a veterinarian survey 26 of his patrons.e discovers that 14 have dogs ,10have cats ,and 5 have fish .our have dogs and cats ,3 have have dogs and fish ,and one has a cat and fish .if no one has all three kinds of pets ,how many patrons have none of these pets?
14 - 4 - 3 = 7 have only dog
10 - 4 - 1 = 5 have only cat
5 - 3 - 1 = 1 has only fish
so 13 have only one animal
3+4+1 = 8 have two animals
total with animals = 13 + 8 = 21 have animals
so
26 - 21 = 5 have no animals at all
To find out how many patrons have none of these pets, we need to subtract the number of patrons who have at least one of these pets from the total number of patrons surveyed.
Let's break it down step-by-step:
Total number of patrons surveyed = 26
Number of patrons with dogs = 14
Number of patrons with cats = 10
Number of patrons with fish = 5
Number of patrons with both dogs and cats = 4
Number of patrons with both dogs and fish = 3
Number of patrons with both cats and fish = 1
To find the number of patrons who have at least one of these pets, we sum up the individual counts above:
Number of patrons with at least one pet = (Number of patrons with dogs) + (Number of patrons with cats) + (Number of patrons with fish) - (Number of patrons with both dogs and cats) - (Number of patrons with both dogs and fish) - (Number of patrons with both cats and fish)
Number of patrons with at least one pet = 14 + 10 + 5 - 4 - 3 - 1
Number of patrons with at least one pet = 21
Now, to find the number of patrons with none of these pets:
Number of patrons with none of these pets = (Total number of patrons surveyed) - (Number of patrons with at least one pet)
Number of patrons with none of these pets = 26 - 21
Number of patrons with none of these pets = 5
Therefore, 5 patrons have none of these pets.
To find out how many patrons have none of these pets, we need to use the principle of inclusion-exclusion.
First, we can add up the number of patrons who have dogs, cats, or fish:
Number of patrons with dogs = 14
Number of patrons with cats = 10
Number of patrons with fish = 5
Now let's consider the overlaps:
Number of patrons with dogs and cats = 4 (since 14 - 4 = 10)
Number of patrons with dogs and fish = 3
Number of patrons with cats and fish = 1
Next, we can find the total number of patrons with at least one of these pets:
Total number of patrons with at least one of these pets = Number of patrons with dogs + Number of patrons with cats + Number of patrons with fish - Number of patrons with dogs and cats - Number of patrons with dogs and fish - Number of patrons with cats and fish
Total number of patrons with at least one of these pets = 14 + 10 + 5 - 4 - 3 - 1 = 21
Finally, to find the number of patrons with none of these pets, we subtract the total number of patrons with at least one of these pets from the total number of patrons surveyed:
Number of patrons with none of these pets = Total number of patrons surveyed - Total number of patrons with at least one of these pets
Number of patrons with none of these pets = 26 - 21 = 5
Therefore, 5 patrons have none of these pets.