In an opera theater, there are 300 seats available.

Expert replies

gmat_winter: Senior | Next Rank: 100 Posts; Posts: 34; Joined: Tue Oct 14, 2014 8:23 am; Thanked: 1 times

by gmat_winter » Fri Apr 03, 2015 11:24 am

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 the maximum profit?

A)87 $

b)95 $

c)105 $

d)120 $

e)127 $

oa

c

Like this post

Source: Beat The GMAT — Problem Solving | Fri Apr 03, 2015 11:24 am

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Fri Apr 03, 2015 6:07 pm

gmat_winter wrote: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 the maximum profit?

A)87 $

b)95 $

c)105 $

d)120 $

e)127 $

We can PLUG IN THE ANSWERS, which represent the price required to maximize the total revenue.
Since 2 fewer seats are sold for every $1 increase in the ticket price, the seat decrease is twice the price increase.
Test the three middle answer choices and LOOK FOR A PATTERN.

Answer choice D: 120
Price increase = 120-60 = 60.
Seat decrease = 2*60 = 120.
Sold seats = 300-120 = 180.
Total revenue = (sold seats)I(ticket price) = 180*120 = 21,600.

Answer choice C: 105
Price increase = 105-60 = 45.
Seat decrease = 2*45 = 90.
Sold seats = 300-90 = 210.
Total revenue = (sold seats)(ticket price) = 210*105 = 22,050.

Answer choice B: 95
Price increase = 95-60 = 35.
Seat decrease = 2*35 = 70.
Sold seats = 300-70 = 230.
Total revenue = (sold seats)(ticket price) = 230*95 = 21,850.

The resulting totals -- 21600, 22050, 21850 -- imply that the total revenue is maximized by answer choice C.

The correct answer is C.

Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3

Like this post

GMAT/MBA Expert

Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Sat Apr 04, 2015 7:02 am

gmat_winter wrote: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 the maximum profit?

A)87 $
b)95 $
c)105 $
d)120 $
e)127 $

I'm not a big fan of this question. The only approach (without using techniques that are out of scope for the GMAT) is TEST THE ANSWERS, and the great thing about most GMAT math problems is that they can be solved using more than 1 approach.

To see what I mean, consider this algebraic approach:

Let x = the number of dollars added to the price of the ticket.
So, the new ticket price = 60+x dollars.

For every $1 increase in the price of the ticket, the number of seats sold decreases by 2.
So, 300-2x = the number of seats sold.

Revenue = (ticket price)(# seats sold)
So, revenue = (60+x)(300-2x)
Expand to get: revenue = 18,000 + 300x - 120n - 2xÂ²
Simplify and rearrange to get: revenue = -2xÂ² + 180x + 18,000
So our goal is to find the value of x that MAXIMIZES the value of -2xÂ² + 180x + 18,000

1) One approach is to use calculus and find the derivative of -2xÂ² + 180x + 18,000 and blah blah blah...(OUT OF SCOPE!)

2) Another approach is to recognize that y = -2xÂ² + 180x + 18,000 represents a downward parabola AND the coordinates of the vertex (at the TOP of the parabola) represent the information that MAXIMIZES profits.
So, let's use approach #2 and try to determine the vertex of the parabola described by y = -2xÂ² + 180x + 18,000
We'll use a process called COMPLETING THE SQUARE (which is also beyond the scope of the GMAT) [more on completing the square here https://www.mathsisfun.com/algebra/compl ... quare.html]

Start with: y = -2xÂ² + 180x + 18,00
Factor -2 from first two terms to get: y = -2(xÂ² - 90x) + 18,000
Complete the square to get: y = -2(xÂ² - 90x + 2025 - 2025) + 18,000
Expand and remove -2025 from brackets to get: y = -2(xÂ² - 90x + 2025) + 4050 +18,000

NOTE: Through our handwork, xÂ² - 90x + 2025 is a square that can be factored!
Simplify and rewrite to get: y = -2(x - 45)Â² + 22,050
Now that we've rewritten our equation in this nice form, we can see that the vertex of the parabola has coordinates (45, 22050)

22050 is the HIGHEST POINT of the parabola, which means the MAXIMUM revenue is $22,050
45 is the value of x that achieves this maximum revenue
So, to maximize revenue, we must add $45 to the ticket price to get $60 + $45 = [spoiler]$105[/spoiler]
Answer: C

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

Like this post

gmat_winter: Senior | Next Rank: 100 Posts; Posts: 34; Joined: Tue Oct 14, 2014 8:23 am; Thanked: 1 times

by gmat_winter » Sun Apr 05, 2015 5:42 am

Thanks Brent

Like this post

Jeff@TargetTestPrep: GMAT Instructor; Posts: 1462; Joined: Thu Apr 09, 2015 9:34 amFollowed by: 22 members; Location: New York, NY; Thanked: 39 times

by Jeff@TargetTestPrep » Fri Jan 05, 2018 7:25 am

gmat_winter wrote: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 the maximum profit?

A)87 $

b)95 $

c)105 $

d)120 $

e)127 $

We can let x = the number of $1 increments in price (or the number of 2-seat decrements) and y = total revenue. Thus, the expression (60 + x) is the increased ticket price and (300 - 2x) represents the decreased number of tickets sold at that increased price. Recall that revenue is the product of the ticket price and the number of tickets sold. Thus, we have:

y = (60 + x)(300 - 2x)

Expanding the right-hand side, we have:

y = 18000 + 180x - 2x^2

Notice that the above is a quadratic equation. Let's rearrange the terms so that it's in standard ax^2 + bx + c form:

y = -2x^2 + 180x + 18000

We see that the graph of the quadratic equation is a downward parabola, so the maximum revenue is the y-value of the vertex. To get the y-value of the vertex, we must get the x-value of the vertex first, using the formula x = -b/(2a):

x = -180/(2(-2)) = -180/-4 = 45

Thus, to maximize the revenue, the ticket price should be 60 + 45 = $105.

(Note: We are using the term "revenue" instead of "profit" because the amount of money received from sales is technically called "revenue," not "profit." As we know, the term "profit" is the difference between revenue and cost. The problem uses the term "profit" instead of "revenue" perhaps because cost is either zero or negligible.)

Answer: C

Jeffrey Miller
Head of GMAT Instruction
[email protected]

See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews

Like this post

Post new topic Post Reply

• Page 1 of 1

GMAT Course Reviews

560 reviews

1168 reviews

431 reviews

359 reviews

147 reviews

5/5

5 Star (560 Reviews)

"Target Test Prep is the closest to the official version of the GMAT exam, about 99% accuracy in terms of the quality and quantity of information. The course has excellently created singular sets of focused lessons and tests for every possible topic that one could come across in the official GMAT exam."

"The TTP course maximizes the efficiency of the time you spend studying. It will take time and effort but I could almost guarantee that if you complete the course exactly as it is laid out you will get an amazing score. They also have a very responsive team willing to help with any questions you might have."

"TTP has two things that I think no other test prep company offers: A teaching approach that reinforces understanding and an attitude that will give you the mental preparedness needed to succeed on the test. TTP gives you a deep understanding of the concept you need to know while teaching you how to think."

Write a Review