Let's analyze the information provided and answer each question. I'll start by identifying each segment of the Venn diagram and then calculating the specific values required.

Information Given:

1. 132 teachers plan to grade papers.
2. 102 teachers plan to attend a football game.
3. 120 teachers plan to do household chores.
4. 15 teachers plan to grade papers and attend a football game.
5. 67 teachers plan to grade papers and do household chores.
6. 43 teachers plan to attend a football game and do household chores.
7. 9 teachers plan to grade papers, attend a football game, and do household chores.

Venn Diagram Interpretation:

Let's label the sets:

• $G$: Grading papers
• $F$: Football game
• $C$: Household chores

Now, using the principle of inclusion-exclusion, we can answer the questions.

a. How many planned to only grade papers?

The formula for those who planned to only grade papers ($G$ only) is: $G - (G \cap F) - (G \cap C) + (G \cap F \cap C)$ Plugging in the numbers: $132 - 15 - 67 + 9 = 59$ So, 59 teachers planned to only grade papers.

b. How many planned to grade papers or do household chores?

The formula for the union of $G$ and $C$ (those who planned to grade papers or do household chores) is: $G + C - (G \cap C)$ $132 + 120 - 67 = 185$ Thus, 185 teachers planned to grade papers or do household chores.

c. How many did not plan to do any of these activities?

The formula for those not participating in any activity is: $\text{Total} - (G \cup F \cup C)$ To find $G \cup F \cup C$: $G + F + C - (G \cap F) - (G \cap C) - (F \cap C) + (G \cap F \cap C)$ $= 132 + 102 + 120 - 15 - 67 - 43 + 9 = 238$ Therefore, those not planning any activities are: $255 - 238 = 17$ So, 17 teachers did not plan to do any of these activities.

d. How many planned to do at least two of these activities?

This is the sum of those in exactly two groups plus those in all three: $(G \cap F - G \cap F \cap C) + (G \cap C - G \cap F \cap C) + (F \cap C - G \cap F \cap C) + (G \cap F \cap C)$ $= (15 - 9) + (67 - 9) + (43 - 9) + 9 = 6 + 58 + 34 + 9 = 107$ Thus, 107 teachers planned to do at least two of these activities.

e. How many planned to attend a football game and do household chores but did not have any plans to grade?

This is simply those in $F \cap C$ but not in $G \cap F \cap C$: $F \cap C - G \cap F \cap C = 43 - 9 = 34$ So, 34 teachers planned to attend a football game and do household chores, but did not plan to grade papers.

Would you like a more detailed breakdown of any specific part of the solution?

Follow-up Questions:

1. How would the answers change if 10 additional teachers planned to do all three activities?
2. What percentage of teachers planned only one activity?
3. How many teachers planned to do exactly one activity?
4. What is the probability that a randomly chosen teacher planned to attend a football game?
5. How could we represent these answers visually using percentages?

Tip: For Venn diagram problems, always start by filling in the intersection of all three sets and work outward to avoid double-counting.