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 _______
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 _______
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 _______
Show Hint
Split the 45 high BP patients into three non-overlapping parts: only BP, BP with cholesterol, and BP with diabetes, then use the numbers you already have to find the missing part.
Updated On: Jul 21, 2026
- 0
- 15
- 20
- 10
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Step 1: Set up what each group means.
Let C be patients with high cholesterol, B be patients with high BP, and D be patients with diabetes. Every patient with high cholesterol also has high BP, so C sits completely inside B. No patient has both high cholesterol and diabetes, so C and D do not overlap at all.
Step 2: Break high BP patients into three parts.
Since C sits entirely inside B, and C never overlaps D, the 45 patients with high BP split into exactly three non-overlapping groups: patients with only high BP and nothing else (given as 20), patients with high BP and high cholesterol (all of C, since C sits inside B, so this group is 10), and patients with high BP and diabetes, the group we need, call it x.
\[ |B| = (\text{only BP}) + |C| + x \]
\[ 45 = 20 + 10 + x \]
Step 3: Solve for x.
\[ x = 45 - 20 - 10 = 15 \]
If x were 0, 10, or 20 instead, the three parts of B would add up to 30, 40, or 50, not 45, so those values do not fit the given high BP count and are ruled out.
Step 4: Check against the total of 75.
Since every cholesterol patient already has high BP, patients with at least one condition are just the union of B and D. Patients with diabetes only (no BP) come to \(75 - 45 = 30\), so \(|D| = 15 + 30 = 45\), and \(|B \cup D| = 45 + 45 - 15 = 75\), which matches the given total and confirms the answer is consistent.
Final Answer:
The number of patients with both diabetes and high BP is 15. \[ \boxed{15} \]
Let C be patients with high cholesterol, B be patients with high BP, and D be patients with diabetes. Every patient with high cholesterol also has high BP, so C sits completely inside B. No patient has both high cholesterol and diabetes, so C and D do not overlap at all.
Step 2: Break high BP patients into three parts.
Since C sits entirely inside B, and C never overlaps D, the 45 patients with high BP split into exactly three non-overlapping groups: patients with only high BP and nothing else (given as 20), patients with high BP and high cholesterol (all of C, since C sits inside B, so this group is 10), and patients with high BP and diabetes, the group we need, call it x.
\[ |B| = (\text{only BP}) + |C| + x \]
\[ 45 = 20 + 10 + x \]
Step 3: Solve for x.
\[ x = 45 - 20 - 10 = 15 \]
If x were 0, 10, or 20 instead, the three parts of B would add up to 30, 40, or 50, not 45, so those values do not fit the given high BP count and are ruled out.
Step 4: Check against the total of 75.
Since every cholesterol patient already has high BP, patients with at least one condition are just the union of B and D. Patients with diabetes only (no BP) come to \(75 - 45 = 30\), so \(|D| = 15 + 30 = 45\), and \(|B \cup D| = 45 + 45 - 15 = 75\), which matches the given total and confirms the answer is consistent.
Final Answer:
The number of patients with both diabetes and high BP is 15. \[ \boxed{15} \]
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