There are $115,226,802$ households in the US. $58$% of households have at least $1$ musician. $43$% have $2$ or more musicians. How many musicians are there?
Any help would be appreciated...
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How big is a household? – John Habert Feb 11 '14 at 16:15
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census data shows 2.61 persons per households – brendan Feb 11 '14 at 16:18
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The 58 are 43 are percentages, guess I forgot to add symbol – brendan Feb 11 '14 at 16:19
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You didn't forget. They accidental got edited out. – John Habert Feb 11 '14 at 16:21
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thanks guys, is it possible to get the answer from this data? – brendan Feb 11 '14 at 16:22
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It is possible to get an estimate because the 2 or more along with average people per household mean the answer won't be perfectly accurate. – John Habert Feb 11 '14 at 16:24
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ok, what is the best method for getting that estimate? – brendan Feb 11 '14 at 16:28
2 Answers
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For a minimum, we have $15\%$ of the households with exactly one musician, so there are at least $115,226,802*(0.15+2*0.43)=116,379,070.02$ musicians. I'm not sure what to do with $0.02$ musician. There could be more, as there could be some households with more than two.
Ross Millikan
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Your units are off. In the comments to the OP, it is mentioned to use 2.61 people per household. – John Habert Feb 11 '14 at 16:33
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@JohnHabert: I don't think we care how many people there are per household. We would need that if we wanted to assess what fraction of the total population is musicians. As it is, my calculation would apply even if there were $10$ people per household-there would still be at least that many musicians. – Ross Millikan Feb 11 '14 at 16:36
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Good point. And the answer you get comes out close enough to an integer that it makes it look like a book solution. – John Habert Feb 11 '14 at 16:39
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ok, so 15% shows households with exactly 1 musician because it is the difference between 58% with at least 1 and 43% with 2 or more. You found that proportion and added to it the 43% with at least 2 – brendan Feb 11 '14 at 16:41
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@JohnHabert: I think the closeness to the integer is an artifact. We have $1.01$ musicians per household, so the $.02$ is just the lower two digits of the number of households. – Ross Millikan Feb 11 '14 at 16:41
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You can only narrow it down.
Assuming: * "2.61 persons per household" * 58% of household with at least 1 musician is exact-- 42% have no musician at all and likewise, * 43% percent have 2+ musicians and thus 57% have exactly 0 or 1 musician
Let T=115,226,802.
a.) 15%*T houses have exactly 1 musician --> 15%*T musicians
b.) 43%*T houses have at least 2 --> 43%*T*2 musicians
c.) 42%*T non-musicians
min.#musicians is 101%*T.
min.#non-musicians is 42%*T.
Total population is 100%*T*2.61=T*2.61=261%T
[261-(101+42)]%T=117%T=1.17*115,226,802 people can be anything.
In the expected case, it's all evenly split: there's 2.61 persons living in each and every household.
Then, it gets narrower:
From (a) above, 15%T musicians and 15%*T*1.61 non-musicians.
From (c) 42%*T*2.61 non-musicians.
the remaining 0.61*43%*T persons-- the rest of the household in 2+musician houses is vague, can't be told without more on how it is distributed beyond 2+/household.
Roam
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