Ratio of Ratios

Date: 06/05/2002 at 22:14:17
From: Tereasa Holland
Subject: WORD PROBLEM

A particular condominium complex is an adults-only complex. In this 
complex, 3/5 of the men are married to 2/3 of the women. What 
fraction of the residents are married? Explain the solution.

Date: 06/06/2002 at 15:03:11
From: Doctor Ian
Subject: Re: WORD PROBLEM

Hi Tereasa,

Suppose there are 5 men.  Then we have this situation:

  married     m m m 

  unmarried   m m

We have to have at least 3 women, to be married to the men... but 
3 isn't 2/3 of any integer. So we have to have at least 6 women:

  married     m m m 
              w w w w w w

  unmarried   m m
              w w w

Which means we need at least 3 more men:

  married     m m m m m m
              w w w w w w

  unmarried   m m m m
              w w w

The entire population has to be some multiple of this population, 
right?  

We could have arrived here more quickly by noting that if m is 
the number of men, and w is the number of women, then the numbers 
of married men and women have to be equal (unless you want to 
start considering gay marriage, polygamy, and other alternative 
arrangements... in which case there is no way to answer the 
question), so 

     3     2
     - m = - w
     5     3

   3*3 m = 2*5 w             Cross multiply

     9m = 10w

So there has to be a ratio of 9 men to 10 women, which is what 
our picture shows.  

Can you take it from here?

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