Overlapping Sets

[manochsa]

Eighty percent of the lights at Hotel California are switched on at 8 p.m. one evening. However, forty percent of the lights that are supposed to be switched off are actually switched on, and ten percent of the lights that are supposed to be switched on are actually switched off. What percent of the lights that are switched on are supposed to be switched off?
A 22(2/9)%
B 16(2/3)%
C 11(1/9)%
D 10%
E 5%


Could you explain as far as overlapping sets are concerned, how we should know whether to use a double-set matrix or a venn diagram? Am I mistaken that a double-set matrix is useful only for problems involving two sets, but not for those involving more? The solution to this problem, taken from MGMAT's CAT 3, uses a double-set matrix, but it seems to me that we're dealing with four sets here. What am I missing? What should I be looking for in overlapping set problems that tells me whether to use the matrix, and how to set it up?

Thanks for you help!

[Ben Ku]

We use a double set matrix when there are two factors, and each factor has only two distinct possibilities.

There's one factor: lights that were supposed to be switch on or off.
The other factor is: lights that actually were switched on or off.

Assuming 100 lights, our double set matrix would look like:

Code: Select All Code

-----------| supposed to be off | supposed to be on  |  total
actual off |                    |  .1(100-x)         |
--------------------------------------------------------------
actual on  |    .4x             |                    |  80
--------------------------------------------------------------
total      |           x        |      100-x         |  100


We can continue to fill out the table.

Code: Select All Code

-----------| supposed to be off | supposed to be on  |  total
actual off |    .6x             |  .1(100-x)         |  20
--------------------------------------------------------------
actual on  |    .4x             |  .9(100-x)         |  80
--------------------------------------------------------------
total      |           x        |      100-x         |  100


Our question is: What percent of the lights that are switched on are supposed to be switched off?

So we are looking for the numerical value of (.4x/80) *100% or (x/2) %

We can figure out using the lights actually on that
.4x + .9(100 - x) = 80
.4x + 90 - .9x = 80
.5x = 10
x = 20
(Alternately, you can also use the lights actually off)

so (x/2)% = 10%

I got (D)

[paula.tudorof]

Hi Ben, I chose the following matrix instead of your correct one, however I can't tell how to recognize the correct route?

here's mine:

----------| supposed to be off | supposed to be on | total
actual off | .6(100-x) | .1x | 20
--------------------------------------------------------------
actual on | .4(100-x) | .9x | 80
--------------------------------------------------------------
total | 100-x | x | 100


And the final step as:
numerical value of (.4(100-x)/80) *100%

.4(100-x) + .9x = 80
40 - .4x +.9x = 80
.5x = 40
x = 80

(.4(100-x)/80) *100% = 80/80 = 1 --> Not Correct!

It's also not an answer option.. Do you know how I can prevent going down this path and losing time? It looks like a 50/50 chance between choosing your correct matrix and this one, and I don't understand the rationale?

Thanks very much!

[dschneid2010]

All your work is correct, you just messed up your math slightly

(.4(100-x)/80) *100% = 80/80 = 1 --> Not Correct!

if x = 80 =>

(.4(100-80)/80) x 100% =
(.4(20)/80) x 100% =
(8/80) x 100% = 10% (correct)

[RonPurewal]

All your work is correct, you just messed up your math slightly

(.4(100-x)/80) *100% = 80/80 = 1 --> Not Correct!

if x = 80 =>

(.4(100-80)/80) x 100% =
(.4(20)/80) x 100% =
(8/80) x 100% = 10% (correct)

yep -- good correction.

[pmmalkan+gmat]

I understand how you got the 10%. However, I believe the question stem is a little confusing.

What percent of the lights that are switched on are supposed to be switched off?

10% of the lights that are on are supposed to be off. (i.e. 10% of 80 = 8 are on but supposed to be off)
but at the same time
8% of the total number of lights are switched on but are supposed to be off (i.e. 8% of 100 = 8 are on but supposed to be off)

Thanks

[mschwrtz]

Whatever comes after the "of" is the whole of which the perecnt expresses a part. So there is no ambiguity here.

X% of Y means (x/100)(y).

So, "what percent of the lights that are switched" means (x/100)(the # of lights that are switched on)=....

[himanshu.hpr]

EDIT: Understood the problem.

Could anyone pl. explain what is done in the last step i.e x/2 but as per the question it should be -> 0.4*x= 8 , which is not in ans choices... pl. help ????

We use a double set matrix when there are two factors, and each factor has only two distinct possibilities.

There's one factor: lights that were supposed to be switch on or off.
The other factor is: lights that actually were switched on or off.

Assuming 100 lights, our double set matrix would look like:

Code: Select All Code

-----------| supposed to be off | supposed to be on  |  total
actual off |                    |  .1(100-x)         |
--------------------------------------------------------------
actual on  |    .4x             |                    |  80
--------------------------------------------------------------
total      |           x        |      100-x         |  100


We can continue to fill out the table.

Code: Select All Code

-----------| supposed to be off | supposed to be on  |  total
actual off |    .6x             |  .1(100-x)         |  20
--------------------------------------------------------------
actual on  |    .4x             |  .9(100-x)         |  80
--------------------------------------------------------------
total      |           x        |      100-x         |  100


Our question is: What percent of the lights that are switched on are supposed to be switched off?

So we are looking for the numerical value of (.4x/80) *100% or (x/2) %

We can figure out using the lights actually on that
.4x + .9(100 - x) = 80
.4x + 90 - .9x = 80
.5x = 10
x = 20
(Alternately, you can also use the lights actually off)

so (x/2)% = 10%

I got (D)

[tim]

your equation is fine, it gives you x = 20 just like Ben's work does. notice though that the way he set up the problem the answer we are looking for is x/2..