Ratios, Rates, and Proportions
A ratio compares two quantities; a rate compares quantities with different units; a proportion says two ratios are equal. Scale the recipe below and watch both quantities move together, always in the same ratio.
A ratio like 3 : 2 compares two quantities of the same kind. A rate is a ratio between different units — miles per hour, dollars per pound — and a unit rate simplifies that to a denominator of 1 (like 50 miles per one hour).
A proportion is simply a statement that two ratios are equal, like 3/2 = 6/4. The fastest way to solve for a missing value in a proportion is cross-multiplication: if a/b = c/d, then a × d = b × c.
Watch for word problems that scale more than one quantity at once — like the factory question above, where both the number of machines and the number of hours change. Find the base rate first (per one machine, per one hour), then scale up from there, rather than trying to scale everything simultaneously.