Unit Rates and Unit Conversion

Unit Rate

A rate with a denominator of 1. To find a unit rate, divide:

$\frac{240 \text{ miles}}{4 \text{ hours}} = 60 \text{ miles per hour}$

Dimensional Analysis

Convert units by multiplying by conversion factors written as fractions:

Example: Convert 5 miles to feet (1 mile = 5,280 feet).

$5 \text{ mi} \times \frac{5{,}280 \text{ ft}}{1 \text{ mi}} = 26{,}400 \text{ ft}$

Multi-Step Conversions

Chain multiple conversion factors:

Convert 72 km/h to m/s:

$72 \frac{\text{km}}{\text{h}} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3600 \text{ s}} = 20 \text{ m/s}$

Rate = Slope

A unit rate is the slope of the linear relationship. "$3 per pound" means slope = 3 on a cost vs. weight graph.

Comparing Rates

To compare, convert both to the same unit rate.

Store A: 12 oz for $3.60 = $0.30/oz. Store B: 16 oz for $4.48 = $0.28/oz.

Store B is cheaper.

Tip 1: Cancel Units Like Variables

Set up conversion factors so unwanted units cancel diagonally.

Tip 2: "Per" Means Division

"Miles per hour" = $\frac{\text{miles}}{\text{hour}}$.

Tip 3: Be Careful with Area/Volume Conversions

Converting square units requires squaring the linear factor. $1 \text{ ft}^2 = 144 \text{ in}^2$ (not 12).

Tip 4: Read Units in the Question Carefully

The SAT may give speed in km/h but ask for m/s, or give area in square feet but ask for square inches.