1. **State the problem:** We have a survey with adults watching two programs: the Big Game and the New Movie. - 178 watched neither program. - 419 watched the New Movie. - 211 who watched the New Movie did not watch the Big Game. We want to find: 1. Number who watched both programs. 2. Number who watched at least one program. 3. Number who watched the Big Game. 4. Number who watched the Big Game but not the New Movie. 2. **Define variables and formulas:** Let: - $B$ = number who watched the Big Game. - $N$ = number who watched the New Movie = 419. - $B \cap N$ = number who watched both. - $B \cup N$ = number who watched at least one. - $\text{Neither}$ = 178. Total surveyed $T = B \cup N + \text{Neither}$. From the problem: - Number who watched New Movie but not Big Game = $N - B \cap N = 211$. 3. **Find number who watched both programs:** $$B \cap N = N - 211 = 419 - 211 = 208$$ 4. **Find total surveyed $T$:** $$T = B \cup N + 178$$ 5. **Find number who watched at least one program $B \cup N$:** Using the formula: $$B \cup N = B + N - B \cap N$$ We need $B$ to find $B \cup N$, so let's find $B$ first. 6. **Find number who watched the Big Game $B$:** Number who watched Big Game but not New Movie is: $$B - B \cap N$$ Number who watched neither is 178, so total surveyed is: $$T = B \cup N + 178$$ But we don't know $T$ or $B$ yet. Let's find $B$ using the fact that: Number who watched New Movie but not Big Game = 211 Number who watched both = 208 Number who watched New Movie = 419 Number who watched Big Game but not New Movie = $B - 208$ Total who watched at least one program: $$B \cup N = (B - 208) + 208 + 211 = B + 211$$ Total surveyed: $$T = B \cup N + 178 = B + 211 + 178 = B + 389$$ Since $T$ must be an integer, but unknown, we need more info. However, the problem only gives these data, so we assume $T$ is total surveyed adults. 7. **Find number who watched Big Game but not New Movie:** We can find $B$ by noting that: Number who watched New Movie but not Big Game = 211 Number who watched both = 208 Number who watched New Movie = 419 So, total who watched New Movie = 419 Number who watched Big Game but not New Movie = $B - 208$ Number who watched neither = 178 Total surveyed $T = B + 211 + 178$ But $T$ is unknown, so we cannot find $B$ exactly without $T$. Assuming $T$ is the sum of all groups: $$T = (B - 208) + 208 + 211 + 178 = B + 389$$ Since $T$ is unknown, we cannot find $B$ exactly. **However, the problem likely expects answers in terms of given data:** - Both programs: $208$ - At least one program: $T - 178 = B + 211$ - Big Game: $B$ - Big Game but not New Movie: $B - 208$ Since $T$ is not given, we cannot find numeric values for $B$, $B \cup N$, or $B - 208$. **But we can find $B$ using the fact that:** Number who watched New Movie but not Big Game = 211 Number who watched both = 208 Number who watched New Movie = 419 Number who watched Big Game but not New Movie = $B - 208$ Total who watched at least one program: $$B \cup N = (B - 208) + 208 + 211 = B + 211$$ Total surveyed: $$T = B \cup N + 178 = B + 211 + 178 = B + 389$$ Since $T$ is total surveyed adults, and $178$ watched neither, the rest watched at least one program. **Therefore, the answers are:** 1. Both programs: $208$ 2. At least one program: $T - 178$ 3. Big Game: $B$ 4. Big Game but not New Movie: $B - 208$ Without $T$ or $B$, we cannot find numeric answers for 2, 3, and 4. **But since $T$ is not given, we assume the problem wants the numeric answers for 1 and 2 as:** - At least one program = Total surveyed - Neither = $T - 178$ If $T$ is unknown, we cannot find numeric value. **Summary:** - Both programs: $208$ - At least one program: Unknown without $T$ - Big Game: Unknown - Big Game but not New Movie: Unknown **If the problem expects numeric answers, it is missing total surveyed $T$.**