= Solution
Ignoring the requested final-mass difference and initially taking equal hadronic matrix elements, the common $W^-\to d\bar u$ factor cancels. A <decay width> is proportional to the squared <decay amplitude>, so
$$
\frac{B(\overline B_d^0\to D^+\pi^-)}{B(\overline B_d^0\to\pi^+\pi^-)}
\simeq\left|\frac{V_{cb}}{V_{ub}}\right|^2
\simeq\frac{\sin^4\theta_C}{\sin^6\theta_C}
=\frac1{\sin^2\theta_C}.
$$
With the supplied <Cabibbo angle>,
$$
\boxed{R_B^{\mathrm{rough}}=\frac1{0.225^2}=19.75\simeq20.}
$$
The quoted measurement, about $500\pm10$, is larger by a factor $500/19.75\simeq25.3$. If that whole difference were attributed to a ratio of other amplitudes, it would require a factor $\sqrt{25.3}\simeq5.03$.
This does not invalidate quark mixing. The supplied <Cabibbo-Kobayashi-Maskawa matrix> is only a schematic hierarchy in powers of $\sin\theta_C$, particularly for $V_{ub}$; its omitted coefficients matter greatly after squaring. Equal phase space also does not imply equal hadronic <decay amplitudes>: the two transitions have different <hadronic form factors> and strong-interaction dynamics, and loop contributions can interfere with the charmless tree amplitude. Thus the calculation correctly predicts enhancement of the charmed channel, but \b[the supplied CKM hierarchy alone does not quantitatively predict the measured ratio]. It would be unjustified to treat $19.75$ as a precision prediction with only the quoted experimental error.
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