Here is the posted problem, along with the answer choices:
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%
This is an EITHER/OR group problem.
Every light is EITHER actually on OR actually off.
Every light is EITHER supposed to be on OR supposed to be off.
To organize the data, use a GROUP GRID:
In the grid above, every row has to add up to the total, as does every column.
Looking at the top row, (supposed to be on and actually on) + (supposed to be on but actually off) = total supposed to be on.
Looking at the left-most column, (actually on and supposed to be on) + (actually on but supposed to be off) = total actually on.
Now let's fill in the data step by step.
Whenever we have 2 entries in a row or column, we can determine the remaining entry, since everything has to add up horizontally and vertically.
Let the total = 100.
Since 80% of the lights are actually on, we get:
What percent of the lights that are on are supposed to be off?:
We can PLUG IN THE ANSWERS for the lights that are actually on but are supposed to be off.
Since all of the percentages in the problem (80%, 40%, 10%) are multiples of 10, the correct answer choice is almost certainly itself a multiple of 10.
Answer choice D: 10% of the lights that are actually on are supposed to be off
Since 80 lights are actually on, .1(80) = 8 of these lights are supposed to be off.
Thus:
Forty percent of the lights that are supposed to be off are actually on.
Since the grid above indicates that (supposed to be off but actually on) = 8, we get:
8 = .4 * (total supposed to be off)
Total supposed to be off = 20.
Thus:
Ten percent of the lights that are supposed to be on are actually off.
If we have plugged in the correct answer choice, this condition will be satisfied by the values in the grid above.
In the grid above:
(supposed to be on but actually off) / (total supposed to be on) = 8/80 = 1/10 = 10%.
Success!
The correct answer is
D.
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