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  • Twenty four people are part of three committees which are to look at research, teaching, and administration respectively. No two committees have any member in common. No

Twenty four people are part of three committees which are to look at research, teaching, and administration respectively. No two committees have any member in common. No two committees are of the same size. Each committee has three types of people: bureaucrats, educationalists, and politicians, with at least one from each of the three types in each committee. The following facts are also known about the committees:

1. The numbers of bureaucrats in the research and teaching committees are equal, while the number of bureaucrats in the research committee is 75% of the number of bureaucrats in the administration committee. 

2. The number of educationalists in the teaching committee is less than the number of educationalists in the research committee. The number of educationalists in the research committee is the average of the numbers of educationalists in the other two committees. 

3. 60% of the politicians are in the administration committee, and 20% are in the teaching committee. 

Question:

Based on the given information, which of the following statements MUST be FALSE? 

 

 

Option: 1

The size of the research committee is less than the size of the administration committee 

 

 


Option: 2

In the teaching committee the number of educationalists is equal to the number of politicians 
 


Option: 3

In the administration committee the number of bureaucrats is equal to the number of educationalists 

 


Option: 4

The size of the research committee is less than the size of the teaching committee 


Answers (1)

best_answer
  Research Teaching Administration
Bureaucrats 3X 3x 4x
Educationalist m>n n 0
Politicians y y 3y
       

Given: There are a total of 24 people on the committees.

The number of bureaucrats in the committees is in the ratio of 3:3:4, meaning there are 3 bureaucrats in the research and teaching committees, and 4 bureaucrats in the administration committee. Therefore, we can assign the value x=1 to represent the number of bureaucrats.

For educationalists, we have the conditions n > m and m < 0, but the exact values are not specified in the given information.

The ratio of politicians in the committees is 1:1:3, indicating that there is 1 politician in the research and teaching committees, and 3 politicians in the administration committee.

Possible values for the educationalists could be either 3, 2, 4 or 3, 1, 5

Below are the cases that can be drawn:

                        Case (i)

  R T A  
B 3 3 4 10
E 3 2 4 9
P 1 1 3 5
  7 6 11 24

                       Case(ii)

  R T A  
B 3 3 4 10
E 3 1 5 9
P 1 1 3 5
  7 5 12 24

Let's construct a table based on the given information.

  Research Teaching Administration Total
Bureaucrats        
Educationalists        
Politicians        
Total       24

The research and teaching committees have an equal number of bureaucrats, and the research committee has 75% of the bureaucrats in the administration committee. Let's assign '4x' as the number of bureaucrats in the administration committee.

  Research Teaching Administration Total
Bureaucrats 3x 3x 4x 10x
Educationalists        
Politicians        
Total       24

The number of educationalists in the teaching committee is fewer than in the research committee, and the number of educationalists in the research committee is the average of the other two committees. Let's assume 'y' represents the number of educationalists in the research committee, and 'd' represents the difference in educationalists between the research and teaching committees.

  Research Teaching Administration Total
Bureaucrats 3x 3x 4x 10x
Educationalists y y-d y+d 3y
Politicians        
Total       24

60% of the politicians are in the administration committee, and 20% are in the teaching committee. Let's assign '5z' as the total number of politicians.

  Research Teaching Administration Total
Bureaucrats 3x 3x 4x 10x
Educationalists y y-d y+d 3y
Politicians z z 3z 5z
Total       24

We can express the following equation: \mathrm{10x + 3y + 5z = 24}

To find integer solutions for x, y, and z, we observe that if x > 1, y and z cannot be natural numbers.

Hence, we can conclude that x = 1. For x = 1, we have \mathrm{3y + 5z = 14}. If y = 1 or 2, z is not an integer. However, at x = 1 and y = 3, z = 1, which is a valid solution.

  Research Teaching Administration Total
Bureaucrats 3 3 4 10
Educationalists 3 3-d 3+d 9
Politicians 1 1 3 5
Total       24

We also find that 'd' can have two possible values: d = 1 or 2.

  Research Teaching Administration Total
Bureaucrats 3 3 4 10
Educationalists 3 2/1 4/5 9
Politicians 1 1 3 5
Total 7 6/5 11/12 24

Now let's evaluate each option:

Option 1: The size of the research committee is smaller than the size of the administration committee. We can see that the minimum size of the administration committee can be '9', which is larger than the size of the research committee. Hence, this statement is correct.
Option 2: The number of educationalists on the teaching committee is equal to the number of politicians. We can see that this is possible when both numbers are '1', so this statement could be correct.
Option 3: The number of bureaucrats in the administration committee is equal to the number of educationalists. We can see that this is possible when both numbers are '4', so this statement could be correct.
Option 4: The size of the research committee is smaller than the size of the teaching committee. We can see that the maximum size of the teaching committee can be '6', which is smaller than the size of the research committee. Therefore, this statement is incorrect.
Hence, option 4 is the answer.



 

Posted by

Ajit Kumar Dubey

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