How do I determine which unit of measure should go into a denominator and which to go into the numerator? An example of converting 60kms/ph to ms (meters per sec) is:
$$\frac{60km}{1hr}\:=\:\frac{60km}{1hr}\:\cdot \frac{1000m}{1km}\:\cdot \frac{1hr}{3600s}\:$$
Looking at it, I can see they went opposites, like km as the numerator so the next is a denominator and the final fraction is the denominator from the first as the numerator.
Why or how did the $\frac{1000m}{1km}\:$ become the second fraction? Such as, why is it not $\frac{1km}{1000m}\:$?
What knowledge do I need to know to use this for any unit conversion (such as $2.7gcm^-3$ to $kg m^-3$) rather than just guessing by the pattern?
Units act just like variables. If you divide $m^3$ by $m^2$ you get $m$. In your example there is one copy of $km$ and $hr$ each in the numerator and denominator on the right, so the total exponent is $0$ for both and they need not appear on the left.