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29 Jul 2024, 03:38
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To be allowed to participate in the Olympic Games, an athlete must pass doping tests conducted by three different agencies. Out of 50 athletes, 40 passed the test by the first agency, 35 by the second agency, and 30 by the third agency. If all athletes passed the test by at least one agency, which of the following represent the lowest and highest numbers of athletes who might have passed the test by all three agencies?
A. 0 and 25
B. 0 and 30
C. 5 and 25
D. 5 and 27
E. 5 and 30
A. 0 and 25
B. 0 and 30
C. 5 and 25
D. 5 and 27
E. 5 and 30
29 Jul 2024, 03:38
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Official Solution:
To be allowed to participate in the Olympic Games, an athlete must pass doping tests conducted by three different agencies. Out of 50 athletes, 40 passed the test by the first agency, 35 by the second agency, and 30 by the third agency. If all athletes passed the test by at least one agency, which of the following represent the lowest and highest numbers of athletes who might have passed the test by all three agencies?
A. 0 and 25
B. 0 and 30
C. 5 and 25
D. 5 and 27
E. 5 and 30
Let's start by minimizing the number of athletes who might have passed the test by all three agencies:
The least number of athletes who could have passed both the first and the second tests can be calculated as follows: (Test 1) + (Test 2) - (Total) = 40 + 35 - 50 = 25. This scenario occurs when all athletes are passed either by the first agency (40), the second agency (35), or both (25), meaning no athlete is failed by both the first and the second agencies.
Using the same logic, the least number of athletes who could have passed the first and the second, AND the third tests can be calculated as: (Test 1 and Test 2) + (Test 3) - (Total) = 25 + 30 - 50 = 5.
Or consider this. After the first step, we'd have two groups of athletes: the first group consists of 25 athletes who were passed by both the first and the second agencies, and the second group consists of the remaining 25 athletes who were passed by either the first or the second agency but NOT by both. The third agency passed 30 athletes. To minimize those passed by all three, the third agency could have passed all 25 from the second group, who were NOT passed by both the first and the second agencies. We are then left with 5 athletes from the 30 passed by the third agency, who must fall into the first group (those passed by both the first and the second agencies). Thus, these 5 athletes are those who were passed by all three agencies.)
Therefore, the least number of athletes who must have passed all three agencies is 5.
_______________________
As for the maximum number of athletes who might have passed the test by all three agencies:
Since the third agency passed the lowest number of athletes, 30, the number of athletes passed by all three agencies cannot possibly be more than that.
Let's check whether 30 is possible.
If all three agencies passed 30 athletes, we'd be left with 20 athletes to accommodate. The first agency, which passed 40 athletes, could have passed 10 (40 - 30 = 10) athletes out of those 20 athletes, leaving 10 athletes to accommodate. However, the second agency, which passed 35 athletes, could have passed only 5 (35 - 30 = 5) athletes out of those 10 athletes, leaving 5 athletes who were passed by neither of the agencies. Since we are given that all athletes passed the test by at least one agency, this scenario is not possible.
Essentially, if x is the maximum number of athletes that could have been passed by all three agencies, then:
• \(40 - x\) would be the number left from the first agency to accommodate other athletes,
• \(35 - x\) would be the number left from the second agency to accommodate other athletes,
• \(30 - x\) would be the number left from the third agency to accommodate other athletes.
Thus, we need such a maximum \(x\) that: \(x + (40 - x) + (35 - x) + (30 - x) \geq 50\). This gives \(x \leq 27.5\). Since \(x\) must be an integer, the maximum \(x\) is 27.
Hence, the maximum number of athletes who might have passed the test by all three agencies is 27.
Answer: D
To be allowed to participate in the Olympic Games, an athlete must pass doping tests conducted by three different agencies. Out of 50 athletes, 40 passed the test by the first agency, 35 by the second agency, and 30 by the third agency. If all athletes passed the test by at least one agency, which of the following represent the lowest and highest numbers of athletes who might have passed the test by all three agencies?
A. 0 and 25
B. 0 and 30
C. 5 and 25
D. 5 and 27
E. 5 and 30
Let's start by minimizing the number of athletes who might have passed the test by all three agencies:
The least number of athletes who could have passed both the first and the second tests can be calculated as follows: (Test 1) + (Test 2) - (Total) = 40 + 35 - 50 = 25. This scenario occurs when all athletes are passed either by the first agency (40), the second agency (35), or both (25), meaning no athlete is failed by both the first and the second agencies.
Using the same logic, the least number of athletes who could have passed the first and the second, AND the third tests can be calculated as: (Test 1 and Test 2) + (Test 3) - (Total) = 25 + 30 - 50 = 5.
Or consider this. After the first step, we'd have two groups of athletes: the first group consists of 25 athletes who were passed by both the first and the second agencies, and the second group consists of the remaining 25 athletes who were passed by either the first or the second agency but NOT by both. The third agency passed 30 athletes. To minimize those passed by all three, the third agency could have passed all 25 from the second group, who were NOT passed by both the first and the second agencies. We are then left with 5 athletes from the 30 passed by the third agency, who must fall into the first group (those passed by both the first and the second agencies). Thus, these 5 athletes are those who were passed by all three agencies.)
Therefore, the least number of athletes who must have passed all three agencies is 5.
_______________________
As for the maximum number of athletes who might have passed the test by all three agencies:
Since the third agency passed the lowest number of athletes, 30, the number of athletes passed by all three agencies cannot possibly be more than that.
Let's check whether 30 is possible.
If all three agencies passed 30 athletes, we'd be left with 20 athletes to accommodate. The first agency, which passed 40 athletes, could have passed 10 (40 - 30 = 10) athletes out of those 20 athletes, leaving 10 athletes to accommodate. However, the second agency, which passed 35 athletes, could have passed only 5 (35 - 30 = 5) athletes out of those 10 athletes, leaving 5 athletes who were passed by neither of the agencies. Since we are given that all athletes passed the test by at least one agency, this scenario is not possible.
Essentially, if x is the maximum number of athletes that could have been passed by all three agencies, then:
• \(40 - x\) would be the number left from the first agency to accommodate other athletes,
• \(35 - x\) would be the number left from the second agency to accommodate other athletes,
• \(30 - x\) would be the number left from the third agency to accommodate other athletes.
Thus, we need such a maximum \(x\) that: \(x + (40 - x) + (35 - x) + (30 - x) \geq 50\). This gives \(x \leq 27.5\). Since \(x\) must be an integer, the maximum \(x\) is 27.
Hence, the maximum number of athletes who might have passed the test by all three agencies is 27.
Answer: D
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01 Aug 2024, 23:53
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Bunuel
Bunuel
I have a doubt what will happen in case of minimizing if we start with Test 2 and Test 3. I want to understand the logic for minimizing because I am always making mistake in these type of questions. Is there any rule that we have to follow while minimizing and maximising the values in 3 set questions.
