Question: In an opera theater. there are 300 seats available. When all the seats in the theater are sold out, the price of each ticket is $60. For every $1 increase in the price of the ticket, the number of seats sold decreases by 2. How much should the theater owner charge for each seat to make maximum profit?

(A) $87
(B) $95
(C) $105
(D) $120
(E) $127

“In an opera theater. there are 300 seats available” - is a topic that is covered in the quantitative reasoning section of the GMAT. To successfully execute the GMAT Problem Solving questions, a student must possess a wide range of qualitative skills. The entire GMAT Quant section consists of 31 questions. The problem-solving section of the GMAT Quant topics requires the solution of calculative mathematical problems.

Solutions and Explanation

Approach Solution : 1

As Brent demonstrated above, we must maximize the revenue provided by (60 + x)* (300 - 2x)
To enlarge, 18000 + 180x - 2x^2
Recognize that this quadratic has a negative x^2 coefficient and that it represents a parabola with a downward slope. It will be at its highest value at x = -b/2a.
x = -180/2*(-2) = 45
For each seat, the owner should therefore charge 60 + x = 60 + 45 = $105 in total.

Correct Answer: (C)

Approach Solution : 2

Knowing the concept of "function derivative" makes it simple to find the answer to this problem.
Simply take the first-level derivative to find the maximum of the defined function.
Then, the following problem can be resolved in under a minute.
Revenue will be calculated as (60+x)*(300-2x)=60*300+180x-2x2
The first order derivative is as follows, x=45 or -4x+180=0.
The New price is 45+60 = 105

Correct Answer: (C)

Approach Solution : 3

Maxima for a quadratic equation can be used to determine the answer to this question.
The equation will be as follows based on the conditions stated in the question.
Let x represent the increase in the ticket's cost.
Total income: (60 + x) (300 - x) −2x^2 + 180x+ 18000
Given that this is an inverted parabola equation,
its maximum value will be at -b/2a ==> -180/2*(-2)==> 45
The cost of the ticket will therefore be 60 + 45 = 105.

Correct Answer: C

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