Step by step
The question
Question: 45 students: 28 walk, 22 cycle, 0 neither.
How many both?
1 Draw the Venn diagram
We don't know the middle.
We call it x.
walk: 28 in all.
x of them are in the middle.
So 28 − x only walk.
In the same way, 22 − x only cycle.
0 neither: outside the circles.
2 Write the equation
All four parts together make 45.
Careful!
50 is more than 45.
The ones in the middle were counted twice: once in each circle.
Put the parts in: (28 − x) + x + (22 − x) + 0 = 45.
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.
28 + 22 + 0 = 50.
−x + x − x = −x.
Now we have 50 − x = 45.
4 Remove the 50
We want x by itself.
The 50 is in the way.
To remove 50, we take away 50.
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.
50 and − 50 make zero.
Cross them out.
45 − 50 = −5.
Now we have −x = −5.
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.
−5 divided by −1 is 5.
So x = 5!
6 Check
Let's check: put 5 in place of x.
We get 45, the same as the right side.
So x = 5 is right!
So 5 both.
In the picture: 23 only walk, 5 both, 17 only cycle, 0 neither.
Together 45.
So it fits!
Let's look back.
First: draw the Venn diagram.
Then: write the equation.
Then: group like terms.
Then: remove the 50.
Then: remove the −1.
Last: check.