Question

question 1 of 6 (1 point) | question attempt: 1 of unlimited
a college food court surveyed students to gather information about the types of drinks they like.

• 689 students were surveyed in all
• 271 of the students drink soda
• 93 of the students drink coffee, but drink neither soda nor milk
• 36 of the students drink soda, but drink neither milk nor coffee
• 62 of the students drink milk, but drink neither soda nor coffee
• 86 of the students drink both soda and coffee, but dont drink milk
• 23 of the students drink both soda and milk, but dont drink coffee
• 585 of the students drink at least one of these three types of drinks (soda, milk, coffee)

construct a venn diagram illustrating these results. then answer the question.
image: venn diagram with three circles (drink soda, drink milk, drink coffee), \all students in the survey\ label, and a text box: \how many of the students drin 370 students\

Answer:

To solve the problem of filling in the Venn diagram, we analyze each region based on the given data:

Step 1: Only Soda (Soda ∩ ¬Milk ∩ ¬Coffee)

Given: 36 students drink soda but neither milk nor coffee.
So, the region for only soda is \( 36 \).

Step 2: Only Milk (Milk ∩ ¬Soda ∩ ¬Coffee)

Given: 62 students drink milk but neither soda nor coffee.
So, the region for only milk is \( 62 \).

Step 3: Only Coffee (Coffee ∩ ¬Soda ∩ ¬Milk)

Given: 93 students drink coffee but neither soda nor milk.
So, the region for only coffee is \( 93 \).

Step 4: Soda and Coffee, not Milk (Soda ∩ Coffee ∩ ¬Milk)

Given: 86 students drink both soda and coffee, but not milk.
So, the region for soda and coffee (not milk) is \( 86 \).

Step 5: Soda and Milk, not Coffee (Soda ∩ Milk ∩ ¬Coffee)

Given: 23 students drink both soda and milk, but not coffee.
So, the region for soda and milk (not coffee) is \( 23 \).

Step 6: Total in Soda Circle

The total number of students who drink soda is 271. We know the regions: only soda (\( 36 \)), soda & coffee (not milk) (\( 86 \)), soda & milk (not coffee) (\( 23 \)), and the intersection of all three (soda, milk, coffee). Let the intersection of all three be \( x \).
So, \( 36 + 86 + 23 + x = 271 \).
Solving: \( 145 + x = 271 \) ⟹ \( x = 271 - 145 = 126 \).
Thus, the region for all three (soda, milk, coffee) is \( 126 \).

Step 7: Total in Milk Circle (at least one drink)

The total number of students who drink at least one of the three drinks is 585. We know the regions: only soda (\( 36 \)), only milk (\( 62 \)), only coffee (\( 93 \)), soda & coffee (not milk) (\( 86 \)), soda & milk (not coffee) (\( 23 \)), all three (\( 126 \)), and the region for milk & coffee (not soda). Let the region for milk & coffee (not soda) be \( y \).
So, \( 36 + 62 + 93 + 86 + 23 + 126 + y = 585 \).
Calculating the sum: \( 36 + 62 = 98 \); \( 98 + 93 = 191 \); \( 191 + 86 = 277 \); \( 277 + 23 = 300 \); \( 300 + 126 = 426 \).
Thus, \( 426 + y = 585 \) ⟹ \( y = 585 - 426 = 159 \).
Wait, no—actually, the milk circle includes only milk (\( 62 \)), soda & milk (not coffee) (\( 23 \)), all three (\( 126 \)), and milk & coffee (not soda) (\( y \)). Let's re-express:
Total in milk circle: \( 62 + 23 + 126 + y \). But we can also use the total at least one drink. Alternatively, let's check the coffee circle.

Step 8: Coffee Circle (at least one drink)

The coffee circle includes only coffee (\( 93 \)), soda & coffee (not milk) (\( 86 \)), all three (\( 126 \)), and milk & coffee (not soda) (\( y \)).
Total in coffee circle: \( 93 + 86 + 126 + y \). But we know the total at least one drink is 585. Let's verify with the total:
Only soda: 36, Only milk: 62, Only coffee: 93, Soda&Coffee:86, Soda&Milk:23, All three:126, Milk&Coffee: \( y \).
Sum: \( 36 + 62 + 93 + 86 + 23 + 126 + y = 585 \).
As before, \( 36+62=98 \), \( 98+93=191 \), \( 191+86=277 \), \( 277+23=300 \), \( 300+126=426 \). So \( 426 + y = 585 \) ⟹ \( y = 159 \). Wait, but let's check the milk total. The milk circle has only milk (62), soda&milk (23), all three (126), and milk&coffee (159). So total in milk circle: \( 62 + 23 + 126 + 159 = 370 \), which matches the given "370 students" (probably the total in milk circle).

Step 9: Outside All Three Circles

Total students surveyed: 689. Students who drink at least one drink: 585. So students who drink none: \( 689 - 585 = 104 \).

Filling the Venn Diagram (from left to right, top to bottom, as per the diagram):

1. Only Soda: \( 36 \)
2. Soda & Milk (not Coffee): \( 23 \)
3. Only Milk: \( 62 \)
4. All Three (Soda, Milk, Coffee): \( 126 \)
5. Soda & Coffee (not Milk): \( 86 \)
6. Milk & Coffee (not Soda): \( 159 \)
7. Only Coffee: \( 93 \)
8. Outside All Circles: \( 104 \)

Final Answer (for the Venn diagram regions, as per the order in the diagram):

• Only Soda: \( \boldsymbol{36} \)
• Soda & Milk (not Coffee): \( \boldsymbol{23} \)
• Only Milk: \( \boldsymbol{62} \)
• All Three: \( \boldsymbol{126} \)
• Soda & Coffee (not Milk): \( \boldsymbol{86} \)
• Milk & Coffee (not Soda): \( \boldsymbol{159} \)
• Only Coffee: \( \boldsymbol{93} \)
• Outside: \( \boldsymbol{104} \)

(Note: The diagram’s regions are filled as per the above calculations. If the question was about a specific region, adjust accordingly. For example, if the question was "How many drink all three?", the answer is \( 126 \); if "How many drink milk only?", it’s \( 62 \), etc.)