Let x be the (total) amount of the automobile loans.

Let y be the (total) amount of the home loans.


One restriction is 

    y >= 4x                         (1)


Another restriction is     

    x + y = 20  millions dollars.   (2)


The return rate of the mixture of these loans is  

    R(x,y) =      (dimensionless quantity)              (3)


There are also standard non-negativity restrictions x >= 0, y>= 0.       (4)


    +-----------------------------------------------------------------------------+
    |    The problem wants we find the maximum R(x,y) under given restrictions.   |
    +-----------------------------------------------------------------------------+


Let's simplify the expression for R(X,y).  

For it, we express y = 20-x from (2) and substitute it into (3).  We get then R(x,y) as the function of x, ONLY:

    R(x) =  =  =  = 0.001x + 0.1.



So, R(x) is a linear function of x.   It will facilitate finding the maximum GREATLY.


Now we should determine, on which set of real numbers {x} we should look for the maximum of R(x).



To determine it, substitute y = 20-x into (1).  You will get

    20-x >= 4x,  or  20 >= 4x + x,  or  20 >= 5x,  or  x <= 20/5 = 4.


Thus the set of {x}, where we look  for the maximum of R(x)  is  0 <= x <= 4.



Now, function R(x) = 0.001x + 0.1 is a linear monotonically increasing function,

and, for given set of {x},  it gets the maximum at the end-point x = 4  (which is OBVIOUS) .



The value of the maximum is then   = R(4) = 0.001*4 + 0.1 = 0.104.


    +-------------------------------------------------------------------------------------+
    |   So, under given restrictions, the company should extend                           |
    |       4 millions dollars to automobile loans and 16 million dollars to home loans.  |
    |       The maximum return rate is then  0.104 = 10.4%.                               |
    +-------------------------------------------------------------------------------------+