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# Eighty percent of the lights at Hotel California are on at 8

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Eighty percent of the lights at Hotel California are on at 8 [#permalink]  16 Sep 2008, 06:01
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Eighty percent of the lights at Hotel California are on at 8 p.m. a certain evening. However, forty percent of the lights that are supposed to be off are actually on and ten percent of the lights that are supposed to be on are actually off. What percent of the lights that are on are supposed to be off?

A. 22(2/9)%
B. 16(2/3)%
C. 11(1/9)%
D. 10%
E. 5%
[Reveal] Spoiler: OA

Last edited by Bunuel on 28 Jul 2014, 05:42, edited 1 time in total.
Edited the question and added the OA.
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Re: Eighty percent of the lights at Hotel California are on at 8 [#permalink]  16 Sep 2008, 08:13
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dancinggeometry wrote:
Eighty percent of the lights at Hotel California are on at 8 p.m. a certain evening. However, forty percent of the lights that are supposed to be off are actually on and ten percent of the lights that are supposed to be on are actually off. What percent of the lights that are on are supposed to be off?

22(2/9)%
16(2/3)%
11(1/9)%
10%
5%

this kind of the wording is always confusing for me.

a= number of lights supposed to be on
b= number of lights supposed to be off
total = a+b
no of lights that are on = 0.8(a+yb

Light that are supposed to be on but are actually off = 10% of a
Light that are supposed to be on are on = 90% of a

Light that aare supposed to be but are on = 40% of b
Light that aare supposed to be off are off = 60% of b

0.9a + 0.4b = 0.8 (a+b)
a = 4b

Suppose total Lights = 100
so, a = 80 and b = 20

No of lights that are on are on but are supposed to be off = 40% of b = 0.4 (20) = 5

So, the % of lights that are on are supposed to be off = 5/100 = 5%.
//E//
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Re: Eighty percent of the lights at Hotel California are on at 8 [#permalink]  16 Sep 2008, 12:35
Let x be the total no. of lights and y be the no. of lights off.
Therefore, (x-y) is the total no. of "on" lights.
Since 40% of lights that should be off are on, 60% of lights that are supposed to be off are actually off. However, 10% of lights that are supposed to be on are off.
Therefore, 60*y/100 + 10*(x-y)/100 = 20*x/100. Solving this gives us x= 5*y. Therefore the no. of lights on are (x-y) = 5y-y = 4y.
Therefore, % of "on" lights that should have been off = {(40y/100) /4y}*100 = 10%
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Need help overlapping set question [#permalink]  16 May 2010, 09:55
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Eighty percent of the lights at Hotel California are on at 8pm a certain evening. However, forty percent of the lights that are supposed to off are actually on and ten percent of the lights that are supposed to be on are actually off. What percent of the lights that are on are supposed to be off?

A) 22 2/9%
B) 16 2/3%
C) 11 1/9%
D) 10%
E) 5%
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Re: Need help overlapping set question [#permalink]  16 May 2010, 13:03
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valaparla wrote:
Eighty percent of the lights at Hotel California are on at 8pm a certain evening. However, forty percent of the lights that are supposed to off are actually on and ten percent of the lights that are supposed to be on are actually off. What percent of the lights that are on are supposed to be off?

A) 22 2/9%
B) 16 2/3%
C) 11 1/9%
D) 10%
E) 5%

let me try..

Let the light which are supposed to be OFF = SO
Let the light which are supposed to be ON = SN
Let the light which are ACTUALLY OFF = AO
Let the light which are ACTUALLY ON = AN

Let the total no. of lights be 100,
So ACTUALLY ON lights = 80
And ACTUALLY OFF lights = 20

Also given >> forty percent of the lights that are supposed to off are actually on >>> (40/100)*SO are ACTUALLY ON
it means >>> (60/100)*SO are ACTUALLY OFF

Also given >> ten percent of the lights that are supposed to be on are actually off >>> (10/100)*SN are ACTUALLY OFF
it means >>> (90/100)*SN are ACTUALLY ON

So, Total ACTUALLY ON lights = (40/100)*SO + (90/100)*SN = 80
and Total ACTUALLY OFF lights = (60/100)*SO + (10/100)*SN = 80

From here we get SO = 20

we need to find: What percent of the lights that are on are supposed to be off >>> So light ACTUALLY ON are 80 and light which are ACTUALLY ON, which are supposed to be OFF = (40/100)*SO = 8.

