1. **Problem statement:** Find the smallest possible number of people who speak both Spanish and French, and the largest possible number of people who speak neither Spanish nor French in a group of 42 people where 30 speak Spanish and 20 speak French. 2. **Relevant formulas and rules:** - Let $S$ be the set of people who speak Spanish, $F$ be the set of people who speak French. - The number of people who speak both languages is $|S \cap F|$. - The number of people who speak at least one language is $|S \cup F| = |S| + |F| - |S \cap F|$. - The number of people who speak neither language is $|E| - |S \cup F|$, where $|E|=42$ is the total number of people. 3. **Find smallest $|S \cap F|$:** - Since $|S|=30$, $|F|=20$, the smallest intersection occurs when overlap is minimal. - Minimum intersection is $|S| + |F| - |E| = 30 + 20 - 42 = 8$. - If this is negative, minimum intersection would be zero, but here it is positive, so minimum intersection is 8. 4. **Find largest number who speak neither:** - Largest number who speak neither means smallest $|S \cup F|$. - Minimum $|S \cup F| = |S| + |F| - \text{max intersection}$. - Maximum intersection is the smaller of $|S|$ and $|F|$, which is 20. - So minimum $|S \cup F| = 30 + 20 - 20 = 30$. - Number who speak neither = $42 - 30 = 12$. **Final answers:** - Smallest number who speak both Spanish and French: $8$ - Largest number who speak neither Spanish nor French: $12$