Answer
The percentage of population: capitalist $ C=1\%$ %, upper-middle class $ U=15$ %, lower-middle class $ L=34\%$ %and working class $ W=30\%$ %
Work Step by Step
Let C be the capitalist percentage.
Let U be the upper-middle class percentage.
Let L be the lower-middle class percentage.
Let W be the working class percentage.
From the given relation, we get:
$ C+U+W+L=80$ (I)
$ L-\left( C+U \right)=18$ (II)
$ L-W=4$ (III)
$3U-\left( C+L \right)=10$ (IV)
Add (I) and (II) as below:
$\begin{align}
& C+U+W+L+L-\left( C+U \right)=80+18 \\
& W+2L=98
\end{align}$ (V)
Add (III) and (V) as below:
$\begin{align}
& \left( W+2L \right)+L-W=\left( 98 \right)+\left( 4 \right) \\
& 3L=102 \\
& L=34 \\
\end{align}$
Substitute the above value in (V) as below:
$\begin{align}
& W+2L=98 \\
& W+\left( 2 \right)\times \left( 34 \right)=98 \\
& W+68=98 \\
& W=30
\end{align}$
Substitute both the values in (I) as below:
$\begin{align}
& C+U+W+L=80 \\
& C+U+30+34=80 \\
& C+U+64=80 \\
& C+U=16
\end{align}$ (VI)
Add (II) and (IV) as below:
$\begin{align}
& L-\left( C+U \right)+3U-\left( C+L \right)=18+10 \\
& -2C+2U=28 \\
& U-C=14
\end{align}$ (VII)
Add (VI) and (VII) as below:
$\begin{align}
& C+U+U-C=16+14 \\
& 2U=30 \\
& U=15
\end{align}$
And
$\begin{align}
& U-C=14 \\
& 15-C=14 \\
& C=1
\end{align}$
