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Option E and Option D are our answers.

Lets understand the information mentioned in the question before trying to solve it.

So the question starts by telling us that "In a company survey, M employees evaluated both Proposal 1 and Proposal 2, selecting either 'agree' or 'disagree' for each". Then it gives us the information about the fraction of employees who voted for the mentioned Proposals:
  • 2/5 of them agreed with Proposal 1, and of those, 1/4 also agreed with Proposal 2.
  • Of those who disagreed with Proposal 1, 2/3 agreed with Proposal 2.
Now the question asks us Select for Column A the expression that represents the number of employees who did not answer "agree" to both proposals, and select for Column B the expression that represents the number of employees who did not answer "disagree" to both proposals.

Lets assume the total number of employees to be 100 i.e. M = 100. So as per this information 40 employees agreed with Proposal 1 and 10 of them also agreed for Proposal 2 as well. Now moving further out of 60 employees who disagreed with Proposal 1, 40 of them agreed to Proposal 2.
So now we can say that:
Employees aligned with Proposal 1 only = 30
Employees aligned with Proposal 2 only = 40
Employees aligned with both Proposal 1&2 = 10
Employees aligned with neither of the Proposal = 20

Now first part of the question asks us "expression that represents the number of employees who did not answer "agree" to both proposals"
=>Total - Employees aligned with both Proposal 1&2
=> 100 - 10
=>90

Now if we check carefully Only Option E gives us the desire value i.e. '90'.
=Option E: 9M/10
=>(9*100)/10
=>90

From here we can say that Option E satisfies this part of the question


Now for the second part of the question i.e. "expression that represents the number of employees who did not answer "disagree" to both proposals".
=>Total - Employees aligned with neither of the Proposal
=>100 - 20
=>80

Now if we check carefully Only Option D gives us the desire value i.e. '80'.
=Option D: 4M/5
=>(4*100)/5
=>80

From here we can say that Option D satisfies this part of the question


Bunuel
 


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In a company survey, M employees evaluated both Proposal 1 and Proposal 2, selecting either 'agree' or 'disagree' for each.

  • 2/5 of them agreed with Proposal 1, and of those, 1/4 also agreed with Proposal 2.
  • Of those who disagreed with Proposal 1, 2/3 agreed with Proposal 2.

Select for Column A the expression that represents the number of employees who did not answer "agree" to both proposals, and select for Column B the expression that represents the number of employees who did not answer "disagree" to both proposals. Make only two selections, one in each column.
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Lets assume M to be 180
Agree to P1 = Agree P1 = 2/5 ×180=72
Of these, Agree P2 = 1/4×72=18
Disagree P2 = 72−18=54

Disagree P1 = 180−72=108
So Disagree P2 = 108−72 = 36

Did NOT agree with both = Total - agree with both = 180 -18 = 162 which is M*9/10
NOT disagree with both proposals = Total - disagree with both = 180 -36 = 162 which is M*4/5

Column A = NOT agree with both = M*9/10
Column B = NOT disagree with both proposals = M*4/5

MAKING MATRIX TABLE WILL BE USEFUL HERE.
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Make a matrix to visualise the components:
Using the information given to us, we can determine all the fractions in the below matrix -

Agree to P1 Disagree to P1
Agree to P2M/102M/5M/2
Disagree to P23M/10M/5M/2
2M/53M/5M


number of employees who did not answer "agree" to both proposals =
= (Agree to P1 & Disagree to P2) or ( Agree to P2 & Disagree to P1) or ( Disagree to P1 & Disagree to P2)
\(\\
= \frac{3M}{10} + \frac {2M} {5} + \frac{M}{5}\\
= \frac{9M}{10}\\
\)


number of employees who did not answer "disagree" to both proposals
= (Agree to P1 & Disagree to P2) or (Agree to P2 & Disagree to P1) or (Agree to P1 & Agree to P2)
\(\\
= \frac{3M}{10} + \frac {2M} {5} + \frac{M}{10}\\
= \frac{8M}{10}\\
= \frac{4M}{5}\\
\)
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Employees=M
P1 agreed=2M/5
P1 disagree=3M/5
P2 agreed who also agreed P1=1/4* 2M/5=M/10
P2 disagreed who also disagreed P1=3/4*2M/5=3M/10
we know P3 disagreed=3M/5 ,2/3 agreed P2
2/3*3M/5=2M/5
Rest disagree both=1/3*3M/5=M/5
So who did not agree to both=M-M/10=9M/10
people who didn't disagree in both=M-M/5=4M/5
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A1 = 2M/5
It's possible to deduce D1 = M - 2M/5 = 3M/5

