Lesson 4 — Unit Conversions (Factor-Label Method)

Multiply by unit fractions equal to 1 so unwanted units cancel and desired units remain.

Example: 72 km/h × (1000 m / 1 km) × (1 h / 3600 s) = 20.0 m/s

Factor-label refresher

Idea: multiply by “unit fractions” that equal 1 so the units you don’t want cancel and the units you want remain.

Write the given value with its units.
Multiply by conversion ratios equal to 1 (e.g., 1000 m / 1 km, 1 h / 3600 s), flipping them so unwanted units cancel.
Cancel units, then multiply numbers straight across. Round at the end to match the least precise data.

Why it works: a conversion ratio like 100 cm / 1 m is exactly 1, just written in different units. Multiplying by 1 never changes the true quantity—only the label.

Quick reference

1 km = 1000 m → 1000 m / 1 km or 1 km / 1000 m
1 m = 100 cm → 100 cm / 1 m or 1 m / 100 cm
1 m = 1000 mm
1 in = 2.54 cm (exact)
1 ft = 0.3048 m (exact)
1 h = 3600 s, 1 min = 60 s
1 L = 1000 mL
1 J = 1 N·m
1 mph = 0.44704 m/s (derived)

Worked examples

Example 1 — Speed: Convert 72 km/h to m/s

\(72~\text{km/h} \times \frac{1000~\text{m}}{1~\text{km}} \times \frac{1~\text{h}}{3600~\text{s}} = 72 \times \frac{1000}{3600}~\text{m/s} = 20.0~\text{m/s}\)
Units: km cancels, h cancels → m/s remains.

Example 2 — Length: Convert 145 cm to m

\(145~\text{cm} \times \frac{1~\text{m}}{100~\text{cm}} = 1.45~\text{m}\)

Example 3 — Squared units: Convert 12.0 cm² to m²

Square the whole factor. \( \left(\frac{1~\text{m}}{100~\text{cm}}\right)^2 = \frac{1}{10^4} \)
\(12.0~\text{cm}^2 \times \left(\frac{1~\text{m}}{100~\text{cm}}\right)^2 = 12.0 \times 10^{-4}~\text{m}^2 = 0.00120~\text{m}^2\)

Example 4 — Imperial → SI: Convert 35 mph to m/s

\(35~\text{mph} \times 0.44704~\text{(m/s)/mph} = 15.6~\text{m/s}\) (≈ \(35 \times 0.447\))

Example 5 — Mixed rate: Convert 2.5 L/min to mL/s

\(2.5~\text{L/min} \times \frac{1000~\text{mL}}{1~\text{L}} \times \frac{1~\text{min}}{60~\text{s}} = \frac{2500}{60}~\text{mL/s} \approx 41.7~\text{mL/s}\)

Example 6 — Back the other way (area): Convert 3.2 m² to cm²

\( \left(\frac{100~\text{cm}}{1~\text{m}}\right)^2 = 10^4 \)
\(3.2~\text{m}^2 \times 10^4 = 3.20 \times 10^4~\text{cm}^2 = 32{,}000~\text{cm}^2\)

Common pitfalls

For area/volume, square/cube the entire factor. (Don’t just add zeroes.)
Flip the ratio if units don’t cancel. Put unwanted units in the denominator.
Round at the end to the correct significant figures; keep guard digits during the work.

Quick self-check

Does the final unit match the question?
Does the size make sense? (e.g., km → m should be a bigger number.)
Could you reverse the process and get back to the start?

Try it

Variant 1 of many

Convert 72 km/h to m/s. Enter the value (number or number+unit).

Within ~0.5% tolerance

More practice (unlocks as you answer correctly)

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