Step by step
The question
Question: n(ξ) = 40, n(A) = 25, n(B) = 20, n((A ∪ B)′) = 3.
Find n(A ∩ B).
n of A means how many are in A.
A ∩ B means in both.
(A ∪ B)′ means in neither.
1 Draw the Venn diagram
We don't know the middle.
We call it x.
In A: 25 in all.
x of them are in the middle.
So 25 − x are only A.
In the same way, 20 − x are only B.
3 are in neither: outside the circles.
2 Write the equation
All four parts together make 40.
Careful!
48 is more than 40.
The ones in the middle were counted twice: once in each circle.
Put the parts in: (25 − x) + x + (20 − x) + 3 = 40.
The brackets are only to show the parts: a plus in front, so they can just go.
3 Group like terms
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
All the x terms go together.
Plain numbers go together.
25 + 20 + 3 = 48.
−x + x − x = −x.
Now we have 48 − x = 40.
4 Remove the 48
We want x by itself.
The 48 is in the way.
To remove 48, we take away 48.
We do it on both sides.
Why both sides?
The = sign means both sides are the same amount.
If we change only one side, they are not the same any more.
48 and − 48 make zero.
Cross them out.
40 − 48 = −8.
Now we have −x = −8.
5 Remove the −1
−x means −1 times x.
We want just x.
To remove times −1, we divide by −1.
We do it on both sides, so they stay equal.
−1 divided by −1 is 1.
So just x is left.
−8 divided by −1 is 8.
So x = 8!
6 Check
Let's check: put 8 in place of x.
We get 40, the same as the right side.
So x = 8 is right!
So 8 are in both.
In the picture: 17 only A, 8 both, 12 only B, 3 neither.
Together 40.
So it fits!
Let's look back.
First: draw the Venn diagram.
Then: write the equation.
Then: group like terms.
Then: remove the 48.
Then: remove the −1.
Last: check.