really stuck on how to solve minimum and maxiumum venn diagram questions xx

Dazzla16
Hi, sarahdaly7!
Somebody asked me the same question a few months ago, so I've basically copyandpasted the response I gave to the same question. Note that you may have different Maths exam papers as I did back when I was in 3rd year (I'm in 5th year now) so you may not be able to find these examples that follow.
To answer your question, I'm going to use an exam question from 2014 Paper 1, and refer to its answers in answering your question.
So this is the question: In a table quiz, 100 questions were asked. Team M answered 72 correctly. Team N answered 38 questions correctly. (i) Find the minimum number of questions answered correctly by both teams.
In this question, we have to find the smallest value for [M n N]
To make the intersection as small of a value as possible, you have to make [(M u N)'] (complement of M union N) equals to 0. This would allow the overlap (intersection) to be as small as possible.
Here are the workings:
M n N = (#M + #N)  100
= (72 + 38)  100
= 110  100
= 10
Therefore, the minimum number of questions answered by both teams is 10.
(ii) Find the maximum number of questions answered correctly by both teams.
In this question, we have to find the largest value for [M n N]
To make the intersection as large of a value as possible, we will say that all the questions answered correctly by Team N was also answered correctly by Team M. You can already see that the maximum value for [M n N] is 38.
Just in case you need it, here are the workings:
72 = [M u N]
[M n N] = (#M +#N)  72
= (72 + 38)  72
= 110  72
= 38
Therefore, the maximum number of questions answered by both teams is 38.
Use this example to answer similar questions you might get stuck on.
Here's another example, which might be easier to understand. This exam question is from the EDCO Sample D Paper 1.
This is the question: A group of students were asked whether they drink tea or coffee. 22 said that they drink tea, 27 said that they drink coffee and 3 said that they drink neither.
(i) What is the maximum number of people in the group?
To get the maximum number of people, make the amount of people who drink both tea and coffee equal to 0. Then, simply add all the numbers.
22 + 27 + 3 = 52
Therefore, the maximum number of people is 52.
(ii) What is the minimum number of people in the group?
To get the minimum number of people, we have to make the amount of people who drink both tea and coffee as big as possible. This will mean that all people who drink tea also drinks coffee. Then, just add the amount of people who drink coffee with the amount of people who drink neither.
27 + 3 = 30
Therefore, the minimum number of people is 30.
Hope it helps :)

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