Discussion
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 _______
Answer: B
The number with both diabetes and high BP is 15.
Everyone with high cholesterol also has high BP, and nobody has both high cholesterol and diabetes, so the 10 cholesterol patients sit inside the high-BP group and outside the diabetes group. The 45 high-BP patients then split into three parts:
\[ \begin{aligned} 45 &= 10 + 20 + x \\ x &= 15 \end{aligned} \]
where 10 have cholesterol and BP, 20 have only BP, and \(x\) have BP and diabetes. The remaining 75 − 45 = 30 patients have diabetes alone, which is consistent with the total of 75.
0 and 10 ignore part of the high-BP count, and 20 confuses this group with the 'only high BP' patients.
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