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Jeremy made a Venn diagram showing the number of students in his class who own types of pets. There are 32 students in his class. In addition to the information in the Venn diagram, Jeremy knows half of the students have a dog, 3/8 have a cat, 6 have some other pet and 5 have no pet at all. How many students have all 3 types of pets (i.e. they have a cat and a dog as well as some other pet)?

 Mar 19, 2021
 #1
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(3/8) ( 32)  =  12 have  cats  which implies  that

 

x +  z + w  =  3        (1)

 

 

(1/2) (32)  = 16   have dogs  which  implies  that

 

x + z + y =  6      (2)

 

8 have some other pet  which implies  that

 

w + z +  y =  4

 

And  since  5  have  no pets  at all.....this implies  thsat

 

w + x + y + z  + 9 + 2 + 10  =  32 - 5 

w + x + y + z + 21  = 27               subtract 21  from both sides

w + x + y + z  =  6     (4)

 

Sub  (1)  into  (4)   and  we  get  that

y   + (x  + z + w)  =  6

y + 3  =  6

y = 3

 

And  

x + z  + y  =  6

x + z  + 3  =6

x  + z  = 6 - 3

x + z  = 3      (5)

 

And 

w + z + y =  4

w + z +  3  = 4

w + z = 4-3

w + z =  1    

 

So  subbing (5)    into (4)  we  get  that

 

w + x + y  + z =  6

w + (x + z) + y = 6

w  + 3 + 3   = 6

So w  = 0

 

And w  + z  =1   means z  =  1

And   x + z = 3     so    x + 1  = 3    so    x   = 2 

 

 

So

 w = 0

x  = 2

y = 3

z  = 1   =  the  number  that  own all three

 

 

cool cool cool

 Mar 19, 2021

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