Unit Rates
Making prices and speeds comparable by putting everything per one.
The explanation
A unit rate tells you how much per one. 240 km in 3 hours is 80 km per hour. $6 for 4 apples is $1.50 per apple.
You get there by dividing by the second quantity.
This is the practical tool for shopping. A 12-pack for $5.40 is $0.45 each; an 8-pack for $3.44 is $0.43 each. The 8-pack wins, which the sticker prices alone do not tell you.
A rate is a ratio between quantities with different units, and a unit rate is that ratio normalised so the denominator is 1: r = a/b expressed as (a/b) : 1.
Because the units are different, rates carry their units through the arithmetic, which makes dimensional analysis available. Writing "80 km/h" as the fraction 80 km / 1 h lets you cancel units when multiplying: (80 km/h)(2.5 h) = 200 km, with hours cancelling.
Every unit rate has a reciprocal rate that is equally valid but answers a different question: 80 km/h inverts to 0.0125 h/km, or 45 seconds per kilometre. Choosing the right orientation is most of the skill — the denominator should be whatever you are given, so the units you want survive the cancellation.
Worked example
Which is cheaper: 500 mL for $2.40 or 750 mL for $3.45?
- 2.40 ÷ 500 = $0.0048 per mL.
- 3.45 ÷ 750 = $0.0046 per mL.
- Compare: 0.0046 < 0.0048.
Answer: The 750 mL bottle, at $0.0046/mL.
Common mistakes
- Dividing the wrong way round and getting units per dollar when you wanted dollars per unit.
- Comparing totals instead of rates when the sizes differ.