Suppose that 25% of all wise people are nice and half of all the nice people are wise. Suppose further that 25% of all the people are neither wise nor nice. What percent of all the people are both wise and nice?

Answer 1

Draw a Venn diagram and put the percentages there. By trial and error, and with the help of the given information, you will find the answer.
(don't forget symmetry).

If you would like a more rigorous solution, here it is:
Let
E = set of all people
W = set of wise people
N = set of nice people

note:
|E|=cardinality of E,
= number of elements in E, etc.

Half of nice people are wise, and
half of wise people are nice
=>
|W∩N| = |W|/2 = |N|/2
=>
|W|=|N|=2|W∩N|
But by the inclusion exclusion theorem, and that 1/4 of all people are neither nice nor wise,
=>
|W|+|N|-|W∩N|=(1-1/4)|E|

Solve for |W| and |N| to get
|W|=|N|=|E|/2.