Difference between revisions of "2015 AMC 12B Problems/Problem 4"
(→Solution 2) |
(Tag: Undo) |
||
| (12 intermediate revisions by 3 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, 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? | 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? | ||
| Line 5: | Line 9: | ||
==Solution 1== | ==Solution 1== | ||
| − | Let | + | Let --- denote any of the 6 racers not named. Then the correct order looks like this: |
<cmath>-, \text{Nabeel}, \text{Rahul}, -, \text{Rafsan}, \text{Arabi}, -, \text{Marzuq}, -, \text{Lian}, -, -</cmath> | <cmath>-, \text{Nabeel}, \text{Rahul}, -, \text{Rafsan}, \text{Arabi}, -, \text{Marzuq}, -, \text{Lian}, -, -</cmath> | ||
| − | + | Thus the 8th place runner is <math>\boxed{\textbf{(B)}\; \text{Marzuq}}</math>. | |
==Solution 2== | ==Solution 2== | ||
| − | We | + | We list the people out vertically for clarity, starting with Marzuq and working from there. |
<cmath>1 - </cmath> | <cmath>1 - </cmath> | ||
| − | <cmath>2 Nabeel </cmath> | + | <cmath>2 \text{Nabeel} </cmath> |
| − | <cmath>3 Rahul </cmath> | + | <cmath>3 \text{Rahul} </cmath> |
<cmath>4 - </cmath> | <cmath>4 - </cmath> | ||
| − | <cmath>5 Rafsan </cmath> | + | <cmath>5 \text{Rafsan} </cmath> |
| − | <cmath>6 Arabi </cmath> | + | <cmath>6 \text{Arabi} </cmath> |
<cmath>7 - </cmath> | <cmath>7 - </cmath> | ||
| − | <cmath>8 Marzuq </cmath> | + | <cmath>8 \text{Marzuq} </cmath> |
| − | Thus our answer is <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
Contents
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?
Solution 1
Let --- denote any of the 6 racers not named. Then the correct order looks like this:
![\[-, \text{Nabeel}, \text{Rahul}, -, \text{Rafsan}, \text{Arabi}, -, \text{Marzuq}, -, \text{Lian}, -, -\]](http://latex.artofproblemsolving.com/e/c/3/ec364be22d6d93081d6a952a2a475fea025d3694.png)
Thus the 8th place runner is 
.
Solution 2
We list the people out vertically for clarity, starting with Marzuq and working from there.
![\[1 -\]](http://latex.artofproblemsolving.com/1/0/5/105aa601b74d255fdec68cea63a561c4634d492d.png)
![\[2 \text{Nabeel}\]](http://latex.artofproblemsolving.com/9/c/a/9ca6eb51f0291777cc58c5cdc75bf68be108dd8e.png)
![\[3 \text{Rahul}\]](http://latex.artofproblemsolving.com/c/8/2/c82430f899e43f0cad590dcf4eafdf1108b223d1.png)
![\[4 -\]](http://latex.artofproblemsolving.com/3/4/d/34dde0e8318825de00c6c696b1c5e715454d4420.png)
![\[5 \text{Rafsan}\]](http://latex.artofproblemsolving.com/b/d/b/bdb4b9bf130d25b99559442531adf9a9abefb547.png)
![\[6 \text{Arabi}\]](http://latex.artofproblemsolving.com/2/4/f/24f1a33d2d70f22868b48630b47539e4b1a81b2f.png)
![\[7 -\]](http://latex.artofproblemsolving.com/1/4/f/14fa5dbaae18d5eaff20f23b696f5eac2a6254be.png)
![\[8 \text{Marzuq}\]](http://latex.artofproblemsolving.com/0/9/5/0953657fbaedfaf8e896893c01bb26f7173899c9.png)
Thus our answer is .
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. 
