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

(Solution)
(Solution)
Line 4: Line 4:
 
<math>\textbf{(A)}\; \text{David} \qquad\textbf{(B)}\; \text{Hikmet} \qquad\textbf{(C)}\; \text{Jack} \qquad\textbf{(D)}\; \text{Rand} \qquad\textbf{(E)}\; \text{Todd}</math>
 
<math>\textbf{(A)}\; \text{David} \qquad\textbf{(B)}\; \text{Hikmet} \qquad\textbf{(C)}\; \text{Jack} \qquad\textbf{(D)}\; \text{Rand} \qquad\textbf{(E)}\; \text{Todd}</math>
  
==Solution==
+
==Solution 1==
 
Let <math>-</math> denote any of the 6 racers not named. Then the correct order following all the logic looks like:
 
Let <math>-</math> denote any of the 6 racers not named. Then the correct order following all the logic looks like:
  
Line 10: Line 10:
  
 
Clearly the 8th place runner is <math>\fbox{\textbf{(B)}\; \text{Hikmet}}</math>.
 
Clearly the 8th place runner is <math>\fbox{\textbf{(B)}\; \text{Hikmet}}</math>.
 +
 +
==Solution 2==
 +
We can list these out vertically to ensure clarity, starting with Marta and working from there.
 +
 +
 +
<cmath>1 - </cmath>
 +
<cmath>2 R </cmath>
 +
<cmath>3 T </cmath>
 +
<cmath>4 - </cmath>
 +
<cmath>5 J </cmath>
 +
<cmath>6 M </cmath>
 +
<cmath>7 - </cmath>
 +
<cmath>8 H </cmath>
 +
 +
Thus our answer is <math>\fbox{\textbf{(B)}\; \text{Hikmet}}</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}}

Revision as of 23:40, 20 January 2018

Problem

David, Hikmet, Jack, Marta, Rand, and Todd were in a 12-person race with 6 other people. Rand finished 6 places ahead of Hikmet. Marta finished 1 place behind Jack. David finished 2 places behind Hikmet. Jack finished 2 places behind Todd. Todd finished 1 place behind Rand. Marta finished in 6th place. Who finished in 8th place?

$\textbf{(A)}\; \text{David} \qquad\textbf{(B)}\; \text{Hikmet} \qquad\textbf{(C)}\; \text{Jack} \qquad\textbf{(D)}\; \text{Rand} \qquad\textbf{(E)}\; \text{Todd}$

Solution 1

Let $-$ denote any of the 6 racers not named. Then the correct order following all the logic looks like:

\[-, \text{Rand}, \text{Todd}, -, \text{Jack}, \text{Marta}, -, \text{Hikmet}, -, \text{David}, -, -\]

Clearly the 8th place runner is $\fbox{\textbf{(B)}\; \text{Hikmet}}$.

Solution 2

We can list these out vertically to ensure clarity, starting with Marta and working from there.


\[1 -\] \[2 R\] \[3 T\] \[4 -\] \[5 J\] \[6 M\] \[7 -\] \[8 H\]

Thus our answer is $\fbox{\textbf{(B)}\; \text{Hikmet}}$.

See Also

2015 AMC 12B (ProblemsAnswer KeyResources)
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