Balcony theatre seats

[smclean23]

A certain theater has 100 balcony seats. For every $2 increase in the price of a balcony seat above $10, 5 fewer seats will be sold. If all the balcony seats are sold when the price of each seat is $10, which of the following could be the price of a balcony seat if the revenue from the sale of balcony seats is $1,360 ?
(A) $12
(B) $14
(C) $16
(D) $17
(E) $18





Answer is C

[Vignesh.4384]

IMO C.

[parallel_chase]

There are total 100 seats, $10 is the price of each seat.

if the price increases by $2 the seats will be reduced by 5.

if
$10-100seats =$1000
$12-95seats
$14-90seats
$16-85seats = $1360
$18-80seats

You easily eliminate $17 option because it hold no relevance in this question.
Hence C is the answer.

[VP_RedSoxFan]

For those that enjoy a good algebra challenge, here's the other way to the solution. I like using algebra as often as possible because I have less of a chance to make a mistake, but for this problem, it's not the most straightforward application.

Let P=the price above $10 for a ticket. We can then write the following equation for sales revenue (price per ticket x number of seats):

(10 + P)(100 - 5P/2) = 1360 [FOIL]
1000 + 100P - 25P - 5P^2/2 = 1360 [rearrange into quadratic form]
P^2 - 30P + 144 = 0 [factor]

(P - 6)(P - 24) = 0, P = 6, 24

The price increase was denoted with P and since we started with 10, the only answer choice to choose from is $16

It is interesting to note, though, that given the equation, tickets of $34 would also get you the $1,360 in revenue. Maybe that's why they don't lower prices for Kansas City Royals games

[Haris]

Let P = Number of times price increased by $2
S = Number of times seat number decreased by 5

So we can write the equation as:

(10+2P)(100-5S) = 1360

We know that number of times P will increase, the same number of times S will decrease

=> P = S

The main equation becomes

(10+2P)(100-5P) = 1360
1000 - 50P +200P - 10P^2 = 1360
P^2 -15P +36 = 0;

P = 3, 12

Price of the Balcony Seat = 10 + 2P = 10 + 2 (3) = $16 or
Price of the Balcony Seat = 10 + 2P = 10 + 2 (12) =$34

The only answer to choose is C

[Scott@TargetTestPrep]

A certain theater has 100 balcony seats. For every $2 increase in the price of a balcony seat above $10, 5 fewer seats will be sold. If all the balcony seats are sold when the price of each seat is $10, which of the following could be the price of a balcony seat if the revenue from the sale of balcony seats is $1,360 ?
(A) $12
(B) $14
(C) $16
(D) $17
(E) $18





Answer is C

Solution:

Let’s let x = the number of (two-dollar) increases in the ticket price above $10. So (10 + 2x) = the (increased) price of the ticket, and (100 - 5x) = the number of tickets sold at that increased ticket price. We know that (ticket price) x (no. of tickets sold) = total revenue, so we have the following equation:

(10 + 2x)(100 - 5x) = 1360

1000 + 200x - 50x - 10x^2 = 1360

-1000 - 150x +10x^2 = -1360

10x^2 - 150x + 360 = 0

x^2 - 15x + 36 = 0

(x - 3)(x - 12) = 0

x = 3 or x = 12

If x = 3, then the price of a balcony seat is 10 + 2(3) = $16. If x = 12, then the price of a balcony seat is 10 + 2(12) = $34. Since only $16 is given in the answer choices, it’s the correct answer.

Alternate Solution:

Let’s test the answer choices.

A) 1,360 is not divisible by 3; therefore, it is not divisible by 12. Since the price of a balcony seat must be a factor of 1,360, it cannot be $12.

B) 1,360 is not divisible by 7; therefore, it is not divisible by 14. Since the price of a balcony seat must be a factor of 1,360, it cannot be $14.

C) Notice that 1,360/16 = 85. We see that there are three increases in the amount of $2 from $10 to $16; therefore, the number of tickets sold must be 3 x 5 = 15 less than the total number of balcony seats. Since 85 is indeed 15 less than 100, this is a possible value for the price of a balcony seat.

Answer: C