Which country is crossed by the train shown at left?

(Don't give up if you don't recognize it. You can puzzle it out!)

Wait, no we don't. Our birthdays are actually 144 (sometimes 145) days apart. But if you read our birthdays as fractions (month/day), then his birthday is the reduced form of mine. What are our birthdays?

BONUS PUZZLE (I thought this would be too obscure for a regular PotD... prove me wrong!):
In the original illustrations for Alice in Wonderland, a certain "fraction" appears on the Mad Hatter's hat. (It's not really a fraction—it's a price tag in shillings and pence.) If you reduce that "fraction", you get the logo of what Cincinnati-based institution with over 1100 branches?

Solution

Our birthdays are January 6 and May 30.

A good place to start is the "sometimes 145" clue. Why aren't our birthdays always the same number of days apart? One must be before February 29 and the other must be after.

144 days is slightly less than 5 months. The birthday with the smaller month must also have the smaller date (because of how fractions reduce), so we're either looking at early January and late May, or early February and late June. Then the only pair of dates that are far enough apart in their respective months is 1/6 and 5/30. (Potentially, the "144 or 145 days" between our birthdays might have wrapped around past New Year's—I leave it to you to check that this does not create any additional solutions!)

As for the bonus puzzle... the Mad Hatter sports a 10/6, which reduces to 5/3, the logo of Fifth Third Bank.

Congratulations to yesterday's solvers Maddy, Yana, Nico, Cloe, Jessica, Leo S., Jacob C., and Zachary S. The last three also solved the bonus puzzle with a little help from Google. Thanks to everybody who made a guess!