A hare & a tortoise participated in a race of 21 kilometers. Average

My answer:
'Winner win by hours' 1 Hour
"Winner win by kilometers” 3km

The definition for two "winner win by" are not inherently consistent: They are measured at different time point. But we can figure it out.

Because Hare's speed is double that of tortoise, but Hare requires 45-minute after 45-minute race, tortoise and Hare are at the same distance from starting point after 90 minutes. After that, thanks to the Hare's incrementally lengthened rest time, Tortoise gain more and more advantage. We can generally believe that Tortoise will win even if we are unfamiliar with the famed story of the race.

So, tortoise shall complete the race at 21/3 = 7 hours.
Let's see how much distance can Hare run within 7 hours.
45, 45; 45, 60; 45, 75; 45 90; 45, 105...
At the end of 7 hours, Hare is at minute 60 of its 90-minute rest time.
45, 45; 45, 60; 45, 75; 45 90; 45, 105...
Totally running time 45/60*4=3 Hours
Distance: 3 X 6 = 18 km.
Because 21-18=3,
"Winner win by kilometers” 3km

In order to calculate "winner win by hours", Hare has to complete the remaining 3km. For Hare, it will take 3/6=0.5 hours
Hare is at minute 60 of its 90-minute rest time. It will then (1) Rest for another 0.5 hours (2) Run 0.5 hours to complete the race.
For a total of 0.5 + 0.5 = 1 hour extra time.
'Winner win by hours' 1 Hour