= Solution
In the weak-field <reptation> regime of <agarose gel electrophoresis>, the <electrophoretic mobility> of a long linear <DNA> chain scales as $\mu\propto M^{-1}$, where $M$ is its number of <base pairs>. One way to see the scaling is that the constrained-chain <diffusion coefficient> scales as $D\propto M^{-2}$, whereas its effective <electric charge> scales as $M$; the drift response is proportional to $DM$. With the same <electric field> and running time, the migration distance is therefore $x=\mu Et\propto M^{-1}$. The time cancels from the ratio, giving
$$
\frac{M_2}{M_1}=\frac{x_1}{x_2},\qquad
\boxed{M_2=10\frac{4}{4.5}\,\mathrm{kbp}\simeq8.9\,\mathrm{kbp}.}
$$
Here “kbp” means a thousand <base pairs>, not a rate. The second <gel band> has traveled farther and corresponds to the shorter chain under this migration model. The inverse-length law applies only over an appropriate migration regime.https://www.damtp.cam.ac.uk/user/gold/pdfs/teaching/140911-ufk-2-corr.pdf[Electrokinetic Phenomena, gel electrophoresis slides 27–28]{ref}.
At very low <agarose> concentration the mesh becomes too open to sieve these chains effectively. In the free-solution limit the chain's <electric charge> and friction are both approximately proportional to its length, so both chains have nearly the same <electrophoretic mobility> and their <gel bands> cease to separate well. At very high <agarose> concentration the pores become small and transport is strongly hindered; over a fixed experimental time both chains migrate very little, so their absolute spatial separation again becomes small. Increasing gel concentration can improve resolution over an intermediate range, but not indefinitely.
The numerical result is conditional on the inverse-length <reptation> law. Actual <gel electrophoresis size calibration> often uses an empirical approximately linear relation between distance and $\log M$ over a specified range. One known band alone cannot fix that relation's slope. The PDF supplies neither gel concentration nor a calibration curve, so $8.9\,\mathrm{kbp}$ is the standard model estimate rather than a uniquely calibrated experimental size.
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