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Here, when they write $v/(\mathrm{mol\ s^{-1}\ (kg\ \text{protein})^{-1}})$ is $0.30$ for the first row first column, do they mean that the value is $0.30\ \mathrm{mol\ s^{-1}\ (kg\ \text{protein})^{-1}}$ or $0.30/(\mathrm{mol\ s^{-1}\ (kg\ \text{protein})^{-1}})$?

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    $\begingroup$ $v/x = 0.30$ means that $v = 0.30x$; this applies whether $x$ is a number, a unit, or a collection of units. $\endgroup$ – orthocresol Mar 20 at 10:08
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The expression $$v/(\mathrm{mol\ s^{-1}\ kg^{-1}})=30$$ means $$v=30\ \mathrm{mol\ s^{-1}\ kg^{-1}}$$

This notation is used in the table, so that the entries are all just numbers without any unit symbols.

For an explanation, I copy the following section from my meta answer, which is mainly a collection of rules and examples taken from various standards and guides (see the meta answer for references).

Unit symbols are mathematical entities and not abbreviations. In expressing the value of a quantity as the product of a numerical value and a unit, both the numerical value and the unit may be treated by the ordinary rules of algebra.

(…)

In expressing the value of a quantity as the product of a numerical value and a unit, both the numerical value and the unit may be treated by the ordinary rules of algebra.

the equation $T=293\ \mathrm K$ may equally be written $T/\mathrm K=293$

It is often convenient to write the quotient of a quantity and a unit in this way for the heading of a column in a table, so that the entries in the table are all simply numbers.

$$\begin{array}{ll} \hline T/\mathrm K & p/\mathrm{kPa} \\ \hline 300.00&3.5368\\ 310.00&6.2311\\ 320.00&10.546\\ 330.00&17.213\\ \hline \end{array}$$

The axes of a graph may also be labelled in this way, so that the tick marks are labelled only with numbers.

do not write “mass [kg]” for “mass in kilogram”

Also note that expressions for units shall contain nothing else than unit symbols and mathematical symbols. Any attachment to a unit symbol as a means of giving information about the special nature of the quantity or context of measurement under consideration is not permitted. Therefore the use of the unit “$\mathrm{mol\ s^{-1}\ kg^{-1}\ \text{protein}}$” as written in the question is not permissible. The correct unit is $\mathrm{mol\ s^{-1}\ kg^{-1}}$.

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