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: Picture three circles that overlap.
Draw one circle for cholesterol (C), one for blood pressure (BP), and one for diabetes (D).
The passage tells us the cholesterol circle is drawn completely inside the BP circle, since every cholesterol patient already has high BP, and the cholesterol circle never touches the diabetes circle at all.

Step 2: Fill in the numbers we are given directly.
The cholesterol circle holds 10 patients in total, and because it sits fully inside BP, all 10 of them are also inside the BP circle.
We are told 20 patients sit in BP only, meaning inside the BP circle but outside both C and D.
The BP circle in total holds 45 patients.

Step 3: Find the missing slice of the BP circle.
The BP circle is made up of exactly three pieces once you account for cholesterol and diabetes, the 20 patients who are in BP alone, the 10 patients who are in BP together with cholesterol, and whatever number of patients are in BP together with diabetes.
There is no fourth piece, because cholesterol and diabetes never overlap, so nobody can sit in BP with both at once.
Adding the three pieces must give the full BP total, $20 + 10 + (\text{BP and diabetes}) = 45$.

Step 4: Subtract to isolate the unknown piece.
$20 + 10 = 30$, so the BP and diabetes overlap is $45 - 30 = 15$.

Step 5: Sanity check using the total of 75.
All patients with at least one condition number 75. Since cholesterol contributes nothing beyond what is already inside BP, this total is just BP patients plus the diabetes patients who are not already counted in BP.
$75 - 45 = 30$ patients have diabetes but not BP, so diabetes in total is $30 + 15 = 45$ patients, a clean whole number that fits the story without contradiction.

Step 6: Rule out the other choices.
15 is not 0 because the question says some BP patients do have diabetes.
15 is not 20, that 20 is the count of patients with BP alone, a different group.
15 is not 10, that 10 is the whole cholesterol count, unrelated to the diabetes overlap once cholesterol is placed inside BP.

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