Lesson 2: Ratio and Proportion

New Concept

This lesson introduces ratios and rates for comparing quantities. You'll learn to set up and solve proportions—equations where two ratios are equal—to find unknown values and recognize when variables are proportional.

What’s next

This is just the foundation. Soon, you'll tackle interactive practice cards on setting up ratios, solving proportions, and graphing proportional variables.

Property

A ratio is a type of quotient used to compare two numerical quantities. The ratio of $a$ to $b$ is written $\frac{a}{b}$. A ratio can be expressed as a decimal instead of a common fraction.

Examples

• In a class with 15 boys and 12 girls, the ratio of boys to girls is $\frac{15}{12}$, which simplifies to $\frac{5}{4}$.

• A recipe calls for 2 cups of sugar for every 5 cups of flour. The ratio of sugar to flour is $\frac{2}{5}$.

Property

A rate is a ratio that compares two quantities with different units.

Examples

• A car travels 180 miles in 3 hours. Its rate of speed is $\frac{180 \text{ miles}}{3 \text{ hours}}$, or 60 miles per hour.

• A phone plan costs 30 dollars for 5 gigabytes of data. The rate is $\frac{30 \text{ dollars}}{5 \text{ GB}}$, or 6 dollars per gigabyte.

Property

A proportion is a statement that two ratios are equal. We can clear the fractions from a proportion by cross-multiplying.

If $\frac{a}{b} = \frac{c}{d}$, then $ad = bc$.

Examples

• To solve $\frac{x}{10} = \frac{3}{5}$, we cross-multiply to get $5x = 10(3)$. This gives $5x=30$, so $x=6$.

Property

Two variables are proportional if their ratio is always the same. If two variables are proportional, they are related by the equation

where $k$ is the constant of proportionality.