Scarves, gloves and winter outfit

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Scarves, gloves and winter outfit

Postby Guest » Wed Jan 03, 2018 10:08 am

We have carried out a research among the school students on a snowy and chilly day, as to wether they are adequately dressed for the cold. Below are the results:
No kid was wearing all three of scarf, gloves and hat. 19 in 100 kids did not wear exactly one of the three accessories. At least 67 in every 100 kids did not wear scarf. At least 83 in every 100 kids did not wear gloves. At least 73 in every 100 kids did not wear hat.
What is the minimum number of kids (in every 100) that did not wear any of the three accessories?

Below are my attempts:
I tried to first get the boundaries: First condition says that 81% of the kids were wearing exactly 2 accessories, since 3 is not possible. Similarly, at most 33% of the kids were wearing scarf, at most 17% gloves and at most 27% hat. If they were all wearing different accessories each, then at most 33+17+27=77 would be wearing something, so minimum 23 would not be wearing anything and this is the lower boundary. But we know that 81 kids were wearing exactly 2 accessories. So at maximum 17 are wearing both scarf and gloves, at maximum 27 scarf and hat and at maximum 17 gloves and hat.
How do we continue?
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