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200 people responded to a survey that asked them to rank three differe

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Bunuel
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200 people responded to a survey that asked them to rank three differe [#permalink] New post   05 May 2015, 02:38
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Question Stats:

36% (03:00) correct 64% (02:50) wrong based on 172 sessions
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200 people responded to a survey that asked them to rank three different brands of soap. The percentage of respondents that ranked each brand 1st, 2nd, and 3rd are listed above. If no respondents rated the soaps in the order Y, Z, X, how many respondents rated the soap in the following order: X, Y, Z?

A. 100
B. 60
C. 50
D. 30
E. 25

Kudos for a correct solution.

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Brand_Rankings.png
Brand_Rankings.png [ 34.63 KiB | Viewed 15252 times ]
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C
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EgmatQuantExpert
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Re: 200 people responded to a survey that asked them to rank three differe [#permalink] New post   08 May 2015, 04:44
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The question looks a bit data heavy, but one strike at the right place immediately gives you the answer.

Given
We are given a set of data with user's preferences of three brands of soaps X, Y & Z. We are told that 200 people responded to the survey with none of them having preference in the order Y,Z,X. We are asked to find the number of respondents who rated the soap in the order of X,Y,Z.

Approach
We are given total number of responses of people at each rank. The key to note here is that people can only vote for soaps in a rank for which they have not voted earlier. For example, if a person has voted soap X as rank 1, he can't vote for soap X again in any of the subsequent ranks. As an extension to this, we can say that people who voted for soap x as rank 2 would have voted for either soap Y or soap Z as rank 1.

We will use this concept along with the fact that no one ranked the soaps in the order of Y, Z, X to find out the number of people who ranked the soaps in the order of X, Y, Z.

Working Out
We are given that no one ranked the soaps in the order Y,Z,X. We also know that 60 i.e. 30% of 200 people ranked soap Y as 1. People who ranked soap Y as 1 did not rank soap Z as 2 which implies that all of them ranked soap X as 2. These people then further ranked soap Z as 3. Note here that these people can't vote for soap X or soap Y as rank 3 because they have earlier ranked soap X & soap Y.

We also know the total number of people who ranked soap Z as 3 is 55% of 200 i.e. 110. Soap Z can be ranked 3 by people who voted for soap X or soap Y as rank 1. Out of these we know that 60 people who voted for soap Y as rank 1, voted for soap Z as rank 3. This leaves 110 - 60 = 50 people who voted for soap Z as rank 3. This can only come from people who voted for soap X as rank 1 and then soap Y as rank 2.

Thus, we know that there were 50 people who ranked in the soaps in the order X, Y, Z :-D




The other blocks in the tree structure do not need to be calculated (although they hardly take any time). They have been calculated here just for your reference.

Hope its clear!

Regards
Harsh
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Mike03
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Re: 200 people responded to a survey that asked them to rank three differe [#permalink] New post   19 Sep 2018, 23:43
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A very good question. Here is my approach:

Order of choices people can make -

X Y Z
X Z Y
Y X Z
Y Z X = 0 people made this choice
Z X Y
Z Y X

People who ranked X number 1 = 40% of 200 = 80. This implies that the sum of X Y Z & X Z Y is equal to 80.

People who ranked Y number 1 = 30% of 200 = 60. This implies that the sum of Y X Z & Y Z X is equal to 60. Since Y Z X is 0, this implies that the number of people who picked the order Y X Z is 60.

People who ranked Z number 3 = 55% of 200 = 110. This implies that the sum of Y X Z & X Y Z is equal to 110. Since Y X Z is 60, this implies that X Y Z =50. Therefore, the correct answer is C.
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Bunuel
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Re: 200 people responded to a survey that asked them to rank three differe [#permalink] New post   11 May 2015, 05:17
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Bunuel wrote:

200 people responded to a survey that asked them to rank three different brands of soap. The percentage of respondents that ranked each brand 1st, 2nd, and 3rd are listed above. If no respondents rated the soaps in the order Y, Z, X, how many respondents rated the soap in the following order: X, Y, Z?

A. 100
B. 60
C. 50
D. 30
E. 25

Kudos for a correct solution.

Show Spoiler
Attachment:
Brand_Rankings.png


VERITAS PREP OFFICIAL SOLUTION:

In a way this problem is a permutations problem in disguise. When you're given the caveat that no one responded in the order Y, Z, X, you can tell a few things:

Anyone who put Y first did so in the order Y, X, Z (because there are only two ways to arrange X and Z if Y is in the first spot, and one of those was expressly prohibited by the prompt).

And, similarly, anytime that Z was second the order was X, Z, Y (because Y and Z couldn't have been arranged in the other order with Z fixed in the middle); and anytime X was last the order had to be Z, Y, X (because the other "X third" option is prohibited).

