How to do unit conversions -- the easy way

 

Seems like I see a lot of questions by people confused about how to convert units, say from km to m, gallons to liters, etc. I don't know what they're teaching in school, but it seems there is a systemic failure imparting this very simple process.

Here's a way that always works and is so conceptually simple, you'll remember it long after you've forgotten many other school facts (its "commonly" known as the unit factor method, the factor-label method, or dimensional analysis). I'm going to take it right from square one, so be patient if it seems elemental at first (it ought to!).

1.
It's OK to multiply any value by 1. This does not change the original value.
example:
1 · 23 = 23
or, in general: m · 1 = m
This is the Identity Property of Multiplication.

2.
Any value divided by an equal value equals 1.
example:
15 / 15 = 1
or, in general: m · 1/m = 1, or even just: m / m = 1
This is the Inverse Property of Multiplication.

3.
Units (like kg, m, etc.) are part of the value, and they multiply and cancel just like numbers.
example:
4 kg / 2 kg = 4 / 2 = 2
5 m · 4 m = 20 m²

This is the "secret" that seems to be overlooked by people doomed to always have trouble with conversions.

4.
Conversion factors = 1!
examples:
• 1 m = 100 cm
so
(1 m / 100 cm) = 1, and (100 cm / 1 m) = 1

• 1 gm = 1000 mg
so
(1 g / 1000 mg) = 1, and (1000 mg / 1 g) = 1

5.
Set-up conversion formulas so that unwanted units cancel out to leave the wanted one(s) in the correct position(s) (numerator or denominator).
examples:
• Convert 56 cm into m:
56 cm · (1 m / 100 cm) = 56 m / 100 = 0.56 m

• Convert 7 g into mg:
7 g · (1000 mg / 1 g) = 7000 mg / 1 = 7000 mg.

6.
Many calculations can be treated as unit conversions.
example: calculate liters of 5 mol of Helium (at STP).
• 1 mol of He ≈ 4.002 g
• Density He ≈ 0.1785 kg/m³ (STP)
• 1000 g = 1 kg
• 1000 L = 1 m³
so
5 mol · 4.002 g/mol · m³/0.1785 kg · (kg ·L)/(g ·m³) ≈ 112.1 L

And that's it.