Algebraic Proportions & Ratios

What You’ll Learn

In this lesson you’ll learn what ratios and proportions are and how to solve for unknowns in proportion problems.

A ratio compares two quantities (e.g., 3:4 or 3/4).

A proportion is two equal ratios:

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

To solve a proportion for an unknown:

Example:

$\frac{3}{5} = \frac{x}{20}$

$\begin{aligned} 3 \times 20 &= 5 \times x \\ 60 &= 5x \\ x &= 12 \end{aligned}$

Also useful: unit rates (e.g., 120 miles / 2 hours = 60 miles per hour).

Solve $\frac{4}{9} = \frac{12}{x}$

$\begin{aligned} 4 \times x &= 9 \times 12 \\ 4x &= 108 \\ x &= 27 \end{aligned}$

Check: 4/9 = 12/27 (both simplify to 4/9, correct).

Another: A recipe uses 2 cups flour for 12 cookies. How much for 30 cookies?

$\begin{aligned} \frac{2}{12} &= \frac{x}{30} \\ 2 \times 30 &= 12x \\ 60 &= 12x \\ x &= 5 \text{ cups} \end{aligned}$

Proportions solve scaling problems:

Recipe adjustment (2 cups sugar for 8 servings → how much for 20 servings?)
Maps (1 inch = 50 miles → 3 inches = 150 miles)
Rates (45 dollars for 5 hours → rate per hour)
Mixing solutions (3 parts water to 1 part juice → how much juice for 12 parts water?)

These are common in cooking, travel, work, and budgeting.

Solve $\frac{5}{8} = \frac{x}{24}$.

If 3 apples cost 1.50 dollars, how much for 10 apples?

A map scale is 1 inch = 40 miles. How far is 2.5 inches?

Solve $\frac{7}{4} = \frac{21}{y}$.

A car travels $180$ miles on $6$ gallons. How many miles per gallon?