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

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by BTGmoderatorDC » Mon Oct 01, 2018 11:42 pm

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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

Source: Princeton Review
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by GMATGuruNY » Tue Oct 02, 2018 3:15 am
BTGmoderatorDC wrote: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
Total = Group 1 + Group 2 - Both + Neither

Here:
Total = 190.
Group 1 = 95 who buy chocolate chip.
Group 2 = 75 who buy peanut butter.
Plugging these values into the equation above, we get:
190 = 95 + 75 - Both + Neither
20 + Both = Neither
Neither = 20 + Both.

To minimize the value of Neither, we must minimize the value of Both.
Since the least possible number who buy both types of cookies = 12, we get:
Least possible number who buy neither = 20 + 12 = 32.

The correct answer is D.
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by Jay@ManhattanReview » Tue Oct 02, 2018 10:38 pm
BTGmoderatorDC wrote: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

Source: Princeton Review
Say,

the number of students who buy chocolate chip cookies = C;
the number of students who buy peanut butter cookie = P;
the number of students who buy both chocolate chip cookies and peanut butter cookie = B;
the number of students who buy none of the cookies = N

Thus, 195 = C + P - B + N

195 = 95 + 75 - B + N
B = N - 20

Since it is given that B ≥ 12, we have

N - 20 ≥ 12
N ≥ 32

Thus, the minimum value of B = 32.

The correct answer: D

Hope this helps!

-Jay
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by Scott@TargetTestPrep » Wed Oct 03, 2018 4:59 pm
BTGmoderatorDC wrote: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
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 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

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