Converting Units

Simple Conversions

If 1 km = 1000 m, then

3 km is 3000 m (multiply by 1000)

5000 m is 5 km (divide by 1000)

100 m is 0.1 km (divide by 1000)

These examples are pretty straightforward, but what do you do when things get more complicated?

Simple Conversions

What if I want to know what 3 km is in inches?

How many inches are there in a kilometer....well, I don't know.

But I do know how many meters there are in a km (1000)

And how many cm are in a meter (100)

And how many cm are in an inch (2.54)

But....which of these should get multiplied or divided?

Simple Conversions

And what do you do if you have to turn 3 km/s into in/hour?

This could get ugly

One OK Method

You may have seen conversions done as a proportion:

1 km = 3 km 1000 m x m

This will work, and I will never stop you from using a successful method, but it has two downsides:

1. It's very slow if you need to do multiple steps

2. It gets very ugly if you have complicated units like km/s

One OK Method

Hmmm, is there a better way?

(ok, obviously there's a better way, or I wouldn't be setting it up like this)

First, Some Background Info

When you do a math problem with units, you are essentially doing two separate problems: the values and the units.

The math you do depends on whether you are adding/subtracting, or multiplying/dividing.

First, Some Background Info

When you do a math problem with units, you are essentially doing two separate problems: the values and the units.

The math you do depends on whether you are adding/subtracting, or multiplying/dividing.

Addition/SubtractionJust add/subtract the numbers, and leave the units alone:

3.3 g + 15 g = 18.8 g

BUT....they must be the same unit to begin with:

3.3 kg + 15 g ≠ 18.8 of anything

We can fix this by putting them both in either g or kg:

3300 g + 15 g = 3315 g

Of course, some things are just unfixable:

3.3 kg + 15 m = ?!??

Multiplication/DivisionRegardless of what the units are, just do the math...then do that same thing with the units.

3.3 m * 15 m = 49.5 m*m = 49.5 m2

If the units don't match...who cares?

3.3 m * 15 cm = 49.5 cm*m

Sure cm*m are a weird unit (the area of a box one meter by one centimeter), but that's ok. We can even get weirder:

3.3 m * 15 g = 49.5 m*g

I don't know what a meter-gram is, but it's a valid unit.

Multiplication/DivisionDivision is just the same, except that—just like with numbers—units in both the numerator and denominator cancel:

15 g / 3.3 mL = 4.55 g/mL

But

15 g / 3.3 g = 4.55 g/g = 4.55 ...and there are no units!

WHOA!

Back to Unit ConversionsWe're going to follow a method called 'dimensional analysis' to convert between units. We'll start with our original number:

3 km

Back to Unit ConversionsNow multiply by a fraction that is set up with the unit we don't want anymore in the denominator

3 km * km

Back to Unit ConversionsAnd our new unit will be in the numerator

3 km * m km

And now we'll fill in the numbers (1 km = 1000 m)

Back to Unit ConversionsNow we can do the math

3 km * 1000 m 1 km

Back to Unit ConversionsThe kilometers cancel

3 km * 1000 m = 3000 km*m = 3000 m 1 km km

You might notice that since 1000 m and 1 km are the same thing, what we've really done is multiply by a really sneaky form of 1.

In fact, this only works if the top and bottom are equivalent.

Back to Unit ConversionsHere's where it gets really useful—multiple conversions at once

3 km * 1000 m * 100 cm * 1 inch = 118110 km*m*cm*in 1 km 1 m 2.54 cm km*m*cm

All the units cancel except inches, so 3 km = 118110 in

(this is a very good time to do a sanity check: 3 km is a pretty long way, so it should be a lot of inches)

...BUT WAIT, THERE'S MORE!

Complex UnitsRemember 3 km/s into in/hr?

3 km * 1000 m * 100 cm * 1 inch * 60 s * 60 min = 4.25x108 km*m*cm*in*s*min s 1 km 1 m 2.54 cm 1 min 1 hr s*km*m*cm*min*hr

And only inches in the numerator and hours in the denominator don't cancel:

4.25x108 in/hr

If you use other methods, you will be very likely to screw up the conversion of the seconds to hours.

Remember, multiplication and division are commutative; you can do them in any order.

SummaryTo convert units, multiply by a fraction...arranged so that the units you don't want anymore cancel.

If you have multiple conversion steps, just multiply by multiple ratios.

If you want to convert units in the denominator, just cancel them with ones in the numerator.