Step by step
The question
Question: 32 people: 16 like jazz, 20 like rock, 2 like neither.
How many like both?
1 Draw the Venn diagram
We don't know the middle.
We call it x.
Like jazz: 16 in all.
x of them are in the middle.
So 16 − x like only jazz.
In the same way, 20 − x like only rock.
2 like neither: outside the circles.
2 Write the equation
All four parts together make 32.
Careful!
38 is more than 32.
The ones in the middle were counted twice: once in each circle.
Put the parts in: (16 − x) + x + (20 − x) + 2 = 32.
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.
16 + 20 + 2 = 38.
−x + x − x = −x.
Now we have 38 − x = 32.
4 Remove the 38
We want x by itself.
The 38 is in the way.
To remove 38, we take away 38.
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.
38 and − 38 make zero.
Cross them out.
32 − 38 = −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 32, the same as the right side.
So x = 6 is right!
So 6 like both.
In the picture: 10 only jazz, 6 both, 14 only rock, 2 neither.
Together 32.
So it fits!
Let's look back.
First: draw the Venn diagram.
Then: write the equation.
Then: group like terms.
Then: remove the 38.
Then: remove the −1.
Last: check.