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)?
+128475 (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
