A certain bakery baked a batch of 500 cookies one day. Of those, 320

Cookies which have both nuts and chocolate chips = 135
Let fewest possible number of cookies with both chocolate chips and nuts that would need to be added to that
batch so that cookies with both nuts and chocolate chips represented more than 3/5 of all the cookies in the batch = x

(135+x)/(500+x) = 6/10
=>1350 + 10x = 3000 + 6x
=> 4x = 1650
=> x = 412.5
Therefore x = 413

Answer C

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Thanks in advance for any reply !!
Please help me in understanding this question correctly!!

Since the question asks for "fewest possible cookies", shouldn't it be that we consider maximum overlap between 320 and 230 i.e. 230 rather 135 and then try to find the "fewest possible number of cookies with both chocolate chips and nuts that would need to be added to that batch so that cookies with both nuts and chocolate chips represented more than 3/5 of all the cookies in the batch"

Considering, 230 as the maximum overlap to find fewest possible .... 3/5 of all the cookies in the batch, as stated above in the solution by Skywalker18. The equation would become as below:

(230+x)/(500+x)=6/10
x=175

why is this not correct ? or why is it wrong to consider 230 already having both nuts and choco chips rather than 135?

Reply to Itisallinurhead:
You are overlooking a basic stipulation of the problem: Of the total 500 cookies, only 85 have neither nuts nor chocolate chips so the number of cookies with nuts or chocolate chips or both are 500-85=415 from which we get the 135 overlap. As per your suggestion, we will have 180 (500 - 320) cookies without chocolate chips or nuts. Hope this clears up the confusion.

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Hey Itisallinurhead

Let's go with your reasoning - If there were 230 cookies which had both nuts and choco chips.
There would be 320 - 230 = 90 cookies which had only nuts and 0 who had only choco chips.

We are also told that the number of cookies which contained neither nuts nor chocolate chips is 85.
If we to add all of these, we should get the total number of cookies(500).

However, we end up short with sum of cookies = 90 + 230 + 85 = 405

Formulae for 2 overlapping set(s)
1. P(Total) = P(A) + P(B) - P(Both) + P(Neither)
2. P(Total) = P(Only A) + P(Only B) + P(Both) + P(Neither)

In this problem, we have
P(Total) = 500 | P(A) = 230 | P(B) = 320 | P(Neither) = 85

Substituting these values in forumula 1, we get 500 = 230 + 320 - P(Both) + 85

Therefore, P(Both) = 635 - 500 = 135

Hope this helps you!

You are given that of 500, only 85 have neither nuts nor chocolate chips. This means 415 MUST have at least one of the two things.
If you have an overlap of 230, you are putting the chocolate chips circle inside the 320 nuts circle. Then you have 500 - 320 = 180 cookies which have neither chocolate chips nor nuts. That is not acceptable. There are only 85 such cookies. Because of this, the overlap is fixed. You do not have a maximum/minimum overlap scenario here.

n(C or N) = n(C) + n(N) - Both
415 = 320 + 230 - Both
Both = 135 (no other value possible)


Actually the confusion arises because of the term "fewest possible number of cookies". Why is that required? Why couldn't the question just ask "the number of cookies needed..."
It is not because the overlap can vary; it is because you can achieve "more than 3/5" in a variety of ways. 413 makes it just more than 3/5. 414 also makes it more than 3/5. 430 also makes it more than 3/5. 1000 also makes it more than 3/5 and so on...
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