So (8/80)*100 = 10%

OA plz.
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Re: Need help overlapping set question [#permalink]  10 Aug 2010, 08:56
Let

off = # supposed to be off
on = # supposed to be on

Then from the information given we have:

.4*off + (1-.1)*on = .8*(off + on)

We are trying to find the number that should be off (but are on) as a portion of everything that is on, e.g.:

$$\frac{.4*off}{.8*(off+on)}$$

Solving the first equation for on, we get

on = 4*off

Substituting this into the second equation, we can eliminate the dependence on the variable off and we solve the fraction to be 1/10 (d).
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Re: Need help overlapping set question [#permalink]  10 Aug 2010, 15:39
something like in the file attached
Attachments

GMAT.xlsx [9.18 KiB]

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Re: Eighty percent of the lights at Hotel California are on at 8 [#permalink]  22 Sep 2011, 14:47
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KASSALMD wrote:
Let x be the total no. of lights and y be the no. of lights off.
Therefore, (x-y) is the total no. of "on" lights.
Since 40% of lights that should be off are on, 60% of lights that are supposed to be off are actually off. However, 10% of lights that are supposed to be on are off.
Therefore, 60*y/100 + 10*(x-y)/100 = 20*x/100. Solving this gives us x= 5*y. Therefore the no. of lights on are (x-y) = 5y-y = 4y.
Therefore, % of "on" lights that should have been off = {(40y/100) /4y}*100 = 10%

Agree with you. My approach:

just pick a number (e. 100) for lights.
80 lights are supposed to be on, but 0.1 (that is 8 lights) are off.
20 lights are supposed to be off, but 0.4 (that is 8 lights) are on.

Totally, 80 - 8 + 8=80 lights are on
and the percentage of lights which are on (and not supposed to be on) is 10% (8 out of 80)
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Re: Eighty percent of the lights at Hotel California are on at 8 [#permalink]  22 Sep 2011, 15:44
All this is under the assumption that there are no lights which are neither supposed to be on or off.
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Re: Eighty percent of the lights at Hotel California are on at 8 [#permalink]  22 Sep 2011, 17:55
shahideh wrote:
KASSALMD wrote:
Let x be the total no. of lights and y be the no. of lights off.
Therefore, (x-y) is the total no. of "on" lights.
Since 40% of lights that should be off are on, 60% of lights that are supposed to be off are actually off. However, 10% of lights that are supposed to be on are off.
Therefore, 60*y/100 + 10*(x-y)/100 = 20*x/100. Solving this gives us x= 5*y. Therefore the no. of lights on are (x-y) = 5y-y = 4y.
Therefore, % of "on" lights that should have been off = {(40y/100) /4y}*100 = 10%

Agree with you. My approach:

just pick a number (e. 100) for lights.
80 lights are supposed to be on, but 0.1 (that is 8 lights) are off.
20 lights are supposed to be off, but 0.4 (that is 8 lights) are on.

Totally, 80 - 8 + 8=80 lights are on
and the percentage of lights which are on (and not supposed to be on) is 10% (8 out of 80)

I really didn't understand why did you pick the numbers 80 and 20?
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Re: Eighty percent of the lights at Hotel California are on at 8 [#permalink]  23 Sep 2011, 03:29
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Quote:
I really didn't understand why did you pick the numbers 80 and 20?

when you encounter with a question related to "percentage", it's easier for you to just intellectually pick a number that covers the percentages of the question, and then solve the problem with that number. This approach makes the question tangible.

So, for this question, I pick the number 100 for lights. that is, i suppose the total lights are 100. According to the question, 80% of lights are on at a certain time. So:
80% * 100=80
That is in my approach, 80 lights are on, and 20 other lights, (100-80=20) are off.

