Unitary Conversions

This is a method of converting different units.

The units are first written as a unit and corresponding conversion.

The unit can be written on top, \[ \frac{1\ \text{day}}{24\ \text{hours}} \] or on the bottom, \[ \frac{24\ \text{hours}}{1\ \text{day}} \]

They are then laid out top to bottom so that they can be cancelled as in multiplying fractions.

Animated Examples

These animations show how unitary conversions work by cancelling units step by step.

Worked Examples

\[ 5\ \text{days} \times \frac{24\ \text{hours}}{1\ \text{day}} \times \frac{60\ \text{minutes}}{1\ \text{hour}} = 7200\ \text{minutes} \]

\[ 3\ \text{years} \times \frac{365\ \text{days}}{1\ \text{year}} \times \frac{24\ \text{hours}}{1\ \text{day}} = 26280\ \text{hours} \]

Convert hours to millennia

\[ 1\ \text{hour} \times \frac{1\ \text{day}}{24\ \text{hours}} \times \frac{1\ \text{year}}{365\ \text{days}} \times \frac{1\ \text{millennium}}{1000\ \text{years}} = 1.14155 \times 10^{-7}\ \text{millennia} \]

This method works for any conversion that can be expressed as a chain of “1 to something” ratios.

Converting Between Kilometres per Hour and Metres per Second

To convert from kilometres per hour to metres per second:

Since there are 1000 metres in a kilometre, 60 seconds in a minute, and 60 minutes in an hour:

\[ 1\ \text{km/h} = 1 \times \frac{1000\ \text{m}}{1\ \text{km}} \times \frac{1\ \text{hour}}{3600\ \text{s}} = \frac{1000}{3600}\ \text{m/s} = \frac{5}{18}\ \text{m/s} \]

To convert from metres per second to kilometres per hour:

\[ 1\ \text{m/s} = 1 \times \frac{1\ \text{km}}{1000\ \text{m}} \times \frac{3600\ \text{s}}{1\ \text{hour}} = \frac{3600}{1000}\ \text{km/h} = 3.6\ \text{km/h} \]

Converting Between Miles and Kilometres

To convert from miles to kilometres:

\[ 1\ \text{mile} = 1.609\ \text{km} \] \[ \text{km} = \text{miles} \times 1.609 \]

To convert from kilometres to miles:

\[ 1\ \text{km} = 0.621\ \text{miles} \] \[ \text{miles} = \text{km} \times 0.621 \]

Speed Conversion Tables

These tables show common conversions between kilometres per hour and metres per second.