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?

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.

To find the percentage of people who are both wise and nice, we can solve this step-by-step:

Step 1: Let's assume there are 100 people in total.

Step 2: According to the given information, 25% of all wise people are nice. Therefore, the number of nice people is (25/100) * 100 = 25.

Step 3: It is also given that half of all the nice people are wise. Hence, the number of wise people is (1/2) * 25 = 12.5 ≈ 13.

Step 4: Now, let's find the percentage of people who are neither wise nor nice. It is given that 25% of all people are neither wise nor nice. Therefore, the number of people who are neither wise nor nice is (25/100) * 100 = 25.

Step 5: Finally, let's calculate the percentage of people who are both wise and nice. The total number of people who are both wise and nice is 13. As we assumed 100 people in total, the percentage is (13/100) * 100 = 13%.

Therefore, 13% of all the people are both wise and nice.

To find the percent of people who are both wise and nice, we can use a technique called Venn diagrams. Let's break down the information given:

We are told that 25% of all wise people are nice. This means that if we divide all wise people into four equal groups, one of those groups will be nice. Let's label this region of the Venn diagram as "Nice Wise."

Next, we are told that half of all nice people are wise. This means that if we divide all nice people into two equal groups, one of those groups will be wise. Let's label this region as "Wise Nice."

Now, we know that 25% of all people are neither wise nor nice. Let's label this region as "Neither Wise nor Nice."

To find the percentage of people who are both wise and nice, we need to find the intersection between the "Nice Wise" and "Wise Nice" regions.

Since the regions "Nice Wise" and "Wise Nice" share the same group, we can think of it as the total percentage of nice people or the total percentage of wise people.

Let's assume there are 100 people in total for easy calculations. Then, 25% of them (25 people) are neither wise nor nice. This means that the remaining 75 people are either wise or nice.

Since half of all nice people are wise, we can take half of the nice people to find the number of people in the "Wise Nice" region. So, half of 75 nice people is 37.5 people.

Therefore, 37.5 people out of the total 100 are both wise and nice, which means the percentage of people who are both wise and nice is 37.5%.

In conclusion, the percent of all people who are both wise and nice is 37.5%.