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Of 500 candidates who were interviewed for admission, 106 got offers from at least one of three colleges, X, Y and Z. The number of candidates who got offers from colleges X only, Y only and Z only were in the ratio 1 : 3 : 4. The number of candidates who got offers from colleges X and Z only, X and Y only and Y and Z only were in the ratio 1 : 2 : 3. Twice as many candidates received offers from college Y only as received offers from all three colleges. Of the candidates who got offers from college Z, half received offers from at least one other college.
Q. What was the total number of offers made by the three colleges?(numerical value)
  • a)
    106
  • b)
    130
  • c)
    150
  • d)
    160
Correct answer is '160'. Can you explain this answer?
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Solve the following question and fillthe answer.Of 500 candidates who ...
Suppose the number of candidates getting offers from colleges X only, Y only and Z only are 2a, 6a and 8a respectively and the number of candidates getting calls from colleges X and Z only, X and Y only and Y and Z only are b, 2b and 3b.
Consider the following Venn diagram.
The number of candidates getting calls from all 3 colleges will be 3a. Since half the number of candidates who got calls from college Z also got calls from at least one other college, we know that the number of candidates who got calls from college Z only is half the number of candidates who got calls from college Z. So, 8a = 3a + 4b, which gives us 5a = 4b. The total number of candidates is 19a + 6b = 106. Solving these expressions simultaneously, we get a = 4 and b = 5. Substituting these values, we get the following Venn diagram.
35 candidates received calls from college X; 61 candidates received calls from college Y; 64 candidates received calls from college Z. Thus the total number of offers made is 35 + 61 + 64 = 160.
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Most Upvoted Answer
Solve the following question and fillthe answer.Of 500 candidates who ...
Given information:
- Total number of candidates interviewed for admission = 500
- Number of candidates who received offers from at least one of three colleges (X, Y, Z) = 106
- Ratio of candidates who received offers from colleges X only, Y only, and Z only = 1 : 3 : 4
- Ratio of candidates who received offers from colleges X and Z only, X and Y only, and Y and Z only = 1 : 2 : 3
- Twice as many candidates received offers from college Y only as received offers from all three colleges
- Half of the candidates who received offers from college Z also received offers from at least one other college

Solution:
Let's assume the number of candidates who received offers from colleges X only, Y only, and Z only as X, Y, and Z respectively.

Ratio of candidates receiving offers:
From the given information, we have the ratios:
X : Y : Z = 1 : 3 : 4
(X + Z) : (X + Y) : (Y + Z) = 1 : 2 : 3

Equations:
We can set up the following equations based on the given information:
X + Y + Z = 106 (Sum of candidates receiving offers)
Y = 2(X + Y + Z) (Twice as many candidates received offers from college Y only as received offers from all three colleges)
Z/2 = X + Y (Half of the candidates who received offers from college Z also received offers from at least one other college)

Solving the equations:
From the third equation, we can substitute X + Y as Z/2:
Z/2 = Z/2 + 2Z
Z/2 = 3Z/2
Z = 0

If Z = 0, then X + Y = 0 from the third equation. But we know that the sum of candidates receiving offers is 106, so X + Y cannot be 0. Hence, Z cannot be 0.

Hence, the assumption that Z = 0 is incorrect.

Revising the equations:
Let's revise the equations by assuming the number of candidates who received offers from colleges X only, Y only, and Z only as X, Y, and Z respectively.

X + Y + Z = 106
Y = 2(X + Y + Z)
Z/2 = X + Y

Solving the revised equations:
From the third equation, we can substitute X + Y as Z/2:
Z/2 = Z/2 + 2Z
Z/2 = 3Z/2
Z = 0

If Z = 0, then X + Y = 0 from the third equation. But we know that the sum of candidates receiving offers is 106, so X + Y cannot be 0. Hence, Z cannot be 0.

Hence, the assumption that Z = 0 is incorrect.

