In a population, patients who have high cholesterol also have high blood-pressure (BP). Some patients with high BP also have diabetes. There are no patients who have both high cholesterol and diabetes. Furthermore, Then the number of patients who have both diabetes and high BP is _______
Let H represent the set of patients with high cholesterol, B represent the set of patients with high blood pressure (BP), and D represent the set of patients with diabetes.
Understanding the Given Conditions
- High cholesterol implies high BP: $H \subseteq B$. This means all patients in H are also in B.
- No patients have both high cholesterol and diabetes: $H \cap D = \emptyset$.
- Total patients with at least one condition: $|H \cup B \cup D| = 75$.
- Number of patients with high cholesterol: $|H| = 10$.
- Number of patients with high BP: $|B| = 45$.
- Number of patients with only high BP (and no other conditions): $|B \setminus (H \cup D)| = 20$.
We need to find the number of patients who have both diabetes and high BP, which is $|B \cap D|$.
Calculating Patients with BP but No Cholesterol
Since all patients with high cholesterol (H) also have high BP (B), the set B can be divided into two parts:
- Patients with high cholesterol (and thus high BP): $|H| = 10$.
- Patients with high BP but *not* high cholesterol: $|B \setminus H|$.
We calculate $|B \setminus H|$:
$ |B \setminus H| = |B| - |H| = 45 - 10 = 35 $
So, there are 35 patients who have high BP but do not have high cholesterol.
Determining Patients with BP and Diabetes
The 35 patients with high BP but no high cholesterol ($B \setminus H$) can be further divided into two groups:
- Those with only high BP (no diabetes): This is given as $|B \setminus (H \cup D)| = 20$.
- Those with high BP, no high cholesterol, *and* diabetes. Let this number be $x$.
Therefore:
$ |B \setminus H| = |B \setminus (H \cup D)| + x $
$ 35 = 20 + x $
Solving for $x$:
$ x = 35 - 20 = 15 $
This value, $x=15$, represents the number of patients who have high BP, diabetes, but *not* high cholesterol.
Finding the Number of Patients with Both Diabetes and High BP
We are looking for $|B \cap D|$.
We know that no patients have both high cholesterol and diabetes ($H \cap D = \emptyset$). This implies that any patient who has diabetes (D) cannot have high cholesterol (H).
Therefore, the set of patients with both high BP and diabetes ($B \cap D$) cannot include anyone with high cholesterol. This means $B \cap D$ must be a subset of patients who have BP and Diabetes but *not* High Cholesterol.
The group represented by $x$ (15 patients) exactly fits this description: they have BP, Diabetes, and no High Cholesterol.
Thus, the number of patients who have both diabetes and high BP is:
$ |B \cap D| = x = 15 $
Important Questions from Venn Diagrams
Each row of Column-I has three items and each item is represented by a circle in Column-II. The arrangement of circles in Column-II represents the relationship among the items in Column-I.
Identify the option that has the most appropriate match between Column-I and Column-II.
Note: The figure shown are representative.
Column-I Column-II (1) Animal, Zebra, Giraffe (P) 
(2) Director, Producer, Actor (Q) 
(3) Word, Sentence, Novel (R) 
(4) Pianist, Guitarist, Instrumentalist (S) 
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In the given diagram, teachers are represented in the triangle, researchers in the circle and administrators in the rectangle. Out of the total number of the people, the percentage of administrators shall be in the range of ___________.
