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%
Ans is D i.e 10%
Here is the solution that I feel is correct.
Number of ligts = x
Number of ligts supposed to be on = y
Number of lights supposed to be off = x-y
At the moment 08x lights are on. This contains
0.9y (because 10% of the lights that are supposed to be on are off) +
0.4 (x-y) (because 40% of the lights that are supposed to be off are on)
so,
0.8x = 0.9y + 0.4 (x-y)
or
y=(4/5)x
Now, number of lights that are on but are supposed to be off =
0.4 (x-y) = 0.4 (x-4x/5) = 0.4x/5 = 0.08x
Percentage = 0.08x/0.8x = 1/10= 10%
Dear D’Zire
I agree upto where y=(4/5)x.
But the question is “What percent of the lights that are on are supposed to be off?”. i.e. we are supposed to find lights that are supposed to be off [ i.e. (x-y) = x - (4/5)x = x/5], as a % of lights that are actually on [i.e. 0.8x]. In other words, we are supposed to find the lights that are ‘supposed’ to be off – MAY ACTUALLY BE ON OR MAY ACTUALLY BE OFF- as be % of lights that are on. The answer, then is 25% (no option matches this)
Whereas, if one were to follow the second equation given by you, the question would have to be: “What percent of the lights that are on are THOSE THAT ARE ON but supposed to be off?” i.e. 0.4(x-y) as % of 0.8X.
Very confusing indeed!
IMO D
Let lights to be switched off be X
Let lights to be switched on be y
Also assume that the total lights be 100 (for sake of simplicity)
Acc to ques,
0.6x + 0.1y = 20
0.4x + 0.9y = 80
Solving we get 0.4x = 8
Hence ans is 10%
Here’s another approach:
x= Number of lights supposed to be ON.
y= Number of lights supposed to be OFF.
40% of the ones that are supposed to be OFF are ON = 0.4y
“10% of the ones that are supposed to be ON are OFF ” this implies 90% of the ones that are supposed to be ON are ON = 0.9x
Total that are actually ON = 0.4y + 0.9x
Total lights is the same, irrespective of supposed or actual = (x+y)
Since 80% of the total is actually on -
0.4y + 0.9x = 0.8 (x + y)
0.1 x= 0.4 y
x/y = 4
percent of the lights that are on are supposed to be off
= (lights that are on but are supposed to be off) / (Total Lights)
= 0.4 y / 0.8 * (x + y) = 10 %
Percentage supposed to be off: X
Percentage Supposed to be on: 100 -X
Thus if the number on at a particular time are 80
and of these 40 % are what were supposed to be off and 10 % of which were supposed to be on are actually off, then this can be represented as :
80 = 0.4X +0.9(100 – X)
80 = 0.4 X +90 – 0.9 X
80 = 90 – 0.5X
X = 10/0.5
X = 20 => % Supposed to be Off
Now since 10% of the lights supposed to be on are actually off, this works out to…
0.1(100 – X) = 0.1 *80 = 8
and the percentage of lights that are on, but supposed to be off thus is (100*8/80 ) = 10 %
Choice D
I would go with answer choice D- 10%.
@ deedee
I got your confusion. The question asked is what percentage of lights on are supposed to be off. So we need to calculate the percentage with respect to lights that are on and hence 10%
D: