Question

When it was found that 150 more tickets for the school play were sold than the seating capacity of the auditorium. It was decided to have two performances. if the total number of tickets sold was equal to the total number who attended and if the auditorium was \(\frac{2}{3}\) full for each of the two performances, what is the seating capacity of the auditorium?

• A. 100
• B. 200
• C. 225
• D. 300
• E. 450

Hint:

Break down complex word problems sentence by sentence. Assign variables to unknown quantities and translate each piece of information into a mathematical equation or expression. This systematic approach helps prevent confusion and errors.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This is a word problem that requires translating the given information into a system of algebraic equations to solve for an unknown variable, the seating capacity.
Step 2: Key Formula or Approach:
1. Define a variable for the seating capacity (e.g., \(C\)).
2. Create an expression for the total number of tickets sold in terms of \(C\).
3. Create an expression for the total attendance across both performances in terms of \(C\).
4. Equate these two expressions and solve for \(C\).
Step 3: Detailed Explanation:
Let \(C\) be the seating capacity of the auditorium.
Let \(T\) be the total number of tickets sold.
From the problem statement, "150 more tickets... were sold than the seating capacity":
\[ T = C + 150 \] There were two performances, and for each one, the auditorium was \(\frac{2}{3}\) full. So, the number of people who attended each performance is:
\[ \text{Attendance per performance} = \frac{2}{3} \times C \] The total number of people who attended the two performances is:
\[ \text{Total Attendance} = 2 \times \left(\frac{2}{3} C\right) = \frac{4}{3} C \] The problem states that "the total number of tickets sold was equal to the total number who attended". Therefore:
\[ T = \text{Total Attendance} \] We can now set our two expressions for \(T\) and Total Attendance equal to each other:
\[ C + 150 = \frac{4}{3} C \] To solve for \(C\), first get all the \(C\) terms on one side. Subtract \(C\) from both sides:
\[ 150 = \frac{4}{3} C - C \] \[ 150 = \frac{4}{3} C - \frac{3}{3} C \] \[ 150 = \frac{1}{3} C \] Multiply both sides by 3 to find \(C\):
\[ C = 150 \times 3 = 450 \] Step 4: Final Answer:
The seating capacity of the auditorium is 450.