50 students: 28 take art, 20 take music, 10 take both, P(neither)

Step by step

The question

Question: 50 students were asked.

28 take art, 20 take music, and 10 take both.

One is picked at random.

What is the probability they take neither?

1 Start in the middle

10 take both.

They go in the middle, where the circles overlap.

2 Fill the rest

28 take art, but 10 of them are already in the middle.

28 minus 10 is 18.

Music only: 20 minus 10 is 10.

Neither: everyone else.

50 minus 18, 10 and 10 is 12.

Careful!

That takes away the 10 who take both twice.

Start in the middle.

3 Find the part we want

Neither: outside both circles, 12.

4 Write the fraction

So the probability is 12/50.

Both 12 and 50 can be divided by 2.

That makes it simpler.

So it is 6/25.

5 Check

Let's check: the four parts add up to 18 plus 10 plus 10 plus 12.

That is 50, everyone.

So the answer is right!

Let's look back.

First: start in the middle.

Then: fill the rest.

Then: find the part we want.

Then: write the fraction.

Last: check.