SOLUTION: A survey of fruit tree growers showed the following: 59 grew raspberries 23 grew both raspberries and blueberries 50 grew blueberries 11 grew both raspberries and str

A survey of fruit tree growers showed the following:
 
59 grew raspberries     23 grew both raspberries and blueberries 
50 grew blueberries     11 grew both raspberries and strawberries
27 grew strawberries    14 grew both blueberries and strawberries
9 grew all three types       13 grew none of these three
 
 
Create a Venn diagram to reflect the above data, label your diagram clearly
and submit to the Week 2, assignment 3 dropbox. Use your diagram to answer the following questions here.
 
(a)   How many grew only blueberries?
(b)   How many grew both raspberries and blueberries, but not strawberries?
(c)   How many grew only raspberries?
(d)   How many grew none of these three or only strawberries?
(e)   How many fruit tree growers were surveyed? 

There are 8 types of growers here.
 
Grower       Growing       Growing          Growing  
type       Raspberries   Blueberries      Strawberries 
 
Type #1         YES           NO           NO
Type #2         YES          YES           NO 
Type #3          NO          YES           NO  
Type #4         YES           NO          YES 
Type #5         YES          YES          YES 
Type #6          NO          YES          YES
Type #7          NO           NO          YES 
Type #8          NO           NO           NO
 
Let's draw three mutually overlapping circles:
 

 
Notice that there are 8 regions.
 
Regions #1, #2, #4, and #5 make up the set R, of raspberry growers.
        
       Region #1 consists only of those growing raspberries, but who are
       NOT growing blueberries and NOT growing strawberries. 
 
       Region #2 consists only of those growing BOTH raspberries and
       blueberries, but who are NOT growing strawberries.
       
       Region #3 consists only of those growing blueberries, but who are
       NOT growing raspberries and NOT growing strawberries.
 
       Region #4 consists only of those growing BOTH raspberries and
       strawberries, but who are NOT growing blueberries.
 
       Region #5 consists only of those growing ALL THREE kinds of berries.  
       
       Region #6 consists only of those growing BOTH blueberries and
       strawberries, but who are NOT growing raspberries.
 
       Region #7 consists only of those growing strawberries, but who are
       NOT growing raspberries and NOT growing blueberries.
 
       Region #8 consists only of those NOT growing ANY of the  
       kinds of berries.
 
Let's start with the most inclusive, i.e., those that grew some
of all three types of berries:

>>...9 grew all three types...<<
 
That's the growers of type #5, so we erase #5 and put 9 in its place:
 
 
 
>>...23 grew both raspberries and blueberries...<<
 
Regions #2 and #5 grew raspberries and blueberries, and since region #5
has 9 of the 23 already accounted for, the other 23-9 or 14 growers are in
region #2, so erase #2 and put 14 in its place

           

>>...11 grew both raspberries and strawberries...<<
 
Regions #4 and #5 grew raspberries and strawberries, and since region #5
has 9 of the 11 already accounted for, the other 11-9 or 2 growers are in
region #4, so erase #4 and put 2 in its place

 

>>...14 grew both blueberries and strawberries...<<
 
Regions #5 and #6 grew blueberries and strawberries, and since region #5
has 9 of the 14 already accounted for, the other 14-9 or 5 growers are in
region #6, so erase #6 and put 5 in its place

 

>>...59 grew raspberries...<<
 
Regions #1, #2, #5, and #4 grew raspberries, and regions #2, #5, and #4
have respectively 14, 9 and 2 of the 59 already accounted for, so we
add 14+9+2 and get 25 of the 59 already accouted for, that leaves
59-25 or 34 in region #1, so erase #1 and put 34 in its place



>>...50 grew blueberries...<<
 
Regions #2, #3, #5, and #6 grew raspberries, and regions #2, #5, and #6
have respectively 14, 9 and 5 of the 50 already accounted for, so we
add 14+9+5 and get 28 of the 50 already accouted for, that leaves
50-28 or 22 in region #3, so erase #3 and put 22 in its place



>>...27 grew strawberries...<<
 
Regions #4, #5, #6, and #7 grew strawberries, and regions #4, #5, and #6
have respectively 2, 9 and 5 of the 27 already accounted for, so we
add 2+9+5 and get 16 of the 27 already accouted for, that leaves
27-16 or 11 in region #7, so erase #7 and put 11 in its place



>>...13 grew none of these three...<<
 
These 13 are all in region #8, so erase #8 and put 13 in its place



(a)   How many grew only blueberries?

These are in region #3, so the answer is 22

(b)   How many grew both raspberries and blueberries, but not strawberries?

These are in regions #2 and #5, so the answer is 14+9 or 23

(c)   How many grew only raspberries?

These are in region #1, so the answer is 34

(d)   How many grew none of these three or only strawberries?

These are in regions #8 and #1, do the answer is 13+34 or 47. 

(e)   How many fruit tree growers were surveyed?

We add the numbers is all the eight regions:

34 + 14 + 22 + 2 + 9 + 5 + 11 + 13 = 110.

Edwin