KA-0069Inclusion–exclusion with two sets
4 points, difficulty 2 of 3Level 5–6about 75sIn a class of 30, 18 play football and 14 play chess. 6 play both. How many play neither?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Fill in the middle of the two circles first.
2The strategy
Draw two overlapping circles and write the 6 who do both in the overlap. The rest of each circle is what is left after taking the overlap out, and everything outside is the answer.
3The full solution
Playing at least one is 18 + 14 - 6 = 26. So 30 - 26 = 4 play neither.
Solution— no shortcut on this one
Adding 18 and 14 counts the 6 who do both twice, so the number playing at least one is 18 + 14 - 6 = 26. Everyone else plays neither: 30 - 26 = 4. Drawing it confirms the parts: 12 football only, 6 both, 8 chess only, 4 neither — and 12 + 6 + 8 + 4 = 30. The transferable idea: fill the overlap in first, then the rest follows.
Why this is on the test: Two overlapping groups is one of the most reliably examined shapes at this level, and the double-counted overlap is the whole difficulty.