How to Safely Convert Between Units

We can convert from km/h (kilometers per hour) to m/s (meters per second) like this:

A kilometer has 1000 meters, and an hour has 3600 seconds, so a kilometer per hour is:

10003600 = 0.2777... m/s

How did I know to make it 10003600 and not 36001000 (the other way around)?

The trick is to do the conversions as fractions!

Example 1:

Let's start with a simple example: convert 3 km to m (3 kilometers to meters). There are 1000 m in 1 km, so the conversion is easy, but let's follow a system.

The system is:

Write the conversion as a fraction that equals 1
Multiply it out (leaving all units in the answer)
Cancel any units that are both top and bottom

We can write the conversion as a fraction that equals 1:

1000 m1 km = 1

And it is safe to multiply by 1 (does not affect the answer):

3 km × 1 = 3 km

so we can do this:

3 km × 1000 m 1 km = 3000 km · m 1 km

The answer looks strange! But we aren't finished yet ... we can "cancel" any units that are both top and bottom:

3000 km · m 1 km = 3000 m

So, 3 km equals 3000 m. Well we knew that, but we want to follow a system, so that when things get harder we know what to do!

And when we do it correctly we get to cancel units that are both top and bottom, and get a neat answer.

Note: if we do it wrong (with the conversion upside down) we get this:

3 km × 1 km 1000 m = 3 km · km 1000 m

And that doesn't let us do any cancelling!

Example 2:

Let's use this method to solve the km/h to m/s conversion from the top of the page.

We will do it in two stages:

from km/h (kilometers per hour) to m/h (meters per hour), then
from m/h (meters per hour) to m/s (meters per second).

1. From km/h (kilometers per hour) to m/h (meters per hour)

1 km h × 1000 m1 km = 1000 km · m 1 h · km

Now "cancel out" any units that are both top and bottom:

1000 km · m 1 h · km = 1000 m 1 h

It is now in meters per hour.

2. From m/h (meters per hour) to m/s (meters per second)

To go from m/h (meters per hour) to m/s (meters per second) we put the "3600 seconds in an hour" conversion "upside down" because we want an "h" on top (so they will cancel later) :

1000 m 1 h × 1 h 3600 s = 1000 m · h 3600 h · s

Then "cancel out" any units that are both top and bottom:

1000 m · h 3600 h · s = 1000 m 3600 s

And so our anwer is:

1000 m 3600 s = 0.2777... m/s

Doing it wrong (with the the 3600 seconds/hour the other way around) gets this:

1000 m 1 h × 3600 s 1 h = 1000 × 3600 m · s 1 h · h

And there is nothing to cancel!

So we know we made a mistake, and can correct it.

All In One Go

With experience we can do it in one line like this:

1 km 1 h x 1000 m 1 km × 1 h 3600 s

= 1000 km · m · h 3600 h · km · s

= 1000 m 3600 s

Or even "all in one go" (crossing out as we go) like this:

1 km 1 h x 1000 m 1 km × 1 h 3600 s = 1000 m 3600 s

Example 3

Now let's use this method to do a real-world conversion.

What is 60 mph (miles per hour) in m/s (meters per second) ?

60 mile h × 1609 m mile × 1 h 3600 s

= 60 × 1609 mile · m · h 3600 h · mile · s

= 26.82 m/s

Summary

The important points are:

Write the conversion as a fraction (that equals one)
Multiply it out (leaving all units in the answer)
Cancel any units that are both top and bottom