Conclusion:
Based on the given information and the revised equations, we conclude that it is not possible to determine the total number of offers made by the three colleges with the given data. Therefore, the correct answer cannot be determined.
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Community Answer
Solve the following question and fillthe answer.Of 500 candidates who ...
Suppose the number of candidates getting offers from colleges X only, Y only and Z only are 2a, 6a and 8a respectively and the number of candidates getting calls from colleges X and Z only, X and Y only and Y and Z only are b, 2b and 3b.
Consider the following Venn diagram.
The number of candidates getting calls from all 3 colleges will be 3a. Since half the number of candidates who got calls from college Z also got calls from at least one other college, we know that the number of candidates who got calls from college Z only is half the number of candidates who got calls from college Z. So, 8a = 3a + 4b, which gives us 5a = 4b. The total number of candidates is 19a + 6b = 106. Solving these expressions simultaneously, we get a = 4 and b = 5. Substituting these values, we get the following Venn diagram.
35 candidates received calls from college X; 61 candidates received calls from college Y; 64 candidates received calls from college Z. Thus the total number of offers made is 35 + 61 + 64 = 160.
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60 students were shortlisted for their Personal Interview by the Indian School of Management. But the interview process was to be conducted online and hence students were sent the KOOM meeting link for the same. All the 60 students were to join the KOOM common room at 9:00 AM sharp and then the admission co-ordinators would send in a certain number of students to different waiting rooms- X, Y and Z prior to their one on one interviews with the panellists. There were 3 panellists- A, B and C who were conducting interviews for the candidates waiting in the rooms X, Y and Z respectively. The capacity for the waiting rooms X, Y and Z were 5, 10 and 15 respectively.The 60 students were given a rank from 1 to 60 based on their composite score and it is known that no two students got the same score. To start off the interviews, the ones with top 5 ranks are sent to panellist A. The 10 below them are sent to B and students ranked 16 to 30 were sent to C. When the last person from the waiting room X, Y and Z gives an interview, the next set of students are immediately sent to the waiting room as soon as the interview gets over and the priority is maintained as students with higher ranks are sent to X, Y and Z respectively if the interviews of the previous batch end simultaneously.It is known that:1. Panellist A takes an interview for 15 minutes and takes a break of 5 minutes only after interviewing 3 candidates2. Panellist B takes an interview for 10 minutes and takes a break of 5 minutes after interviewing 5 students.3. Panellist C takes an interview of 5 minutes and takes a break of 5 minutes after interviewing 4 students.4. Lunch break is scheduled between 12:00 noon to 12:30 PM and students are not cut abruptly if their interview began before 12:00 noon, i.e. the interviewers take the break only after finishing the interview if it started before 12:00 noon.5. Batches of 5, 10 and 15 students enter the waiting room together such that students who enter the waiting room at a time have continuous ranks. For example, if the interviewer X is ending up the interview of the last student from the previous batch, 5 students with continuous ranks (say 21-25) enter the waiting room X together.6. Students allotted to a particular waiting room is only interviewed by the designated interviewer and is not interviewed by anyone else even if the other interviewer is free.7. If panellist A goes into the lunch break after having interviewed 1 student, he will again take his 5 minute break after interviewing 2 students post the lunch break so as to satisfy the condition 1 always. The same applies for panellist B and C who follow conditions 2 and 3 respectively.Which of the following ranked students was interviewed by panellist C?

60 students were shortlisted for their Personal Interview by the Indian School of Management. But the interview process was to be conducted online and hence students were sent the KOOM meeting link for the same. All the 60 students were to join the KOOM common room at 9:00 AM sharp and then the admission co-ordinators would send in a certain number of students to different waiting rooms- X, Y and Z prior to their one on one interviews with the panellists. There were 3 panellists- A, B and C who were conducting interviews for the candidates waiting in the rooms X, Y and Z respectively. The capacity for the waiting rooms X, Y and Z were 5, 10 and 15 respectively.The 60 students were given a rank from 1 to 60 based on their composite score and it is known that no two students got the same score. To start off the interviews, the ones with top 5 ranks are sent to panellist A. The 10 below them are sent to B and students ranked 16 to 30 were sent to C. When the last person from the waiting room X, Y and Z gives an interview, the next set of students are immediately sent to the waiting room as soon as the interview gets over and the priority is maintained as students with higher ranks are sent to X, Y and Z respectively if the interviews of the previous batch end simultaneously.It is known that:1. Panellist A takes an interview for 15 minutes and takes a break of 5 minutes only after interviewing 3 candidates2. Panellist B takes an interview for 10 minutes and takes a break of 5 minutes after interviewing 5 students.3. Panellist C takes an interview of 5 minutes and takes a break of 5 minutes after interviewing 4 students.4. Lunch break is scheduled between 12:00 noon to 12:30 PM and students are not cut abruptly if their interview began before 12:00 noon, i.e. the interviewers take the break only after finishing the interview if it started before 12:00 noon.5. Batches of 5, 10 and 15 students enter the waiting room together such that students who enter the waiting room at a time have continuous ranks. For example, if the interviewer X is ending up the interview of the last student from the previous batch, 5 students with continuous ranks (say 21-25) enter the waiting room X together.6. Students allotted to a particular waiting room is only interviewed by the designated interviewer and is not interviewed by anyone else even if the other interviewer is free.7. If panellist A goes into the lunch break after having interviewed 1 student, he will again take his 5 minute break after interviewing 2 students post the lunch break so as to satisfy the condition 1 always. The same applies for panellist B and C who follow conditions 2 and 3 respectively.At what time does Interviewer A finish all his interviews?

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Solve the following question and fillthe answer.Of 500 candidates who were interviewed for admission, 106 got offers from at least one of three colleges, X, Y and Z. The number of candidates who got offers from colleges X only, Y only and Z only were in the ratio 1 : 3 : 4. The number of candidates who got offers from colleges X and Z only, X and Y only and Y and Z only were in the ratio 1 : 2 : 3. Twice as many candidates received offers from college Y only as received offers from all three colleges. Of the candidates who got offers from college Z, half received offers from at least one other college.Q.What was the total number of offers made by the three colleges?(numerical value)a)106b)130c)150d)160Correct answer is '160'. Can you explain this answer?
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