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.

GMAT Club Official Explanation: Column A The number of employees who did not answer "agree" to both proposals means we must exclude those who answered {agree, agree} . Total employees: M Agreed with Proposal 1: 2/5 * M = (2M)/5 Of these, agreed with Proposal 2: 1/4 * (2M)/5 = (2M)/20 = M/10 So, M/10 employees agreed with both proposals. Therefore, the number who did not agree with both = M - M/10 = (9M)/10 Column B The number of employees who did not answer "disagree" to both proposals means we must exclude those who answered {disagree, disagree} . Disagreed with Proposal 1: M - (2M)/5 = (3M)/5 Of these, agreed with Proposal 2: 2/3 * (3M)/5 = (6M)/15 = (2M)/5 So, disagreed with Proposal 2: (3M)/5 - (2M)/5 = (M)/5 Thus, M/5 employees disagreed with both proposals. So, the number who did not disagree with both = M - M/5 = (4M)/5

Think of All of the M employees as belonging to one of four possible answer‐pairs. 1) First, notice that 2/5 of the staff agreed with the first proposal, and 1/4 of that subgroup also agreed with the second proposal. so 1/10 of the total agreed with both. Therefore, everyone else (9/10 of the total) did not answer agree to both. That’s the choice you make for Column A. 2) Next, notice that 3/5 of the staff disagreed with the first proposal, and 2/3 of those went on to agree with the second leaving 1/3 of that 3/5 (1/5 of the total) who disagreed with both proposals. Hence, the remaining 4/5 did not answer disagree to both, and 4/5 is the choice for Column B.

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. Proposal 1- Agree Proposal 1- Disagree Total Proposal 2- Agree 1/4*2M/5 = M/10 2/3*3M/5 = 2M/5 M/2 Proposal 2- Disagree 3M/10 M/5 M/2 Total 2M/5 3M/5 M 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. Column A Column B M/5 M/10

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GMAT Club Olympics 2025 (Day 10): In a company survey, M employees : Two-part Analysis (TPA)

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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.

	Proposal 1- Agree	Proposal 1- Disagree	Total
Proposal 2- Agree	1/4*2M/5 = M/10	2/3*3M/5 = 2M/5	M/2
Proposal 2- Disagree	3M/10	M/5	M/2
Total	2M/5	3M/5	M

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.

Column A	Column B
M/5	M/10

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P1A = Employees that agree with Proposal 1
P2A = Employees that agree with Proposal 2
P1D = Employees that disagree with Proposal 1
P2D = Employees that disagree with Proposal 2

So we have :

P1A = 2M/5
P1D = 3M/5
P1AP2A = M/10
P1DP2A = 2M/5
P1AP2D = 3M/10
P1DP2D = M/5

So number of employees who did not answer "agree" to both proposals :
= P1DP2A + P1AP2D + P1DP2D
= 2M/5 + 3M/10 + M/5
= 9M/10

So number of employees who did not answer "disagree" to both proposals :
= P1AP2A + P1DP2A + P1AP2D
= M/10 + 2M/5 + 3M/10
= 8M/10
= 4M/5

Answer is : 9M/10 , 4M/5

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11 Jul 2025, 09:25

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Here's the breakdown of the calculations:

1. Agreed with Proposal 1:
(2/5)M employees agreed with Proposal 1.

2. Agreed with Proposal 1 and Proposal 2:
(1/4) of those who agreed with Proposal 1 also agreed with Proposal 2.
So, $(1/4) \* (2/5)M = (2/20)M = (1/10)M$ employees agreed with both.

3. Disagreed with Proposal 1:
The remaining employees disagreed with Proposal 1: M−(2/5)M=(3/5)M employees.

4. Disagreed with Proposal 1 and Agreed with Proposal 2:
Of those who disagreed with Proposal 1, (2/3) agreed with Proposal 2.
So, $(2/3) \* (3/5)M = (6/15)M = (2/5)M$ employees disagreed with Proposal 1 and agreed with Proposal 2.

5. Disagreed with Proposal 1 and Disagreed with Proposal 2:
These are the employees who disagreed with Proposal 1 and did not agree with Proposal 2:
(3/5)M−(2/5)M=(1/5)M employees.

