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### How Many are in the Group?

```
Date: 10/17/96 at 14:58:25
From: Paul Benedict
Subject: How Many in Group?

Mr. Jackson polled the class to see how many students had been to one
or both of the nearby state parks. He found that everyone in Katie's
group had been to at least one of the parks.  When he asked how many
had been to Punta de las Cuevas, four children raised their hands.
When he asked how many had been to Salt Bay, three raised their hands.
When he asked how many had been to both state parks, two raised their
hands.

How many people were in Katie's group?
```

```
Date: 10/18/96 at 17:53:7
From: Doctor Dennis
Subject: Re: How Many in Group?

Questions like this are often very easy using Venn Diagrams.

Imagine you have a big rectangle. In that big rectangle are two
circles which are partially overlapping. Now, say that all the
students at the table are represented by the stuff within the
rectangle. Call one circle people who have gone to Punta de Cuevas and
the other circle people who have gone to Salt Bay.

Now, since two have been to both parks there are two students in the
area that is the intersection of the circles. The Salt bay circle
should have 3 people in it, but there are already two people in the
part of the circle that intersects with the other circle, so there is
just one person in the part of the Salt Bay Circle that is not within
the other circle. Similarly, there will be 2 people in the punta de
cuevas circle that are not in the part that overlaps the Salt Bay
circle. so there are 2 + 2+ 1 = 5 kids total.

-Doctor Dennis,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```

```
Date: 10/18/96 at 17:52:11
From: Doctor Keith
Subject: Re: How Many in Group?

Hi Paul,

This is an example of everyone's favorite type of problem, the word
problem. The real trick to word problems is trying to turn the words
into equations. Let's summarize what we have in a table for easy
reference:

Number of students
Place                   who have been there

Punta de las Cuevas            4
Salt Bay                       3
Both                           2
Neither                        0

We have several ways of looking at this but probably the easiest is a
logical argument.  The two students who have been to both parks would
have raised their hands for both Punta de las Cuevas and Salt Bay.
Thus two of the students in the Punta de las Cuevas group are also in
the Salt Bay group.  So let's find out how many have been to only one.

For Punta de las Cuevas, four students have been but two also went to
Salty Bay, so we have to remove them (remember we want students who
have only been to Punta de las Cuevas).

4 - 2 = 2  no. of students who have only been to Punta de las Cuevas

Thus only two students have only been to Punta de las Cuevas.
Similarly, for Salt Bay we have three students who have gone, but two
also went to Punta de las Cuevas so we must remove those two from the
three who have been.

3 - 2 = 1  no. of students who have only been to Salt Bay

Thus only one student has gone to only Salt Bay.  To get the final
answer we note that the sum of the students who have only been to
Punta de las Cuevas, plus those who have only been to Salty Bay, plus
those who have been to both must be the total in the group.  So:

2 + 1 + 2 = 5

Thus there are 5 students in the group.

If this seemed hard to follow you can always name the kids in small
groups like this and count the distinct names.  This is a good
visualization tool, but I suggest you not get used to it, because it
could be a real problem if the numbers got large.  To help you see
this problem, though, let's run through it. We will revise our table
by naming students in the first group only.

Place                      Students who have been there

Punta de las Cuevas         Katie, Billy, Tom, Sarah
Salt Bay                         3
Both                             2
Neither                          0

Now we know that two of the kids from the first group must be in the
"Both" group.  Let's pick the first two for simplicity.

Place                      Students who have been there
Punta de las Cuevas          Katie, Billy, Tom, Sarah
Salt Bay                         3
Both                         Katie, Billy
Neither                          0

We also know that the two from the "Both" group  must also be in the
Salt Bay group, with one student left over to be named as we choose.

Place                      Students who have been there
Punta de las Cuevas          Katie, Billy, Tom, Sarah
Salt Bay                     Katie, Billy, Fred
Both                         Katie, Billy
Neither                          0

Thus since we don't want to count Katie and Billy in all three groups,
we remove them form the first two and get

Place                      Students who have been there
only Punta de las Cuevas     Tom, Sarah
only Salt Bay                Fred
Both                         Katie, Billy
Neither                          0

And by counting the names we get 5 students.  As you can see, while
this will help you to see what we did and why, if we were to follow
this for a whole class or school, it could easily become a nightmare.

Hope this helps you out. Good luck!

-Doctor Keith,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Discrete Mathematics
High School Logic
High School Sets
Middle School Logic
Middle School Word Problems

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