2016 AMC 10A Problems/Problem 13
Problem
Five friends sat in a movie theater in a row containing seats, numbered
to
from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?
Solution 1
Bash: we see that the following configuration works.
Bea - Ada - Ceci - Dee - Edie
After moving, it becomes
Ada - Ceci - Bea - Edie - Dee.
Thus, Ada was in seat .
Solution 2
Process of elimination of possible configurations.
Let's say that Ada=, Bea=
, Ceci=
, Dee=
, and Edie=
.
Since moved more to the right than
did left, this implies that
was in a LEFT end seat originally:
This is affirmed because , which there is no new seats uncovered. So
are restricted to the same
seats.
Thus, it must be
, and more specifically:
So , Ada, was originally in seat
.
Solution 3
The seats are numbered 1 through 5, so let each letter () correspond to a number.
Let a move to the left be subtraction and a move to the right be addition.
We know that . After everyone moves around, however, our equation looks like
because
and
switched seats,
moved two to the right, and
moved 1 to the left.
For this equation to be true, has to be -1, meaning
moves 1 left from her original seat. Since
is now sitting in a corner seat, the only possible option for the original placement of
is in seat number
.
See Also
2016 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
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 10 Problems and Solutions |
2016 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 9 |
Followed by Problem 11 |
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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