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 _______

Question

In the 2020 summer Olympics' Javelin throw finals, Neeraj Chopra exhibited a spectacular performance to win the gold medal. The silver medal was won by Jakub Vadlejch and the bronze medal was won by Vitezlav Vesely. There were six rounds of throws with each athlete having one throw per round. The best of all the throws of each athlete is considered for the medal. Following were the observations about the throws:
(i) The first and second rounds were dominated by Neeraj Chopra with a gold medal performance in his second throw, while the other two athletes did not have any medal winning throws in these rounds.
(ii) The throws in the last round by both Jakub Vadlejch and Vitezlav Vesely were fouls and were not considered for scoring.
(iii) After four rounds, Vitezlav Vesely was in the second position and could not improve upon his best throw in the succeeding rounds.
(iv) In the fourth round, the throw by Jakub Vadlejch was the best in that round.
In which round did Vitezlav Vesely have his best throw?

Answer 1

This question is a set-overlap problem, so let's place the numbers directly onto a Venn diagram in words instead of writing the equation first.

High cholesterol patients are all inside the high BP group, and high cholesterol never overlaps diabetes. So on the diagram, the cholesterol circle sits fully inside the BP circle, in a region that does not touch the diabetes circle at all. That region holds 10 patients.
The BP circle also has a region marked "only BP" (no cholesterol, no diabetes), which holds 20 patients.
The rest of the BP circle, the part that overlaps diabetes, is what the question calls "both diabetes and high BP". Since the BP circle totals 45, and 10 (cholesterol overlap) and 20 (only BP) are already placed inside it, the remaining part of the BP circle is $45 - 10 - 20 = 15$.
As a check, add up everyone with at least one condition: only BP (20) plus BP-and-cholesterol (10) plus BP-and-diabetes (15) plus diabetes-only. Since the grand total is 75, diabetes-only must be $75 - 20 - 10 - 15 = 30$, a sensible positive number, so the diagram is consistent.

Let's summarize:

The high BP circle of 45 splits into three parts: only BP (20), BP with cholesterol (10), and BP with diabetes (the unknown part).
Subtracting the known parts from 45 gives the BP-with-diabetes part as 15.
The overall total of 75 checks out once diabetes-only works out to 30, confirming the split is correct.

So the number of patients with both diabetes and high BP is 15.

Answer 2

This question is a set-overlap problem, so let's place the numbers directly onto a Venn diagram in words instead of writing the equation first.

High cholesterol patients are all inside the high BP group, and high cholesterol never overlaps diabetes. So on the diagram, the cholesterol circle sits fully inside the BP circle, in a region that does not touch the diabetes circle at all. That region holds 10 patients.
The BP circle also has a region marked "only BP" (no cholesterol, no diabetes), which holds 20 patients.
The rest of the BP circle, the part that overlaps diabetes, is what the question calls "both diabetes and high BP". Since the BP circle totals 45, and 10 (cholesterol overlap) and 20 (only BP) are already placed inside it, the remaining part of the BP circle is $45 - 10 - 20 = 15$.
As a check, add up everyone with at least one condition: only BP (20) plus BP-and-cholesterol (10) plus BP-and-diabetes (15) plus diabetes-only. Since the grand total is 75, diabetes-only must be $75 - 20 - 10 - 15 = 30$, a sensible positive number, so the diagram is consistent.

Let's summarize:

The high BP circle of 45 splits into three parts: only BP (20), BP with cholesterol (10), and BP with diabetes (the unknown part).
Subtracting the known parts from 45 gives the BP-with-diabetes part as 15.
The overall total of 75 checks out once diabetes-only works out to 30, confirming the split is correct.

So the number of patients with both diabetes and high BP is 15.

Show Hint

The Correct Option is B

Solution and Explanation

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