# Problem of the Week

Problem D and Solution

How Many in the House?

## Problem

The POTW Theatre has four levels of seating: gold, silver, red, and
black.

One night, the manager of the theatre was asked how many patrons are
in the theatre. The manager replied that \(\frac{1}{6}\) of the patrons in the theatre
that night are in the gold seating, \(\frac{1}{4}\) of the patrons are in either
the red seating or the black seating, there are three times as many
patrons in the the silver seating as in the red seating, and there are
\(138\) patrons in the black
seating.

How many patrons were in the theatre that night?

## Solution

Let \(n\) be the total number of
patrons in the theatre that night.

Let \(g\) be the number of patrons in
the gold seating, \(s\) be the number
of patrons in the silver seating, \(r\)
be the number of patrons in the red seating, and \(b\) be the number of patrons in the black
seating.

Therefore, \(n = g + s + r +
b\).

Since \(\frac{1}{6}\) of the patrons
in the theatre are in the gold seating, \(g =
\frac{1}{6}n\).

Since \(\frac{1}{4}\) of the patrons
are either in the black seating or the red seating,

\(r + b = \frac{1}{4}n\).

It is given that \(b = 138\).
Therefore, \(r + b = \frac{1}{4}n\)
becomes \(r + 138 = \frac{1}{4}n\), or
\(r = \frac{1}{4}n-138\).

Since there are three times as many patrons in the silver seating as
patrons in the red seating, \(s = 3r = 3
(\frac{1}{4}n-138)\).

Substituting these expressions for \(g\), \(r\), and \(s\), and the value for \(b\) into \(n = g
+ s + r + b\), we have \[\begin{aligned}
n&= \left(\frac{1}{6}n\right) + 3 \left(\frac{1}{4}n-138\right) +
\left(\frac{1}{4}n-138\right) + 138\\
n&= \frac{1}{6}n + \frac{3}{4}n-414 + \frac{1}{4}n-138 + 138\\
n&= \frac{1}{6}n + n-414\\
n&= \frac{7}{6}n-414\\
\frac{1}{6}n &= 414\\
n &= 2484
\end{aligned}\]

Therefore, there were \(2484\)
patrons in the theatre that night.

Although it is not required, we could further determine that the
number of patrons in the silver seating is \(1449\), the number of patrons in the gold
seating is \(414\), and the number of
patrons in the red seating is \(483\).
We could then use these numbers to verify the given information.