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

Step 1: Set up the three groups.
Let C stand for patients with high cholesterol, B for patients with high BP, and D for patients with diabetes. The question tells us every patient with high cholesterol also has high BP, so C sits fully inside B. It also tells us no patient has both cholesterol and diabetes, so C and D never overlap.

Step 2: Break high BP into its non overlapping parts.
Since C is completely inside B, and C never touches D, the group B splits into three clean, non overlapping parts: patients with only high BP, patients with high BP and cholesterol together, and patients with high BP and diabetes together.
We are told the "only high BP" part has 20 patients.
Because every cholesterol patient already has high BP, the "high BP and cholesterol" part is just the full cholesterol group, which has 10 patients.
Call the "high BP and diabetes" part y, the value we want to find.

Step 3: Write the equation for the total high BP count.
Adding the three parts of B gives the total high BP count of 45:
\[ 20 + 10 + y = 45 \]
\[ y = 45 - 30 = 15 \]

Step 4: Check with the total of 75.
Since every cholesterol patient is already counted inside high BP, the group with at least one condition is really just high BP together with diabetes. Using this, the diabetes only group works out to \(75 - 45 = 30\), which does not clash with anything given, so \(y = 15\) is consistent.

Final Answer:
The number of patients with both diabetes and high BP is 15, so the correct option is (B). Option (A), 0, ignores the given 45 high BP total. Option (C), 20, comes from wrongly leaving out the cholesterol overlap. Option (D), 10, mixes up the diabetes and BP overlap with the cholesterol count.
\[ \boxed{15} \]