Difference between revisions of "2001 AMC 8 Problems/Problem 20"

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==Solution==
 
==Solution==
Since the only other score Quay knows is Kaleana's, and he knows that two of them have the same score, Quay and Kaleana must have the same score, and <math> K=Q </math>. Marty knows that he didn't get the lowest score, and the only other score he knows is Kaleana's, so Marty must know that Kaleana must have a lower score than him, and <math> M>K </math>. Finally, Shana knows that she didn't get the highest score, and the only other score she knows is Kaleana's, so Shana must know that Kaleana must have a higher score than her, and <math> S<K </math>. Putting these together and substituting <math> Q </math> for <math> K </math>, we have <math> S<Q<M </math>, and from least to greatest this is <math> \text{S, Q, M}, \boxed{\text{A}} </math>.
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Since the only other score Quay knows is Kaleana's, and he knows that two of them have the same score, Quay and Kaleana must have the same score, and <math> K=Q </math>. Marty knows that he didn't get the lowest score, and the only other score he knows is Kaleana's, so Marty must know that Kaleana must have a lower score than him, and <math> M>K </math>. Finally, Shana knows that she didn't get the highest score, and the only other score she knows is Kaleana's, so Shana must know that Kaleana must have a higher score than her, and <math> S<K </math>. Putting these together and substituting <math> Q </math> for <math> K </math>, we have <math> S<Q<M </math>, and from least to greatest this is <math> \text{S, Q, M}</math>, therefore the answer is <math>\boxed{\text{A}} </math>.
  
 
==See Also==
 
==See Also==
 
{{AMC8 box|year=2001|num-b=19|num-a=21}}
 
{{AMC8 box|year=2001|num-b=19|num-a=21}}
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{{MAA Notice}}

Latest revision as of 13:49, 31 October 2015

Problem

Kaleana shows her test score to Quay, Marty and Shana, but the others keep theirs hidden. Quay thinks, "At least two of us have the same score." Marty thinks, "I didn't get the lowest score." Shana thinks, "I didn't get the highest score." List the scores from lowest to highest for Marty (M), Quay (Q) and Shana (S).

$\text{(A)}\ \text{S,Q,M} \qquad \text{(B)}\ \text{Q,M,S} \qquad \text{(C)}\ \text{Q,S,M} \qquad \text{(D)}\ \text{M,S,Q} \qquad \text{(E)}\ \text{S,M,Q}$

Solution

Since the only other score Quay knows is Kaleana's, and he knows that two of them have the same score, Quay and Kaleana must have the same score, and $K=Q$. Marty knows that he didn't get the lowest score, and the only other score he knows is Kaleana's, so Marty must know that Kaleana must have a lower score than him, and $M>K$. Finally, Shana knows that she didn't get the highest score, and the only other score she knows is Kaleana's, so Shana must know that Kaleana must have a higher score than her, and $S<K$. Putting these together and substituting $Q$ for $K$, we have $S<Q<M$, and from least to greatest this is $\text{S, Q, M}$, therefore the answer is $\boxed{\text{A}}$.

See Also

2001 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 19
Followed by
Problem 21
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 AJHSME/AMC 8 Problems and Solutions

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