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In a school, 30 teachers teach either French or Math. Out of these, 15 teach French, and 7 teach both subjects. How many teachers teach only Math?
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Solution: Step 1: Let's denote F as the set of teachers teaching French and M as the set of teachers teaching Math. Step 2: Given that |F| = 15, |F ∩ M| = 7, and |F ∪ M| = 30. Step 3: We need to find |M| - |F ∩ M|, which represents teachers teaching only Math. Step 4: Using the principle of inclusion-exclusion: |F ∪ M| = |F| + |M| - |F ∩ M|. Step 5: Rearrange to solve for |M|: |M| = |F ∪ M| - |F| + |F ∩ M|. Step 6: Substitute known values: |M| = 30 - 15 + 7 = 22. Step 7: Teachers teaching only Math = |M| - |F ∩ M| = 22 - 7 = 15. Step 8: Therefore, 15 teachers teach only Math.
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