Dilution: One Equation, and Knowing When to Do It in Steps

Diluting does not change how much solute you have — only how much solvent it is spread through. That single fact is the whole of dilution arithmetic, and it is why C1V1 = C2V2 works: both sides are the same number of moles. The practical question is not the arithmetic but whether to do it in ONE step or several. A hundredfold dilution in one step means measuring 1 mL accurately and making it up to 100 mL, and 1 mL is right at the limit of what most glassware measures well. Two steps of ten give the same result while measuring 10 mL twice — a volume that is measured far more accurately. That is the entire reason serial dilution exists, and the arithmetic below shows exactly how much it buys.