190 students go to a school bake sale. 95 buy a chocolate

Hi All,

We're told that 190 students go to a school bake sale. Of those students, 95 buy a chocolate chip cookie, 75 buy a peanut butter cookie, and AT LEAST 12 buy both. We're asked for the LEAST number of students who could have bought NEITHER type of cookie. This question is a variation on a standard Overlapping Sets question (although there is a 'twist'; AT LEAST 12 students bought both types of cookie), but we can still use the Overlapping Sets Formula:

Total = (1st group) + (2nd group) - (Both) + (Neither)

Based on the given information, the equation would look like this:

190 = (95) + (75) - (AT LEAST 12) + (Neither)
190 = 170 - (AT LEAST 12) + (Neither)
20 = (Neither) - (AT LEAST 12)

To minimize the "neither group", we have to make the "both" group as SMALL as possible. In this case, that would be exactly 12 people...

20 + 12 = Neither
32 = Neither

Final Answer: D

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AAPL wrote:Princeton Review

190 students go to a school bake sale. 95 buy a chocolate chip cookie, 75 buy a peanut butter cookie, and at least 12 buy both. What is the least number of students who could have bought neither type of cookie?

A. 10
B. 24
C. 30
D. 32
E. 45

OA D.

We can use the equation:

#total = #chocolate chip + #peanut butter - #both + #neither

#neither = #total - #chocolate chip - #peanut butter + #both

As we can see from the equation, keeping everything else constant, the number of students who purchase neither kind of cookie decreases as the number of students who purchase both kinds decreases. Therefore, to minimize #neither, we should minimize #both:

190 = 95 + 75 - 12 + neither

190 = 158 + neither

32 = neither

Answer: D