## Waiting in Line

In a movie theater line, $x$ people are behind Mark, who is $y$ places in front of Sam. If there are $z$ people in front of Sam, how many people are in line?
Source: NCTM Mathematics Teacher, October 2006

Solution

The figure illustrates the specifics of the line. The dots represent the people waiting in line starting from the left and Mark and Sam are randomly placed in the line.
$x$ = number of people behind Mark
$y$ = Mark is $y$ places in front of Sam
$z$ = number of people in front of Sam
Let $u$ represent Sam and the number of people behind him.
Number of people in line = $z+u$
$u=x-y+1$
$z+u=z+x-y+1$

Answer: $z+x-y+1$

Alternative solution
$z$ = number of people in front of Sam excluding Sam
$x-y$ = number of people behind Sam excluding Sam
Total number of people waiting in line = $z+x-y+1$.

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## About mvtrinh

Retired high school math teacher.
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