Okay, let’s humor this new lGas unit and see if it works. If my engine consumes 10lGas/100km when traveling at 100km/h, that means it uses 10lGas/h, or .002778

Okay, let’s humor this new lGas unit and see if it works. If my engine consumes 10lGas/100km when traveling at 100km/h, that means it uses 10lGas/h, or .002778 lGas/s. If the pipe feeding my engine has a cross sectional area of 10^-6m^2, how fast is the gasoline flowing through that pipe? Naively I would expect I could divide flow rate by area to get mean velocity. But I’m expecting the result to be in m/s. But if I divide .002778 lGas/s by 10^-6m^2 I get 2778lGas/sm^2. Unless an lGas is 10^-3mGas^3 and I can measure gas area in mGas^2, so I can get my gas speed in mGas/s? If you want a mathematical way to capture ‘of gasoline’ the right way to think of it is not as a unit dimension but rather as a basis vector, kinda like ‘up’ or ‘across’. 1m * up is the vector quantity ‘1m up’, which has dimension ‘length’. 1m * across is the vector quantity ‘1m across’ which also has dimension length (we call it a displacement, but it’s a vector in the length dimension). We can add them together even though they point in different directions because they share the same dimension. The result is the vector quantity ‘1m up and 1m across’ and it is also a length. Similarly 1l * of gas is a vector quantity ‘1l of gas’ with dimension ‘volume’. I can add it to ‘1l of air’ to get ‘1l of gas plus 1l of air’ which is still a vector volume. It might describe the contents of my fuel tank when it’s half empty for example.