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:

Draw three overlapping sets for cholesterol, BP and diabetes, use the fact that the cholesterol set sits entirely inside the BP set, and split the BP set into its three non-overlapping parts.

Answer 1

The Correct Option is B

Step 1: Set up the three groups.
Let C be the set of patients with high cholesterol, B be the set of patients with high BP, and D be the set of patients with diabetes.
The question tells us every patient with high cholesterol also has high BP, so C is completely inside B, in symbols \( C \subseteq B \).
It also tells us no patient has both high cholesterol and diabetes, so C and D do not overlap at all, \( C \cap D = \varnothing \).

Step 2: Split the BP group into three non-overlapping parts.
Because C sits entirely inside B, every patient in B falls into exactly one of three groups, patients in B with cholesterol too (this is simply all of C, since C is inside B), patients in B with diabetes too (call this x, the quantity we want), and patients in B with neither of the other two conditions (given as only high BP, which is 20).
Since C and D do not overlap, a patient in B cannot be counted in both the cholesterol part and the diabetes part at the same time, so these three parts really are separate and add up to the whole of B.

Step 3: Write this as one equation.
\[ |B| = (\text{only BP}) + |C| + x \]
Fill in the known numbers, the size of B is 45, only BP is 20, and the size of C is 10.
\[ 45 = 20 + 10 + x \]

Step 4: Solve for x.
\[ 45 = 30 + x \]
\[ x = 15 \]
So 15 patients have both high BP and diabetes.

Step 5: Why the other options are wrong.
Option (A), 0, would mean high BP and diabetes never occur together, but the question directly states some patients with high BP also have diabetes, so 0 cannot be right.
Option (C), 20, is actually the count of patients with only high BP and no other condition, not the overlap with diabetes, so it answers a different question.
Option (D), 10, is the total number of patients with high cholesterol, a number that has nothing to do with the BP-diabetes overlap once cholesterol is accounted for separately.

Step 6: Check the total of 75 is consistent.
This number is not even needed to answer the question, since B, C and the only-BP count already pin down x on their own, but it is worth checking the story holds together.
The union of all three sets equals \( |B| + |D| - |B \cap D| \) here, because C contributes nothing extra outside B. Setting this equal to 75 with \( |B|=45 \) and \( |B \cap D| = 15 \) gives \( |D| = 45 \), a sensible whole number, so the figures are consistent.

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