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A certain theater has 100 balcony seats. For every $2

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A certain theater has 100 balcony seats. For every $2 [#permalink] New post 28 Apr 2012, 20:39
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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


I solved it as follows:

10+2(x) = 100 -5(x)
x= 12

Why is my approach wrong?
[Reveal] Spoiler: OA

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Re: A certain theater has 100 balcony seats. For every $2 [#permalink] New post 29 Apr 2012, 03:51
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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


I solved it as follows:

10+2(x) = 100 -5(x)
x= 12

Why is my approach wrong?


Equation should be (10+$2*x)(100-5x)=1,360, where x is the # of times we increased the price by $2. (10+$2*x)(100-5x)=1,360 --> (5+x)(20-x)=136 --> x=3 or x=12 --> price=10+$2*3=$16 or price=10+$2*12=$34.

Answer: C.

But the easiest way to solve this problem would be to write down a chart:
$10*100=$1,000;
$12*95=$1,140;
$14*90=$1,260;
$16*85=$1,360.

Or do the other way around and plug the answer choices.

Answer: C.
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Re: A certain theater has 100 balcony seats. For every $2 [#permalink] New post 30 Apr 2012, 02:53
Here's a long winded approach!

Think about a supply and demand chart, but put price along the x axis.

We are told:
Q = mP + C
(10,100)
(12,95)

Which is enough to work out the equation:

Work out m, m = (100-95) / (10 - 12) = -5 / 2
Solve for C

Hence Q = -5/2P + 125

Great. So for any given P, we now know Q seats were sold, but we are interested in total revenue

TR = Q * P = 1360, so

Q*P = -5/2P^2 + 125P
1360 = -5/2P^2 + 125P
2720 = -5P^2 + 250P
5P^2 - 250P + 2720 = 0
P^2 - 50P + 544 = 0

(P - 16)(P - 34) = 0
P = 16, P = 34

16.
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Re: A certain theater has 100 balcony seats. For every $2 [#permalink] New post 01 May 2012, 06:54
The official answer given here is wrong, it is given E. Moderators please correct it, it'll be C.
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Re: A certain theater has 100 balcony seats. For every $2 [#permalink] New post 12 May 2012, 04:16
Bunuel wrote:
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


I solved it as follows:

10+2(x) = 100 -5(x)
x= 12

Why is my approach wrong?


Equation should be (10+$2*x)(100-5x)=1,360, where x is the # of times we increased the price by $2. (10+$2*x)(100-5x)=1,360 --> (5+x)(20-x)=136 --> x=3 or x=12 --> price=10+$2*3=$16 or price=10+$2*12=$34.

Answer: C.

But the easiest way to solve this problem would be to write down a chart:
$10*100=$1,000;
$12*95=$1,140;
$14*90=$1,260;
$16*85=$1,360.

Or do the other way around and plug the answer choices.

Answer: C.


@Bunuel,

in this chart, $17*80 = 1360.
why should we avoid 'D' as answer.?
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Re: A certain theater has 100 balcony seats. For every $2 [#permalink] New post 12 May 2012, 10:56
Expert's post
kashishh wrote:
Bunuel wrote:
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


I solved it as follows:

10+2(x) = 100 -5(x)
x= 12

Why is my approach wrong?


Equation should be (10+$2*x)(100-5x)=1,360, where x is the # of times we increased the price by $2. (10+$2*x)(100-5x)=1,360 --> (5+x)(20-x)=136 --> x=3 or x=12 --> price=10+$2*3=$16 or price=10+$2*12=$34.

Answer: C.

But the easiest way to solve this problem would be to write down a chart:
$10*100=$1,000;
$12*95=$1,140;
$14*90=$1,260;
$16*85=$1,360.

Or do the other way around and plug the answer choices.

Answer: C.


@Bunuel,

in this chart, $17*80 = 1360.
why should we avoid 'D' as answer.?


We can have ONLY one correct answer, so if C is correct then D is automatically wrong.

Also, we are told that "for every $2 increase in the price 5 fewer seats will be sold" and from $16 to $17 there is only $1 increase, so you cannot write $17*80.

Hope it's clear.
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Re: A certain theater has 100 balcony seats. For every $2 [#permalink] New post 31 May 2013, 05:22
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Re: A certain theater has 100 balcony seats. For every $2 [#permalink] New post 02 Jun 2013, 20:49
Hello Bunuel,
I solved it the following way. Let me know if I am correct in the logic and the approach.

R1 = 1000; revenue when price per seat = $10 and no. of seats are 100
R2 = 1360; revenue when price increase every time by $2 and 5 lesser no of seats sold that every time.

Hence, for R2,
x = no. of times price of each seat increase by $2
n = no. of times 5 lesser seats are sold when each time the price is increased by $2

Incremental revenue = $ 360

Now,
essentially x = n (since everytime the price is increased, lesser no. of seats are sold) and this is attributed to the incremental revenue

the number of times the price is increased by $2 = 2x
the number of times 5 lesser seats are sold = 5n

hence,
2x * 5n = 360
10xn = 360
xn = 36

now, as mentioned above, x = n
hence x^2= 36
giving, x = + / - 6

x cannot be -6 since the price is increased, thus making x = 6

hence the new price during R2 (1360) = 10+6 = $16


thanks,
Rohan
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Re: A certain theater has 100 balcony seats. For every $2 [#permalink] New post 02 Jun 2013, 20:56
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


I solved it as follows:

10+2(x) = 100 -5(x)
x= 12



Equation should be (10+$2*x)(100-5x)=1,360, where x is the # of times we increased the price by $2. (10+$2*x)(100-5x)=1,360 --> (5+x)(20-x)=136 --> x=3 or x=12 --> price=10+$2*3=$16 or price=10+$2*12=$34.

Answer: C.

But the easiest way to solve this problem would be to write down a chart:
$10*100=$1,000;
$12*95=$1,140;
$14*90=$1,260;
$16*85=$1,360.

Or do the other way around and plug the answer choices.

Answer: C.[/quote]


Hello Bunuel,
I solved it the following way. Let me know if I am correct in the logic and the approach.

R1 = 1000; revenue when price per seat = $10 and no. of seats are 100
R2 = 1360; revenue when price increase every time by $2 and 5 lesser no of seats sold that every time.

Hence, for R2,
x = no. of times price of each seat increase by $2
n = no. of times 5 lesser seats are sold when each time the price is increased by $2

Incremental revenue = $ 360

Now,
essentially x = n (since everytime the price is increased, lesser no. of seats are sold) and this is attributed to the incremental revenue

the number of times the price is increased by $2 = 2x
the number of times 5 lesser seats are sold = 5n

hence,
2x * 5n = 360
10xn = 360
xn = 36

now, as mentioned above, x = n
hence x^2= 36
giving, x = + / - 6

x cannot be -6 since the price is increased, thus making x = 6

hence the new price during R2 (1360) = 10+6 = $16


thanks,
Rohan
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Re: A certain theater has 100 balcony seats. For every $2 [#permalink] New post 06 Jul 2014, 20:46
(final price)*(final seats sold) = 1360

so either (final price) or (final seats sold) should be a factor of 1360

A. 1360/12 indivisible
B. 1360/14 indivisible
C. 1360/16 = 85 (final seats sold) divisible, correct
Re: A certain theater has 100 balcony seats. For every $2   [#permalink] 06 Jul 2014, 20:46
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