Question 1006385
There is a total of {{{100*4=400}}} assistants,
and each will serve in exactly {{{3}}} committees,
so we will prepare {{{3*400=1200}}} assistant place cards.
{{{x}}}= number of committees made up of 2 governors and 12 assistants.
{{{y}}}= number of committees made up of 4 governors and 2 assistants.
{{{12x+4y=1200}}} number of assistant place cards prepared for the {{{x+y}}} committees with assistants.
Since each of the {{{100}}} governors serve in exactly {{{5}}} committees,
we will prepare {{{5*100=500}}} governor place cards.
For those place cards to suffice for the committees with assistants, we need
{{{2x+4y<=500}}} .
We have the system {{{system(12x+4y<=1200,2x+4y=500)}}} ,which graphs as
{{{drawing(300,300,-200,800,-200,800,
grid(0),green(line(0,125,500,0)),
green(line(50,0,50,112.5)),
green(line(100,0,100,100)),
green(line(150,0,150,87.5)),
green(line(200,0,200,75)),
green(line(250,0,250,62.5)),
green(line(300,0,300,50)),
green(line(350,0,350,37.5)),
green(line(400,0,400,25)),
green(line(450,0,450,12.5)),
red(line(0,500,166.67,0))
)}}} We need to be in the green-dashed region and on the red line.
If we do not set up any 5-governor committees,
{{{system(12x+4y=1200,2x+4y=500)}}}-->{{{system(3x+y=300,2x+4y=500)}}}-->{{{system(x=70,y=90)}}} .
So, forming {{{70+90=160}}} committees, including
{{{70}}} committees made up of 2 governors and 12 assistants, and
{{{90}}} committees made up of 4 governors and 2 assistants,
seems a possibility.
We could rearrange the committees differently, getting rid 
Setting up 5-governor meetings sounds like it would lead to less meetings.
However, as shown by common sense , and by {{{3x+y=300}}}<-->{{{y=300-3x}}} ,
for every extra 12-assistant committee, we must give up {{{3}}} 4-assistant committees,
for a total of {{{3-1=2}}} less committees with assistants.
using {{{2}}} more governor place cards for the extra 12-assistant committee,
and requiring {{{3*4=12}}} less governor place cards for the {{{3}}} eliminated 4-assistant committees.
That would leave us with {{{12-2=10}}} governor place cards,
which would be used in {{{10/5=2}}} 5-governor committee.
That way, every time we do that shift,
we eliminate a total of {{{2}}} committees with assistants,
but create {{{2}}} 5-governor committees,
leaving the total number of committees always at {{{highlight(160)}}} .