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To solve this problem, I drew a graph shown below and tried to solve the problem using the graph. I found it very complicated, so I don't believe that I am on right track. What is the easy way to solve the problem? (I try to include the graph, but the system does not allow me to do so because I am new here.)

In a small town of 100 men, 85 are married, 70 have a telephone, 75 own a car, and 80 own their own home. On this basis, what is the least possible number of men who are married, have their own telephone, own their own car, and own their own house?

math
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  • Well, it is complicated, have a look at this page: https://www.mbatious.com/prepnote/how-deal-four-sets-set-theory – zar Oct 17 '16 at 15:38

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Suppose that we're deliberately trying to distribute wives spouses, telephones, cars and homes such that the overlap is minimized.

First, $85+70=155$ so we need to give a telephone to at least $55$ of the married men.

$55+75 = 130$, so we need to give a car to at least $30$ of the married men with telephones.

$30+80 = 110$, so we need to let at least $10$ of the married men with telephones and cars own their home.

Alternatively (and more straightforwardly): Start by assuming that everyone have everything. Then, to get the numbers right, we need to take away 15 wives (let's face it, if phone ownership is only 70%, this clearly takes place in a time where the spouse of a man is a wife), 30 phones, 25 cars and 20 homes.

There's only $15+30+25+20=90$ things in total we can take away, so the exercise leaves at least $10$ men having everything -- or possibly more, if there are men that had two or more things taken from them.

hmakholm left over Monica
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In order to make the least possible men with everything, we need to maximize the amount of men who don't have everything.

That is, the 15 men who are not married do have a telephone, car, and house. Of the other 85 men, 55 have a telephone, 60 have a car, and 65 have their own home.

The 30 men who don't have a telephone do have a car and house. Of the other 55 men, 30 have a car, and 35 have their own home.

The 25 men who don't have a car do have a home. Of the other 30 men, 10 have their own home.

And there we have it, we are forced to give everything to 10 people.

Kaynex
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