Question

Dr. Hawk works in an allergy clinic, and his patients have the following allergies: 71 are allergic to dairy products, 98 are allergic to pollen, 96 are allergic to animal dander, 33 are allergic to all three, 30 are allergic only to pollen, 10 are allergic only to dairy products, 43 are allergic to dairy products and pollen. (Round your answers to one decimal place.)

(a) What percent of Dr. Hawk's patients are allergic to animal dander?

(b) What percent of Dr. Hawk's patients are allergic only to animal dander?

Answer 1

Answer: a) 65.75% and b) 13.7%

Explanation:

Since we have given that

Number of people who are allergic to dairy products = n(A) = 71

Number of people who are allergic to pollen = n(B) = 98

Number of people who are allergic to animal dander = n(C) = 96

Number of people who are allergic to all n(A∩B∩C) = 33

Number of people who are allergic only to pollen = 30

Number of people who are allergic only to dairy product = 10

Number of people who allergic to dairy products and pollen n(A∩B)= 43

Now, we first find a, c, d, e from the Venn Diagram:

Since n(A∩B)= 43

`a+b=43\\\\a+33=43\\\\a=43-33\\\\a=10`

For Diary product,

`10+a+b+c=71\\\\a+b+c=61\\\\10+33+c=61\\\\c=18`

For pollen,

`a+30+b+e=98\\\\10+30+33+e=98\\\\e=98-73=25`

For animal dander,

`c+b+d+e=96\\\\18+33+25+d=96\\\\d=96-76=20`

So, Total allergic persons = 10+18+20+25+30+10+33=146

So, (a) What percent of Dr. Hawk's patients are allergic to animal dander?

Percentage that patients are allergic to animal dander is given by

`(96)/(146)* 100=65.75\%`

(b) What percent of Dr. Hawk's patients are allergic only to animal dander?

Percentage that patients are allergic to only animal dander is given by

`(20)/(146)* =13.7\%`

Hence, a) 65.75%

b) 13.7%