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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]

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New post 28 Apr 2012, 21:39
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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?

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Re: A certain theater has 100 balcony seats. For every $2  [#permalink]

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New post 29 Apr 2012, 04:51
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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]

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New post 30 Apr 2012, 03:53
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]

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New post 01 May 2012, 07:54
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]

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New post 12 May 2012, 05:16
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]

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New post 12 May 2012, 11:56
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]

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Re: A certain theater has 100 balcony seats. For every $2  [#permalink]

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New post 02 Jun 2013, 21:49
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]

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New post 02 Jun 2013, 21:56
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]

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New post 06 Jul 2014, 21:46
(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
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Re: A certain theater has 100 balcony seats. For every $2  [#permalink]

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New post 16 Oct 2014, 05:42
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.



Hey Bunuel,
What should we do with this piece of information?
If all the balcony seats are sold when the price of each seat is $10
shouldn't it be included in the total revenue from sale of tickets? something like
10y+(10+$2*x)(100-5x)=1,360 where y is the number of tickets sold @10$
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Re: A certain theater has 100 balcony seats. For every $2  [#permalink]

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New post 16 Oct 2014, 07:51
1
saggii27 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.



Hey Bunuel,
What should we do with this piece of information?
If all the balcony seats are sold when the price of each seat is $10
shouldn't it be included in the total revenue from sale of tickets? something like
10y+(10+$2*x)(100-5x)=1,360 where y is the number of tickets sold @10$


No. The number of tickets sold at $10 is 100. We are told that the theater has 100 balcony seats and all the balcony seats are sold when the price of each seat is $10. Check the red portion of my solution above.
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Re: A certain theater has 100 balcony seats. For every $2  [#permalink]

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New post 16 Oct 2014, 10:46
Bunuel wrote:
saggii27 wrote:
Bunuel 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.



Hey Bunuel,
What should we do with this piece of information?
If all the balcony seats are sold when the price of each seat is $10
shouldn't it be included in the total revenue from sale of tickets? something like
10y+(10+$2*x)(100-5x)=1,360 where y is the number of tickets sold @10$


No. The number of tickets sold at $10 is 100. We are told that the theater has 100 balcony seats and all the balcony seats are sold when the price of each seat is $10. Check the red portion of my solution above.[/quote]


got it thanks.

kudos!!!
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Re: A certain theater has 100 balcony seats. For every $2  [#permalink]

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New post 16 Oct 2014, 23:15
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



This seems to be another brilliant question, which seems easy, but has a small trap. If we start from option E, we would realize that even 17 divides 1360. But, that can not be the correct choice, since the increase in price d be $2
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Re: A certain theater has 100 balcony seats. For every $2  [#permalink]

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New post 09 Dec 2015, 20:14
solved it as everyone else:
1360=1000+150x-10x^2
or (x-3)(x-12)

now, either x is 3 or 12. if 12, then the price would be 12*2+10 => 34$. Since we don't have it in our answer choices, it must be the other case, when x is 3.

if x is 3, then the price is 3*2+10 = 16.
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A certain theater has 100 balcony seats. For every $2  [#permalink]

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New post 20 Jan 2016, 15:37
let 10+x=seat price
100-2.5x=number of seats sold
(10+x)(100-2.5x)=1360
x=6
10+x=$16
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A certain theater has 100 balcony seats. For every $2  [#permalink]

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New post 21 Jan 2016, 05:09
Suppose the price of a balcony seat = $x
So, seats sold = 100 - [(x-10)/2]*5

But x*{100 - [(x-10)/2]*5} = 1360
-> x*{100 - (x/2 - 5)*5} = 1360
-> x*{100 - 5x/2 + 25} = 1360
-> x*{125 - 5x/2} = 1360
-> 125x - 5*x^2/2 = 1360
-> 250x - 5*x^2 = 2720
-> 50x - *x^2 = 544
-> x^2 - 50x + 544 = 0
-> x^2 - 34x - 16x + 544 = 0
-> x(x - 34) - 16(x - 34) = 0
-> (x - 34)(x - 16) = 0

So, x could be 34 or 16

Since 16 is given in the answer choices, answer = C
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Re: A certain theater has 100 balcony seats. For every $2  [#permalink]

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New post 31 Mar 2017, 19:23
Hi Guys,

If found a different way to find the value. What do you think???

We can se that 1360=P*Q

then P and Q MUST be a factor of 1360

---Factors of 1360 = 2x2x2x2x5x17..... thus, P can [b]ONLY have as factors 2, 5 and 17[/b]

If you see the answers we have:

A. $12 <-- has factor 3 WRONG
B. $14 <-- has factor 7 WRONG
C. $16
D. $17
E. $18 <-- has two times the factor 3 and we can hace onlye one at most WRONG


Now that we only have 2 answers we can "plug" the remainin factors of 1360 for each answers
C. $16 =2x2x2x2<-- Q= 5x17 = 85
D. $17 =17<-- Q= 2x2x2x2x5 =16x5>100 NOT POSSIBLE WRONG
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