30 students: 18 play football, 15 play tennis, 8 play both. How many… Easy

The question

Question: 30 students: 18 play football, 15 play tennis, 8 play both.

How many play neither?

1 Draw the Venn diagram

8 play both: they go in the middle first.

2 Fill the circles

Play football: 18 in all, 8 of them are in the middle.

18 minus 8 is 10.

Careful!

The 8 in the middle are inside both circles.

Taking away 18 and 15 takes them away twice.

Play tennis: 15 minus 8 is 7.

3 Find the outside

The circles hold 10 plus 8 plus 7: 25.

The rest of the 30 are outside: 5.

4 Check

Let's check: the four parts add up to 30.

So it fits!

Let's look back.

First: draw the Venn diagram.

Then: fill the circles.

Then: find the outside.

Last: check.

▶ Watch it solved

30 students: 18 play football, 15 play tennis, 5 play neither. How… Medium

The question

Question: 30 students: 18 play football, 15 play tennis, 5 play neither.

How many play both?

1 Draw the Venn diagram

We don't know the middle.

We call it x.

Play football: 18 in all.

x of them are in the middle.

So 18 − x play only football.

In the same way, 15 − x play only tennis.

5 play neither: outside the circles.

2 Write the equation

All four parts together make 30.

Careful!

38 is more than 30.

The ones in the middle were counted twice: once in each circle.

Put the parts in: (18 − x) + x + (15 − x) + 5 = 30.

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.

18 + 15 + 5 = 38.

−x + x − x = −x.

Now we have 38 − x = 30.

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.

30 − 38 = −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 30, the same as the right side.

So x = 8 is right!

So 8 play both.

In the picture: 10 only football, 8 both, 7 only tennis, 5 neither.

Together 30.

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.

▶ Watch it solved

n(ξ) = 40, n(A) = 25, n(B) = 20, n((A ∪ B)′) = 3, find n(A ∩ B) Harder

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.

▶ Watch it solved