Step by step
The question
Question: n(ξ) = 36, n(A) = 15, n(B) = 21, n((A ∪ B)′) = 6.
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: 15 in all.
x of them are in the middle.
So 15 − x are only A.
In the same way, 21 − x are only B.
6 are in neither: outside the circles.
2 Write the equation
All four parts together make 36.
Careful!
42 is more than 36.
The ones in the middle were counted twice: once in each circle.
Put the parts in: (15 − x) + x + (21 − x) + 6 = 36.
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.
15 + 21 + 6 = 42.
−x + x − x = −x.
Now we have 42 − x = 36.
4 Remove the 42
We want x by itself.
The 42 is in the way.
To remove 42, we take away 42.
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.
42 and − 42 make zero.
Cross them out.
36 − 42 = −6.
Now we have −x = −6.
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.
−6 divided by −1 is 6.
So x = 6!
6 Check
Let's check: put 6 in place of x.
We get 36, the same as the right side.
So x = 6 is right!
So 6 are in both.
In the picture: 9 only A, 6 both, 15 only B, 6 neither.
Together 36.
So it fits!
Let's look back.
First: draw the Venn diagram.
Then: write the equation.
Then: group like terms.
Then: remove the 42.
Then: remove the −1.
Last: check.