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Each light bulb at Hotel California is either incandescent or fluoresc

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Each light bulb at Hotel California is either incandescent or fluoresc [#permalink]

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Each light bulb at Hotel California is either incandescent or fluorescent. At a certain moment, forty percent of the incandescent bulbs are switched on, and ninety percent of the fluorescent bulbs are switched on. If eighty percent of all the bulbs are switched on at this moment, what percent of the bulbs that are switched on are incandescent?

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

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Re: Each light bulb at Hotel California is either incandescent or fluoresc [#permalink]

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Let total number of bulbs be 100
x be the number of incandescent bulbs.
100-x is the number of fluorescent bulbs.

forty percent of the incandescent bulbs are switched on == \(\frac{x}{100}\) * 40 = 0.4x
ninety percent of the fluorescent bulbs are switched on == \(\frac{(100-x)}{100}\) * 90 = 0.9(100-x)

eighty percent of all the bulbs are switched on == 80
So 0.4x + 0.9(100-x) = 80
x=20

so incandescent bulbs = 20 and fluorescent = 80
0.4x incandescent bulbs are ON, which is equal to 0.4*20 = 8

what percent of the bulbs that are switched on are incandescent?
\(\frac{8}{80}\)* 100 = 10%

Ans:D

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Re: Each light bulb at Hotel California is either incandescent or fluoresc [#permalink]

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let I be incandescent bulbs and F fluorescent bulbs
therefore we are given
0.4I + 0.9F = 0.8 (I+F)
so we get 0.4I = 0.1F

we need [0.4I / 0.8(I+F) ] *100 ->percent of the bulbs that are switched on are incandescent
substituting 0.4I as 0.1F we get 0.4I / 0.8(I+F) = 0.1F/1F so the ans is 10%

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Re: Each light bulb at Hotel California is either incandescent or fluoresc [#permalink]

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New post 24 Jul 2015, 09:39
Bunuel wrote:
Each light bulb at Hotel California is either incandescent or fluorescent. At a certain moment, forty percent of the incandescent bulbs are switched on, and ninety percent of the fluorescent bulbs are switched on. If eighty percent of all the bulbs are switched on at this moment, what percent of the bulbs that are switched on are incandescent?

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

Kudos for a correct solution.


Number of Incandescent bulbs = a --> Incandescent + ON = \(0.4a\)
Number of Fluorescent bulbs = b --> Fluorescent + ON = \(0.9b\)
Eighty percent of all the bulbs are switched on = \(0.8(a+b)\) --> \(0.4a+0.9b=0.8(a+b)\) --> \(b=4a\)
Percent of the bulbs that are switched on are incandescent = \(\frac{0.4a}{0.8(a+b)}=\frac{0.4a}{0.8(a+4a)}=\frac{1}{10}=10percent\) --> Answer D

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Re: Each light bulb at Hotel California is either incandescent or fluoresc [#permalink]

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Hi All,

This question is essentially a 'Weighted Average' question with a couple of extra steps.

We're told that 40% of the Incandescent bulbs and 90% of the Fluorescent bulbs are switched on; we're also told that 80% of the TOTAL bulbs are switched on.

N = # of Incandescent bulbs
F = # of Fluorescent bulbs

(.4N + .9F)/(N + F) = .8

.4N + .9F = .8N + .8F
.1F = .4N
F = 4N

This means that for every 1 incandescent bulb, there are 4 fluorescent bulbs. This ratio is important - you can use it to TEST VALUES or do the remaining algebra.

We're THEN asked what percent of the bulbs that are SWITCHED ON are INCANDESCENT.

TESTing VALUES can help to make this math easier, but it's not necessary. We already know that 40% of the incandescent and 90% of the fluorescent bulbs are turned on.....

(.4)(1) + .9(4) = .4 + 3.6 = 4

So, for every 4 bulbs that are turned on, 0.4 of them are incandescent.

.4/4.0 = 1/10 = 10%

Final Answer:
[Reveal] Spoiler:
D


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Re: Each light bulb at Hotel California is either incandescent or fluoresc [#permalink]

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New post 24 Jul 2015, 21:14
Hi Bunuel / other maths champs,

I am wondering how can we solve this one using real numbers?

For xample lets consider out of 100 incandecent lamps 40 are on and out of 100 another flourscent bulbs 90 are on. Hence there are 130 bulbs on out of 200, making it 65% bulbs on at a given moment.

