Step by step
The question
Question: n(ξ) = 60, n(A) = 35, n(B) = 30, n((A ∪ B)′) = 10.
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: 35 in all.
x of them are in the middle.
So 35 − x are only A.
In the same way, 30 − x are only B.
10 are in neither: outside the circles.
2 Write the equation
All four parts together make 60.
Careful!
75 is more than 60.
The ones in the middle were counted twice: once in each circle.
Put the parts in: (35 − x) + x + (30 − x) + 10 = 60.
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.
35 + 30 + 10 = 75.
−x + x − x = −x.
Now we have 75 − x = 60.
4 Remove the 75
We want x by itself.
The 75 is in the way.
To remove 75, we take away 75.
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.
75 and − 75 make zero.
Cross them out.
60 − 75 = −15.
Now we have −x = −15.
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.
−15 divided by −1 is 15.
So x = 15!
6 Check
Let's check: put 15 in place of x.
We get 60, the same as the right side.
So x = 15 is right!
So 15 are in both.
In the picture: 20 only A, 15 both, 15 only B, 10 neither.
Together 60.
So it fits!
Let's look back.
First: draw the Venn diagram.
Then: write the equation.
Then: group like terms.
Then: remove the 75.
Then: remove the −1.
Last: check.