johng2016 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 people ranked vanilla first?
A. 2
B. 6
C. 14
D. 16
E. 24
Let C represent chocolate, V represent vanilla, and C represent strawberryLet a = the number of people who ranked the flavors (from best to worst) as V-C-S
Let b = the number of people who ranked the flavors (from best to worst) as V-S-C
Let c = the number of people who ranked the flavors (from best to worst) as C-S-V
Let d = the number of people who ranked the flavors (from best to worst) as C-V-S
Let e = the number of people who ranked the flavors (from best to worst) as S-C-V
Let f = the number of people who ranked the flavors (from best to worst) as S-V-C
We know that
a + b + c + d + e + f = 60 [since there are 60 people in the survey]
3/5 of the people ranked vanilla lastThis refers to people who had any of the following rankings: C-S-V (c) or S-C-V (e)
3/5 of 60 = 36
So, we can write:
c + e = 36 1/10 of the people ranked vanilla before chocolateThis refers to people who had any of the following rankings: V-C-S (a), V-S-C (b) or S-V-C (f)
1/10 of 60 = 6
So, we can write:
a + b + f = 61/3 of the people ranked vanilla before strawberryThis refers to people who had any of the following rankings: V-C-S (a), V-S-C (b) or C-V-S (d)
1/3 of 60 = 20
So, we can write:
a + b + d = 20How many people ranked vanilla first? This refers to people who had any of the following rankings: V-C-S (a) or V-S-C (b)
So, our GOAL is to find the value of
a + bSo, we have the following equations:
c + e = 36a + b + f = 6a + b + d = 20ADD them all to get:
2a + 2b + c + d + e + f = 62We also know that
a + b + c + d + e + f = 60If we SUBTRACT the bottom equation from the top equation, we get:
a + b = 2Answer: A
Cheers,
Brent
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