Now, let's determine the expressions for Column A and Column B:

Column A: The expression that represents the number of employees who did not answer "agree" to both proposals.
This is the total number of employees minus those who agreed to both proposals.
M−(1/10)M=(9/10)M

Column B: The expression that represents the number of employees who did not answer "disagree" to both proposals.
This is the total number of employees minus those who disagreed with both proposals.
M−(1/5)M=(4/5)M

Therefore:
Column A: 9M/10
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.

make 2x2 matrix

-------P1-------NP1------total
P2----- M 1/10 ------- M 2/5---- M 1/2
NP2---- M 3/10 ------ M 1/5 ---- M 1/2
total-- M 2/5----------M 3/5------- M 1
1 is M

Column A the expression that represents the number of employees who did not answer "agree" to both proposals

M- (1/10)* M = (9/10)M

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

M(1/10)+ M(3/10)+ M(2/5) = (8/10)M = 4/5 M

correct option

(9/10)M & (4/5)M

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.

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11 Jul 2025, 09:28

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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.

agree to proposal 1 is 2/5 ,agree to proposal 2 is 2/5*1/4=1/10 ,agree to proposal 2 is 3/5*2/3=2/5 total agree is 4/5+1/10=9/10 hence column A is M/10 and column B is 9M/10

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From question, let A be agree and D be Disagree

Let proposal1 be (P1) and proposal2 be (P2)
A(P1)=2/5M[tells us the number of people who agreed proposal 1]
D(P1)=3/5M[tells us the number of people who disagreed proposal 1]

For column A where number of employees who did not answer "agree" to both proposals

it can be A(P1)*D(P2), D(P1)*D(P2), D(P1)*A(P2) are okay for column A but not A(P1)*A(P2)
now calculate
A(P1)*D(P2)=(2/5*3/4)M=3/10M

D(P1)*D(P2), D(P1)*A(P2) are all cases of D(P1) so its 3/5M.

add both=3/10M+3/5M=9/10M

Now for column B where number of employees who did not answer "disagree" to both proposals
it can be A(P1)*D(P2), A(P1)*A(P2), D(P1)*A(P2) are okay for column A but not D(P1)*D(P2)

D(P1)*A(P2)=(3/5*2/5)M=2/5M
A(P1)*A(P2), A(P1)*D(P2) are all cases of A(P1) so its 2/5M.

add both=2/5M+2/5M=4/5M

So the answers are
Column A=9/10M
Column B=4/5M

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.

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Let, M=20
Agreed with P1= (2/5)*20=8
Disagreed with P1= 20-8=12
Of those 8 who agreed for P1, people also agreed with P2= (1/4)*8=2
Agreed P1 & Agreed P2= 2
Agreed P1 & Disagreed P2=6

Of those 12 who disagreed with P1, 2/3 agreed with P2= (2/3)*12=8
12-8=4 disagreed with both P1 & P2
Disagreed P1 & Agreed P2= 8
Disagreed P1 & Disagreed P2= 4

Column A : Disagreed to both = 20- agreed to both= 20-2=18
Put in options value of M and see which option gives us 18
= 9M/10

Column B: did not disagree to both = 20-4=16
Put in options value of M and see which gives us 16
= 4M/5

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11 Jul 2025, 09:39

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2/5 of M employees agreed with proposal,

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11 Jul 2025, 09:47

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Total no of employees = M
Agree to proposal 1 = 2M/5
Agree to proposal 2 = 2M/5*1/4 = M/10

No of employees who did not answer agree to both are M-M/10 = 9M/10

Number of employees disagreed to proposal 1 = 3M/5
Number of employees still agreed to proposal 2 = 2/3*3M/5 = 2M/5
Number of employees who disagreed with proposal 2 = (1-2/3)*3M/5 = 1M/5

So number of employees who did not answer disagree to both = 1-M/5 = 4M/5

Hence answer is:
Column A : 9M/10
Column B : 4M/5

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.

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11 Jul 2025, 09:52

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P1=2/5 M
both=1/4*2/5M=M/10

P2+NONE=1-2/5M=3/5M
P2=2/3*3/5M=2/5M
None=1/3*3/5M=1/5M

Ans M/5 & M/10

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[*]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.

P1	P2	Employees
A	A	=2/51/4M=M/10
A	N	=2/53/4M=3M/10
N	A	=3/52/3M =M/10
N	N	=3/51/3M=M/5

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

Column B the expression that represents the number of employees who did not answer "disagree" to both proposals.
1-diagree to both = M-M/5=4M/5

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.

