Question 1194404
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A = person owns a fish
B = person owns a bird
C = person owns a cat
D = person owns a dog


There are 4 pets, so there are 2^4 = 16 different possibilities.


Here are the 16 possibilities<ol><li>Person doesn't own any of the four pets mentioned</li><li>A - persons owns a fish only</li><li>B - they own a bird only</li><li>C - owns a cat only </li><li>D - owns a dog only</li><li>AB - fish and bird only</li><li>AC - fish and cat only</li><li>AD - fish and dog only</li><li>BC - bird and cat only</li><li>BD - bird and dog only</li><li>CD - cat and dog only</li><li>BCD - bird, cat, and dog (i.e. everything but a fish)</li><li>ACD - fish, cat and dog (i.e. everything but a bird)</li><li>ABD - fish, bird and dog (everything but a cat)</li><li>ABC - fish, bird and cat (everything but a dog)</li><li>ABCD - person owns all four pets mentioned</li></ol>
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Let's draw out the Venn Diagram.


Start by drawing two ovals like so. Make sure they overlap.
<img src = "https://i.imgur.com/5t7uRoN.png">
Create a mirror copy of those two ovals to have a total of four ovals.
<img width = "45%" src = "https://i.imgur.com/KR11SNu.png">
Then slide the ovals over until we get this
<img width = "45%" src = "https://i.imgur.com/KOPAE27.png">
It may seem like a bit of a mess, so be sure you can break apart the various ovals (or be able to spot the four ovals). 
This is why I recommend following these steps to slide the ovals over.
Unfortunately we can't have the ovals separated or else we won't be able to form the overlapped regions needed.


Next, we'll add the 15 labels mentioned in the previous section above. The first label of "none of the 4 pets mentioned" is ignored for now, as it goes outside the four ovals.
<img width = "45%" src = "https://i.imgur.com/36IxHV8.png">
Color-coding things helps us separate the various regions (it may be easy to get lost)
<img width = "45%" src = "https://i.imgur.com/RjpNNX0.png">
<ul><li>Regions in white are for the people who own exactly one pet only</li><li>Regions in blue are for the people who own exactly two pets only</li><li>Regions in red are for the people who own exactly three pets only</li><li>The region in green is for the people who own all four pets mentioned</li></ul>


Take note how something like region ABC is in ovals A, B and C at the same time. Region ABC is outside oval D.
Similar logic applies to how the other regions are set up.
The region outside the ovals represents people who own none of the four pets mentioned.


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Let's fill out the Venn Diagram with numeric values.


To do so, we'll need to use the given facts:<table border = "1" cellpadding = "5"><tr><td>Number</td><td>Statement</td></tr><tr><td><font color=blue>1</font></td><td>47 own fish</td></tr><tr><td><font color=blue>2</font></td><td>53 own a bird</td></tr><tr><td><font color=blue>3</font></td><td>50 own a cat</td></tr><tr><td><font color=blue>4</font></td><td>64 own a dog</td></tr><tr><td><font color=blue>5</font></td><td>2 own all four</td></tr><tr><td><font color=blue>6</font></td><td>11 own only fish</td></tr><tr><td><font color=blue>7</font></td><td>14 own only a bird</td></tr><tr><td><font color=blue>8</font></td><td>10 own fish and a bird</td></tr><tr><td><font color=blue>9</font></td><td>21 own fish and a cat</td></tr><tr><td><font color=blue>10</font></td><td>24 own a bird and a dog</td></tr><tr><td><font color=blue>11</font></td><td>27 own a cat and a dog</td></tr><tr><td><font color=blue>12</font></td><td>3 own fish, a bird, a cat, and no dog</td></tr><tr><td><font color=blue>13</font></td><td>1 owns fish, a bird, a dog, and no cat</td></tr><tr><td><font color=blue>14</font></td><td>9 own fish, a cat, a dog, and no bird</td></tr><tr><td><font color=blue>15</font></td><td>10 own a bird, a cat, a dog, and no fish</td></tr></table>


We start with statement 5, which says "2 own all four"
This number goes inside the region labeled ABCD.
In other words, this number goes in the region that is inside all four ovals.


Now move onto statement 6. It says "11 own only fish", which means 11 goes in the region labeled "A". 
This is the region inside oval A, but outside any other oval.


