Here are two more ways to do it:

The least common multiple of 15 men and 25 men is 75 men

Then 5 times as many men, or 75 men, can do it in 1/5th the time, or 4 days

25 men is 1/3rd of 75 men so it will take them 3 times as long, or 12 days.

---------------------
OR
---------------------
The greatest common factor of 15 men and 25 men is 5 men

Then 1/3 as many men, or 5 men, will take 3 times as long, or 60 days

25 men is 5 times as many men as 5 men, so it will take them 1/5 as long, 
or 12 days.

Edwin

Here is the way your teacher most probably wants you to
solve it:

The number of workers and the number of days required
are inversely proportional (one increases if the other
decreases), so we set up the proportions in reverse
order:



Let the number of days the SECOND time be N



Cross-multiply:



Divide both sides by 25





So it will take 12 days.

Edwin