Question 165788
<font size = 7 color = "red"><b>Edwin's solution:</b></font>
<pre><font size = 4 color = "indigo"><b>
Ivan, Zoe, Denys and Ethan wanted to divided
 up a big old pickle 30 feet long. So,
{{{I + Z + D + E = 30}}}

Denys wants twice as much as Ivan does. So,
{{{D = 2I}}}

Zoe wants twice as much as Denys does
{{{Z= 2D}}}

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

{{{Z = (1/2)E}}}

So we have these equations:

{{{I + Z + D + E = 30}}}
{{{D = 2I}}}
{{{Z= 2D}}}
{{{Z = (1/2)E}}}

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
{{{Z = (1/2)E}}}, by multiplying both sides by 2,
{{{2Z = E}}}

So we have something to substitute for 
{{{E}}} in terms of Z, namely {{{2Z}}}.

From 
{{{Z= 2D}}}, by multiplying both sides by {{{1/2}}},
{{{(1/2)Z=D}}}

So we have something to substitute for 
{{{D}}} in terms of Z, namely {{{(1/2)Z}}}.

From
{{{D=2I}}}, by mutiplying both sides by {{{1/2}}},
{{{(1/2)D=I}}}.  Then we can substitute
{{{(1/2)Z}}} for {{{D}}} in that, and get

{{{(1/2)(1/2)Z=I}}} which is the same as
{{{(1/4)Z=I}}}

So we have something to substitute for 
{{{I}}} in terms of Z, namely {{{(1/4)Z}}}.

So let's substitute all those in

{{{I + Z + D + E = 30}}}, leaving {{{Z}}} as it is:

{{{(1/4)Z + Z + (1/2)Z + 2Z = 30}}}

Then we can multiply everything by {{{4}}} and get

{{{Z + 4Z + 2Z + 8Z = 120}}}

{{{15Z=120}}}

Then we can divide both sides by {{{15}}} and get

{{{Z=8}}}

Then we can get E from 
{{{2Z = E}}}
{{{2(8)=E}}}
{{{16=E}}}

Then we can get D from

{{{(1/2)Z=D}}}
{{{(1/2)(8) = D}}}
{{{4=D}}}

Finally we can get I from

{{{(1/4)Z=I}}}
{{{(1/4)(8)=I}}}
{{{2=I}}}

So

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

Edwin</pre>