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Difference between revisions of "2015 AMC 12B Problems/Problem 4"

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(Undo revision 205386 by Beishexi (talk))
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===Note===
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This problem is also problem number 5 on the 2015 AMC 10B, just with different names.
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[https://artofproblemsolving.com/wiki/index.php/2015_AMC_10B_Problems/Problem_5 This is the link.]
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==Problem==
 
==Problem==
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Lian, Marzuq, Rafsan, Arabi, Nabil, and Rahul were in a 12-person race with 6 other people. Nabil finished 6 places ahead of Marzuq. Arabi finished 1 place behind Rafsan. Lian finished 2 places behind Marzuq. Rafsan finished 2 places behind Rahul. Rahul finished 1 place behind Nabil. Arabi finished in 6th place. Who finished in 8th place?
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Lian, Marzuq, Rafsan, Arabi, Nabeel, and Rahul were in a 12-person race with 6 other people. Nabeel finished 6 places ahead of Marzuq. Arabi finished 1 place behind Rafsan. Lian finished 2 places behind Marzuq. Rafsan finished 2 places behind Rahul. Rahul finished 1 place behind Nabeel. Arabi finished in 6th place. Who finished in 8th place?
  
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<math>\textbf{(A)}\; \text{Lian} \qquad\textbf{(B)}\; \text{Marzuq} \qquad\textbf{(C)}\; \text{Rafsan} \qquad\textbf{(D)}\; \text{Nabil} \qquad\textbf{(E)}\; \text{Rahul}</math>
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<math>\textbf{(A)}\; \text{Lian} \qquad\textbf{(B)}\; \text{Marzuq} \qquad\textbf{(C)}\; \text{Rafsan} \qquad\textbf{(D)}\; \text{Nabeel} \qquad\textbf{(E)}\; \text{Rahul}</math>
  
 
==Solution 1==
 
==Solution 1==
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Let <math>-</math> denote any of the 6 racers not named. Then the correct order following all the logic looks like:
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Let --- denote any of the 6 racers not named. Then the correct order looks like this:
  
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<cmath>-, \text{Nabil}, \text{Rahul}, -, \text{Rafsan}, \text{Arabi}, -, \text{Marzuq}, -, \text{Lian}, -, -</cmath>
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<cmath>-, \text{Nabeel}, \text{Rahul}, -, \text{Rafsan}, \text{Arabi}, -, \text{Marzuq}, -, \text{Lian}, -, -</cmath>
  
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Clearly the 8th place runner is <math>\fbox{\textbf{(B)}\; \text{Marzuq}}</math>.
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Thus the 8th place runner is <math>\boxed{\textbf{(B)}\; \text{Marzuq}}</math>.
  
 
==Solution 2==
 
==Solution 2==
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We can list these out vertically to ensure clarity, starting with Marta and working from there.
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We list the people out vertically for clarity, starting with Marzuq and working from there.
  
  
 
<cmath>1 - </cmath>
 
<cmath>1 - </cmath>
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<cmath>2 Nabil </cmath>
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<cmath>2 \text{Nabeel} </cmath>
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<cmath>3 Rahul </cmath>
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<cmath>3 \text{Rahul} </cmath>
 
<cmath>4 - </cmath>
 
<cmath>4 - </cmath>
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<cmath>5 Rafsan </cmath>
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<cmath>5 \text{Rafsan} </cmath>
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<cmath>6 Arabi </cmath>
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<cmath>6 \text{Arabi} </cmath>
 
<cmath>7 - </cmath>
 
<cmath>7 - </cmath>
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<cmath>8 Marzuq </cmath>
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<cmath>8 \text{Marzuq} </cmath>
  
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Thus our answer is <math>\fbox{\textbf{(B)}\; \text{Marzuq}}</math>.
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Thus our answer is <math>\boxed{\textbf{(B)}\; \text{Marzuq}}</math>.
  
 
==See Also==
 
==See Also==
 
{{AMC12 box|year=2015|ab=B|num-a=5|num-b=3}}
 
{{AMC12 box|year=2015|ab=B|num-a=5|num-b=3}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 13:57, 23 November 2023

Note

This problem is also problem number 5 on the 2015 AMC 10B, just with different names. This is the link.

Problem

Lian, Marzuq, Rafsan, Arabi, Nabeel, and Rahul were in a 12-person race with 6 other people. Nabeel finished 6 places ahead of Marzuq. Arabi finished 1 place behind Rafsan. Lian finished 2 places behind Marzuq. Rafsan finished 2 places behind Rahul. Rahul finished 1 place behind Nabeel. Arabi finished in 6th place. Who finished in 8th place?

$\textbf{(A)}\; \text{Lian} \qquad\textbf{(B)}\; \text{Marzuq} \qquad\textbf{(C)}\; \text{Rafsan} \qquad\textbf{(D)}\; \text{Nabeel} \qquad\textbf{(E)}\; \text{Rahul}$

Solution 1

Let --- denote any of the 6 racers not named. Then the correct order looks like this:

\[-, \text{Nabeel}, \text{Rahul}, -, \text{Rafsan}, \text{Arabi}, -, \text{Marzuq}, -, \text{Lian}, -, -\]

Thus the 8th place runner is $\boxed{\textbf{(B)}\; \text{Marzuq}}$.

Solution 2

We list the people out vertically for clarity, starting with Marzuq and working from there.


\[1 -\] \[2 \text{Nabeel}\] \[3 \text{Rahul}\] \[4 -\] \[5 \text{Rafsan}\] \[6 \text{Arabi}\] \[7 -\] \[8 \text{Marzuq}\]

Thus our answer is $\boxed{\textbf{(B)}\; \text{Marzuq}}$.

See Also

2015 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 3 		Followed by
Problem 5
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 12 Problems and Solutions

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