Seventeen Horses/General Problem 2

Theorem
A man dies, leaving $n$ indivisible and indistinguishable objects to be divided among $m$ heirs.

They are to be distributed in the ratio $\dfrac 1 {a_1} : \dfrac 1 {a_2} : \cdots : \dfrac 1 {a_m}$.

Let $t = \dfrac q r = \ds \sum_{k \mathop = 1}^m \dfrac 1 {a_k}$ expressed in canonical form.

Let $t \ne 1$.

Then it is possible to achieve the required share by adding $s$ objects to the existing $n$ such that:
 * $s + q = r$

when $q = n$.

This still works whether $q$ is positive or negative.