= Solution
In the Earth <inertial frame>, the return journey takes $2d/v$. Bob is at rest in this frame, while Alice's <proper time> is reduced by the <Lorentz factor> $\gamma=(1-v^2/c^2)^{-1/2}$. Neglecting the short turnaround interval, \b[their increases in age] are
$$
\boxed{\Delta\tau_B=\frac{2d}{v},\qquad
\Delta\tau_A=\frac{2d}{v}\sqrt{1-\frac{v^2}{c^2}}}.
$$
The <twin paradox> is the apparent contradiction from applying reciprocal <time dilation> to claim that each twin must be younger than the other. Alice changes inertial frames at the turnaround, so a single comoving <inertial frame> cannot describe her whole journey; Bob uses one inertial frame throughout. The twins follow different <world lines>, and integrating <proper time> along those world lines gives the displayed, consistent ages. The change in Alice's notion of simultaneity at turnaround removes the apparent contradiction. Acceleration distinguishes the paths; its magnitude is not an additional local rule for aging.
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