college food court surveyed students to gather information about the types of drinks they like. Here are the results. - 699 students were surveyed in all - 375 of the students drink soda - 322 of the students drink tea - 205 of the students drink both soda and tea - 255 of the students drink both tea and milk - 270 of the students drink both soda and milk - 163 of the students drink all three types of drinks (soda, tea, milk) - 123 of the students don't drink any of these three types of drinks (soda, tea, milk) Construct a Venn diagram illustrating these results. Then answer the question. All students in the survey

We are told that 699 students were surveyed and given the following information: • 375 drink soda • 322 drink tea • (An unknown total) drink milk • 205 drink both soda and tea • 270 drink both soda and milk • 255 drink both tea and milk • 163 drink all three (soda, tea, milk) • 123 drink none of these drinks Our goal is to “fill in” a Venn diagram showing the number of students in each region and then check that all parts add up to 699. Step 1. Find the number of students who drink at least one of the drinks. Since 123 drink none, the number who drink at least one is   576 = 699 – 123 Step 2. Label the regions of the three-set Venn diagram. We denote:   S = set of students who drink soda   T = set of students who drink tea   M = set of students who drink milk The typical Venn diagram regions are:  1. Only S  2. Only T  3. Only M  4. S ∩ T (but not M)  5. S ∩ M (but not T)  6. T ∩ M (but not S)  7. S ∩ T ∩ M (all three) We are directly told that the number in the triple overlap (all three) is 163. Since the pair counts given include those in the triple overlap, we remove the triple count to get the “only” pair parts. Step 3. Compute the regions that are in exactly 2 of the sets. • For soda and tea:  Given S ∩ T = 205, so the part that is only soda and tea is   42 = 205 – 163 • For soda and milk:  Given S ∩ M = 270, so the part that is only soda and milk is   107 = 270 – 163 • For tea and milk:  Given T ∩ M = 255, so the part that is only tea and milk is   92 = 255 – 163 Step 4. Find the “only” parts by subtracting the overlaps from the total for that drink. For soda (S):  Total soda drinkers = 375. This set includes:   – Those who drink only soda   – Those in S ∩ T "only" (42)   – Those in S ∩ M "only" (107)   – Those who drink all three (163) Thus, the number who drink only soda is   Only S = 375 – (42 + 107 + 163)     = 375 – 312     = 63 For tea (T):  Total tea drinkers = 322. This set includes:   – Those who drink only tea   – Those in S ∩ T "only" (42)   – Those in T ∩ M "only" (92)   – Those who drink all three (163) Thus, only tea is   Only T = 322 – (42 + 92 + 163)     = 322 – 297     = 25 For milk (M):  We are not directly given the total number of milk drinkers, but we can find it using the fact that the total number of students who drink at least one drink is 576. The sum of the regions within the three sets must equal 576. Currently we have:   Only S = 63   Only T = 25   Only M = (unknown, call this x)   S ∩ T "only" = 42   S ∩ M "only" = 107   T ∩ M "only" = 92   All three = 163 So,   63 + 25 + x + 42 + 107 + 92 + 163 = 576 Add the known numbers:   63 + 25 = 88   88 + 42 = 130   130 + 107 = 237   237 + 92 = 329   329 + 163 = 492 Thus:   492 + x = 576   x = 576 – 492   x = 84 So, the number of students who drink only milk is 84. Step 5. (Optional) Find the total number of milk drinkers. This isn’t required to fill in the diagram, but for completeness:  Total milk drinkers = (Only M) + (S ∩ M "only") + (T ∩ M "only") + (all three)   = 84 + 107 + 92 + 163   = 446 Step 6. Check that all parts add up correctly: Sum of all parts in the circles =  63 (only S)  + 25 (only T)  + 84 (only M)  + 42 (S and T only)  + 107 (S and M only)  + 92 (T and M only)  + 163 (all three) = 63 + 25 = 88; 88 + 84 = 172; 172 + 42 = 214; 214 + 107 = 321; 321 + 92 = 413; 413 + 163 = 576 Then add those who drink none: 576 + 123 = 699, which is correct. Step 7. Draw the Venn Diagram. You would draw three overlapping circles labeled S (soda), T (tea), and M (milk). Then fill in the numbers as follows:  • In the intersection of all three (the very center), write 163.  • In the area where S and T overlap but outside M, write 42.  • In the area where S and M overlap (but not T), write 107.  • In the area where T and M overlap (but not S), write 92.  • In the S-only (no overlap with T or M) region, write 63.  • In the T-only region, write 25.  • In the M-only region, write 84.  • Outside all three circles, note that 123 students chose none of these drinks. Step 8. Final Answer Thus, the final numbers in each section of the Venn diagram are:  • Only soda: 63  • Only tea: 25  • Only milk: 84  • Soda and tea only: 42  • Soda and milk only: 107  • Tea and milk only: 92  • All three: 163  • None: 123 And checking:   63 + 25 + 84 + 42 + 107 + 92 + 163 + 123 = 699 This completes the construction of the Venn diagram and verifies that the numbers match the survey of all 699 students.