Hotel California incandescent vs. fluorescent

[cgomezblank]

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%

Why can't I use Smart Numbers for this problem? I said that there are 100 each of incandescent and fluorescent bulbs. So 40 incandescents and 90 fluorescents are turned on at this moment. Total bulbs is 200 so 80% of that is 160. I put 40/160 and get 1/4, which isn't one of the answer choices. Why doesn't this method work for this problem? Thanks!

[tim]

The main reason you can't use smart numbers is because the problem effectively asks for the ratio of one type of bulb to the other, and when you assign numbers you are forcing your own ratio that does not necessarily agree with the facts. In general, if assigning numbers directly contradicts the facts given in a problem (as you've done here), you cannot use smart numbers.

The correct approach to this problem is weighted averages. When two separate percentages balance out to a certain overall percentage (as we have here), weighted averages is the way to go.

[NickS62]

I'm having some trouble on this problem. I understand how to get the answer, but logically, it doesn't make sense to me. If the ratio of Fluorescent bulbs to Incandescent bulbs is 1F:4I, then 10% is too low for this to be possible. Smart numbers don't work because you only know the ratio, which I get. But I've listed out and tried various combinations, keeping the 4:1 ratio constant, and none of them work. Whether there were 5 bulbs, 50, 100, 250, 500, 1000, 2500, or 5000...splitting the total up to match the ratio of 4:1, then applying 40% and 90% to each type and adding them will only amount to a constant 62.5% of the total...which does not equal 80%. I can't see how 10% is possible given the constraints in the problem.

Another way to look at it is, 10% of the bulbs that are on are Incandescent. We'll say it's 8 bulbs. That would mean 80 bulbs are on (80% of the total). If the ratio is 4I:1F, then the amount of Fluorescent bulbs on is 2, when it should be 72. The ratio just doesn't work.

90% of x will always be < 40% of (4x)

So how can the answer be I<F?

[PetriF258]

Hi Nick

So you are happy that the ratio is 4I:1F

Two ways of calculating this
1) 0,4I + 0,9F = 0,8(I + F) ---> 4I = 1F
2) Weighted average method: The average is 80%. I is 40% and F is 90%. Thus the "gap" between the 90 and 80 is 10, and the gap between the 40 and 80 is 40. The F is closer to the 80 and therefore there should be more F's than I's. Thus the ratio is 4:1 and as there should be more F's than I's, the ratio is 4I = 1F.

Ok next, let's use 50. There should be 40 F's and 10 I's. That is 50 total. We know that 80% of all lights are on and therefore 40 lights are on. If we look at the I's, 40% of the I's are switched on, which equals 4 (40% x 10), similarly 36 F's are on (90% x 40). The total lights on is therefore 40 (36 + 4). To answer the question, what percentage of the lights that are on are I's? 4 / 40 = 10%. Is this the correct answer.

In you example of 8 I's, I think you made 2 mistakes: Switched the ratio and used the 4:1 ratio for the lights on, while the 4:1 ratio represents the total lights. Let's use your example. 8 I's that are on means a total of 80 lights are on. 8 I's that are on translates to 20 total I's, as 40% of the I's are on. If we have 20 I's, we must have 80 F's (ratio of 4 to 1). We therefore have 100 lights (80 + 20). How much is on? I's: 20 x 40% = 8; F's: 80 x 90% = 72. Total on = 80 which is 80% of total lights.

I hope it helps. You mentioned that you were able to get to the answer, how did you get to the answer?

[PetriF258]

Did you get the answer by using the following formula:
I's on / Total lights on
0,4I / 0,8(I+F)
0,4I / 0,8(5I)
0,4I / 4I = 10%

I think this is the fastest way after getting 4I = 1F

[RonPurewal]

nick—

• according to the weighted-average thingy, the overall ratio of F to I—including the bulbs that are off, as well as those that are on—is 4:1.

• then, %90 and %40 of those numbers, respectively, are switched on.

e.g., say there are 40 F bulbs and 10 I bulbs.
then, 0.9(40) = 36 F bulbs are on, and 0.4(10) = 4 I bulbs are on. – overall, 40/50 = %80 of the bulbs are on.
– of the bulbs that are on, only 4/40 = %10 are I-bulbs.

[TamerA271]

Hi,
Would I be able to use the teeter totter method for this one?

If I do, I get 1/5 for I and 4/5 for F but this does not match the result of 10%. Can you please tell me where I went wrong?

1/5 4/5
<--------------------->
40% 80% 90%
I F

Thanks,
Tamer

[tim]

What you have done so far is correct: You got the same 4:1 ratio that Ron did in the post above yours. You just didn't finish the problem! See Ron's post if you can't figure out what else you need to do to solve the problem.

[RonPurewal]

...not only that, but this is exactly the same point at which the original poster stopped, too.

please read the entire thread before posting! the answers you're seeking are, quite often, already there.