Step by step
The question
Question: 32 students: 20 play football, 14 play tennis, 9 play both.
How many play neither?
1 Draw the Venn diagram
The box holds all 32 students.
One circle for football.
One circle for tennis.
The middle is both.
9 play both: they go in the middle first.
2 Fill the circles
Play football: 20 in all, 9 of them are in the middle.
20 minus 9 is 11.
play only football = 20 − 9 = 11
Careful!
The 9 in the middle are inside both circles.
If we just take 20 and 14 from 32, the 9 in the middle are taken away twice.
Careful: not 32 − 20 − 14 = −2 ✗
Play tennis: 14 minus 9 is 5.
play only tennis = 14 − 9 = 5
3 Find the outside
The circles hold 11 plus 9 plus 5: 25.
The rest are outside: 32 minus 25 is 7.
neither = 32 − (11 + 9 + 5) = 7
4 Check
Let's check: the four parts add up to 32.
So it fits!
11 + 9 + 5 + 7 = 32
Let's look back.
First: draw the Venn diagram.
Then: fill the circles.
Then: find the outside.
Last: check.
Answer
neither = 32 − (11 + 9 + 5) = 7