A candidate applying for a research program must undergo up to three

Question

A candidate applying for a research program must undergo up to three interview cycles. The first interview is conducted by A, followed by B, and, if necessary, by C. A candidate is selected as soon as exactly two interviewers provide a recommendation. For example, if both A and B recommend a candidate, the candidate is selected and does not proceed to an interview with C. C interviews only those candidates recommended by exactly one of A or B.
 
A total of 132 candidates applied for the research program, of which C interviewed 82. The number of candidates recommended by A, B, and C were 70, 86, and 50, respectively.
 
Based on the information provided, select for Both A and B the number of candidates recommended by both A and B, and select for Rejected the total number of candidates out of 132 who were not selected for the research program.

Bunuel · Accepted Answer

A candidate applying for a research program must undergo up to three interview cycles. The first interview is conducted by A, followed by B, and, if necessary, by C. A candidate is selected as soon as exactly two interviewers provide a recommendation. For example, if both A and B recommend a candidate, the candidate is selected and does not proceed to an interview with C. C interviews only those candidates recommended by exactly one of A or B.

A total of 132 candidates applied for the research program, of which C interviewed 82. The number of candidates recommended by A, B, and C were 70, 86, and 50, respectively.

Based on the information provided, select for Both A and B the number of candidates recommended by both A and B, and select for Rejected the total number of candidates out of 132 who were not selected for the research program.

Official Solution:

Denoting the candidates recommended by both A and B as "Both," the number of candidates recommended by only A is 70 - Both, and the number of candidates recommended by only B is 86 - Both. Since C interviewed 82 candidates, and C only interviews those recommended by exactly one of A or B, we have (70 - Both) + (86 - Both) = 82, which gives Both = 37.

Therefore, both A and B recommended 37 candidates, and since C (together with A or B) recommended 50 candidates, the total number of candidates selected for the program is 37 + 50 = 87. Thus, the number of candidates not selected for the program is 132 - 87 = 45.

Correct answer:

Both A and B "37"

Rejected "45"

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Attachment:

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Bismuth83 · Answer

1. There are 6 different scenarios or groups for each person:

- Group 1: were not recommended by both A and B, therefore not even going to C.
 - Group 2: were recommended by A, not by B, therefore going to C and not getting recommended.
 - Group 3: were recommended by A, not by B, therefore going to C and getting recommended.
 - Group 4: were recommended by A and B, therefore going to C.
 - Group 5: were not recommended by A, recommended by B, therefore going to C and not getting recommended.
 - Group 6: were not recommended by A, recommended by B, therefore going to C and getting recommended.

2. Let each group have a number of people G1, G2, G3,.... Then, we can build the following table:

Groups 
 A 
 B 
 C 
 
  1 
 0 
 0 
 Not interviewed (0) 
 
  2 
 G2 
 0 
 0 
 
  3 
 G3 
 0 
 G3 
 
  4 
 G4 
 G4 
 Not interviewed (0) 
 
  5 
 0 
 G5 
 0 
 
  6 
 0 
 G6 
 G6 
 
  # of recommended 
 70 
 86 
 50

3. Based on the given information, the following two equations can be said:
 
 - (1): G1 + G2 + G3 + G4 + G5 + G6 = 132
 - (2): G2 + G3 + G5 + G6 = 82

4. The question asks us to find G4 and the sum G1 + G2 + G5.

5. From the table, we can add the two equations: G2 + G3 + G4 = 70, G4 + G5 + G6 = 86  \rightarrow  G2 + G3 + G4 + G5 + G6 + G4 = 156 = G1 + G2 + G3 + G4 + G5 + G6 - G1 + G4 = 132 - G1 + G4  \rightarrow  G4 - G1 = 24.

6. Using (2) and (1), 82 = G2 + G3 + G5 + G6 = (G1 + G2 + G3 + G4 + G5 + G6) - (G1 + G4) = 132 - (G1 + G4)  \rightarrow  G1 + G4 = 132 - 82 = 50.

7. Let's add the two resulting equations: (G4 - G1) + (G1 + G4) = (24) + (50)  \rightarrow 2 * G4 = 74  \rightarrow   G4 = 37 . Also, G1 = 37 - 24 = 13.

8. Using equation (2) and that G3 + G6 = 50, it can be said that: 82 = G2 + G3 + G5 + G6 = 50 + G2 + G5  \rightarrow  G2 + G5 = 32.

9. That means  G1 + G2 + G5  = 13 + 32 =  45 .

10. Our answer will be:   Both A and B - 37 and Rejected - 45  .

vikash905 · Answer

https://gmatclub.com/forum/download/file.php?id=83501 GMAT-Club-Forum-wdcyi6wh.png

Adit_ · Answer

Bunuel  I was actually calculating the fact that:
Both accept + Both reject = 50 (since 82 went into C)
Now I also know 32 were rejected by C alone so the total count to the above equation is added by 32

50+32 = Both accept + Both reject = 82

So now I know the summation has to be 82.

Now I knew it has to be a combination of 37 and 45 only and ended up choosing the other way and yeah the method is very efficient that you mentioned but in my solution how can i quickly prove if Rejection>Both accept or the other way? Because that is all I would need to find out the answer in this case. Or would I have to work out Both properly and work it out?
I just wanna know if I could deduce the inequality from my equation somehow which im unable to

Thankyou!

egmat · Answer

Unfortunately, no.  Your equation alone doesn't contain enough information to determine the split. Here's why:

Your equation tells you:  x + Rejected = 82

But this has multiple solutions ( 37 + 45  or  45 + 37 ). To break the tie, you  must  use the given recommendation numbers.

Quick calculation to find "Both A and B":

We know:
- A recommended =  70 
- B recommended =  86 
- C interviewed =  82  (exactly one of A or B)

Add A's and B's totals:  70 + 86 = 156

This counts "Both A and B"  twice  (once in A's count, once in B's count).
The rest (exactly one of A or B) is counted once =  82

So:  2x + 82 = 156 
 x = 37

Therefore:  Both A and B =  37 , Rejected =  82 - 37 = 45

A candidate applying for a research program must undergo up to three

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Groups	A	B	C
1	0	0	Not interviewed (0)
2	G2	0	0
3	G3	0	G3
4	G4	G4	Not interviewed (0)
5	0	G5	0
6	0	G6	G6
# of recommended	70	86	50

GMAT Two-part Analysis (TPA) Questions

A candidate applying for a research program must undergo up to three

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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My Rewards