How Many People Went on the Cruise?

Date: 12/03/2001 at 17:26:34
From: Jessica Morris
Subject: 7th grade word problem

At the end of a special cruise, the employees could not remember the 
total number of people who were on board. However, they had the 
following data from the passenger list: 

   520 European females
   340 North American women
   340 North American males
   580 girls
   880 European citizens of which 580 were women
   480 European adults

Determine the total number of people on the cruise.

I've spent a lot of time trying to figure this problem out. It seems 
as if the females would include the girls and women, and the adults 
would include women and men, but I am totally stuck. I have even had 
my parents try to help me and we are all confused. Please help if 
possible. Thank you!

Date: 12/03/2001 at 21:04:04
From: Doctor Peterson
Subject: Re: 7th grade word problem

Hi, Jessica.

I'll just get you started by showing you how I approached the problem. 
If you think visually, this should work for you. If not, you can 
convert what I do to simultaneous equations and solve them by whatever 
means works best for you.

First, I like to turn words into symbols, so I devised these:

    A = American    B = British
    F = Female      M = Male
    a = adults      c = children

There are two special terms:

    Women = Female adults = Fa
    Girls = Female children = Fc

"Citizens" is a throw-away word, assuming that all the "British" 
people are British citizens.

Now I can write the given numbers this way:

    BF =  26
    AFa = 17
    AM =  17
    Fc =  29
    B =   44
    Fa =  29
    Ba =  24

Now we can write these on a diagram. There are three categories being 
used to divide people up (A/B, F/M, and a/c), so we can use either the 
traditional three overlapping circles (which are hard to draw in 
text), or a cross-and-circle diagram:

          A             |             B
                        |
                        |
                        |
F        AFa        *********        BFa
                 ***    |    ***
               **       |       **
              *   AFc   |  BFc    *
             *          |          *
            *           |           *
------------*-----------+-----------*------------
            *           |           *
             *          |        c *
              *   AMc   |  BMc    *  a
               **       |       **
                 ***    |    ***
M        AMa        *********         BMa
                        |
                        |
                        |
                        |

The left side is Americans, the right side British; the top half is 
female, the bottom male; and the inside is children, the outside 
adults. Each region is labeled with a complete set of three letters; 
an incomplete set, such as Fa, means "all Female adults, British or 
American", so it is the union of two (or four) regions, in this case 
Fa = AFa + BFa.

Now I can fill in the numbers, writing the number for a specific 
region (where all three categories are known) in the middle of the 
region, and the number for a combination of regions at their 
intersection:

          A             |           B
                      Fa=29
                        |
                        |
F        AFa=17     *********        BFa
                 ***    |    ***
               **     Fc=29     **
              *   AFc   |  BFc    *
             *          |        BF=26
            *           |           *
------------*-----------+---------B=44---Ba=24---
            *           |           *
           AM=17        |          *
              *   AMc   |  BMc    *
               **       |      c**
                 ***    |    ***
M        AMa        *********  a      BMa
                        |
                        |
                        |
                        |

For example, I wrote Fa=29 straddling the border of AFa and BFa, 
because Fa= AFa + BFa. I put B=44 where the line and circle cross, 
because B = BFc + BFa + BMa + BMc. (Actually, I just wrote the 
numbers, not the names, so the numbers stood out more.)

Now you can look for numbers to subtract. We can start with the one 
specific number we are given, AFa=17. Look for a number on the border 
of that region: Fa=29. Subtract those to get BFa=12, since Fa = AFa + 
BFa. Keep going like this, and you'll find that you know every 
specific number except for one pair, for which you know the sum. (This 
makes sense, because you have 7 equations for 8 variables, so you 
wouldn't expect to be able to solve for all of them.) From this you 
can get the total number of people.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   

Date: 12/04/2001 at 10:40:51
From: Jessica Morris
Subject: Re: 7th grade word problem

Dr. Peterson,

Thank you for your help, but I am still struggling with part of this
problem. I made symbols for all of the people in my problem, as you
did, and I also drew it out on a graph, as you did. This is my
problem: How can I have 580 European adult women, when the problem
also says I only have 480 European adults? This doesn't make sense to
me.  

Thank you!  
Jessica

Date: 12/04/2001 at 11:12:39
From: Doctor Peterson
Subject: Re: 7th grade word problem

Hi, Jessica.

I knew I recalled doing a problem like this, and with the reminder
that there was a bad number in there, I searched again and found this,
an answer I sent before:

=====================================================================
> Millennium Cruise
>
> Every week I give my students a problem of the week from a variety 
> of sources which I make note of in my plans, however I have 
> forgotten where this came from and believe I've made a typo 
> somewhere.  If I haven't made a typo, then I guess(along with my 
> colleagues), I am stumped at the following problem:
> 
> At the end of a special millenium cruise, the employees could not 
> remember the total number of people on board. They had the following 
> information from the passenger list: 520 European females, 340 North 
> American women, 340 North American males, 580 girls, 880 European 
> citizens of which 580 were women, and 480 European adults. Determine 
> the total number of people on board the Millennium Cruise.

The 580 European women is too many; there are only 520 European
females, a category that includes all the European women! Similarly,
there are only 480 European adults, which would also include all 580
European women. Can we determine what the number should be? 

Here's a way to analyze the problem, and perhaps find a valid number
for X, the number of European women. I find three categories, which
I'll label by using capitals for the first and lower case for the
second type in each: European (E) vs. North American (n); Female (F)
vs. male (m); and Adult (A) vs. juvenile (j). There are eight
combinations of these categories, and each of the numbers you give
includes one or more of them:

    (a) 520 European females       EFA+EFj
    (b) 340 North American women   nFA
    (c) 340 North American males   nmA+nmj
    (d) 580 girls                  EFj+nFj
    (e) 880 European citizens      EFA+EFj+EmA+Emj
    (f)   X European women         EFA
    (g) 480 European adults        EFA+EmA

We're assuming that   EFA = X.
From (fg) we have (h) EmA = 480-X, and X = <480.
From (af) we have (i) EFj = 520-X, and X = <520.
From (di) we have (j) nFj = 580-EFj = 580-(520-X) = 60+X.
From (ea) we have (k) EmA+Emj = 880-520 = 360.
From (hk) we have (l) Emj = 360-(480-X) = X-120, and X >= 120.
Now the total of all eight groups is

EFA +   EFj   +   EmA   +   Emj   + nFA +   nFj  + (nmA+nmj) = 
 X  + (520-X) + (480-X) + (X-120) + 340 + (60+X) +    340    = 1640+X

So you can replace the 580 with any value of X between 120 and 480,
and the total number of people will be X+1640. I strongly suggest you
try X = 360, which gives a nice solution to a Millennium cruise
problem! (Or would you prefer X = 361 for the same reason?) On the
other hand, what crew would have been unable to remember that number?
There must have been too much to drink ...

Sometimes figuring out what the problem should be can be more fun than 
actually solving the problem itself!
======================================================================

I think that should answer your question precisely. It's interesting
that the word "millennium" was left out of your version.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/