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:

Use the given overlap and only-BP values to isolate the region representing both BP and diabetes from the total high-BP count of 45.

Answer 1

The Correct Option is B

Step 1: Note that every high-cholesterol patient is also a high-BP patient.
The problem states that patients with high cholesterol also have high BP, so the 10 high-cholesterol patients form a subset of the 45 high-BP patients rather than a separate group.

Step 2: Break the 45 high-BP patients into non-overlapping groups.
Since there are no patients with both high cholesterol and diabetes, a high-BP patient falls into exactly one of three non-overlapping groups: those who also have high cholesterol, those who have only high BP and nothing else, and those who also have diabetes. This gives 45 = (high BP and cholesterol) + (only high BP) + (high BP and diabetes).

Step 3: Substitute the known values.
We are told high BP and cholesterol together number 10, and only high BP (no other condition) numbers 20. Substituting into the equation from Step 2: 45 = 10 + 20 + (high BP and diabetes).

Step 4: Solve for the unknown group.
Rearranging gives (high BP and diabetes) = 45 - 10 - 20 = 15.

Step 5: Sanity-check using the total of 75.
The total number of patients with at least one condition is made up of: only high BP (20), high BP and cholesterol (10), high BP and diabetes (15), and any patients with diabetes but no high BP. Adding the three known groups gives 20+10+15=45, leaving 75-45=30 patients who have diabetes without high BP, which is a perfectly consistent, non-negative number, confirming the answer is valid.

Conclusion: The number of patients with both diabetes and high BP is 15, which is option 2.