SOLUTION: In a condo complex, 2/3 of the men were married to 3/4 of the women. What is the ratio of married people to the total adult population of the condo complex? Explain how you can obt

1.  From the condition,

         = .     (1)


    Multiply both sides by 12. You will get

        8M = 9W.           (2)

    or

        M =           (3)



2.  They ask you to evaluate the ratio of   +   (the amount of married people) to

    (M+W)  (the total adult population of the condo complex).  


    The numerator is   +  =    (4)


    In (4), replace 8M by 9W,  based on (2).  Then you can continue for the numerator

         +  =  =  =  = .



     The denominator is  M + W =  = .       ( <<<---=== I replaced  here M  by   based on (3). )



3.  Now the ratio under the question

        =  =  =  =  = .


Answer.  The ratio under the question = .

    You are given two linear forms (functions) of two variable x and y:  p(x,y) = ax + by  and  q(x,y) = cx + dy.

    You are given also linear dependence between these two variables   = e.

    Then they ask you evaluate the ratio  .


The solution is very simple:

    you express x = ey  and substitute it into each of the two linear functions:

        p(x,y) = ax + b*ex = (a + be)*x,

        q(x,y) = cx + d*ex = (c + de)*x.


Thus each of the two functions p(x,y)  and  q(x,y)  becomes the linear function of one variable (linear proportionality).


Then the ratio of the forms  p(x,y) = ax + by  and  q(x,y) = cx + dy  is

        =  =  = constant not depending on x and y.


That's all.