In continue, the question says: forty percent of the lights that are supposed to be off are actually on
40% * 20 = 8
That is, 20 lights are supposed to be off, but 8 of them are on now.

again in continue, the question says: ten percent of the lights that are supposed to be on are actually off
10% * 80 = 8
that is, 80 lights are supposed to be on, but 8 lights of them are off now.

so the total number of lights which are on is: 80 + 8 - 8 = 80

To calculate the percentage: (8/80)*100=10

let me know if my explanation is clear : )
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Re: Eighty percent of the lights at Hotel California are on at 8 [#permalink]  03 Nov 2011, 01:27
shahideh wrote:
Quote:
I really didn't understand why did you pick the numbers 80 and 20?

when you encounter with a question related to "percentage", it's easier for you to just intellectually pick a number that covers the percentages of the question, and then solve the problem with that number. This approach makes the question tangible.

So, for this question, I pick the number 100 for lights. that is, i suppose the total lights are 100. According to the question, 80% of lights are on at a certain time. So:
80% * 100=80
That is in my approach, 80 lights are on, and 20 other lights, (100-80=20) are off.

In continue, the question says: forty percent of the lights that are supposed to be off are actually on
40% * 20 = 8
That is, 20 lights are supposed to be off, but 8 of them are on now.

again in continue, the question says: ten percent of the lights that are supposed to be on are actually off
10% * 80 = 8
that is, 80 lights are supposed to be on, but 8 lights of them are off now.

so the total number of lights which are on is: 80 + 8 - 8 = 80

To calculate the percentage: (8/80)*100=10

let me know if my explanation is clear : )

So clear
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Re: Eighty percent of the lights at Hotel California are on at 8 [#permalink]  28 Jul 2014, 05:32
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: Eighty percent of the lights at Hotel California are on at 8 [#permalink]  30 Jul 2014, 00:01
Using Double matrix method:

Answer $$= \frac{8}{80} * 100 = 10%$$

Refer diagram below:
Attachments

mat.png [ 5.64 KiB | Viewed 2058 times ]

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Re: Need help overlapping set question [#permalink]  09 Sep 2014, 18:29
Hello from the GMAT Club BumpBot!

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Re: Need help overlapping set question [#permalink]  09 Sep 2014, 20:15
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Solved using Matrix method. Refer below:

Attachment:

over.png [ 4.01 KiB | Viewed 1689 times ]

1. Let the total number of lights = 100

2. 80% of the lights are ON = 80

3. So, lights which are OFF = 100-80 = 20

4. Actual OFF = Suppose OFF = 20

5. 40% of "Suppose OFF" are "Actual ON" $$= \frac{40}{100}* 20 = 8$$

6. Percentage of the lights that are on are supposed to be off $$= \frac{8}{80} * 100 = 10%$$

Bunuel: Kindly update the OA

Also, in the problem, they have used words like "Eighty percent" (instead of 80%) etc.... is this normal in GMAT?
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Re: Need help overlapping set question [#permalink]  09 Sep 2014, 21:17
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valaparla wrote:
Eighty percent of the lights at Hotel California are on at 8pm a certain evening. However, forty percent of the lights that are supposed to off are actually on and ten percent of the lights that are supposed to be on are actually off. What percent of the lights that are on are supposed to be off?

A) 22 2/9%
B) 16 2/3%
C) 11 1/9%
D) 10%
E) 5%

You can do it using algebra to get an equation with a single variable:

Say, total 100 lights. 80 are ON.

Say L are supposed to be on and 100-L are supposed to be off.

Lights that are on = 40% of (100 - L) + 90% of L = 80
L = 80 = Number of lights supposed to be on.
20 = Number of lights supposed to be off. 40% of these are on so should be switched off. 40% of 20 = 8

Of the lights that are on, 8/80 = 10% should be switched off.

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Re: Need help overlapping set question   [#permalink] 09 Sep 2014, 21:17
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