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Question 1123090: A survey of a group of 113 tourists was taken in St. Louis. The survey showed the following:
62 of the tourists plan to visit Gateway Arch;
48 plan to visit the zoo;
11 plan to visit the Art Museum and the zoo, but not the Gateway Arch;
14 plan to visit the Art Museum and the Gateway Arch, but not the zoo;
16 plan to visit the Gateway Arch and the zoo, but not the Art Museum;
8 plan to visit the Art Museum, the zoo, and the Gateway Arch;
14 plan to visit none of the three places.
How many plan to visit the Art Museum only
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
Break down what we know into 8 different facts
Fact 1: 113 tourists total
Fact 2: 62 of the tourists plan to visit Gateway Arch
Fact 3: 48 plan to visit the zoo
Fact 4: 11 plan to visit the Art Museum and the zoo, but not the Gateway Arch
Fact 5: 14 plan to visit the Art Museum and the Gateway Arch, but not the zoo
Fact 6: 16 plan to visit the Gateway Arch and the zoo, but not the Art Museum
Fact 7: 8 plan to visit the Art Museum, the zoo, and the Gateway Arch
Fact 8: 14 plan to visit none of the three places
we'll use these facts to help set up the problem (venn diagram) and ultimately to answer the question.
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Draw a venn diagram with 3 overlapping circles. The three circles will be located inside a rectangle. The rectangle represents all of the tourists, which we'll call set T. The numbers in circle A represent those who plan to visit the Art Museum. The numbers in circle G is for the Gateway Arch. Finally, those in circle Z are aiming to visit the zoo
A = Art Museum
G = Gateway Arch
Z = Zoo
T = set of all tourists
We have 3 overlapping circles so there will be 2^3 = 8 different regions. I'll label the regions R1 through R8
Definition of each region:
R1 = region inside circle A but outside of the other circles
R2 = region inside circle A and circle G but not inside circle Z
R3 = region inside circle G but outside of the other circles
R4 = region inside circle A and circle Z but not inside circle G
R5 = region inside all three circles
R6 = region inside circle G and circle Z but not inside circle A
R7 = region inside circle Z but outside of the other circles
R8 = region outside all of the circles
The expanded definitions (based on the context of the problem) would be:
R1 = set of people who want to see the art museum only (no arch, no zoo)
R2 = people who want to see the art museum and gateway arch, but not the zoo
R3 = set of people who want to see the gateway arch only (no art museum, no zoo)
R4 = people who want to see the art museum and zoo, but not the art museum
R5 = set of people who want to see all three locations
R6 = people who want to see the gateway arch and zoo, but not the art museum
R7 = set of people who want to see the zoo only (no art museum, no arch)
R8 = set of people who don't want to see any of the three locations
All regions R1 through R8 are inside the rectangle T, to represent the set of all tourists
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Now follow these steps (refer to the blue color coded facts that I wrote above at the top of the problem)
| Number | Instruction | Reason |
|---|
| 1 | Replace R8 with 14 | Fact 8 | | 2 | Replace R5 with 8 | Fact 7 | | 3 | Replace R2 with 14 | Fact 5 | | 4 | Replace R4 with 11 | Fact 4 | | 5 | Replace R6 with 16 | Fact 6 |
After completing those steps, you should have this venn diagram
Since we have 14, 8 and 16 people in the "G" circle, this means that 14+8+16 = 38 people plan to go to the Gateway Arch in some way. This isn't the full amount of people that want to go to the Gateway Arch since Fact 2 states the total count for the Gateway Arch is actually 62. So we're short 62-38 = 24 people. This value 24 will replace R3. The value 24 represents those who only want to see the gateway arch (and nothing else).
Using similar logic, we have 11+8+16 = 33 people so far in the "Zoo" circle, but according to Fact 3, that grand total for people who want to go to the Zoo should be 48. We're short 48-33 = 15 people, so this value (15) will go in region R7. The 15 is inside circle Z but outside the other circles, meaning that 15 people only want to see the zoo (and nothing else).
The only region left to fill in is R1, which is the number of people who only want to go to the art museum (skipping out on the gateway arch and not going to the zoo either).
From Fact 1, we know that 113 tourists were surveyed. If x represents the value that replaces R1, then we can add up all the numbers, set it equal to 113 and solve for x
x+14+24+11+8+16+15+14 = 113
x+102 = 113
x+102-102 = 113-102
x = 11
So 11 will replace R1
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We now have the completed venn diagram. Now onto the question
Question: How many plan to visit the Art Museum only?
Answer: 11
This is simply the value that is in region A but outside the other circles. Once you have the full completed venn diagram, it's simply a matter of locating the proper region and writing down the value. In this case, the proper region is R1.
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