A conversion factor is a fraction equal to 1.
Since 1 gal = 3.785 L, the fraction 3.785 L1 gal equals exactly 1. Multiplying by it doesn't change how much milk you have, only the units you describe it in. Put the unit you want to get rid of on the bottom so it cancels.
Build the chain
Pick a problem, tap the conversion factors to add them, and use ⇅ to flip one so the unit you're getting rid of ends up on the bottom. Units that cancel get crossed out. Some factors are decoys.
SI prefixes worth memorizing
| Prefix | Symbol | Means | Example |
|---|---|---|---|
| kilo | k | 10³ = 1000 | 1 kg = 1000 g |
| (base) | — | 10⁰ = 1 | g, L, m, s |
| deci | d | 10⁻¹ | 1 L = 10 dL |
| centi | c | 10⁻² | 1 m = 100 cm |
| milli | m | 10⁻³ | 1 L = 1000 mL |
| micro | μ | 10⁻⁶ | 1 g = 10⁶ μg |
| nano | n | 10⁻⁹ | 1 m = 10⁹ nm |
Exact, defined
1 in = 2.54 cm · 1 gal = 3.785 L · 1 min = 60 s · 1 kg = 1000 gThese never limit sig figs.
Measured or rounded
1 kg = 2.205 lb · milk ≈ 8.6 lb/gal · 200 ppm = 200 mg/LThese count toward sig figs.
Set it up and solve
Units are where expensive mistakes hide.
In 1999 NASA lost the Mars Climate Orbiter because one team's software output thrust in pound-force seconds and another team's software read it as newton-seconds. In 1983 an Air Canada 767 ran out of fuel in midair, the "Gimli Glider," because the fuel load was calculated in pounds when the plane needed kilograms.
Neither one was hard math. In both cases someone multiplied by a number without writing its units. If every number in your chain carries its units, a wrong factor shows up as units that refuse to cancel.
Common mistakes
- Flipping the factor the wrong way. If the unit doesn't cancel, flip it.
- Squared and cubed units: 1 m = 100 cm, but 1 m³ = 100³ cm³ = 10⁶ cm³. Use the factor three times.
- Dropping units halfway through. Write them on every number, every time.
- Forgetting that a density or concentration is also a conversion factor (8.6 lb = 1 gal of milk).