Question

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 _______

• A. 0
• B. 15
• C. 20
• D. 10

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.

Answer 1

The Correct Option is B

Step 1: Set up what the conditions tell us.
Let Ch = patients with high cholesterol, BP = patients with high blood pressure, and D = patients with diabetes. Every patient with high cholesterol also has high BP, so Ch sits entirely inside BP. Also, no patient has both high cholesterol and diabetes, so Ch and D never overlap.

Step 2: Break the 45 high-BP patients into three non-overlapping groups.
Because Ch lies completely inside BP, every one of the 45 high-BP patients belongs to exactly one of three groups: only high BP (no other condition), high BP with high cholesterol, or high BP with diabetes. A patient cannot be in the "BP with cholesterol" group and the "BP with diabetes" group at the same time, since cholesterol and diabetes never occur together in the same patient.

Step 3: Fill in the two groups we already know.
We are told the number of patients with only high BP is 20. We are also told the number with high cholesterol is 10, and since every cholesterol patient also has high BP, all 10 of them fall inside the "high BP with high cholesterol" group.

Step 4: Solve for the "high BP with diabetes" group.
The three groups must add up to the full high-BP count of 45:
\[ \text{only BP} + \text{BP with cholesterol} + \text{BP with diabetes} = 45 \]
\[ 20 + 10 + \text{BP with diabetes} = 45 \]
\[ \text{BP with diabetes} = 45 - 30 = 15 \]

Step 5: Check the answer against the total of 75.
Since every cholesterol patient is already counted inside BP, the total of 75 patients with at least one condition is the number with BP, or diabetes, or both. Using the inclusion-exclusion idea, \[ 75 = |BP| + |D| - |BP \cap D| = 45 + |D| - 15 \], which gives \(|D| = 45\). Out of these 45 diabetic patients, 15 also have high BP, leaving 30 with diabetes alone, no BP and no cholesterol. Adding everyone up, 20 (only BP) + 10 (BP and cholesterol) + 15 (BP and diabetes) + 30 (diabetes only) = 75, which matches the given total, so the figures are consistent.

Step 6: Why the other options are traps.
Option (A), 0, would only be correct if high BP and diabetes never occurred together, which contradicts the statement that some BP patients also have diabetes. Option (C), 20, is actually the "only BP" count, not the overlap with diabetes. Option (D), 10, is the total cholesterol count, which has nothing to do with diabetes since cholesterol and diabetes never occur in the same patient.

Final Answer:
The number of patients who have both diabetes and high BP is 15. \[ \boxed{15} \]