Now onto statement 7. It says "14 own only a bird", so we'll write "14" in region B. This is the region inside oval B, but outside any other oval.


Let's jump ahead to statement 12.
It says "3 own fish, a bird, a cat, and no dog".
We'll write the number 3 inside the region labeled "ABC". 
This is the region inside ovals A,B,C but outside oval D.
To be more accurate, I'll write it off to the side since region ABC is very small.
I'll be doing the same with regions ABD, AC and BD as well.


The next statement (statement 13) says "1 owns fish, a bird, a dog, and no cat"
The number "1" goes with region ABD.


Up next is the statement "9 own fish, a cat, a dog, and no bird" (statement 14). It tells us that the number 9 goes inside region ACD.


Up next is the statement "10 own a bird, a cat, a dog, and no fish" (statement 15). 
It tells us that the number 10 goes inside region BCD.


Here's a mini recap so far:
A = 11
B = 14
ABC = 3
ABD = 1
ACD = 9
BCD = 10
ABCD = 2
That takes care of 7 regions, so there are 16-7 = 9 more to go.


This is what your Venn Diagram should look like at this point (after filling in the proper regions)
<img width = "45%" src = "https://i.imgur.com/aSbbeiD.png">


We'll move to statement 8 now.
It says "10 own fish and a bird"
Use a pen tool to highlight oval A and oval B as shown below.
<img width = "45%" src = "https://i.imgur.com/xkWAjHL.png">
The two ovals highlighted overlap in this teardrop shaped region marked below
<img width = "45%" src = "https://i.imgur.com/V68B5CJ.png">
Everything in this teardrop region is in oval A and oval B at the same time. The subregions involved in this teardrop are:
Region AB
Region ABC
Region ABD
Region ABCD
Each of the four strings starts with "AB", and we either leave it as is, or we tack on various combos of C and/or D.
This is a methodical way to list out all the possibilities of someone owning a fish and a bird (and possibly a cat and/or dog).


Algebraically we can form the equation
AB+ABC+ABD+ABCD = 10
since 10 people own a fish and a bird (and possibly a cat and/or dog).


Be careful and try not to think about something like "ABC" as "A*B*C".
ABC is one variable name, and not the product of three separate variables.


We'll apply the following substitutions
ABC = 3
ABD = 1
ABCD = 2
to go from this
AB+ABC+ABD+ABCD = 10
to this
AB+3+1+2 = 10
and it solves to 
AB = 4
This says "4 people have a fish and a bird, but not a cat and not a dog".
That was all for statement 8. 


Now onto statement 9.
We'll have this equation
AC+ABC+ACD+ABCD = 21
Each item on the left hand side has "A" and "C" in it somewhere.
Plug in these items
ABC = 3
ACD = 9
ABCD = 2
to go from this
AC+ABC+ACD+ABCD = 21
to this
AC+3+9+2 = 21
which solves to 
AC = 7
Therefore, 7 people have a fish and a cat, but none of the other two mentioned pets.


Onto statement 10.
We'll have this equation
BD+ABD+BCD+ABCD = 24
Each item on the left hand side has "B" and "D" in it somewhere.
Plug in these items
ABD = 1
BCD = 10
ABCD = 2
to go from this
BD+ABD+BCD+ABCD = 24
to this
BD+1+10+2 = 24
which solves to 
BD = 11
Therefore, 11 people have a bird and a dog, but none of the other two mentioned pets.


Onto statement 11.
We'll have this equation
CD+ACD+BCD+ABCD = 27
Each item on the left hand side has "CD" in it.
Plug in these items
ACD = 9
BCD = 10
ABCD = 2
to go from this
CD+ACD+BCD+ABCD = 27
to this
CD+9+10+2 = 27
which solves to 
CD = 6
Therefore, 6 people have a cat and a dog, but none of the other two mentioned pets.


Let's update that previous recap:
ABCD = 2
A = 11
B = 14
ABC = 3
ABD = 1
ACD = 9
BCD = 10
<font color=red>AB = 4
AC = 7
BD = 11
CD = 6</font>
The new items are in red.


