# In a mixed company of Poles, Italians, Greeks, Turks and Germans.... How many Germans are there?

In a mixed company of Poles, Italians, Greeks, Turks and Germans. The Poles are one less than 1/3 of the Germans; and 3 less than half the Italians; the Germans and Turks outnumber the Greeks and Italians by 3; the Greeks and the Germans form one less than half the company and the Greeks and the Italians form 7/16 of the company. How many Germans are there?

I have come up with the following equations but I'm not sure how to proceed.

P = (Gm / 3) - 1
P = (I / 2) -3
Gm + T = Gk + I + 3
Gk + Gm = (Total / 2) -1
Gk + I = (7/16) Total
P + I + Gk + T + Gm = Total

• Rewrite the first two equations as $Gm=3P+3$ and $I=2P+6$ Substitute in the other equations and proceed from there. Dec 19, 2020 at 3:00

In[1]:=Solve[{P == (Gm / 3) - 1,P == (It / 2) -3,Gm + T == Gk + It + 3,Gk + Gm == (Total / 2) -1,Gk + It == (7/16) Total,P + It + Gk + T + Gm == Total },{P,It,Gk,T,Gm, Total}]
Out[1]={{P->7,It->20,Gk->15,T->14,Gm->24,Total->80}}