boomtangboy wrote:
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
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 _________________
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