SI - Converting Units of Distance

Converting units of distance is fairly straightforward when all units are expressed within the metric system — provided that one has memorized the prefixes. The following example shows how to convert from kilometers to meters:

Q) How many meters is 8.5 kilometers?

Remembering that 1 kilometer = 1,000 meters:

A) 8.5 km. x 1,000 m./km. = 8,500 m.

But how was this done?

If 1 kilometer equals 1,000 meters, then it stands to reason that there are 1,000 meters per kilometer:

$$ 1 km. = 1,000 m.$$

In mathematics, the word "per" signifies division, allowing one to rewrite 1,000 meters per kilometer as a ratio:

$$ \frac{1,000 m.}{1 km.}$$

Multiplying the number of kilometers given with this ratio allows us to convert between kilometers and meters:

$$8.5 km. \times \frac{1,000 m.}{1 km.} = 8,500 m.$$

Or:

$$ \frac{8.5 km.}{1} \times \frac{1,000 m.}{1 km.} = 8,500 m.$$

Note how the units of kilometers (km.) in the two terms cancel each other out when multiplied with one another — leaving us with meters (m.).

$$ \frac{8.5 \cancel{km.}}{1} \times \frac{1,000 m.}{1 \cancel{km.}} = 8,500 m.$$

Getting into the habit of writing units down and canceling them out is one of quickest ways to check your work.

Note:
Even if you understand the concepts modeled by the equations you are using, the units must line up with one another to calculate the result.