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 "2015 AMC 12B Problems/Problem 4"

(→‎Problem)
(Undo revision 205386 by Beishexi (talk))
(Tag: Undo)
 
(18 intermediate revisions by 4 users not shown)
Line 1: Line 1:
 +
===Note===
 +
This problem is also problem number 5 on the 2015 AMC 10B, just with different names.
 +
[https://artofproblemsolving.com/wiki/index.php/2015_AMC_10B_Problems/Problem_5 This is the link.]
 +
 
==Problem==
 
==Problem==
−
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 Rafsab. 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?
+
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?
  
−
<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>
+
<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==
−
Let <math>-</math> denote any of the 6 racers not named. Then the correct order following all the logic looks like:
+
Let --- denote any of the 6 racers not named. Then the correct order looks like this:
  
−
<cmath>-, \text{Rand}, \text{Todd}, -, \text{Jack}, \text{Marta}, -, \text{Hikmet}, -, \text{David}, -, -</cmath>
+
<cmath>-, \text{Nabeel}, \text{Rahul}, -, \text{Rafsan}, \text{Arabi}, -, \text{Marzuq}, -, \text{Lian}, -, -</cmath>
  
−
Clearly the 8th place runner is <math>\fbox{\textbf{(B)}\; \text{Hikmet}}</math>.
+
Thus the 8th place runner is <math>\boxed{\textbf{(B)}\; \text{Marzuq}}</math>.
  
 
==Solution 2==
 
==Solution 2==
−
We can list these out vertically to ensure clarity, starting with Marta and working from there.
+
We list the people out vertically for clarity, starting with Marzuq and working from there.
  
  
 
<cmath>1 - </cmath>
 
<cmath>1 - </cmath>
−
<cmath>2 R </cmath>
+
<cmath>2 \text{Nabeel} </cmath>
−
<cmath>3 T </cmath>
+
<cmath>3 \text{Rahul} </cmath>
 
<cmath>4 - </cmath>
 
<cmath>4 - </cmath>
−
<cmath>5 J </cmath>
+
<cmath>5 \text{Rafsan} </cmath>
−
<cmath>6 M </cmath>
+
<cmath>6 \text{Arabi} </cmath>
 
<cmath>7 - </cmath>
 
<cmath>7 - </cmath>
−
<cmath>8 H </cmath>
+
<cmath>8 \text{Marzuq} </cmath>
  
−
Thus our answer is <math>\fbox{\textbf{(B)}\; \text{Hikmet}}</math>.
+
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

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=2015_AMC_12B_Problems/Problem_4&oldid=205389"