Question

In a group of 40 students, 522 are taking Algebra, 18are taking Biology, 14 aretaking Chemistry, 9 aretaking Algebra andBiology, 7 are takingAlgebra and Chemistry, 5are taking Biology andChemistry and 2 aretaking all three courses.How many students arenot taking any of thesecourses?​

Answer 1

Explanation:

I'm told there are 40 students, so that means my universal set is going to contain

40 elements.

So the size of universal set

is 40.

There are 22 students taking algebra, so we'll us an 'A'

for this set of students in algebra,

and the size of that set

is

22.

18 are taking biology,

so the size of what we'll call set B

is 18.

14 are taking chemistry, so that gonna be set C.

And that has 14 students.

Okay, now we've got some intersections.

We've got

7 students taking algebra and chemistry, so that's the intersection of sets

A and C.

I'm sorry, I skipped one.

9 students are taking algebra and biology, biology

so that's the intersection of sets A and B.

The size of that intersection,

of

A intersect B

is 9.

Then I've got 7 taking algebra and chemistry, so the size of

the intersection of A and C

is

7.

What else do I have? 5 are taking biology and chemistry,

so the size of that intersection

biology

and chemistry

is 5,

and then

2 are taking all three courses.

So that would be the intersection of

A and B

as well as the intersection of that with C.

So here's the intersection of A and B, and we

intersect that

with C.

And there are

2 students in there.

So let's draw a rectangle for the universal set and start filling this in.

So let's see what we have.

I'm gonna have 3 circles,

one for set A,

one for set B,

and one for set C.

And they all overlap.

And usually the easiest thing to do is to start with the smallest area. The smallest are we have is

that intersection of A and B

and C.

and we know there are 2 students in there.

And now,

here's what we're going do from here.

i've got the intersection of A and B. day

Well the intersection of A and B is supposed to contain 9 students,

but 2 of those students

are already accounted for, they're in the place where all three sets overlap.

So that means I've got 7 more students

who are going to fit in that intersection.

I'll do the same thing for A and C.

My intersection of A and C

has

7 students in it.

But I've already accounted for 2 of them in that little triangle in the middle.

So that means I've got 5 more students

over there.

For the intersection of B and C

I've got

5 students.

2 of them are accounted for,

so that means there are 3 students there.

I can find out how many students are taking

only algebra.

There's a total of 22 students in all who are taking algebra.

I want to subtract the ones I've already

accounted for. So that's 7

plus 2 is 9,

plus 5 is 14.

I'll subtract that from 22

and that's going to give me 8.

Then I've got a total of 18 students

who are taking

biology,

and I've already accounted for

let's see... 7 and 3 is 10,

and 2 is 12. So I've accounted for 12 of them. There's 18 altogether,

so there's 6 left who are taking only biology.

In the chemistry set, there's a total of 14 students.

I've got 5 plus 2 plus 3 accounted for. That's ten, so that means

there are 4 students left

who are taking only chemistry.

And I've also got the fact that the universal set

contains

2o students....

I'm sorry, contains 40 students.

And my question was: How many students are not taking any of these courses?

So all I have to do

is add up

the total number of students that I have

who are in any of these

courses - algebra, biology,

or chemistry,

and subtract that from the universal set,

which is 40.

So let's see... I've got 8 plus 7 is 15,

plus 5 is 20,

plus 2 more is 22,

plus 6 over here is 28,

plus 3 is

31,

and 4 more would be 35.

So there are 35 students

who are taking algebra or biology or chemistry or some combination of them.

I've got 40 students in my universal set,

so that means there will be 5 students

left

who are not taking

algebra or biology

or chemistry.

So this is a bit of work,

but mostly its a mac of getting yourself set up, getting all your facts

straight,

entering the facts you know,

and then using

what's basically just arithmetic

to get the remaining facts.

So that's it for now.

Take care. I'll see you next time.