One of the "light bulb went on" moments for me was in chemistry class where I learned how cancellation of units was involved in unit conversions.
For example, to convert from inches to feet, multiply the quantity of inches by the inch-to-feet conversion factor in the form of a ratio. The conversion factor is 12 inches per 1 foot. So that conversion factor can be written in a ratio in two ways - either 1 ft/12 in or 12 in/1 ft. The way to figure out which ratio to use is that we want to use the ratio that has the starting units in the denominator. So if we are converting, say, 144 inches into feet, we will use the ratio that has inches in the denominator so that the "in" units in the numerator and denominator will cancel each other out, leaving only the "ft" units in the numerator.
So 144 in = (144 in)##(\frac {1 ft} {12 in})##
Dividing 144 by 12 results in 12. And the "in" units in the numerator and denominator cancel each other out, leaving units of feet in the numerator.
So the answer is 12 ft.
If I want to convert back to inches, I use the other ratio: 12 in/1 ft so that the "ft" units in the numerator and denominator cancel out, leaving the "in" units.
12 ft = (12 ft)##(\frac {12 in} {1 ft})## = 144 in
Another example:
Say we want to convert 15.3 cm3 to mm3.
What is the conversion between centimeter and millimeter? It is 1 cm = 10 mm
We are starting with cm3 in the numerator, so the ratio we use will need to have cm3 in the denominator and mm3 in the numerator so that the cm3 units cancel out, leaving only units of mm3 in the numerator. But how do we get cm3 and mm3? Well we have to start with the original conversion factor of 1 cm = 10 mm and cube both sides.
So (1 cm)3 = (10 mm)3 --> 1 cm3 = 1000 mm3
Then 15.3 cm3 = (15.3 cm3)##(\frac {1000 mm^3} {1 cm^3})## = 15300 mm3
The cm3 units in the numerator and the denominator cancel out, leaving only mm3 units. And multiplying 15.3 by 1000 = 15300
So 15.3 cm3 = 15300 mm3
