Unit Conversions
The Factor-Label Method
Convert units within and across systems using conversion factors that equal \( 1 \), so units cancel cleanly.
What Is a Unit?
A unit tells you what kind of quantity a number describes. The value \( 12 \) is meaningless alone, but \( 12 \) inches or \( 12 \) ounces measures a real thing.
The Conversion Factor
A conversion factor is a fraction built from a known equality. Since \( 12 \) inches \( = 1 \) foot, both of these fractions equal \( 1 \):
Why Units Cancel
Treat units like algebra variables. When the same unit appears in a numerator and a denominator, it cancels out — exactly like \( \frac{x}{x} = 1 \).
The Factor-Label Rule
To convert a quantity, multiply by a conversion factor arranged so the old unit cancels and the new unit remains:
Put the unit you want to cancel in the opposite position (top vs. bottom) from where it starts.
Facts Worth Memorizing
\( 3 \) ft \( = 1 \) yd
\( 5280 \) ft \( = 1 \) mi
\( 2000 \) lb \( = 1 \) ton
\( 60 \) min \( = 1 \) hr
\( 24 \) hr \( = 1 \) day
\( 1000 \) m \( = 1 \) km
\( 1000 \) g \( = 1 \) kg
Within vs. Across Systems
Some conversions stay inside one system (feet to miles). Others cross between systems (inches to centimeters) and use a bridge factor.
| Within a System | Across Systems | |
|---|---|---|
| Example | ft → mi | in → cm |
| Bridge fact | \( 5280 \) ft \( = 1 \) mi | \( 1 \) in \( = 2.54 \) cm |
| System | U.S. only | U.S. ↔ Metric |
How to Apply It
Example 1: Feet to Inches
Convert \( 7 \) feet to inches. We want feet to cancel, so put feet on the bottom of the factor.
Example 2: Two-Step Chain
Convert \( 3 \) hours to seconds. No single fact links hours to seconds, so chain two factors.
Hours cancel with the first factor, then minutes cancel with the second. \( 3 \times 60 \times 60 = 10800 \).
Example 3: Across Systems
Convert \( 10 \) inches to centimeters using the bridge fact \( 1 \) in \( = 2.54 \) cm.
To reverse it (cm → in), flip the factor so centimeters cancel: \( \frac{1\,\text{in}}{2.54\,\text{cm}} \).
Common Mistakes
Quick Reference
| Goal | Put old unit... | Result |
|---|---|---|
| Cancel old unit | opposite to its position | units divide out |
| To smaller unit | multiply by larger # | bigger number |
| To larger unit | divide by larger # | smaller number |
| No direct link | chain factors | step through middle unit |
Question 1 of 5
How many inches are in \( 5 \) feet?
Question 2 of 5
A package weighs \( 48 \) ounces. How many pounds is that? (\( 16 \) oz \( = 1 \) lb)
Question 3 of 5
How many seconds are in \( 2 \) hours?
Question 4 of 5
Convert \( 4 \) inches to centimeters. (\( 1 \) in \( = 2.54 \) cm)
Question 5 of 5
Which arrangement correctly converts \( 90 \) minutes into hours?
Key Takeaways
You're Ready to Convert
Set up the factor, cancel the units, and let the math handle the rest. The same method works for any conversion — within or across systems.