Question: On a specific day, a campus cafeteria kept track of the overall number of purchases made, along with how many had entr\'e\'es, salads, or desserts. The data is organized in the Venn diagram below. Entr\'e\'e Salad 45 40 96 41 75 91 80 58 Dessert How many purchases were made in the campus cafeteria that day? What does the number 40 in the diagram mean? 40 purchases included an entr\'e\'e, but neither a salad nor a dessert. 40 people preferred the entr\'e\'e to anything else. 40 entr\'e\'es were purchased that day. Neither of these statements properly describe what 40 means in the diagram. Which of these items was purchased the most that day? It may be necessary to select more than one. The salad The dessert The entr\'e\'e None of these items was purchased the most because they all had the same number of purchases. graph: a three-circle Venn diagram inside a rectangle, with Entr\'e\'e in the top-left circle, Salad in the top-right circle, and Dessert in the bottom circle; counts are 40 in entr\'e\'e-only, 96 in salad-only, 80 in dessert-only, 45 in entr\'e\'e-salad overlap only, 41 in entr\'e\'e-dessert overlap only, 91 in salad-dessert overlap only, 75 in the center triple-overlap, and 58 outside all circles; position_hint=center

1. **State the problem:** We have a Venn diagram showing purchases of entr\'e\'es, salads, and desserts with various counts in each region. We need to find: - The total number of purchases made that day. - The meaning of the number 40 in the diagram. - Which item(s) was purchased the most. 2. **Calculate total purchases:** The total purchases include all regions inside and outside the circles. The numbers given are: - Entr\'e\'e only: $40$ - Salad only: $96$ - Dessert only: $80$ - Entr\'e\'e and Salad only: $45$ - Entr\'e\'e and Dessert only: $41$ - Salad and Dessert only: $91$ - All three (Entr\'e\'e, Salad, Dessert): $75$ - Outside all circles: $58$ Total purchases = $40 + 96 + 80 + 45 + 41 + 91 + 75 + 58$ 3. **Sum the total:** $$ 40 + 96 = 136 \\ 136 + 80 = 216 \\ 216 + 45 = 261 \\ 261 + 41 = 302 \\ 302 + 91 = 393 \\ 393 + 75 = 468 \\ 468 + 58 = 526 $$ So, total purchases = $526$. 4. **Interpret the number 40:** The number 40 is in the Entr\'e\'e-only region, meaning these purchases included an entr\'e\'e but neither a salad nor a dessert. - It does NOT mean 40 people preferred entr\'e\'e to anything else. - It does NOT mean 40 entr\'e\'es were purchased in total (since other regions include entr\'e\'es too). Therefore, the correct interpretation is: **40 purchases included an entr\'e\'e, but neither a salad nor a dessert.** 5. **Determine which item was purchased the most:** Calculate total purchases for each item by summing all regions that include that item: - Entr\'e\'e total = Entr\'e\'e only + Entr\'e\'e-Salad + Entr\'e\'e-Dessert + All three $$40 + 45 + 41 + 75 = 201$$ - Salad total = Salad only + Entr\'e\'e-Salad + Salad-Dessert + All three $$96 + 45 + 91 + 75 = 307$$ - Dessert total = Dessert only + Entr\'e\'e-Dessert + Salad-Dessert + All three $$80 + 41 + 91 + 75 = 287$$ 6. **Compare totals:** - Salad: $307$ - Dessert: $287$ - Entr\'e\'e: $201$ The salad was purchased the most. **Final answers:** - Total purchases: $526$ - The number 40 means purchases with entr\'e\'e only (no salad or dessert). - The salad was purchased the most that day.