How the dilution equation works
Dilution conserves the amount of solute: whatever you take out is replaced by solvent, so the moles (or milligrams) in before must equal the moles in after:
C₁ × V₁ = C₂ × V₂
Solve for whichever variable you need. If you are making a working solution from a stock, keep the units consistent — both volumes in the same unit, both concentrations in the same unit — and the units cancel on their own. The dilution factor is C₁ ÷ C₂: a 10× dilution from a 10 mg/mL stock gives 1 mg/mL.
Worked example
Prepare 100 mL of a 2 mg/mL standard from a 10 mg/mL stock: V₁ = (2 × 100) ÷ 10 = 20 mL. So you pipette 20 mL of stock and add 80 mL of diluent to reach 100 mL total. Both sides of the equation check: 10 × 20 = 200 and 2 × 100 = 200, the same amount of drug in both cases. (This is exactly how a 100 mL of 2 mg/mL solution is made from 20 mL of a 10 mg/mL stock — remember that a lab "dilution of 1:5" means total volume five times the stock, i.e. 1 part plus 4 parts diluent, which trips up a lot of newcomers.)
Serial dilutions and the practical checks
A serial dilution steps down in series: each tube dilutes the previous one, and the factors multiply. A 1:10 dilution done three times gives a 1:1000 total, because 10 × 10 × 10 = 1000 — this is how standards spanning many orders of magnitude get built, since no single 1000× dilution is practical. Two practical rules keep the arithmetic honest: you can only ever dilute down (a target more concentrated than the stock is a mistake, which the calculator flags), and the final volume is always the sum of what you add, never less than the volume you started with.