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Ratio, proportion & rates of change · Compound units & rates

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Converting compound units and rates of flow

Why converting a compound unit means converting both of its parts, where the ÷3.6 between km/h and m/s comes from, why g/cm³ to kg/m³ is ×1000, and how rates of flow work.

Ratio, proportion & rates of change · Compound units & rates

Converting compound units and rates of flow

Why converting a compound unit means converting both of its parts, where the ÷3.6 between km/h and m/s comes from, why g/cm³ to kg/m³ is ×1000, and how rates of flow work.

Why it works

A compound unit like km/h is built from two simple units — so converting it means converting both parts, top and bottom. Convert only one and the answer is silently wrong.

km/h to m/s, from first principles. Take 54 km/h. Convert the top: 5454 km =54×1000=54 000= 54 \times 1000 = 54\,000 m. Convert the bottom: 1 hour =3600= 3600 s. So

54 km/h=54 000 m3600 s=15 m/s.54 \text{ km/h} = \frac{54\,000 \text{ m}}{3600 \text{ s}} = 15 \text{ m/s}.

The combined effect is ×1000÷3600\times 1000 \div 3600, i.e. ÷3.6. Going the other way, m/s to km/h, is ×3.6. Don't memorise which is which — sense-check it: 15 m/s and 54 km/h describe the same motion, and the km/h number is always the bigger one (an hour is much longer than a second, so more ground is covered).

Units with squares and cubes scale by the power. 1 m = 100 cm, but 1 m³ =1003=1 000 000= 100^3 = 1\,000\,000 cm³ — the factor applies once per dimension. This is why density conversions surprise people:

1 g/cm3=0.001 kg0.000001 m3=1000 kg/m3.1 \text{ g/cm}^3 = \frac{0.001 \text{ kg}}{0.000001 \text{ m}^3} = 1000 \text{ kg/m}^3.

Top gets ÷1000 (g → kg), bottom gets ÷1 000 000 (cm³ → m³), net effect ×1000. So water at 1 g/cm³ is 1000 kg/m³ — a cubic metre of water weighs a tonne, which is a good sanity anchor to remember.

Rates of flow are the same grammar. Litres per second is a division: rate=volumetime\text{rate} = \frac{\text{volume}}{\text{time}}, so time=volumerate\text{time} = \frac{\text{volume}}{\text{rate}} — a tank of 720 litres filling at 0.3 litres per second takes 7200.3=2400\frac{720}{0.3} = 2400 s. As always with rates: match the units first (m³ → litres, seconds → minutes at the end), and sense-check the direction — a faster rate must give a shorter time.

The general recipe for any rate conversion:
  1. Write the rate as a fraction of units: kmh\frac{\text{km}}{\text{h}}, gcm3\frac{\text{g}}{\text{cm}^3}, Ls\frac{\text{L}}{\text{s}}.
  2. Convert the top and the bottom separately.
  3. Recombine, and sanity-check against a fact you know (walking pace ≈ 1.5 m/s ≈ 5 km/h; water = 1 g/cm³ = 1000 kg/m³).