The using these figures can get to an answer when all of 80% (160 out of 200) bulbs are on?

If not, where my approach / understanding wrong in solving this way?

You help and guidance is always valuable and will be highly appreciated

Thanks
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Re: Each light bulb at Hotel California is either incandescent or fluoresc [#permalink]

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Hi amrishsoni86,

In this prompt, starting off by TESTing VALUES is a problem since there IS a relationship between the total number of incandescent bulbs and the total number of fluorescent bulbs (so you CANNOT just choose 100 for each).

If you read through my explanation, you'll see that "for every 1 incandescent bulb, there are 4 fluorescent bulbs. This ratio is important - you can use it to TEST VALUES..."

The 'clue' that you can't just start off by TESTing VALUES is the final piece of information in the prompt: "80% of ALL the bulbs are switched on..." - since we don't know WHICH bulbs are switched on, we have to do a big of algebra at the beginning of this question to make the proper deductions about THAT information.

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New post 25 Jul 2015, 22:11
Hmm. Now I think I am getting the point. The prompt tricked me a bit. Thanks for clarifying Rich! By the way, I was amazed to see you score (800). Truly a feat!

Thanks again
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Re: Each light bulb at Hotel California is either incandescent or fluoresc [#permalink]

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Hi Amrish,

Your instinct to TEST VALUES was a good one. As you continue to study and improve, you're going to come across questions that come with 'twists' or are quirky. While they're rarer issues, you do have to learn to spot the inherent 'patterns' involved in the wording of these questions, so that you can make the necessary adjustments to how you approach these prompts (and thus get the correct answer and do so faster).

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Re: Each light bulb at Hotel California is either incandescent or fluoresc [#permalink]

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This is how I solved it, found that to be simpler than the existing solutions in the thread. See attachment.
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Re: Each light bulb at Hotel California is either incandescent or fluoresc [#permalink]

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New post 01 Aug 2015, 11:42
Hi noTh1ng,

You're going to find that most GMAT questions can be approached in a variety of ways, so having the flexibility to solve problems using different methods will be quite beneficial on Test Day. Your approach here is just as valid as any, so the only other thing to consider is how long it took you to solve the problem. On any given question, you have 2 goals:

1) Get the question correct, in a reasonable amount of time (if possible).
2) Use the most efficient method possible to answer the question (so that you can save time).

I recall an old post of yours (from a month or so ago) mentioning that you were planning to start taking CATs. Did you start taking them? How have you been scoring?

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Re: Each light bulb at Hotel California is either incandescent or fluoresc [#permalink]

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2*2 Matrix method:
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Re: Each light bulb at Hotel California is either incandescent or fluoresc [#permalink]

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New post 25 Aug 2015, 18:32
It seems when the total isn't stated it might be best to reduce the system of equations to one variable, whether or not it's a matrix, where \(x\) is the number of one variable and \(100-x\) is another variable. The weighted average approach has quite a few fractions slash ratios than the one-variable systems. Just my thoughts.

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Re: Each light bulb at Hotel California is either incandescent or fluoresc [#permalink]

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New post 20 Jan 2016, 23:57
There is some kind of blip in my brain happening here...

I treated this like a mixture problem and got 1 incandescent = 4 fluorescent, which left me wondering where the 20% answer was.

I understand that 20% represents ALL incandescent bulbs. What I don't understand is if we are plugging in only the numbers of "turned on" bulbs into the mixture equation, why does the 20% answer account for ALL bulbs and not only those that are turned on?

Many thanks

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Re: Each light bulb at Hotel California is either incandescent or fluoresc [#permalink]

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Anonamy wrote:
There is some kind of blip in my brain happening here...

I treated this like a mixture problem and got 1 incandescent = 4 fluorescent, which left me wondering where the 20% answer was.

I understand that 20% represents ALL incandescent bulbs. What I don't understand is if we are plugging in only the numbers of "turned on" bulbs into the mixture equation, why does the 20% answer account for ALL bulbs and not only those that are turned on?

Many thanks


Take a parallel example:
Mixture A (incandescent bulbs) has 40% alcohol (bulbs on) while mixture B (fluorescent bulbs) has 90% alcohol (bulbs on). When you mix them, if you get 80% alcohol (total bulbs on), what percent of total alcohol (on bulbs) is obtained from mixture A (incandescent bulbs)?

