# 3 newspapers- a question about Inclusion–exclusion principle

#### lola19991

##### New member
There is a city with 100,000 people, which has 3 newspapers: A, B and C. 10% read A, 30% read B, 5% read C. 8% read A and B, 2% read A and C, 4% read B and C and only 1% read all of them.
a) How much people read only one newspaper?
b) How much people read at least two newspapers?
c) If A and C are morning newspapers and B is an evening newspaper, how much people read at least one morning newspaper and one evening newspaper?
d) How much people read one morning newspaper and one evening newspaper?
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I did a&b and the answers that I got are:
a) 20,000
b) 12,000
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I would like to know how to solve the other parts of the question.

#### Klaas van Aarsen

##### MHB Seeker
Staff member
There is a city with 100,000 people, which has 3 newspapers: A, B and C. 10% read A, 30% read B, 5% read C. 8% read A and B, 2% read A and C, 4% read B and C and only 1% read all of them.
a) How much people read only one newspaper?
b) How much people read at least two newspapers?
c) If A and C are morning newspapers and B is an evening newspaper, how much people read at least one morning newspaper and one evening newspaper?
d) How much people read one morning newspaper and one evening newspaper?
--------------
I did a&b and the answers that I got are:
a) 20,000
b) 12,000
--------------
I would like to know how to solve the other parts of the question.
Hey Lola!

Can you clarify what '10% read A' means exactly?
Does it mean that '10% read at least A'? Or '10% read only A'?

Anyway, for (c) we want to know:
$$\#(\text{at least 1 morning paper} \land \text{at least 1 evening paper}) =\#\Big((A \cup C) \cap B\Big)$$
Do you know how to calculate that (and what it means)?
Typically we draw a so called Venn Diagram to figure out something like that.

#### tkhunny

##### Well-known member
MHB Math Helper
There is a city with 100,000 people, which has 3 newspapers: A, B and C. 10% read A, 30% read B, 5% read C. 8% read A and B, 2% read A and C, 4% read B and C and only 1% read all of them.
a) How much people read only one newspaper?
b) How much people read at least two newspapers?
c) If A and C are morning newspapers and B is an evening newspaper, how much people read at least one morning newspaper and one evening newspaper?
d) How much people read one morning newspaper and one evening newspaper?
--------------
I did a&b and the answers that I got are:
a) 20,000
b) 12,000
--------------
I would like to know how to solve the other parts of the question.
How did you solve the first two?
Why is Part c any different? (A or C) and B
Why is Part d any different? Subset of the answer to Part c?

Translation Hint:
How MANY people? People are countable.
How MUCH sugar? Sugar is in countable, but it is measurable.
How MANY frogs? Countable.
How MUCH air? Measurable.

#### lola19991

##### New member
Hey Lola!

Can you clarify what '10% read A' means exactly?
Does it mean that '10% read at least A'? Or '10% read only A'?

Anyway, for (c) we want to know:
$$\#(\text{at least 1 morning paper} \land \text{at least 1 evening paper}) =\#\Big((A \cup C) \cap B\Big)$$
Do you know how to calculate that (and what it means)?
Typically we draw a so called Venn Diagram to figure out something like that.
It means that 10% read at least A and I would like to know how to calculate that and what it means and I know that part d is related to part c, so I would like to understand them both.

#### Klaas van Aarsen

##### MHB Seeker
Staff member
It means that 10% read at least A and I would like to know how to calculate that and what it means and I know that part d is related to part c, so I would like to understand them both.
Ok. So that means we have the following Venn Diagram.
\begin{tikzpicture}
\begin{scope}[blend group = soft light]
\fill[red!30!white] ( 90:2) circle (3);
\fill[green!30!white] (210:2) circle (3);
\fill[blue!30!white] (330:2) circle (3);
\end{scope}
\node at (90:5) {$A$};
\node at (210:5) {$B$};
\node at (330:5) {$C$};
\node at (90:3) {1\%};
\node at (210:3) {19\%};
\node at (330:3) {0\%};
\node {1\%};
\node at (30:2) {1\%};
\node at (150:2) {7\%};
\node at (270:2) {3\%};
\end{tikzpicture}
We can see that the people reading exactly 1 news paper are 1% + 19% + 0% = 20% of 100,000.
That is indeed 20,000 people. Good!

For (c) we want $(A∪C)∩B$.
That is, we look at the $A$ and $C$ combined.
And from those parts only the ones that are within $B$.
That is 7% + 1% + 3% isn't it?