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In the the formula

$$C = 10P\frac{D}{M}$$

$C$ is the concentration of a substance in a solution, $P$ is the percent concentration (purity) of the substance, $D$ is the relative density of the solution and $M$ is the molar mass of the substance.

Say $P = 60\%$, do we substitute $60$ or $60/100$ in the formula? The proofs I've read suggest $60$.

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It all depends on the units other variables are expressed in. By dimensional analysis it's obvious that you don't expect percent to appear in the final result, so $P$ is expressed as a fraction. Using percentage number can be justified if $100\%$ is conveniently cancelled out.

The $10$-multiplier suggests that this might be case here, e.g. if $D$ is expressed in $\pu{g cm-3}$, $M$ — in $\pu{g mol-1}$, then in order to obtain $C$ in $\pu{mol dm-3}$ (or $\pu{mol L-1}$, which is the same), you must plug in the value for $P$ in percents:

$$[C] = \frac{\pu{g cm-3}}{\pu{g mol-1}} = 10^3\cdot\pu{g dm-3} = 10\cdot [P~\text{in}~\%]\cdot\pu{g dm-3}$$

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