Ratios, Rates, and Proportions — SAT Math

Introduction

Ratios, rates, and proportions appear throughout everyday life — recipe scaling, map reading, speed, density — and they make up a reliable slice of the SAT's Problem-Solving and Data Analysis domain (roughly 15% of the section). This note focuses on setting up and solving proportions, unit rates, scale drawings and maps, and part-to-part versus part-to-whole ratios. For converting between different units (e.g., miles to kilometers), see the sibling note Unit Conversion and Derived Units.

Core Concept

Ratios

A ratio compares two quantities. It can be written as $a:b$ , $\dfrac{a}{b}$ , or " $a$ to $b$ ."

Key idea

Critical distinction:

Part-to-part ratio — compares one part of a whole to another part. Example: "almonds to cashews = 3:5."
Part-to-whole ratio — compares one part to the total. Example: "almonds to total nuts = 3:(3+5) = 3:8."

The SAT frequently tests whether you use the right type. If a ratio $a:b$ is part-to-part, then the parts are $\dfrac{a}{a+b}$ and $\dfrac{b}{a+b}$ of the whole.

Rates and Unit Rates

A rate is a ratio of two quantities with different units (e.g., miles per hour, dollars per pound). A unit rate has a denominator of 1 — it's the rate per single unit (e.g., 35 miles per gallon).

Finding a unit rate: divide the numerator quantity by the denominator quantity.

\text{unit rate} = \frac{\text{total amount}}{\text{number of units}}

Proportions and the Scale-Factor Principle

A proportion is an equation stating that two ratios are equal:

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

Cross-multiply to solve: $a \cdot d = b \cdot c$ .

The deep insight the SAT tests: in any proportional relationship, if one quantity changes by a scale factor $k$ , the other changes by the same scale factor $k$ . This means you can always multiply (or divide) both sides of the relationship by the same number without breaking it.

Scale Drawings and Maps

A scale drawing uses a constant ratio (the scale) between drawn lengths and actual lengths:

\frac{\text{drawn length}}{\text{actual length}} = \text{scale (constant)}

Set up a proportion and solve for the unknown length.

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