Step by step
The question
Question: 40 children: 22 have a cat, 15 have a dog, 6 have a both.
How many have a neither?
1 Draw the Venn diagram
6 have a both: they go in the middle first.
2 Fill the circles
Have a cat: 22 in all, 6 of them are in the middle.
22 minus 6 is 16.
have a only cat = 22 − 6 = 16
Careful!
The 6 in the middle are inside both circles.
Taking away 22 and 15 takes them away twice.
Careful: not 40 − 22 − 15 = 3 ✗
Have a dog: 15 minus 6 is 9.
have a only dog = 15 − 6 = 9
3 Find the outside
The circles hold 16 plus 6 plus 9: 31.
The rest of the 40 are outside: 9.
neither = 40 − (16 + 6 + 9) = 9
4 Check
Let's check: the four parts add up to 40.
So it fits!
16 + 6 + 9 + 9 = 40
Let's look back.
First: draw the Venn diagram.
Then: fill the circles.
Then: find the outside.
Last: check.
Answer
neither = 40 − (16 + 6 + 9) = 9