Question

In a group of 50 people, two tests were conducted, one for diabetes and one for blood pressure. 30 people were diagnosed with diabetes and 40 people were diagnosed with high blood pressure. What is the minimum number of people who were having diabetes and high blood pressure?

- No copied answers please
- Well explanation needed ​

Answer 1

Ven diagram:-

$\setlength{\unitlength}{.5in}\begin{picture}(0,0)\thicklines\put(0,0){\circle{1}{\bf }}\put(0.5,0){\circle{1}{\bf B}}\put(-0.3,0){\bf A}\put(0,0){\scriptsize{\bf {A{\cap} B} }}\put(0,0.7){\bf { A{\cup}B }}\end{picture}$

Solution:-

Let

$\bull\sf A\cup B=Set\:of\:all\:People$

$\bull\sf A=Set\:of\:people\:diagnosed\:with\:Diabetes$

$\bull\sf B=Set\:of\:People\:diagnosed\:with\:high\:blood\:Pressure$

$\bull\sf A\cap B=Set\:of\:people\:having\:both\:Diabetes\:and\:High\:Blood\:Pressure$

Now

$\bull\sf n(A\cup B)=50$

$\bull\sf n(A)=30$

$\bull\sf n(B)=40$

$\bull\sf n(A\cap B)=?$

We know that

$\boxed{\sf n(A\cup B)=n(A)+n(B)-n(A\cap B)}$

$\\ \sf\longmapsto n(A\cap B)=n(A)+n(B)-n(A\cup B)$

$\\ \sf\longmapsto n(A\cap B)=30+40-50$

$\\ \sf\longmapsto n(A\cap B)=70-50$

$\\ \sf\longmapsto\underline{\boxed{\bf{ n(A\cap B)=20}}}$

$\\ \therefore{\underline{\underline{\footnotesize{\sf{Minimum\:number\:of\:people\:who\:were\:having\:Diabetes\:and\:High\:Blood\:Pressure\:are\:20.}}}}}$

Answer 2

Answer:

20 people

Step-by-step explanation:

Let A be the set containing people diagnosed with diabetes and B be the set containing people diagnosed with Blood pressure.

We know set formula

n(A∪B)+n(A∩B)=n(A)+n(B)

n(A∩B)=x are people having both

n(A∪B) are total people

n(A) is people having diabetes and n(B) is having blood pressure.

⇒50+x=40+30

x=20

Thus, there should be minimum of 20 people having both.