# Solving a Set Problem

Here is my question

• In a class that has $40$ students, there are $10$ students who could not pass the math exam and $30$ students who passed the english exam. If there are $6$ students who could not pass the both exam, How many are there students who passed the both exam?

I'm currently not able to build the correct equation. Might I get help? It seems like a bit confused. That's why I need get help.

• Hint: use the inclusion/exclusion principle. – user370967 Oct 31 '17 at 16:37
• @Math_QED What is it, sir? I'm already trying on it. Before using inclusion/exclusion principle, I ought to build the correct equation. – user440404 Oct 31 '17 at 16:38
• "group theory" is an incorrect tag; this is a question about sets. – Andrew Tindall Oct 31 '17 at 16:39
• I believe that It seems better now. – user440404 Oct 31 '17 at 16:42
• @Bobtrollsten The inclusion/exclusion principle "gives you the equation," so it would be silly to build the equation before applying the principle. – rschwieb Oct 31 '17 at 16:43

There are $4$ types of people:

Let $A$ be the people who passed no exam.

Let $B$ be the people who passed only Math.

Let $C$ be the people who passed only English.

Let $D$ be the people who passed both.

We know $A+B+C+D=40$

We know $A+C=10$

We know $C+D=30$

We know $A=6$.

So this is just a system of equations.

• I think I got it now. – user440404 Oct 31 '17 at 16:54

I find a Venn Diagram to always be very helpful for these kinds of questions:

$A,B,C,D$ are the regions according to Jorge's answer

So we know the following:

Once you figure out how the '10 people did not pass Math' plays out in the diagram, the rest goes quick.

• I was trying to plug it into venn. – user440404 Oct 31 '17 at 16:57
• does the venn diagram help? You get a lot of clumping in this case. – Yorch Oct 31 '17 at 16:59
• @JorgeFernández True ... but I think different approaches work better for different people. So maybe for some the diagram is more distracting than helpful, but for others it may be helpful. Personally, I'm a very visual person. And once you figure out how the '10 people not inside the Passed Math' circle plays out in the diagram, the rest goes quick. – Bram28 Oct 31 '17 at 17:08

I hope my cute Venn diagram can be useful to somebody :) $$...$$