Question 1123090
<font color="black" face="times" size="3">
Break down what we know into 8 different facts
<font color="blue">Fact 1</font>: 113 tourists total</li>
<font color="blue">Fact 2</font>: 62 of the tourists plan to visit Gateway Arch</li>
<font color="blue">Fact 3</font>: 48 plan to visit the zoo</li>
<font color="blue">Fact 4</font>: 11 plan to visit the Art Museum and the zoo, but not the Gateway Arch</li>
<font color="blue">Fact 5</font>: 14 plan to visit the Art Museum and the Gateway Arch, but not the zoo</li>
<font color="blue">Fact 6</font>: 16 plan to visit the Gateway Arch and the zoo, but not the Art Museum</li>
<font color="blue">Fact 7</font>: 8 plan to visit the Art Museum, the zoo, and the Gateway Arch</li>
<font color="blue">Fact 8</font>: 14 plan to visit none of the three places</li>
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
<img src = "https://i.imgur.com/FRC0BTG.png">
We have 3 overlapping circles so there will be 2^3 = 8 different regions. I'll label the regions <font color="red">R1</font>  through <font color="red">R8</font>
<img src = "https://i.imgur.com/7ubADhm.png">
Definition of each region:
<font color="red">R1</font> = region inside circle A but outside of the other circles
<font color="red">R2</font> = region inside circle A and circle G but not inside circle Z
<font color="red">R3</font> = region inside circle G but outside of the other circles
<font color="red">R4</font> = region inside circle A and circle Z but not inside circle G
<font color="red">R5</font> = region inside all three circles
<font color="red">R6</font> = region inside circle G and circle Z but not inside circle A
<font color="red">R7</font> = region inside circle Z but outside of the other circles
<font color="red">R8</font> = region outside all of the circles


The expanded definitions (based on the context of the problem) would be:
<font color="red">R1</font> = set of people who want to see the art museum only (no arch, no zoo)
<font color="red">R2</font> = people who want to see the art museum and gateway arch, but not the zoo
<font color="red">R3</font> = set of people who want to see the gateway arch only (no art museum, no zoo)
<font color="red">R4</font> = people who want to see the art museum and zoo, but not the art museum
<font color="red">R5</font> = set of people who want to see all three locations
<font color="red">R6</font> = people who want to see the gateway arch and zoo, but not the art museum
<font color="red">R7</font> = set of people who want to see the zoo only (no art museum, no arch)
<font color="red">R8</font> = set of people who don't want to see any of the three locations
All regions <font color="red">R1</font>  through <font color="red">R8</font> 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)
<table border=1><tr><th>Number</th><th>Instruction</th><th>Reason</th></tr><tr><td>1</td><td>Replace <font color="red">R8</font> with 14</td><td><font color="blue">Fact 8</font></td></tr><tr><td>2</td><td>Replace <font color="red">R5</font> with 8</td><td><font color="blue">Fact 7</font></td></tr><tr><td>3</td><td>Replace <font color="red">R2</font> with 14</td><td><font color="blue">Fact 5</font></td></tr><tr><td>4</td><td>Replace <font color="red">R4</font> with 11</td><td><font color="blue">Fact 4</font></td></tr><tr><td>5</td><td>Replace <font color="red">R6</font> with 16</td><td><font color="blue">Fact 6</font></td></tr></table>
After completing those steps, you should have this venn diagram
<img src = "https://i.imgur.com/ktZ2dI4.png">
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 <font color="blue">Fact 2</font> states the total count for the Gateway Arch is actually 62. So we're short 62-38 = 24 people. This value 24 will replace <font color="red">R3</font>. 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 <font color="blue">Fact 3</font>, 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 <font color="red">R7</font>. The 15 is inside circle Z but outside the other circles, meaning that 15 people only want to see the zoo (and nothing else).
<img src = "https://i.imgur.com/755y1Vc.png">
The only region left to fill in is <font color="red">R1</font>, 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 <font color="blue">Fact 1</font>, we know that 113 tourists were surveyed. If x represents the value that replaces <font color="red">R1</font>, 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 <font color="red">R1</font>
<img src = "https://i.imgur.com/yaqNtfh.png">
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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 <font color="red">R1</font>.
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