How to Safely Convert From One Unit to Another

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

A kilometer has 1,000 meters, and an hour has 3,600 seconds, so a kilometer per hour is:

  • 1000 / 3600 = 0.277... m/s

How did I know to make it 1000 / 3600, and not 3600 / 1000 (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 1,000 m in 1 km, so the conversion is obvious, but let's follow a system.

The system is:

  • Write the conversion as a fraction
  • Multiply
  • Cancel any units that are both top and bottom

You can write the conversion as a fraction which equals 1:

1000 m = 1
1 km

And it is safe to multiply by 1 (does not affect the answer) so we can do this:

3 km × 1000 m = 3000 km · m
1 km 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 = 3000 m
km

So, 3 km equals 3000 m. Well, we knew that, but i wanted to show you how to do it systematically, so that when things get harder you will know what to do!

And the trick is to know that you will cancel when you finish, so make sure you write the conversion the correct way around (so you can cancel afterwards).

Doing it wrong (with the conversion upside down) gets this:

3 km × 1 km = 3 km · km
1000 m 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:

  1. from km/h (kilometers per hour) to m/h (meters per hour), then
  2. 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 × 1000 m = 1000 km · m
h 1 km h · km

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

1000 km · m = 1000 m
h · km h

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

Now, 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 = 1000 m · h
h 3600 s 3600 h · s

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

1000 m · h = 1000 m
3600 h · s 3600 s

And so our anwer is:

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

 

Doing it wrong (with the the 3600 s/h upside down) gets this:

1000 m × 3600 s = 1000×3600 m · s
h 1 h 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 you can do it in one line like this:

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

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

1 km × 1000 m × 1 h  =  1000 m
h 1 km 3600 s 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 × 1609 m × 1 h = 60 × 1609 mile · m · h  = 26.82 m/s
h mile 3600 s 3600 h · mile · 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