```Question 462000
Here is a Venn Diagram like the one you will need to draw. The key to solving these problems is knowing where to start (o:

.
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{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),
circle(0,-.5,2),
locate(-2,2,d),
locate(-3.5,-2,k)
locate(0,-2.7,C),
locate(-.3,-1,j),
locate(1.1,.4,i),
circle(sqrt(2),sqrt(2),2),
locate(-3.5,2.5,A),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,B),
locate(-1.3,.5,g),
locate(0,2.5,e),
locate(2,2,h),
locate(-.2,1.1,f) )}}}
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Let's beginning in the center.
Region f represents the people who have read all three books, or 10 people.
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Region f&c is the region where circle A and circle B overlap, so region f&c represents the people who read book A and book B. That's 25 people. We know that there are 10 people who read all three books (region f), so region c must have 15 people in it. Make sure you understand this paragraph...we're going to use similar reasoning to fill in the rest of the Venn Diagram.
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Region f&i is the region where circle B and circle C overlap, so this region represents all people who read book B and book C. That's 15 people. Subtract the 10 people who read all three books and region i has 5 people.
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Region f&g represents the people who read book A and book, or 20 people. So region g has 20 - 10 = 10 people in it.
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Here's what we have so far:
{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),
circle(0,-.5,2),
locate(-2,2,d),
locate(-3.5,-2,k)
locate(0,-2.7,C),
locate(-.3,-1,j),
locate(1.1,.4,5),
circle(sqrt(2),sqrt(2),2),
locate(-3.5,2.5,A),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,B),
locate(-1.3,.5,10),
locate(0,2.5,15),
locate(2,2,h),
locate(-.2,1.1,10) )}}}
Now, we'll fill in remain regions of circle A, B, and C.
Circle A has all the people who read book A in it. To fill in region d, we need to eliminate those who read more than one book. So region d is 51 minus the people in regions e, f, and g. That's 51 - 15 - 10 - 10 = 16.
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Thinking along the same lines, we see that region h is all the people who read only book B. That's 41 - 15 - 10 - 5 = 11 people.
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We find the number of people in region j the same way. That's 31 people who read book B minus people who read more books than B. We have 31 - 10 - 10 - 5 = 6.
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So far our Venn Diagram looks like this:
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{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),
circle(0,-.5,2),
locate(-2,2,16),
locate(-3.5,-2,k)
locate(0,-2.7,C),
locate(-.3,-1,6),
locate(1.1,.4,5),
circle(sqrt(2),sqrt(2),2),
locate(-3.5,2.5,A),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,B),
locate(-1.3,.5,10),
locate(0,2.5,15),
locate(2,2,11),
locate(-.2,1.1,10) )}}}
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The last thing we need to calculate is the number of people who read none of the books. That's region k. The Venn Diagram makes this very easy. If we count the number of people in each distinct region d through j, we will sure sure not to double (or triple) count anyone. So 16 + 15 + 10 + 10 + 11 + 5 + 6 = 73 people who read one, two, or three books. Since 99 people took the survey,  there must be 99 - 73 = 26 people who read none of the books. Region k has 99 people in it.
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Here's the final completed Venn Diagram.:

{{{drawing(300,300,-4,4,-5,4,
rectangle(-4,-3.5,4,4),
circle(0,-.5,2),
locate(-2,2,16),
locate(-3.5,-2,26)
locate(0,-2.7,C),
locate(-.3,-1,6),
locate(1.1,.4,5),
circle(sqrt(2),sqrt(2),2),
locate(-3.5,2.5,A),
circle(-sqrt(2),sqrt(2),2),
locate(3.5,2.5,B),
locate(-1.3,.5,10),
locate(0,2.5,15),
locate(2,2,11),
locate(-.2,1.1,10) )}}}
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Hope this helps!
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Ms.Figgy
math.in.the.vortex```