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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.

Given: Total number of employee = M
and the ones that agree with Proposal 1= 2*M/5
The ones that agree with both of the proposals = 2*M/5 * (1/4) = M/10 Answer.

and then The ones that disagreed with proposal 1 = 3*M/5
and then we can find 3/5 which also disagreed with proposal 2 = 3*M/5 * (1/3) = M/5 Answer.

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Out of the total employees(m), 2m/5 agreed with Proposal 1. Among these,1/4 also agreed with Proposal 2,
Employees who agreed to both = 1m/10 .
The rest who agreed with Proposal 1 but disagreed with Proposal 2 = 3m/10.
Of those who disagreed with Proposal 1 is 3m/5 , two-thirds ,2/5 m agreed with Proposal 2, and one-fifth 1m/5 disagreed with both.
Employees who did not agree to both proposals are everyone except the 1m/10 who agreed to both, which is 9m/10 . Employees who did not disagree with both proposals are everyone except the 1m/5 who disagreed to both, which is 4m/5 .

Answer : Column a is 9m/10
Column b is 4m/5

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11 Jul 2025, 10:19

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We are given:

2/5 agree to P1, of those: 2/5*1/4 = 1/10 also agree to P2
3/5 disagree to P1, of those: 3/5*2/3 = 2/5 also agree to P2

Agree: A, Not Agree: NA

	P1 A	P1 NA
P2 A	1/10	2/5
P2 NA	3/10	1/5
Total	2/5	3/5

Who did not "agree" to both = Total - "agree to both" = 1 - 1/10 = 9/10 (A)
Who did not "disagree" to both = Total - "disagree to both" = 1 - 1/5 = 4/5 (B)

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We start by finding the number of employees who agreed with both proposals: (1/4) of (2/5)M, which is (1/10)M. This is the agree on both group. Next, we find the number who disagreed with Proposal 1 but agreed with Proposal 2: (2/3) of the (3/5)M who disagreed with Proposal 1, which equals (2/5)M. This is the disagree P1 - agree P2 group. Then, we find the agree P1 - disagree P2 group by subtracting the on both from the total who agreed with Proposal 1: (2/5)M - (1/10)M = (3/10)M. Finally, the disagree P1 - disagree P2 group is found by subtracting the disagree P1 - agree P2 from the total who disagreed with Proposal 1: (3/5)M - (2/5)M = (1/5)M. For Column A, "did not answer 'agree' to both proposals" means all employees except the (1/10)M on both group, so M−(1/10)M=(9/10)M. For Column B, "did not answer 'disagree' to both proposals" means all employees except the (1/5)M disagree P1 - disagree P2 group, so M−(1/5)M=(4/5)M.

Regards,
Lucas

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.

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11 Jul 2025, 10:23

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We have M employees

For proposal 1: If $\frac{2M}{5} $ agreed, then $\frac{3M}{5} $ disagreed

For the ones that agreed to proposal 1 ( $\frac{2M}{5} $ employees)

$\frac{2M}{20} $ agreed to proposal 2 [since 1/4 of the ones that agreed to proposal 1 also agreed to proposal 2] [equation1]
Therefore, $\frac{6M}{5} $ disagreed to proposal 2

For the ones that disagreed to proposal 1 ( $\frac{3M}{5} $ employees)

$\frac{6M}{15} $ agreed to proposal 2 [since 2/3 of the ones that disagreed to proposal 1, agreed to proposal 2]
Therefore, $\frac{3M}{15} $ disagreed to proposal 2 [equation 2]

Employees who agreed to both proposals are: The ones who agreed to proposal 1 and then agreed to proposal 2 - from equation 1
we get, $\frac{M}{10} $

Thus, number of employees who did not agree to both = 1 - (who agreed to both) = 1 - $\frac{M}{10} $ = $\frac{9M}{10} $

Employees who disagreed to both proposals are: The ones who disagreed to proposal 1 and then disagreed to proposal 2 - from equation 2
we get, $\frac{M}{5} $

Thus, number of employees who did not disagree to both = 1 - (who disagreed to both) = 1 - $\frac{M}{5} $ = $\frac{4M}{5} $

Answer:

Column A: $\frac{9M}{10} $
Column B: $\frac{4M}{5} $