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15 Dec 2023, 00:06
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To get a job at Company X, an applicant must be recommended by three interviewers. Out of 30 applicants, 15 were recommended by the first interviewer, 17 by the second and 20 by the third interviewer. What is the least number of applicants who would have to have been recommended by all three interviewers?
A. 0
B. 2
C. 3
D. 5
E. 10
A. 0
B. 2
C. 3
D. 5
E. 10
15 Dec 2023, 00:16
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JessicaJ
To get a job at Company X, an applicant must be recommended by three interviewers. Out of 30 applicants, 15 were recommended by the first interviewer, 17 by the second and 20 by the third interviewer. What is the least number of applicants who would have to have been recommended by all three interviewers?
A. 0
B. 2
C. 3
D. 5
E. 10
A. 0
B. 2
C. 3
D. 5
E. 10
The least number of applicants that both the first and the second interviewer could have recommended is 15 + 17 - 30 = 2. This scenario occurs when all applicants are recommended either by the first interviewer, the second interviewer, or by both (in other words, when no applicant is recommended by neither, or None = 0). Now, these 2 applicants, who were recommended by both the first and the second interviewers, could have been overlooked by the third interviewer, who could have recommended any 20 out of the remaining 28 applicants. Therefore, the least number of applicants that must have been recommended by all three interviewers is 0.
Answer: A.
Hope it helps.
19 Feb 2024, 17:45
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Apologies in advance for the very long-winded answer. I think this is an interesting question worth unpacking because there's actually so much going on here: word problems, overlapping sets, even some elementary number properties. And all these concepts are generalizable to questions just like this one.
To begin, I think this question is really a three-group overlapping sets problem in disguise. While it may be worded as a min/max question, it fits very nicely into the Venn diagram and formulaic frameworks. I prefer the latter. Recall that Total = # entities in Group 1 + # entities in Group 2 + # entities in Group 3 - (2 * # entities in all 3) - (# entities in exactly 2) + (# entities in no group). In this case, they are all assigned a group, so that's one less thing to worry about.
Step One, Model a Venn Diagram or Work with the Formula
So, plugging these numbers in, we get the following:
\(30 = 15 + 17 + 20 - 2a - b\), where a is the number of elements belonging to all three groups and b is the number of elements belonging to only two.
With this in mind, we can simplify it down to \(30 = 52 - (2a + b)\). Something should emerge here: that the number of applicants belonging to more than two groups has to equal 22. Algebraically, it means \(2a + b\) = 22. That was the overlapping sets portion.
Now comes the basic number properties and more general exam strategy portion.
Step Two, Test for Values:
This question gets really interesting here because we're being asked for the least number of individuals who are assigned to all three groups. What should stand out to you is that b has to be a multiple of 2 (this is that knowledge of number properties coming into play), because 2a is always going to resolve to a multiple of 2. This is super important when it comes to generalizing to other min/max style overlapping sets questions like this. With this in mind, let's test for answer choices, starting with 0 because we're being asked for the least number.
Recall that \(2a + b = 22\), so if \(a\) is 0, that leaves \(b\) as 22. Is that possible? Certainly. It would just mean that nobody gets a job and 22 people are recommended by two of the interviewers. Since this works, it leaves AC A as the correct answer.
A fun thought experiment with this one is to add the caveat that at least one person is hired, or at most 20 interviewees receive two recommendations. This would mean \(2(1) + b = 20\), making the least number of applicants recommended by all three interviewers equal to 1. I mentioned above that this strategy is generalizable. Here is a very difficult Two-Part Analysis question straight from a GMAT Club mock (and very OG-like) that tests you on this very concept: https://gmatclub.com/forum/at-rocket-brown-elementary-school-there-are-150-students-and-three-sports-teams-hockey-tennis-and-football-rocket-brown-students-are-allowed-to-participate-in-as-many-424579.html
To begin, I think this question is really a three-group overlapping sets problem in disguise. While it may be worded as a min/max question, it fits very nicely into the Venn diagram and formulaic frameworks. I prefer the latter. Recall that Total = # entities in Group 1 + # entities in Group 2 + # entities in Group 3 - (2 * # entities in all 3) - (# entities in exactly 2) + (# entities in no group). In this case, they are all assigned a group, so that's one less thing to worry about.
Step One, Model a Venn Diagram or Work with the Formula
So, plugging these numbers in, we get the following:
\(30 = 15 + 17 + 20 - 2a - b\), where a is the number of elements belonging to all three groups and b is the number of elements belonging to only two.
With this in mind, we can simplify it down to \(30 = 52 - (2a + b)\). Something should emerge here: that the number of applicants belonging to more than two groups has to equal 22. Algebraically, it means \(2a + b\) = 22. That was the overlapping sets portion.
Now comes the basic number properties and more general exam strategy portion.
Step Two, Test for Values:
This question gets really interesting here because we're being asked for the least number of individuals who are assigned to all three groups. What should stand out to you is that b has to be a multiple of 2 (this is that knowledge of number properties coming into play), because 2a is always going to resolve to a multiple of 2. This is super important when it comes to generalizing to other min/max style overlapping sets questions like this. With this in mind, let's test for answer choices, starting with 0 because we're being asked for the least number.
Recall that \(2a + b = 22\), so if \(a\) is 0, that leaves \(b\) as 22. Is that possible? Certainly. It would just mean that nobody gets a job and 22 people are recommended by two of the interviewers. Since this works, it leaves AC A as the correct answer.
A fun thought experiment with this one is to add the caveat that at least one person is hired, or at most 20 interviewees receive two recommendations. This would mean \(2(1) + b = 20\), making the least number of applicants recommended by all three interviewers equal to 1. I mentioned above that this strategy is generalizable. Here is a very difficult Two-Part Analysis question straight from a GMAT Club mock (and very OG-like) that tests you on this very concept: https://gmatclub.com/forum/at-rocket-brown-elementary-school-there-are-150-students-and-three-sports-teams-hockey-tennis-and-football-rocket-brown-students-are-allowed-to-participate-in-as-many-424579.html
