The Core Idea
Dimensional analysis treats units like fractions that can be multiplied and cancelled. To convert 5 miles to kilometers, multiply by a fraction equal to 1: 5 mi × (1.60934 km / 1 mi). The miles cancel, leaving kilometers: 8.0467 km. If your units don't cancel to leave the desired unit, your setup is wrong.
Chaining Conversions
You can chain several factors in a row. To convert feet per second to kilometers per hour, you go through meters: multiply by 0.3048 (m/ft) and by 3.6 (km/h per m/s). Each step cancels the previous unit, which keeps the logic transparent and the arithmetic checkable.
Why It Catches Mistakes
Because units must cancel correctly, dimensional analysis exposes common errors — like forgetting that temperature needs an offset, or confusing miles per gallon (distance per fuel) with liters per 100 km (fuel per distance). If the units don't work out, the answer won't either.