So if you follow that logic, your next step is to pick one of those certainties (for example, for all 60 cases of Y first - Y was first in 30% of the 200 cases - the order was Y, X, Z) and incorporate the statistics. That means that of the 110 cases of Z going last, 60 of them were in the order Y, X, Z. Which then means that, since the only other way to fix Z at the last spot is to go in the asked-about order X, Y, Z, the other 50 cases of Z third have to have come from the X, Y, Z order. Therefore, the answer is 50.
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Re: 200 people responded to a survey that asked them to rank three differe [#permalink] New post   11 Oct 2018, 15:26
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I felt dumb, I couldn t get how you infer "people who ranked Y number 1 = 30% of 200 = 60." from the graphs? maybe cause it's late here
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Re: 200 people responded to a survey that asked them to rank three differe [#permalink] New post   26 Apr 2019, 23:59
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faltan wrote:
I felt dumb, I couldn t get how you infer "people who ranked Y number 1 = 30% of 200 = 60." from the graphs? maybe cause it's late here


Question states:
200 people responded to a survey that asked them to rank three different brands of soap. The percentage of respondents that ranked each brand 1st, 2nd, and 3rd are listed above.
If no respondents rated the soaps in the order Y, Z, X, how many respondents rated the soap in the following order: X, Y, Z?

Given:
The first graph tells up how many people rated each brand of soap number 1:
40% rated X number 1. 30% rated Y number 1 and 30% rated Z number 1.

The second graph tells up how many people rated each brand of soap number 2:
45% rated X number 2. 40% rated Y number 2 and 15% rated Z number 2.

The third graph tells up how many people rated each brand of soap number 1:
15% rated X number 3. 30% rated Y number 3 and 55% rated Z number 3.

YZX = 0. ---- eq (1)

Calculations
15% = 30 people rated Z number 2. Possible combinations with Z in middle is XZY or YZX. So, 30 people voted for XZY because no respondents rated the soaps in the order Y, Z, X.
Basically,
XZY + YZX = 30 ,
From eq(1) we get XZY + 0 = 30 ,
XZY = 30 --- eq (2)

40% = 80 people rated X number 1. These people voted for XYZ or XZY .
Basically,
XYZ + XZY = 80.
From eq (2), we get XZY = 30 .
XYZ + 30 = 80.
So, XYZ = 50.
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Re: 200 people responded to a survey that asked them to rank three differe [#permalink] New post   13 Jun 2019, 08:08
Mike03 wrote:
A very good question. Here is my approach:

Order of choices people can make -

X Y Z
X Z Y
Y X Z
Y Z X = 0 people made this choice
Z X Y
Z Y X

People who ranked X number 1 = 40% of 200 = 80. This implies that the sum of X Y Z & X Z Y is equal to 80.

People who ranked Y number 1 = 30% of 200 = 60. This implies that the sum of Y X Z & Y Z X is equal to 60. Since Y Z X is 0, this implies that the number of people who picked the order Y X Z is 60.

People who ranked Z number 3 = 55% of 200 = 110. This implies that the sum of Y X Z & X Y Z is equal to 110. Since Y X Z is 60, this implies that X Y Z =50. Therefore, the correct answer is C.


Hi Mike03, How is 'People who ranked Z number 3 = 55% of 200 = 110'?
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Re: 200 people responded to a survey that asked them to rank three differe [#permalink] New post   20 Oct 2020, 22:52
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Bunuel wrote:

200 people responded to a survey that asked them to rank three different brands of soap. The percentage of respondents that ranked each brand 1st, 2nd, and 3rd are listed above. If no respondents rated the soaps in the order Y, Z, X, how many respondents rated the soap in the following order: X, Y, Z?

A. 100
B. 60
C. 50
D. 30
E. 25

Kudos for a correct solution.

Show Spoiler
Attachment:
Brand_Rankings.png



There was no YZX order. So who are the 15% people who rated X last? (from fig 3)
They must have rated ZYX (nothing else is possible). So 15% people rated ZYX.

Now, who are the 40% people who rated Y in the middle? (from fig 2) They are those who rated XYZ and ZYX (nothing else is possible). We know that 15% people rated ZYX. Then rest 25% must have rated XYZ. 25% gives us 50 people.
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Re: 200 people responded to a survey that asked them to rank three differe [#permalink] New post   05 Nov 2023, 07:08
Question asks for how many made the choice X Y Z. The total combos are:
I) X Y Z = The unknown we need to find.
II) X Z Y
III) Y Z X = 0 (Given in stem)
IV) Y X Z
V) Z X Y
VI) Z Y X

we know, by looking at the graphs, that those who put X first (I + II) is 40% = 80 people (40% of 200) - Using graph first choice. But we need to know what is the value of I only, without the addition of II. How to find this? Well, we know that those who put Z second consists of II and III, and III is zero, so the 15% (graph 2nd choice) who put Z second consists only of II as III = 0, so II = 0.15*200 = 30. Thus: (I + II) - (II) = I = X Y Z = 80 - 30 = 50.
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Re: 200 people responded to a survey that asked them to rank three differe [#permalink]
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