Step by step
The question
Question: 20 students: 9 study French, 8 study Spanish, 3 study both.
How many study neither?
1 Draw the Venn diagram
3 study both: they go in the middle first.
2 Fill the circles
Study French: 9 in all, 3 of them are in the middle.
9 minus 3 is 6.
study only French = 9 − 3 = 6
Careful!
The 3 in the middle are inside both circles.
Taking away 9 and 8 takes them away twice.
Careful: not 20 − 9 − 8 = 3 ✗
Study Spanish: 8 minus 3 is 5.
study only Spanish = 8 − 3 = 5
3 Find the outside
The circles hold 6 plus 3 plus 5: 14.
The rest of the 20 are outside: 6.
neither = 20 − (6 + 3 + 5) = 6
4 Check
Let's check: the four parts add up to 20.
So it fits!
6 + 3 + 5 + 6 = 20
Let's look back.
First: draw the Venn diagram.
Then: fill the circles.
Then: find the outside.
Last: check.
Answer
neither = 20 − (6 + 3 + 5) = 6