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,
1. the total number of patients with at least one of these conditions is 75,
2. the number of patients with high cholesterol is 10,
3. the number of patients with high BP is 45, and
4. the number of patients with only high BP and no other conditions is 20.
Then the number of patients who have both diabetes and high BP is _______
Show Hint
Split the 45 high-BP patients into three groups: only BP, BP with cholesterol (equal to the whole cholesterol count, since cholesterol patients are always inside BP), and BP with diabetes. Solve for the last group.
Step 1: Understanding the Concept:
Picture a Venn diagram with three circles: cholesterol (Ch), blood pressure (BP), and diabetes (D). Because every cholesterol patient also has high BP, the Ch circle is drawn completely inside the BP circle, not overlapping it partially. Because cholesterol and diabetes never occur together, the Ch circle and the D circle never touch.
Step 2: Key Formula or Approach:
With Ch fully inside BP, the BP circle splits into exactly two regions where Ch is concerned: the part of BP that is also Ch, and the part of BP that is not Ch. The part of BP that is not Ch splits again into "BP only" and "BP and D," since D cannot reach into the Ch region at all. So: $|BP| = |BP \text{ only}| + |BP \cap Ch| + |BP \cap D|$.
Step 3: Detailed Explanation:
We are given $|BP| = 45$, $|BP \text{ only}| = 20$, and $|Ch| = 10$. Since Ch sits entirely inside BP, $|BP \cap Ch| = |Ch| = 10$. Substituting into the equation from Step 2:
$45 = 20 + 10 + |BP \cap D|$
$|BP \cap D| = 45 - 30 = 15$
As a cross-check, use the total count of 75 patients with at least one condition. Since Ch contributes nothing beyond what BP already counts, $75 = |BP| + |D \text{ only}|$, so $|D \text{ only}| = 30$. Adding every region, $20 + 10 + 15 + 30 = 75$, which lines up with the figure given in the question, confirming the value found for $|BP \cap D|$.
Step 4: Final Answer:
The number of patients with both high BP and diabetes is 15, so the correct option is (B). \[ \boxed{|BP \cap D| = 15} \]
