A number is to be divided into three parts such that the sum of the middle part and one of the extreme parts stands in a given ratio to the other extreme part.
$$
\begin{array}{ll}
y+z=p x ; & p x+x=a \\
x+y=q z ; & q z+z=a
\end{array}
$$

from which

$$
\begin{gathered}
x=\frac{a}{1+p} \\
y=\frac{a(p q-1)}{(p+1)(q+1)} \\
z=\frac{a}{1+q} .
\end{gathered}
$$