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incandescent

by j_shreyans » Sun Mar 29, 2015 5:46 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%
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by GMATGuruNY » Sun Mar 29, 2015 6:16 am
j_shreyans 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%
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%.

The correct answer is D.
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by Brent@GMATPrepNow » Sun Mar 29, 2015 7:12 am
j_shreyans 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%
Here's how Mitch's solution looks when we 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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NOTE: This question type is VERY COMMON on the GMAT, so be sure to master the technique.

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by [email protected] » Sun Mar 29, 2015 4:42 pm
Hi j_shreyans,

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 umasarath52 » Mon Mar 30, 2015 1:27 pm
Let X = Incandescent Bulbs
Let Y = Fluorescent Bulbs
Total bulbs = x+y

40% of I bulbs are ON = 0.4x
90% of F bulbs are ON = 0.9y

Total number of bulbs ON = 0.4x+0.9y
Also given that 80% of all bulbs are ON = 0.8(x+y)

0.4x+0.9y = 0.8(x+y)
on solving
y = 4x


Percent of I bulbs ON = 0.4x/(0.8(x+y))
= 0.5 * (x/(x+y))
substituting y = 4x
= 0.5 * x/x+4x = 0.5 * 0.2 = 0.1 = 10%
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by Amrabdelnaby » Sun Nov 29, 2015 4:35 pm
I think it's also possible to solve this using two unknowns.
GMATGuruNY wrote:
j_shreyans 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%
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%.

The correct answer is D.
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by [email protected] » Sun Nov 29, 2015 10:54 pm
Hi Amrabdelnaby,

Yes, you can solve this question using 2 variables (If you read umasarath52's post - the one above yours - you'll see how).

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