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by jain2016 » Mon Apr 11, 2016 9:58 am
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%

OAD

Hi Experts ,

Please explain.

Thanks,

SJ
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by Brent@GMATPrepNow » Mon Apr 11, 2016 10:01 am
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%
One option is to use 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: D

------------------------

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by [email protected] » Mon Apr 11, 2016 10:23 am
Hi jain2016,

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: D

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by Matt@VeritasPrep » Mon Apr 11, 2016 1:04 pm
Here's a short way. Let I = # of incandescent and F = # of fluorescent.

Total bulbs = I + F
Total bulbs on = 80% of (I + F)

But we also know that 40% of I and 90% of F are on, so we can say that

Total bulbs on = 40% of I + 90% of F

We have two different ways of writing the total bulbs that are on, so these two are equal:

80% of (I + F) = 40% of I + 90% of F

.8*I + .8F = .4I + .9F

F = 4I

Now we have a ratio for all the bulbs: the number of fluorescent bulbs = 4 * the number of incandescent.

To finish, we'll set up our equation for the fraction of switched on bulbs that are incandescent:

.4I / (.4I + .9F)

then replace F with 4I

.4I / (.4I + .9*4I)

and we're done!

.4I / 4I =

10%
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by regor60 » Tue Apr 12, 2016 6:46 am
Matt@VeritasPrep wrote:Here's a short way. Let I = # of incandescent and F = # of fluorescent.

Total bulbs = I + F
Total bulbs on = 80% of (I + F)

But we also know that 40% of I and 90% of F are on, so we can say that

Total bulbs on = 40% of I + 90% of F

We have two different ways of writing the total bulbs that are on, so these two are equal:

80% of (I + F) = 40% of I + 90% of F

.8*I + .8F = .4I + .9F

F = 4I

Now we have a ratio for all the bulbs: the number of fluorescent bulbs = 4 * the number of incandescent.

To finish, we'll set up our equation for the fraction of switched on bulbs that are incandescent:

.4I / (.4I + .9F)

then replace F with 4I

.4I / (.4I + .9*4I)

and we're done!

.4I / 4I =

10%
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by DavidG@VeritasPrep » Tue Apr 12, 2016 6:55 am
jain2016 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%

OAD

Hi Experts ,

Please explain.

Thanks,

SJ
We can also do a little fun alligation.

Incandescent bulbs switched on: 40%
Fluorescent bulbs switched on: 90%
Total Bulbs switch on: 80%

40---------80-----90
Gap: -- 40 ---- 10

So we know there is a ratio of 40/10 in favor of the fluorescent bulbs. So let's say there are 40 fluorescent bulbs and 10 incandescent. If 90% of the fluorescent bulbs are on, that's .9*40 = 36. If 40% of the incandescent bulbs are on that's .4*10 = 4. Total bulbs on: 36 + 4 = 40.

So the ratio of incandescent bulbs on to total bulbs on is 4/40 = 10%
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by GMATGuruNY » Tue Apr 12, 2016 7:07 am
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 ten percent of the fluorescent bulbs are switched off. If eighty percent of all the bulbs are switched on at this moment, what percent of the bulbs that are switched on are incandescent?

22 (2/9)%
16 (2/3)%
11 (1/9)%
10%
5%
Like many GMAT problems, the one above can be solved with clever reasoning and perhaps some trial and error.

Let total bulbs = 100.
Number of bulbs on = .8*100 = 80.
Since 80 bulbs are on, and 90% of the fluorescent bulbs are switched on, the number of fluorescent bulbs must be very high.
We know that 90 fluorescent bulbs would be too many, because with 10% of these switched off, the number switched on would be 90-9 = 81, 1 more than the 80 that are supposed to be switched on.

Let fluorescent = 80.
10% of these switched off = 8 off, 72 on.

Incandescent = 100-80 = 20.
40% of these switched on = 8 on, 12 off.

Total on = 72+8 = 80. This works.

Thus, of the 80 bulbs that are switched on, 8 are incandescent.
8/80 = 10%.

The correct answer is D.

Here's an algebraic approach:

Let total bulbs = 100.

Let x = incandescent bulbs.
Let 100-x = fluorescent bulbs.

40% of the incandescent bulbs switched on = .4x
10% of the fluorescent bulbs switched off = 90% switched on = .9(100-x).

Since .8*100 = 80 bulbs are switched on, we get:
.4x + .9(100-x) = 80.
4x + 900 - 9x = 800
-5x = -100
x = 20.

Thus, .4*20 = 8 of the incandescent bulbs are switched on.
Incandescent on/Total on = 8/80 = 10%.
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by Matt@VeritasPrep » Tue Apr 12, 2016 6:21 pm
regor60 wrote:Were you peeking at my paper ?
Great minds peek alike!
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