I found one formula showing to use percentage and multiply by 10000.
So I used (.0405/500)*10000 = 0.81ppm
Then I find another reference saying to convert values and divide milligrams by kilograms.
So .0405 grams is 40.5 mg and 500 grams is .5kg (40.5/.5) = 81ppm
It is true that 1 percent (part per hundred) is equal to 10,000 ppm (parts per million) or 1mg/1kg.
1 percent = 104 ppm
ppm =(g solute/gsolvent)x106
ppm=mg solute/kg solvent
ppm=ml solute/liter solvent
In your first computation you miss to multiply first the (0.0405g/500g) by 100 to get the percent value of 0.0081%. Then you have to multiply it by 10,000 to get 81ppm. Your second computation is right.