Question 464865: A survey of flower gardeners showed the following:
45 grew roses 7 grew both roses and tulips 4 grew all three types
16 grew tulips 5 grew both orchids and tulips 9 grew none of these three
13 grew orchids 10 grew both roses and orchids
Create a Venn diagram to reflect the above data, label your diagram clearly and submit to the Dropbox. Use your diagram to answer the following questions here.
(a) How many grew only roses?
(b) How many grew both roses and tulips, but not orchids?
(c) How many grew only tulips?
(d) How many grew none of these three or only orchids?
(e) How many flower gardeners were surveyed?
Answer by josmiceli(19441)   (Show Source):
You can put this solution on YOUR website! The trick is to get a count for each little
region of the Venn diagram.
Label the big interlocking circles:
R = 45
T = 16
O = 13
A key item is: 4 grew all three types
Look at just the intersection of Roses and Tulips
There are 7 gardeners in this intersection, but 4 grew all 3 types,
so there are 7 - 4 = 3 who grew ONLY Roses and Tulips
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Look at just the intersection of Tulips and Orchids
There are 5 gardeners in this intersection, but 4 grew all 3 types,
so there are 5 - 4 = 1 who grew ONLY Orchids and Tulips
---------------------------------------------
Look at just the intersection of Roses and Orchids
There are 10 gardeners in this intersection, but 4 grew all 3 types,
so there are 10 - 4 = 6 who grew ONLY Roses and Orchids
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Now all that is left is to count how many grew
ONLY Roses
ONLY Tulips
ONLY Orchids
----------
ONLY Roses = 
ONLY Tulips = 
ONLY Orchids = 
--------------
There were 9 gardeners who didn't grow any of these 3 types
Now you can answer any question
(a) How many grew only roses?
32 grew only Roses
(b) How many grew both roses and tulips, but not orchids?
This is the intersection of Roses and Tulips, which is 7,
minus the ones that grew all 3 types, 
(c) How many grew only tulips?
(d) How many grew none of these three or only orchids?
Only Orchids is  , and
none of the 3 types is  , so
(e) How many flower gardeners were surveyed?
Add up all the little areas of the Venn diagram to get
a true count of gardeners who grew SOME of the 3 types
Add this to gardeners who grew NONE of the 3 types
And, I hope I didn't mess something up, but this is the method I use
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