Now, solve it:
w1/w2 = (90 - 80)/(80 - 40) = 1/4
What are w1 and w2? The weights in which mixture A and mixture B were mixed. Is w1 the amount of alcohol obtained from mixture A? No. It is the amount of mix A in total solution. How do you find it then? You say that out of a total 100 ml mix (which has 80 ml alcohol), 20 is mix A and 80 is mix B. Mix A is 40% alcohol so that is 40% of 20 = 8 ml.
So 8/80 i.e. 10% alcohol is from mix A.

Using the parallel structure, the weights must be the total incandescent bulbs and total fluorescent bulbs and you will get 10% as the answer using the same logic.

Now think why... What are the weights in the weighted average formula? Since we are averaging Percent of "bulbs that are ON of each type" out of "total bulbs of each type", the weights will be "total bulbs of each type".

This concept is discussed in detail here: http://www.veritasprep.com/blog/2014/12 ... -averages/
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Each light bulb at Hotel California is either incandescent or fluoresc [#permalink]

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Anonamy wrote:
There is some kind of blip in my brain happening here...

I treated this like a mixture problem and got 1 incandescent = 4 fluorescent, which left me wondering where the 20% answer was.

I understand that 20% represents ALL incandescent bulbs. What I don't understand is if we are plugging in only the numbers of "turned on" bulbs into the mixture equation, why does the 20% answer account for ALL bulbs and not only those that are turned on?

Many thanks


Since you haven't shown your work/'steps', I have to assume that you worked through the initial calculation (which was to figure out the ratio of incandescent bulbs to fluorescent bulbs) and then stopped. However, that is NOT what the question was asking for. To keep this type of 'issue' from occurring again (especially in wordier prompts), it often helps to label your work (and not just put numbers on the pad). Quickly jotting down "1 incand. for every 4 fluor." on your pad provides you with a reference for the NEXT few steps needed to answer the question that is asked.

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noTh1ng wrote:
This is how I solved it, found that to be simpler than the existing solutions in the thread. See attachment.


Hi..would you like to clarify over how you found their intersection to be 50%??

Just starting the prep.. so confused..

Thanks!!
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Bunuel wrote:
Each light bulb at Hotel California is either incandescent or fluorescent. At a certain moment, forty percent of the incandescent bulbs are switched on, and ninety percent of the fluorescent bulbs are switched on. If eighty percent of all the bulbs are switched on at this moment, what percent of the bulbs that are switched on are incandescent?

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

Kudos for a correct solution.


Here's a step-by-step solution using the Double Matrix method.
Here, we have a population of lightbulbs, and the two characteristics of each bulb are:
- incandescent or fluorescent
- on or off

Since the questions asks us to find a certain PERCENT, let's say that there are 100 bulbs altogether.
So, we can set up our matrix as follows:
Image

Eighty percent of ALL the bulbs are switched on at this moment
So, 80 bulbs are turned ON.
This also means that the remaining 20 bulbs are OFF.
Add this to our diagram to get:
Image

Forty percent of the incandescent bulbs are switched on
This one is tough, because we don't know how many incandescent bulbs there are.
So, let's let x = the number of incandescent bulbs.
This means the remaining 100-x bulbs are fluorescent
Let's add this to our diagram first, and THEN tackle the given info:
Image

Okay, if x = the number of incandescent bulbs, and 40% of those bulbs are switched on, then the number of incandescent bulbs that are on = 40% of x = 0.4x
Likewise, if 100-x = the number of fluorescent bulbs, and 90% of those bulbs are switched on, then the number of fluorescent bulbs that are on = 90% of 100-x = 0.9(100 - x)
Add this to our diagram to get:
Image

When we examine the left-hand column, we can see that the sum of the boxes is 80.
In other words: 0.4x + 0.9(100 - x) = 80
Expand: 0.4x + 90 - 0.9x = 80
Simplify: -0.5x = -10
Solve: x = 20
So, there are 20 incandescent bulbs, and 40% of them are on. 40% of 20 = 8, so 8 of the incandescent bulbs are on:
Image

We can see that, of the 80 bulbs that are on, 8 of them are incandescent.
8/80 = 1/10 = [spoiler]10%[/spoiler]

Answer:
[Reveal] Spoiler:
D


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Re: Each light bulb at Hotel California is either incandescent or fluoresc   [#permalink] 04 Nov 2017, 00:00
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