SOLUTION: Assist me with this difficult homework problem: 1. A survey of 51 pet owners revealed that 24 owned dogs, 28 owned cats, 11 owned turtles, 12 owned both dogs and cats, 7 owned b

Question 419831: Assist me with this difficult homework problem:
1. A survey of 51 pet owners revealed that 24 owned dogs, 28 owned cats, 11 owned turtles, 12 owned both dogs and cats, 7 owned both dogs and turtles, 3 owned both cats and turtles, and 2 owned all three. Find the number of pet owners who owned none of the three lovely animals.
The answer is not 26.
Found 4 solutions by sudhanshu_kmr, josmiceli, richard1234, ewatrrr:
Answer by sudhanshu_kmr(1152)   (Show Source): You can put this solution on YOUR website!

no. of owners have at least one pet = 24 + 28 + 11 - 12 - 7- 3 + 2
= 63 - 22 + 2
= 43

so, no. of persons have none of the three = 51 - 43 = 8

answer is : 8

Answer by josmiceli(19441)   (Show Source): You can put this solution on YOUR website!
If you don't know what a Venn diagram is,
or you can't draw one, I don't know how you
could even approach this. But if you're OK
with that, you have a chance.
--------------
The trick is to count pet owners but don't
count any twice. They can be counted twice
where the 3 circles of the Venn diagram
overlap, and in the center where all 3 overlap.
--------------
I struggle with all kinds of approaches, but I
always return to a method that isolates each
area in the Venn diagram so I only count
the exact number that are in that region.
--------------
If there are 3 circles, there are 7 "areas"
The number of pet owners who own
just 1 animal are dog owners
cat owners, and turtle owners
-------------
The overlap areas are = cats & turtles & not dogs
= cats & dogs & not turtles, = dogs & turtles & not cats
and = dogs, cats, & turtles
------------
So, without counting any twice,

owners of these animals

There are 8 pet owners who don't own dogs, cats, or turtles
and I hope I got it right

Answer by richard1234(7193)   (Show Source): You can put this solution on YOUR website!
By the Inclusion-Exclusion principle,

N = Dog + Cat + Turtle - Dog∩Cat - Dog∩Turtle - Cat∩Turtle + Dog∩Cat∩Turtle

where Dog represents the number of dog owners, etc. and N represents the number of people who own at least one of the three.

N = 24 + 28 + 11 - 12 - 7 - 3 + 2 = 43

So, 43 people own at least one of the three animals. Therefore, 51-43 = 8 people do not own any of the three.
Answer by ewatrrr(24785)   (Show Source): You can put this solution on YOUR website!

Hi
survey of 51 pet owners: results described in the chart below:
Find the number of pet owners who owned none of the three lovely animals.
>= 1 D&C D&T C&T 1
Dog 24
Cat 28
Turle 11
ALL 2
65 -12 -7 -3 = 43
51 - 43 = 8 own none
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