Answer
The number of Asians is $122$, the number of Africans is $28$, the number of Europeans is $24$, and the number of Americans is $9$.
Work Step by Step
Step 1. Assume the number of Asians is $x$, the number of Africans is $y$, the number of Europeans is $z$, and the number of Americans is $w$.
Step 2. Based on the given conditions, we have
$\begin{cases} x+y+z+w=183 \\ x- y-z=70 \\ z-w=15 \\ 2y-z-w=23 \end{cases}$
Step 3. Taking the difference between the last two equations, we have $2y-2z=8$ or $y=z+4$
Step 4. Substitute $y=z+4$ and $w=z-15$ in the first two equations; we have
$\begin{cases} x+3z=194 \\ x-2z=74 \end{cases}$
Step 5. Take the difference of the two equations to get $5z=120$ and $z=24$
Step 6. Back-substitute to get $y=28, w=9$, and $x=70+y+z=122$
Step 7. The number of Asians is $122$, the number of Africans is $28$, the number of Europeans is $24$, and the number of Americans is $9$.
