The underlying point is that work in the order concentration, then volume, then units, and the arithmetic stops being confusing. Concentration is milligrams per millilitre and comes from the vial contents and the diluent volume. Volume per dose is dose divided by concentration. Units on a U-100 syringe are volume in millilitres multiplied by one hundred.
The rounding error accumulates if you round too many times — rounding concentration to 5.0, rounding the dose volume to 0.1 mL, rounding the unit reading to 10 — and the safest approach is to work the full precision and round only the final answer.
Stated carefully, number of stopper piercings matters less than the gauge doing the piercing. A 30G or 31G needle through a butyl stopper leaves a track that reseals; a 21G or 18G drawing needle punches a core and can drop it into the solution.
Published data on syringe dead space quantifies low-dead-space designs as retaining under 2 µL against 35 µL or more for conventional detachable-needle syringes.
Write the arithmetic on the vial label. It costs nothing and removes the step where you reconstruct it from memory.
edited 12 Aug 2025 by RP_C18 — corrected a unit error in the worked example