Ratios, Proportions, and Unit Rates

What is a ratio?

A ratio compares two quantities. Three ways to write the ratio "3 to 4":

• $3 : 4$
• $\dfrac{3}{4}$
• "3 to 4"

A ratio with denominator 1 is a unit rate. "$60$ miles in $2$ hours" gives a unit rate of $\dfrac{60}{2} = 30$ miles per hour.

Proportions

A proportion is an equation between two ratios:

$\dfrac{a}{b} = \dfrac{c}{d}$

Cross-multiply to solve for an unknown:

$\dfrac{a}{b} = \dfrac{c}{d} \implies a \cdot d = b \cdot c$

Example. $\dfrac{3}{4} = \dfrac{x}{20}$. Cross-multiply: $3 \cdot 20 = 4 \cdot x \implies 60 = 4x \implies x = 15$.

Unit conversion

Multiply by ratios that equal 1 (because numerator and denominator name the same quantity) — units cancel like variables.

Example. Convert $90$ feet to inches. Since $1$ foot $= 12$ inches:

$90 \text{ ft} \times \dfrac{12 \text{ in}}{1 \text{ ft}} = 1080 \text{ in}$

The "ft" in top and bottom cancel, leaving just "in."

Scale

Maps and models use ratios. A scale of $1 : 50$ means 1 unit on the drawing represents 50 of the same units in reality.

Example. A blueprint shows a wall as $4$ inches at scale $1 : 50$. The actual wall is $4 \times 50 = 200$ inches — about $16.7$ feet.

Worked example — better deal

Brand A: $3.60 for $12$ oz. Brand B: $5.25 for $21$ oz.

• Brand A unit rate: $\dfrac{3.60}{12} = 0.30$, or $0.30 per oz.
• Brand B unit rate: $\dfrac{5.25}{21} = 0.25$, or $0.25 per oz.

Brand B is the better deal.

Key takeaways

• A ratio compares two quantities; a unit rate is a ratio with denominator 1.
• Solve proportions by cross-multiplying.
• For unit conversion, multiply by a fraction equal to 1; let units cancel.
• Convert to unit rates to compare deals on different sizes.