Ratios and Proportions

A ratio compares two quantities by division. 3 cups of flour to 2 cups of sugar is the ratio $3:2$ or $\frac{3}{2}$.

A proportion says two ratios are equal. $\frac{3}{2} = \frac{x}{8}$ — solve by cross-multiplying.

The cross-multiplication shortcut. If $\frac{a}{b} = \frac{c}{d}$, then $a \cdot d = b \cdot c$. Solve $\frac{3}{2} = \frac{x}{8}$:

$3 \cdot 8 = 2 \cdot x \Rightarrow x = 12$

Watch the units. When setting up a proportion, the units must match across both fractions. 2 inches per 5 miles on a map; how many miles is 8 inches?

$\frac{2 \text{ in}}{5 \text{ mi}} = \frac{8 \text{ in}}{x \text{ mi}}$

Inches over miles on both sides — units match. Cross-multiply: $2x = 40 \Rightarrow x = 20$ miles.

Ratios with three or more parts. Boys to girls to teachers is $5:3:2$, and there are 200 students total. Add the parts: $5 + 3 + 2 = 10$. Each part = $\frac{200}{10} = 20$ people.

• Boys: $5 \times 20 = 100$.
• Girls: $3 \times 20 = 60$.
• Teachers: $2 \times 20 = 40$.

Note: "total" here was 200 — but if the problem said 200 students, you'd add only the student parts ($5 + 3 = 8$).

Direct vs. inverse proportionality:

• Direct: $y = kx$. As $x$ doubles, $y$ doubles. (Distance and time at constant speed.)
• Inverse: $y = k/x$. As $x$ doubles, $y$ halves. (Number of workers and time to finish a job.)

SAT-specific tip: when two quantities are proportional, the ratio between them stays constant. Set up the proportion using the constant, then solve.