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:

Cholesterol is a subset of BP and never overlaps diabetes, so split the 45 BP patients into only BP, BP with cholesterol and BP with diabetes.

Answer 1

The Correct Option is B

This problem asks for the overlap between two of three medical conditions, so drawing out a Venn diagram in words is the cleanest way to see it. Call the groups C (cholesterol), B (BP) and D (diabetes).

1. Reading the clues: every cholesterol patient also has high BP, so C is a smaller circle sitting inside B. No patient has both cholesterol and diabetes, so C and D do not overlap at all. That leaves exactly four possible regions with patients in them: only B, B with C, B with D, and only D.
2. Filling in the known regions: "only B" is given directly as 20. "B with C" equals the whole cholesterol group, since every one of those 10 patients is inside B, so this region is 10.
3. Using the BP total: the three regions that make up B are only B, B with C, and B with D. Adding them must give the stated BP total of 45. So $20 + 10 + (\text{B with D}) = 45$, which gives $\text{B with D} = 15$.
4. Cross check with the grand total: the four regions together equal the total of 75 patients with at least one condition, so only D works out to $75 - 20 - 10 - 15 = 30$. Nothing in the question contradicts this, so the answer stands.

The number of patients with both high BP and diabetes is 15, which is option (B). Options (A), (C) and (D) all come from mis-splitting the 45 high BP patients across the wrong regions.

Let's summarize:

• Cholesterol sits fully inside BP, so its whole count of 10 belongs to the "BP and cholesterol" region.
• The high BP total of 45 splits into only BP (20), BP with cholesterol (10) and BP with diabetes (the unknown), which forces the unknown to be 15.

So the number of patients who have both diabetes and high BP is 15.