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/