Difference between revisions of "2021 AMC 10B Problems/Problem 17"
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Ravon, Oscar, Aditi, Tyrone, and Kim play a card game. Each person is given 2 cards out of a set of 10 cards numbered <math>1,2,3, \dots,10.</math> 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? | Ravon, Oscar, Aditi, Tyrone, and Kim play a card game. Each person is given 2 cards out of a set of 10 cards numbered <math>1,2,3, \dots,10.</math> 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? | ||
− | <math> \textbf{(A) | + | <math> \textbf{(A) }\text{Ravon was given card 3.}</math> |
− | + | <math>\textbf{(B) }\text{Aditi was given card 3.}</math> | |
− | Oscar must be given 3 and 1, so we rule out <math>\textbf{(A) \ }</math> and <math>\textbf{(B) \ }</math>. If Tyrone had card 7, then he would also have card 9, and then Kim must have 10 and 7 so we rule out <math>\textbf{(E) \ }</math>. If Aditi was given card 4, then she would have card 3, which Oscar already had. So the answer is <math>\boxed{ \textbf{(C) | + | <math>\textbf{(C) }\text{Ravon was given card 4.}</math> |
+ | |||
+ | <math>\textbf{(D) }\text{Aditi was given card 4.}</math> | ||
+ | |||
+ | <math>\textbf{(E) }\text{Tyrone was given card 7.}</math> | ||
+ | |||
+ | ==Solution 1== | ||
+ | |||
+ | Oscar must be given 3 and 1, so we rule out <math>\textbf{(A) \ }</math> and <math>\textbf{(B) \ }</math>. If Tyrone had card 7, then he would also have card 9, and then Kim must have 10 and 7 so we rule out <math>\textbf{(E) \ }</math>. If Aditi was given card 4, then she would have card 3, which Oscar already had. So the answer is <math>\boxed{ \textbf{(C) }\text{Ravon was given card 4.}}</math> | ||
~smarty101 and smartypantsno_3 | ~smarty101 and smartypantsno_3 | ||
==Solution 2== | ==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 <math>\boxed{ \textbf{(C) | + | 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 <math>\boxed{ \textbf{(C) }\text{Ravon was given card 4.}}</math>. |
-SmileKat32 | -SmileKat32 | ||
+ | |||
+ | ==Solution 3 (Comprehensive)== | ||
+ | Using observations, we consider the scores from lowest to highest. We make the following logical deduction: | ||
+ | |||
+ | <cmath>\begin{align*} | ||
+ | \text{Oscar's score is 4.} &\Longrightarrow \text{Oscar was given cards 1 and 3.} \ | ||
+ | &\Longrightarrow \text{Aditi was given cards 2 and 5.} \ | ||
+ | &\Longrightarrow \text{Ravon was given cards 4 and 7.} \ | ||
+ | &\Longrightarrow \text{Tyrone was given cards 6 and 10.} \ | ||
+ | &\Longrightarrow \text{Kim was given cards 8 and 9.} | ||
+ | \end{align*}</cmath> | ||
+ | |||
+ | Therefore, the answer is <math>\boxed{\textbf{(C) }\text{Ravon was given card 4.}}</math> | ||
+ | |||
+ | Of course, if we look at the answer choices earlier, then we can stop after line 3 of the block of logical statements. | ||
+ | |||
+ | ~MRENTHUSIASM | ||
== Video Solution by OmegaLearn (Using logical deduction) == | == Video Solution by OmegaLearn (Using logical deduction) == |
Revision as of 16:08, 5 March 2021
Contents
[hide]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 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?
Solution 1
Oscar must be given 3 and 1, so we rule out and . If Tyrone had card 7, then he would also have card 9, and then Kim must have 10 and 7 so we rule out . If Aditi was given card 4, then she would have card 3, which Oscar already had. So the answer is
~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 .
-SmileKat32
Solution 3 (Comprehensive)
Using observations, we consider the scores from lowest to highest. We make the following logical deduction:
Therefore, the answer is
Of course, if we look at the answer choices earlier, then we can stop after line 3 of the block of logical statements.
~MRENTHUSIASM
Video Solution by OmegaLearn (Using logical deduction)
~ pi_is_3.14
Video Solution by TheBeautyofMath
https://youtu.be/FV9AnyERgJQ?t=284
~IceMatrix
See Also
2021 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
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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