The large cog has 8 grooves and small cog has 6 grooves .
The large cog makes 15 complete revolutions.
At the same time how many complete revolutions does the small cog make ?

I'm no maths expert but I would do:
For every full rotation the big cog does, the little one has to do 8/6
(It has to do more because it's smaller and the proportion has to be 8:6).
The big cog does 15, so the little one does
8/6 x 15
= 4/3 x 15
= 60/3
= 20