= Solution
Keep both component masses constant during a circular, gravitational-radiation-driven inspiral. Since $J=\mu\sqrt{GMA}$, differentiation gives $\dot J/J=\dot A/(2A)$. Combining this with the supplied angular-momentum loss yields
$$
\boxed{\dot A=-\frac{64}{5}\frac{G^3\mu M^2}{c^5A^3}\equiv-\frac K{A^3}.}
$$
The expression $A/\dot A$ printed in the source is negative during inspiral. A positive characteristic shrinking timescale is
$$
\boxed{\tau_A=\frac{A}{|\dot A|}=\frac{A^4}{K}
=\frac5{64}\frac{c^5A^4}{G^3\mu M^2}\propto A^4.}
$$
The actual remaining coalescence time follows by integrating $A^3\,dA=-K\,dt$:
$$
\boxed{t_{\mathrm{coal}}=\frac{A_0^4}{4K}=\frac{\tau_A(A_0)}4.}
$$
This constant factor does not change the requested separation scaling.
For an old, approximately coeval population, let $dN/dA_0=C$ over the relevant separation interval, and write the positive delay as $t_m=BA_0^4$. Conservation of event counts under this change of variable gives
$$
\nu_m(t_m)=\frac{dN}{dA_0}\left|\frac{dA_0}{dt_m}\right|
=\frac{C}{4B^{1/4}}t_m^{-3/4}.
$$
Therefore
$$
\boxed{\nu_m\propto t_m^{-3/4}.}
$$
The scaling applies between the minimum and maximum delays of the assumed separation distribution. It is a rate per delay interval, not the number per logarithmic delay interval.
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