Ivan, Zoe, Denys and Ethan wanted to divided
 up a big old pickle 30 feet long. So,


Denys wants twice as much as Ivan does. So,


Zoe wants twice as much as Denys does


Zoe wants...half as much as Ethan does. 



So we have these equations:






Let's see if we can get equations for
I, D, and E in terms of Z. Then we'll substitute
them in the first equation.

From
, by multiplying both sides by 2,


So we have something to substitute for 
 in terms of Z, namely .

From 
, by multiplying both sides by ,


So we have something to substitute for 
 in terms of Z, namely .

From
, by mutiplying both sides by ,
.  Then we can substitute
 for  in that, and get

 which is the same as


So we have something to substitute for 
 in terms of Z, namely .

So let's substitute all those in

, leaving  as it is:



Then we can multiply everything by  and get





Then we can divide both sides by  and get



Then we can get E from 




Then we can get D from





Finally we can get I from





So

Ivan gets 2 inches, Zoe gets 8 inches, 
Denys gets 4 inches, and Ethan gets 16 inches

Edwin