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3 newspapers- a question about Inclusion–exclusion principle

lola19991

New member
Apr 14, 2018
6
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.
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,736
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! (Wave)

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. (Thinking)
 

tkhunny

Well-known member
MHB Math Helper
Jan 27, 2012
267
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
Apr 14, 2018
6
Hey Lola! (Wave)

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. (Thinking)
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
Mar 5, 2012
8,736
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? (Wondering)