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Question

In a rehabilitation programme, a group of 50 families were assured new houses and compensation by the government. Number of families who got both is equal to the number of families who got neither of the two. The number of families who got new houses is 6 greater than the number of families who got compensation. How many families got houses?

(a) 22
28
23
25

The correct answer is: 28

Let $A$ and $B$ denote respectively the sets of families who got new houses and compensation
It is given that
$n open parentheses A intersection B close parentheses equals n left parenthesis stack A union B with minus on top right parenthesis$
$rightwards double arrow n open parentheses A intersection B close parentheses equals 50 minus n open parentheses A union B close parentheses$
$rightwards double arrow n open parentheses A close parentheses plus n open parentheses B close parentheses equals 50$
$rightwards double arrow n open parentheses B close parentheses plus 6 plus n open parentheses B close parentheses equals 50 blank left square bracket because n open parentheses A close parentheses equals n open parentheses B close parentheses plus 6 blank left parenthesis g i v e n right parenthesis right square bracket$
$rightwards double arrow n open parentheses B close parentheses equals 22 rightwards double arrow n open parentheses A close parentheses equals 28$

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