A marathon

This sure was a fun little problem.

So, before starting, we need to do some basic conversions:

$10Km = 10000m$

$12Km = 12000m$

Now that we have every data needed, we do some operations:

First 10kms

$t' = time' = \frac{1000}{5} = 2000 secs$ and we'll keep it as seconds for now.

Second 10 kms

$t'' = time'' = \frac{1000}{4.5} = 2222 secs$ (it would actually be $2222.\overline{2}$ , but we don't care about decimals since we're dealing with seconds)

Third 10 kms

$t''' = time''' = \frac{1000}{4} = 2500 secs$

Now that we have all the times, we simply sum them up: $t' + t'' + t'' = 2000 + 2222 + 2500 = 6722 secs$ .

Time for some more conversions: 6722 seconds are 1 hour (3600 secs) and 52 minutes (3120 secs) and 2 seconds.

Now we find the time left for Phidippides to run in his last 12 kms:

$2hr 25' 38'' - 1hr 52' 2'' = 33' 36''$

So, to equal his best time, Phidippides has to run 12kms in 33 minutes and 36 seconds, which are 2016 seconds ( $33 minutes = 1982 seconds, 1980 + 36 = 2016$ ).