Question

At a college football game, \(\dfrac{4}{5}\) of the seats in the lower deck of the stadium were sold. If \(\dfrac{1}{4}\) of all the seating in the stadium is located in the lower deck, and if \(\dfrac{2}{3}\) of all the seats in the stadium were sold, then what fraction of the unsold seats in the stadium was in the lower deck?

• A. \(\dfrac{3}{20}\)
• B. \(\dfrac{1}{6}\)
• C. \(\dfrac{1}{5}\)
• D. \(\dfrac{1}{3}\)

Hint:

Work in terms of total seats x: find lower deck unsold and total unsold separately, then divide.

Answer 1

The Correct Option is A

Step 1: Set up variables for the stadium.
Let the total number of seats be \(x\). Since \(\dfrac{1}{4}\) of all seats are in the lower deck, the lower deck has \(\dfrac{x}{4}\) seats, and the upper deck has \(\dfrac{3x}{4}\) seats.

Step 2: Find sold and unsold seats in the lower deck.
\(\dfrac{4}{5}\) of the lower deck seats were sold, so sold-in-lower-deck \(= \dfrac{4}{5}\times\dfrac{x}{4} = \dfrac{x}{5}\).
Unsold in the lower deck \(= \dfrac{x}{4} - \dfrac{x}{5} = \dfrac{5x-4x}{20} = \dfrac{x}{20}\).

Step 3: Find the total unsold seats in the whole stadium.
\(\dfrac{2}{3}\) of all seats were sold, so total sold \(= \dfrac{2x}{3}\), and total unsold \(= x - \dfrac{2x}{3} = \dfrac{x}{3}\).

Step 4: Compute the required fraction.
\[ \frac{x/20}{x/3} = \frac{3}{20} \]

Final Answer:
The fraction of unsold seats belonging to the lower deck is \(\dfrac{3}{20}\), so option A is correct. \[ \boxed{\frac{3}{20}} \]