Let V = vanilla, C = chocolate, and S = strawberry.
We are asked to determine how many people ranked vanilla first.
In other words:
VCS + VSC.
The sum of all of the rankings is 60:
VCS + VSC + CVS + CSV + SCV + SVC = 60.
Since 3/5 of the 60 people ranked vanilla last, CSV + SCV = (3/5)(60) = 36.
Substituting CSV + SCV = 36 into VCS + VSC + CVS + CSV + SCV + SVC = 60, we get:
VCS + VSC + CVS + 36 + SVC = 60
VCS + VSC + CVS + SVC = 24.
1/10 of the 60 people ranked vanilla before strawberry:
VCS + VSC + CVS = (1/10)(60) = 6.
1/3 of the 60 people surveyed ranked vanilla before chocolate:
VCS + VSC + SVC = (1/3)(60) = 20.
Adding together the two equations in red, we get:
VCS + VSC + CVS + VCS + VSC + SVC = 6+20
VCS + VSC + CVS + SVC + VCS + VSC = 26.
Substituting VCS + VSC + CVS + SVC = 24 into VCS + VSC + CVS + SVC + VCS + VSC = 26, we get:
24 + VCS + VSC = 36
VCS + VSC = 2.
The correct answer is A.
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3/5 ranked vanilla last.girishj wrote:In a marketing survey, 60 people were asked to rank three flavors of ice cream, chocolate, vanilla and strawberry, in order of their preference. All 60 people responded, and no two flavors were ranked equally by any of the people surveyed. If 3/5 of the people ranked vanilla last, 1/10 of them ranked vanilla before chocolate, and 1/3 of them ranked vanilla before strawberry, how many of them ranked vanilla first?
(A) 2
(B) 6
(C) 14
(D) 16
(E) 24
3/5 x 60 = 36
1/10 ranked vanilla before chocolate. 1/10 x 60 = 6
6 ranked vanilla before chocolate. So a maximum of 6 ranked vanilla first, because any who ranked vanilla first ranked vanilla before chocolate.
So we are down to two possible answers, A and B. Try the choices.
If 6 ranked vanilla first, then 60 - 36 - 6 = 24 ranked vanilla second.
There can't be any who ranked chocolate third, because then some of the 24 who ranked vanilla second would have ranked vanilla ahead of chocolate, and we already have 6 who ranked vanilla over chocolate.
We have 36 who ranked vanilla third. To get to 60 third rankings, we need 24 more third rankings. None can have ranked chocolate third. So the other 24 third rankings would have to be all strawberry.
However, only 1/3 x 60 = 20 ranked vanilla ahead of strawberry. So the maximum who could have ranked strawberry third is 20, meaning 24 does not work.
So 6 ranking vanilla first does not work, leaving only 2 as a possibility.
The correct answer is A.
Last edited by MartyMurray on Tue Apr 26, 2016 1:07 pm, edited 1 time in total.
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An alternate -- and perhaps easier -- approach is to use the following formula for two overlapping groups:
Total = Group 1 + Group 2 - Both + Neither.
I discuss this formula here:
https://www.beatthegmat.com/finance-majo ... 67425.html
In the problem above, the formula becomes:
Total = (number who ranked V before C) + (number who ranked V before S) - (number who ranked V before BOTH C AND S) + (number who ranked V before NEITHER C NOR S).
According to the prompt:
Total = 60.
Number who ranked V before C = (1/10)(60) = 6.
Number who ranked V before S = (1/3)(60) = 20.
Number who ranked V before neither S nor C = number who ranked V last = (3/5)(60) = 36.
Plugging these values into the formula, we get:
60 = 6 + 20 - (number who ranked V before both S and C) + 36
60 = 62 - (number who ranked V before both S and C)
Number who ranked V before both S and C = 62-60 = 2.
The correct answer is A.
Total = Group 1 + Group 2 - Both + Neither.
I discuss this formula here:
https://www.beatthegmat.com/finance-majo ... 67425.html
In the problem above, the formula becomes:
Total = (number who ranked V before C) + (number who ranked V before S) - (number who ranked V before BOTH C AND S) + (number who ranked V before NEITHER C NOR S).
According to the prompt:
Total = 60.
Number who ranked V before C = (1/10)(60) = 6.
Number who ranked V before S = (1/3)(60) = 20.
Number who ranked V before neither S nor C = number who ranked V last = (3/5)(60) = 36.
Plugging these values into the formula, we get:
60 = 6 + 20 - (number who ranked V before both S and C) + 36
60 = 62 - (number who ranked V before both S and C)
Number who ranked V before both S and C = 62-60 = 2.
The correct answer is A.
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Maybe an easier way:
To rank vanilla first, you must rank it above strawberry AND above chocolate.
That means our greatest possible answer is (1/10) * Total = 6, assuming that everybody who rated vanilla above chocolate also rated vanilla above strawberry. So we can only have A or B.
Now let's suppose that the answer is (1/10); that is, that everybody who ranked vanilla > chocolate also rated vanilla > strawberry. If that's true, we have
6 rated vanilla first
(1/3 - 1/10)*60 = 14 rated chocolate first and vanilla second
But then we only have 3/5 of the total, or 36 people, rating vanilla last. This only gives us 36 + 14 + 6 = 56 ratings, so it can't be the right answer.
From there we can conclude that B is impossible, and that A must be the answer.
To rank vanilla first, you must rank it above strawberry AND above chocolate.
That means our greatest possible answer is (1/10) * Total = 6, assuming that everybody who rated vanilla above chocolate also rated vanilla above strawberry. So we can only have A or B.
Now let's suppose that the answer is (1/10); that is, that everybody who ranked vanilla > chocolate also rated vanilla > strawberry. If that's true, we have
6 rated vanilla first
(1/3 - 1/10)*60 = 14 rated chocolate first and vanilla second
But then we only have 3/5 of the total, or 36 people, rating vanilla last. This only gives us 36 + 14 + 6 = 56 ratings, so it can't be the right answer.
From there we can conclude that B is impossible, and that A must be the answer.
The sum of the number of people who ranked vanilla should be 60.
Here, 36(third) + 6(before chocolate) + 20(before strawberry) = 62.
The extra count is the number of people who rated vanilla before both chocolate and strawberry i.e. who ranked vanilla first.
This means, that 62 - 60 = 2 people ranked vanilla first.
Here, 36(third) + 6(before chocolate) + 20(before strawberry) = 62.
The extra count is the number of people who rated vanilla before both chocolate and strawberry i.e. who ranked vanilla first.
This means, that 62 - 60 = 2 people ranked vanilla first.
