Fraction Diagrams

Date: 03/13/2002 at 08:53:53
From: Amani Abuhabsah
Subject: Fractions for children

The question that my instructor wanted me to ask students is the 
following:

In an adult condominium complex, 2/3 of the men are married to 3/5 of 
the women. What fraction of the residents are married?

I multiplied and got 2/5 as the answer. I also thought that because 
2/3 of the men are married, 1/3 are not, and because 3/5 of the women 
are married, 2/5 are not, so I multiplied that ....

Can you help me get the answer please? Thanks a lot.

Date: 03/13/2002 at 12:57:49
From: Doctor Peterson
Subject: Re: Fractions for children

Hi, Amani.

You didn't say how old the students who are expected to do this are; 
it's a nice problem that can be solved without much trouble by 
algebra, but I'll try a visual method instead.

Here are all the men, 2/3 of whom are married:

    +--------------+--------------+--------------+
    |              |           married           |
    +--------------+--------------+--------------+

Here are all the women, 3/5 of whom are married:

    +---------+---------+---------+---------+---------+
    |           married           |                   |
    +---------+---------+---------+---------+---------+

But the married men are married to the women, so if we assume no 
polygamy is allowed, we can match up the married men to the married 
women:

    +--------------+--------------+--------------+
    |              |           married           | men
    +--------------+--------------+--------------+
                   +---------+---------+---------+---------+---------+
             women |           married           |                   |
                   +---------+---------+---------+---------+---------+

Now we can cut my bars into thirds and halves respectively, to find a 
common denominator:

    +----+----+----+----+----+----+----+----+----+
    |              |           married           | men
    +----+----+----+----+----+----+----+----+----+
                   +----+----+----+----+----+----+----+----+----+----+
             women |           married           |                   |
                   +----+----+----+----+----+----+----+----+----+----+

However many people are represented by each little chunk, there are 12 
chunks of married people out of 9+10 = 19 chunks in all, so 12/19 of 
the people are married.

We can do the same thing algebraically. Letting X be the number of 
married men and married women, there are 2X married people, 3/2 X men 
residents, and 5/3 X women residents, making a ratio of 
(2X) / (3/2 X + 5/3 X) = 2/(19/6) = 12/19.

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