Theory:

Relative molecular mass (hydrogen scale):
The relative molecular mass of a gas or vapour is the ratio between the mass of one molecule of the gas or vapour to the mass of one atom of hydrogen.
Vapour density:
Vapour density is the ratio of the mass of a certain volume of a gas or vapour, to the mass of an equal volume of hydrogen, measured under the same conditions of temperature and pressure.
$\mathit{Vapour}\phantom{\rule{0.147em}{0ex}}\mathit{density}\phantom{\rule{0.147em}{0ex}}\left(\mathit{VD}\right)=\frac{\mathit{Mass}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}a\phantom{\rule{0.147em}{0ex}}\mathit{given}\phantom{\rule{0.147em}{0ex}}\mathit{volume}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}\mathit{gas}\phantom{\rule{0.147em}{0ex}}\mathit{or}\phantom{\rule{0.147em}{0ex}}\mathit{vapour}\phantom{\rule{0.147em}{0ex}}\mathit{at}\phantom{\rule{0.147em}{0ex}}\mathit{STP}}{\mathit{Mass}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{same}\phantom{\rule{0.147em}{0ex}}\mathit{volume}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}\mathit{hydrogen}}$

According to Avogadro's law, all gases have the same number of molecules in equal volumes.

Thus, let the number of molecules in one volume $$= n$$, then

$\mathit{VD}\phantom{\rule{0.147em}{0ex}}\mathit{at}\phantom{\rule{0.147em}{0ex}}\mathit{STP}=\frac{\mathit{Mass}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}"n"\phantom{\rule{0.147em}{0ex}}\mathit{molecules}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}a\phantom{\rule{0.147em}{0ex}}\mathit{gas}\phantom{\rule{0.147em}{0ex}}\mathit{or}\phantom{\rule{0.147em}{0ex}}\mathit{vapour}\phantom{\rule{0.147em}{0ex}}\mathit{at}\phantom{\rule{0.147em}{0ex}}\mathit{STP}}{\mathit{Mass}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}"n"\phantom{\rule{0.147em}{0ex}}\mathit{molecules}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}\mathit{hydrogen}\phantom{\rule{0.147em}{0ex}}\mathit{gas}}$

Cancelling '$$n$$' which is common, you get,

$\mathit{VD}=\frac{\mathit{Mass}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}\mathit{1}\phantom{\rule{0.147em}{0ex}}\mathit{mole}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}a\phantom{\rule{0.147em}{0ex}}\mathit{gas}\phantom{\rule{0.147em}{0ex}}\mathit{or}\phantom{\rule{0.147em}{0ex}}\mathit{vapour}\phantom{\rule{0.147em}{0ex}}\mathit{at}\phantom{\rule{0.147em}{0ex}}\mathit{STP}}{\mathit{Mass}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}\mathit{1}\phantom{\rule{0.147em}{0ex}}\mathit{molecules}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}\mathit{hydrogen}}$

However, since hydrogen is a diatomic,

$\mathit{VD}=\frac{\mathit{Mass}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}\mathit{1}\phantom{\rule{0.147em}{0ex}}\mathit{mole}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}a\phantom{\rule{0.147em}{0ex}}\mathit{gas}\phantom{\rule{0.147em}{0ex}}\mathit{or}\phantom{\rule{0.147em}{0ex}}\mathit{vapour}\phantom{\rule{0.147em}{0ex}}\mathit{at}\phantom{\rule{0.147em}{0ex}}\mathit{STP}}{\mathit{Mass}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}\mathit{2}\phantom{\rule{0.147em}{0ex}}\mathit{atoms}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}\mathit{hydrogen}}$

When you compare the formula for vapour density and relative molecular mass, you get,

$\mathit{VD}=\frac{\mathit{Mass}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}\mathit{1}\phantom{\rule{0.147em}{0ex}}\mathit{mole}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}a\phantom{\rule{0.147em}{0ex}}\mathit{gas}\phantom{\rule{0.147em}{0ex}}\mathit{or}\phantom{\rule{0.147em}{0ex}}\mathit{vapour}\phantom{\rule{0.147em}{0ex}}\mathit{at}\phantom{\rule{0.147em}{0ex}}\mathit{STP}}{\mathit{2}×\mathit{Mass}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}1\phantom{\rule{0.147em}{0ex}}\mathit{atom}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}\mathit{hydrogen}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}......\left(1\right)$

Relative molecular mass (hydrogen scale)

$\mathit{VD}=\frac{\mathit{Mass}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}\mathit{1}\phantom{\rule{0.147em}{0ex}}\mathit{mole}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}a\phantom{\rule{0.147em}{0ex}}\mathit{gas}\phantom{\rule{0.147em}{0ex}}\mathit{or}\phantom{\rule{0.147em}{0ex}}\mathit{vapour}\phantom{\rule{0.147em}{0ex}}\mathit{at}\phantom{\rule{0.147em}{0ex}}\mathit{STP}}{\mathit{Mass}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}1\phantom{\rule{0.147em}{0ex}}\mathit{atom}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}\mathit{hydrogen}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}......\left(2\right)$

By substituting the equation ($$2$$) into equation ($$1$$), we get,

$\mathit{VD}=\frac{\mathit{Relative}\phantom{\rule{0.147em}{0ex}}\mathit{molecular}\phantom{\rule{0.147em}{0ex}}\mathit{mass}}{2}$

Now, on cross multiplication, we have,

$2×\mathit{Vapour}\phantom{\rule{0.147em}{0ex}}\mathit{density}=\mathit{Relative}\phantom{\rule{0.147em}{0ex}}\mathit{molecular}\phantom{\rule{0.147em}{0ex}}\mathit{mass}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}a\phantom{\rule{0.147em}{0ex}}\mathit{gas}$
or
$\mathit{Relative}\phantom{\rule{0.147em}{0ex}}\mathit{molecular}\phantom{\rule{0.147em}{0ex}}\mathit{mass}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}a\phantom{\rule{0.147em}{0ex}}\mathit{gas}=2×\mathit{Vapour}\phantom{\rule{0.147em}{0ex}}\mathit{density}$