

https://youtu.be/FV9AnyERgJQ?t=284

~IceMatrix

@@ Line 28: / Line 28: @@
 <cmath>\begin{align*}
-\text{Oscar's score is 4.} &\Longrightarrow \text{Oscar was given cards 1 and 3.} \\
+\text{Oscar's score is 4.} &\Longrightarrow \text{Oscar is given cards 1 and 3.} \\
-&\Longrightarrow \text{Aditi was given cards 2 and 5.} \\
+&\Longrightarrow \text{Aditi is given cards 2 and 5.} \\
-&\Longrightarrow \text{Ravon was given cards 4 and 7.} \hspace{13mm} (*) \\
+&\Longrightarrow \text{Ravon is given cards 4 and 7.} \hspace{13mm} (*) \\
-&\Longrightarrow \text{Tyrone was given cards 6 and 10.} \\
+&\Longrightarrow \text{Tyrone is given cards 6 and 10.} \\
-&\Longrightarrow \text{Kim was given cards 8 and 9.}
+&\Longrightarrow \text{Kim is given cards 8 and 9.}
 \end{align*}</cmath>

2021 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 16	Followed by Problem 18
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