How do you know which unit of a conversion factor must be in the denominator?

Question

agent0smith · Answer

It depends. I guess, whatever is in the numerator of the original value. Pretty weird question :/

ipwnbunnies · Answer

Well, the thing you're trying to get rid of what would be in denominator. For example:
$$(1 kilometer)(\frac{1000 meter}{1 kilometer}) = 1000 meters$$

accessdenied · Answer

Your choice of conversion factor is relative to the situation it is necessary for.  In general, you want to use a conversion factor such that the unit you want to convert is eliminated and the unit you are introducing takes its place.  If that isn't clear what I am talking about, try an example.

A simple example is a time conversion.  Let's say I want to convert 1 mile per minute into miles per hour.  This is trivial to do logically, but it will serve to model our case well.
   We start with the original quantity:  $\dfrac{1 \text{ mile}}{1 \text{ minute}} $.
   We are converting minutes into hours.  There are 60 minutes in one hour.
      60 minutes = 1 hour
      ==>   Conversion Factors:   $ \dfrac{60 \text{ minutes}}{1 \text{ hour}} $,    $ \dfrac{1 \text{ hour}}{60 \text{ minutes}} $.
   We are going to multiply our conversion factor onto the original expression (understand that it is the equivalent of multiplying by 1).
   $ \dfrac{1 \text{ mile}}{1 \text{ minute}} \times \color{blue}{\text{Conversion Factor}} $
   We treat units almost like constant variables, we can cancel them if they appear both in the numerator and denominator.  To eliminate minutes which is in the denominator, we have to put minutes in the numerator.  Then hours will be in the denominator, taking its place.
   $ \dfrac{1 \text{ mile}}{1 \cancel{\text{ minute}}} \times \color{blue}{\dfrac{60 \cancel{\text{ minutes}}}{1 \text{ hours}}}  = \dfrac{60 \text{ miles}}{1 \text{ hour}} $

This is a detailed explanation that seems cumbersome, but the practical method is second-nature once you use it so many times.

As a side-note, the same treatment of units is very useful to get a handle for because you can gain a strong understanding of some equations by learning the units inside and out.  Ex) Converting grams into moles.  g (mass) / (g/mol) (molar mass) = mol (number of moles).

gebooors · Answer

An example. 1 mile = 1.609 km

5 miles is 5 mi * 1.609 km/mi

5 km = 5 km / 1.609 km/mi

You have to think how only new unit stays in calculation.

highschoolmom2010 · Answer

ty @agent0smith  @Gebooors @accessdenied @ipwnbunnies