There are 30 students in a class. 15 study woodwork and 13 study metal work. 6 study neither of the 2…

Analyze the question: This is a two-set Venn diagram problem. We need the number who study woodwork only.

Concept: For two sets W and M in a total group:

• Students in at least one subject = Total − Neither
• |W ∪ M| = |W| + |M| − |W ∩ M|
• Woodwork only = |W| − |W ∩ M|

Solution:

1. Students studying at least one subject: 30 − 6 = 24

2. Find the number studying both (let it be x): 15 + 13 − x = 24 28 − x = 24 x = 4

3. Woodwork but not metal work: 15 − 4 = 11

Check: Woodwork only = 11, both = 4, metal only = 13 − 4 = 9, neither = 6. 11 + 4 + 9 + 6 = 30 ✓

Tip: In "neither" problems, subtract the "neither" group from the total first to get the union. Then use |W ∪ M| = |W| + |M| − |W ∩ M| to find the overlap. Always check that all regions add up to the total.