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:

Split the high BP group into cholesterol-overlap, BP-only and diabetes-overlap pieces that must sum to 45.

Answer 1

The Correct Option is B

Step 1: Set up the regions from the given conditions.
Every patient with high cholesterol also has high BP, so the cholesterol group sits entirely inside the BP group. No patient has both cholesterol and diabetes, so cholesterol and diabetes never overlap. This leaves high BP split into three non-overlapping parts: cholesterol and BP (no diabetes), BP only (no other condition), and BP and diabetes (no cholesterol).

Step 2: Write the total-BP equation.
Total high BP patients \(= 45\). Patients with cholesterol and BP \(= 10\) (all \(10\) cholesterol patients also have BP). Patients with BP only \(= 20\). So the BP-and-diabetes group is \[ 45 = 10 + 20 + x \implies x = 45 - 30 = 15. \]

Step 3: Sanity check with the total count.
Let \(y\) be patients with diabetes only (no BP, no cholesterol). The total with at least one condition is \(10 + 20 + 15 + y = 75\), giving \(y = 30\), a valid non-negative count, so the setup is consistent.

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