...1:1,00,000 implies that 1 gram of the drug is in 1,00,000 millilitre (ml) of the solution. Gram on left side and ml on right side of the ratio.

...therefore, 1 gm = 1000 mg = 1,000,000 microgram (ug) in 1,00,000 ml.

...thus, 1,000,000 ug in 1,00,000 ml.

...upon cancelling out the 5 zeros from each side will yield: 10 ug in 1 ml i.e. 10 ug/ml.

...obviously, a drug which has the dilution of 1:1,00,000, the drug concentration in the solution is 10 ug/ml.

...similarly, a drug, which has the dilution of 1:1,000, the drug concentration is 1000 ug/ml = 1 mg/ml

...for simple and direct derivation one can just cancel out the number of Zeros from 1,000,000, the six zeros of a million (1 gm)

...if 1:10, then just cancel out one zero from 1,000,000, so the remaining 1,00,000 is the microgram (ug) per millilitre = 100 mg/ml

...if 1:100, then cancel two zeros because 100 has two zeros from 1,000,000 (1 gm), it will yield 10,000 ug/ml = 10 mg/ml.

...so on and so forth!