Step by step
The question
Question: 50 students: 30 take art, 25 take music, 12 take both.
How many take neither?
1 Draw the Venn diagram
12 take both: they go in the middle first.
2 Fill the circles
Take art: 30 in all, 12 of them are in the middle.
30 minus 12 is 18.
take only art = 30 − 12 = 18
Careful!
The 12 in the middle are inside both circles.
Taking away 30 and 25 takes them away twice.
Careful: not 50 − 30 − 25 = −5 ✗
Take music: 25 minus 12 is 13.
take only music = 25 − 12 = 13
3 Find the outside
The circles hold 18 plus 12 plus 13: 43.
The rest of the 50 are outside: 7.
neither = 50 − (18 + 12 + 13) = 7
4 Check
Let's check: the four parts add up to 50.
So it fits!
18 + 12 + 13 + 7 = 50
Let's look back.
First: draw the Venn diagram.
Then: fill the circles.
Then: find the outside.
Last: check.
Answer
neither = 50 − (18 + 12 + 13) = 7