Notation for rule of thumb, without breaking dimensional homogeneity? I'd like to know how to write rules of thumb in a concise way, without breaking dimensional homogeneity.
For example, if a runner has an average speed of ~10 km / h, an approximation of the covered distance would be
$\mathrm{distance} \approx \mathrm{duration} * 10 \frac{\mathrm{km}}{\mathrm{h}}$
Is there any shorter way to write it? The goal would be to make it clear that you can simply multiply the number of hours by 10, and you'd get the number of kilometers.
$\mathrm{km} = 10 * \mathrm{h}$
is concise, but it's also obviously wrong because it breaks dimensional homogeneity.
There was a question on bicycle.stackexchange ("How to convert calories to watts on Strava rides?"), and one of the answers was Calories(kcal) = Watts * Hours * 4. This rule of thumb doesn't break homogeneity, but it still looks weird because one kcal is 1.163Wh, and not 4Wh. What would be a better way to write it?
 A: If your aim is to improve the equation Calories(kcal) = Watts * Hours * 4, there are a few things to consider:

*

*First of all, you want to take the input quantities (time $t$ , average power delivered to the bicyle $P$) and compute the energy $E$ consumed by the metabolism. (You could add subscripts to distinguish "pedal power" and "food energy", but for the purpose of this answer, this is not necessary.) Clearly, this will also involve the conversion efficiency $\eta$ (from food to pedal). Here, the energy is $$E=P t /\eta\,.\tag{+}$$
This equation is fine as it is, and independent of any units.

*Next, you have specific units in mind (W, h, kcal; $\eta$ has units of 1 or percent), so you'll like to have an easy way to compute the energy in your head while biking. The simplest way (I think) is to make Eq. ($+$) dimensionless (divide everything by the desired overall unit and do some algebra):
$$
\tag{$\times$}
\frac{E}{\text{kcal}}=\frac{P t /\eta}{\text{kcal}}
= \frac{P}{\text{W}}\,\frac{t}{\text{h}}\,\frac{1}{\eta}\underbrace{\frac{\text{W} \cdot \text{h}}{\text{kcal}}}_\lambda
= \frac{P}{\text{W}}\,\frac{t}{\text{h}}\,\frac{1}{\eta}\,\frac{\text{W} \cdot 3600 \text{s}}{4184\text{J}}
= \frac{P}{\text{W}}\,\frac{t}{\text{h}}\,\frac{1}{\eta}\cdot 0.8604 \,.
$$
Note that the factor $\lambda$ is dimensionless from the start (this is a cross-check for such calculations). This equation is still exact (except for rounding the numerical factor), and still valid for all units, but it is most convenient to use for the ones we put in.

*Finally, you assume an average value for the efficiency of $\eta=21.5\%=0.215$.
$$
\tag{$\ast$}
\frac{E}{\text{kcal}}
= \frac{P}{\text{W}}\,\frac{t}{\text{h}}\,\frac{1}{0.215}\cdot 0.8604
= \frac{P}{\text{W}}\,\frac{t}{\text{h}}\cdot 4.002 \,.
$$
So you can express the equation in any of the three ways and it will be exact (except for rounding) and general (except that the last form assumes a specific efficiency).
A: My advice:

*

*Have a symbol for each quantity, including coefficients such as your distance-to-duration ratio; the symbols should represent the quantities, not what they become when nondimensionalized on division by a unit.

*State coefficients' values, where known, in separate equations.

*Trust your reader to remember how the arithmetic of ordinary (dimensionless) numbers translates into that of dimensionful quantities.

Your example is $s\approx vt,\,v=10\text{km/h}$.
A: My preference for such things is
$$\left[\frac{\mathrm{distance}}{1\ \mathrm {km}}\right] = 10\left[\frac{\mathrm{duration}}{1\ \mathrm{hr}}\right]$$
As another example, the electron plasma frequency is given by $\omega = \sqrt{ne^2/\epsilon_0 m}$.  Since all but one of the quantities on the right-hand side are constants, this can be written as a very straightforward rule of thumb:
$$ \left[\frac{\omega}{1\ \mathrm{Hz}}\right] = 5.64 \times 10^{4} \left[\frac{n}{1\ \mathrm{cm}^{-3}}\right]^{1/2}$$
