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?

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Answer 1

Given :-

• ‌Total number of persons = 50
• ‌Number of people diagnosed with diabetes = 30
• ‌Number of people diagnosed with high blood pressure = 40

To find :-

• ‌Minimum number of people having both diabetes and high blood pressure.

Solution :-

In order to find the minimum number of people having both diabetes as well as high blood pressure, we assume that there is not a single person who is not having diabetes or high blood pressure. Everyone is either a patient of diabetes or high blood pressure or both. By applying formula for union, we can find the minimum number of people having diabetes and high blood pressure (or intersection ).

Assume all the people in the group to be the elements of the universal set.

Also assume people who are having diabetes and high blood pressure to be the elements of sets D (for diabetes patients) and set H (for high blood pressure).

We have formula :-

⟹ n(U) = n(D) + n(H) - n(D ∩ H)

Here n(D ∩ H) represents the set of all the people who are having both diabetes and high blood pressure. n(X) is for representing number of elements in a set named X.

⟹ 50 = 30 + 40 - n(D ∩ H)

⟹ 50 = 70 - n(D ∩ H)

⟹ n(D ∩ H) = 70 - 50

⟹ n(D ∩ H) = 20

So the minimum number of people who are having both diabetes and high blood pressure is 20.