Problem Solving

Not sure how to attack this problem

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%

Tame the CAT wrote:Not sure how to attack this problem
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%

Let there be 100 bulbs
80% implies 80 are On and therefore 20 are Off
40% that are supposed to be off are on implies 40% of 20 = 8
ten percent of the lights that are supposed to be on are actually off =10% of 80 = 8

Number of lights that are on but supposed to be Off = 8
Number of lights that are on = 80
Required Percentage = 8/80 = 10%

Hi Neo...I do not agree with the solution for one basic reason....lights which are "off" (20 in your solution) does not represent the total number of lights supposed to be off...How can we use it interchangebly?

I used a table type set-up to try to solve this problem. For each light, there are four possible situations:

(1) Supposed to be on, Is on
(2) Supposed to be on, Is off
(3) Supposed to be off, Is on
(4) Supposed to be off, Is off

If we assume 100 lights, we know the actual on and actual off are 80 and 20. Then if we let X be the number supposed to be on and Y be the number supposed to be off, we can fill in a grid like this...

______________Supposed____________
__________ON__________OFF_______
A
c___ON____0.9X_________0.4Y________80
t
u___OFF___0.1X_________0.6Y________20
a
l__________X_____________Y________100

Now, we can get 2 equations, such as

X + Y = 100
0.1X + 0.6Y = 20,

which we can solve for Y=20, X=80. Then the grid can be filled in as

______________Supposed____________
__________ON__________OFF_______
A
c___ON____72____________8________80
t
u___OFF____8____________12________20
a
l__________80____________20________100

Then, we are interested in the percent of lights that are actually on (80) that are supposed to be off ( 8 ). So 10%.

Beautiful approach to this problem. I like this tabular format.

Let the % of lights that are supposed to be on = x
Then the lights that are supposed to be off = 100-x
Also it says that 40% of lights that are supposed to be
off are on , i.e 40*(100-x)
& 10% of those supposed to be on are off , ie , 90% of them are on
ie, 90x. Totally 80% are on.
That is , 40*(100-x) + 90x = 80

Solving for x , you get x = 80%
ie 80 % are supposed to be on & 20 % are to be off.
Next it says "forty percent of the lights that are supposed to be off are actually on" or 40% of 20% = 8% of total lights.
Question is " What percent of the lights that are on are supposed to be off?", which is equal to 8 % / 80 % = 10%
Option D is the correct answer.