A1A2 = 1/4*2M/5 = M/10
It's possible to deduce A1D2 = (1-1/4)*2M/5 = 3M/10

D1A2 = 2/3*3M/5 = 2M/5
It's possible to deduce D1D2 = (1-2/3)*3M/5 = M/5

A1A2 = M/10
A1D2 = 3M/10
D1A2 = 2M/5
D1D2 = M/5

Column A = A1D2 + D1A2 + D1D2 = 3M/10 + 2M/5 + M/5 = 9M/10
Column B = A1A2 + A1D2 + D1A2 = M/10 + 3M/10 + 2M/5 = 4M/5

Correct answers: Column A = 9M/10 and Column B = 4M/5
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In a company survey, M employees evaluated both Proposal 1 and Proposal 2, selecting either 'agree' or 'disagree' for each.

2/5 of them agreed with Proposal 1, and of those, 1/4 also agreed with Proposal 2.
Of those who disagreed with Proposal 1, 2/3 agreed with Proposal 2.


Agreed to proposal 1 (p1)= 2/5M
Did not agree to proposal 1= M-2/5M= 3/5M
Agreed to Proposal 1 and 2= 1/4*2/5M= 2/20M=1/10M
Agreed to proposal 2= 2/3*3/5M=2/5M
Did not answer agree to proposal 1 & 2= 3/5M-2/5M= 1/5M

Question
Select for Column A the expression that represents the number of employees who did not answer "agree" to both proposals,
M-1/10M= 9/10M

select for Column B the expression that represents the number of employees who did not answer "disagree" to both proposals

M-1/5M= 4/5M
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Prop1 = 2M/5
Prop1 and Prop2 = 1/4 * 2M/5 = M/10

Column A = M - (Prop1 and Prop2) = M - M/10 = 9M/10

DisagreeProp1 = M - 2M/5 = 3M/5
If, of those 3M/5, 2/3 agreed with Proposal 2, 1/3 disagreed with Proposal 2.

DisagreeProp1 and DisagreeProp2 = 1/3 * 3M/5 = M/5

Column B = M - (DisagreeProp1 and DisagreeProp2) = M - M/5 = 4M/5

The right answers:
Column A = 9M/10
Column B = 4M/5
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Best way to do this is to set up the mutually exclusive set table ->

Each row/column descriptor should be mutually exclusive to eachother. Such that both cannot be true at the same time. Example - A person who agreed to proposal 1 cannot disagree with proposal 1.

(1/4)(2/5)M
P1 AgreeP1 DisagreeTotal
P2 Agree(2/3)(3/5)M
P2 Disagree
Total(2/5)M(3/5)MM

P1 AgreeP1 DisagreeTotal
P2 Agree(1/10)M(2/5)M(1/2)M
P2 Disagree(1/2)M
Total(2/5)M(3/5)MM

P1 Agree|P2 Disagree = P1 Agree Total -P1 Agree|P2 Disagree
=(2/5)M-(1/10)M = (3/10)M

P1 Disagree|P2 Disagree = P1 Disagree Total -P1 Disagree|P2 Disagree
=(3/5)M-(2/5)M = (1/5)M

P1 AgreeP1 DisagreeTotal
P2 Agree1/10M2/5M1/2M
P2 Disagree3/10M1/5M1/2M
Total2/5M3/5MM

The number of employees who did not answer "agree" to both proposals = Total number of employees - number of employees who did answer "agree" to both proposals
=M - P1 Agree|P2 Agree
=M-1/10M
=9/10M

The number of employees who did not answer "disagree" to both proposals = Total number of employees - the number of employees who did answer "disagree" to both proposals
=M - P1 Disagree|P2 Disagree
=M - 1/5M
=4/5M
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