the sticky wicket for most non-mathematicians and non-physicists is the notion that an abstract arithmetic operation has meaning to a real-world object such as a unit of time.

The really interesting thing is how often you

*can* find a real-world meaning to the units on some set of measurements.

Example: the fuel economy of a car. In the USA it's usually measured in miles per gallon. It could also be given as gallons per mile, as in many other countries it is given as litres per 100 km. A gallon is a unit of volume, so it has units of length cubed. (You compute volume by multiplying three length measurements, so the result has units of length cubed).

The mile, of course, is just a unit of length. So if you divide gallons by miles you get units of length squared. Does this have a physical meaning? Actually, it does! It's the cross sectional area of the trough of gasoline the car would have to scoop up to continue moving.

**Edited to add:** As an example, a car that gets 30 mpg would have to scoop up a trough of gasoline with a cross-sectional area of 0.0784 mm^2. That's a square 0.28 mm on a side. Doesn't seem like much, but it adds up.

The same works for electric vehicles, which are rated in units of miles per kilowatt-hour, or kilowatt-hours per 100 km. A kilowatt-hour is a unit of energy, which has basic units of kg m

^{2}/s

^{2}, also known as the

*joule*. (1 kWh = 3.6 million joules.) If you divide that by units of distance, you get kg m/s

^{2}, which happens to be the

*newton*, the unit of force. In other words, the mileage rating for an electric car is equivalent to the physical force needed to overcome drag and keep the car going. (This also includes some electrical and mechanical losses that appear as "virtual" drag in the final result.)

There are all sorts of other examples like these; physics can be a lot more intuitive than many people think.