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Go back to the top to statement 1.
47 people own a fish. Some of these 47 people own a fish only, while others may own exactly one other pet along with the fish, or two other pets along with the fish, or own all 4 pets mentioned.


A = fish owners only
AB = fish + bird
AC = fish + cat
AD = fish + dog
ABC = fish + bird + cat
ABD = fish + bird + dog
ACD = fish + cat + dog
ABCD = all four pets mentioned
It may help to use your pen tool to circle all of oval A.


We have the following values:
A = 11 (given from statement 6)
AB = 4 (solved earlier; see statement 8)
AC = 7 (solved earlier; see statement 9)
AD = unknown, we'll be solving for it soon
ABC = 3 (solved earlier; see statement 12)
ABD = 1 (solved earlier; see statement 13)
ACD = 9 (solved earlier; see statement 14)
ABCD = 2 (given; see statement 5)


These values must add up to the 47 fish owners
A+AB+AC+AD+ABC+ABD+ACD+ABCD = 47
11+4+7+AD+3+1+9+2 = 47
AD+37 = 47
AD = 47-37
AD = 10
There are 10 people who have a fish and a dog, but no other pet mentioned.


Move onto statement 2, which says "53 own a bird".
We'll circle the regions that involve the letter B in some fashion, i.e. we'll circle all of oval B.
B = bird owners only
AB = fish + bird
BC = bird + cat
BD = bird + dog
ABC = fish + bird + cat
ABD = fish + bird + dog
BCD = bird + cat + dog
ABCD = all four pets mentioned


We'll be using these values to help solve for variable BC
B = 14 (given; see statement 7)
AB = 4 (solved earlier; see statement 8)
BD = 11 (solved earlier; see statement 10)
ABC = 3 (solved earlier; see statement 12)
ABD = 1 (solved earlier; see statement 13)
BCD = 10 (solved earlier; see statement 15)
ABCD = 2 (given; see statement 5)


So,
B+AB+BC+BD+ABC+ABD+BCD+ABCD = 53
14+4+BC+11+3+1+10+2 = 53
BC = 8
There are 8 pet owners that have a bird and a cat, but no other pets mentioned.


Now onto statement 3.
Circle oval C, i.e. circle each region that involves C in some way
C = cat only
AC = fish and cat only
BC = bird and cat only
CD = cat and dog only
ABC = fish + bird + cat
ACD = fish + cat + dog
BCD = bird + cat + dog
ABCD = all four pets mentioned


We have the following values:
AC = 7 (solved earlier in statement 9)
BC = 8 (solved earlier in statement 2)
CD = 6 (solved earlier in statement 11)
ABC = 3 (see statement 12)
ACD = 9 (see statement 14)
BCD = 10 (see statement 15)
ABCD = 2 (see statement 5)


So,
C+AC+BC+CD+ABC+ACD+BCD+ABCD = 50
C+7+8+6+3+9+10+2 = 50
C+45 = 50
C = 50-45
C = 5
There are 5 people who own a cat only (and none of the other three pets mentioned).


Now onto the last statement left, which is statement 4.
There are 64 people who own a dog which means,
D+AD+BD+CD+ABD+ACD+BCD+ABCD = 64
D+10+11+6+1+9+10+2 = 64
D+49 = 64
D = 64-49
D = 15
There are 15 dog owners who do not have a fish, nor a bird, nor a cat.


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Here's the complete summary of what we found
A = 11
B = 14
C = 5
D = 15
AB = 4
AC = 7
AD = 10
BC = 8
BD = 11
CD = 6
ABC = 3
ABD = 1
ACD = 9
BCD = 10
ABCD = 2
This is what the completed Venn Diagram looks like
<img width = "45%" src = "https://i.imgur.com/G9NXBdh.png">
The 14 outside of all of the ovals is the final answer and explained in the next two paragraphs.


Add up all of those values to find out how many people there are that own at least one of the four pets mentioned.
A+B+C+D+AB+AC+AD+BC+BD+CD+ABC+ABD+ACD+BCD+ABCD
11+14+5+15+4+7+10+8+11+6+3+1+9+10+2
116


There are 116 people who have at least one pet of the four pets mentioned.
There are 130 people total. 
That must mean there are 130-116 = 14 people who do not have any of the four pets mentioned.


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Final Answer: <font color=red>14</font>
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