SAT: Ratios, Rates & Proportions Cheat Sheet

The core ideas of SAT: Ratios, Rates & Proportions distilled into a single, scannable reference — perfect for review or quick lookup.

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Quick Reference

Ratio

A comparison of two quantities expressed as $a:b$ or $\frac{a}{b}$. Ratios can describe part-to-part or part-to-whole relationships and remain equivalent when both terms are multiplied or divided by the same nonzero number.

Unit Rate

A rate in which the denominator is 1 unit. Unit rates make it easy to compare different rates and to scale quantities up or down proportionally.

Proportion

An equation stating that two ratios are equal: $\frac{a}{b} = \frac{c}{d}$. Proportions can be solved by cross-multiplication, yielding $ad = bc$.

Direct Variation

A relationship where $y = kx$ for some constant $k$. As one variable increases, the other increases proportionally. The graph passes through the origin and is a straight line with slope $k$.

Inverse Variation

A relationship where $xy = k$ or equivalently $y = \frac{k}{x}$. As one variable increases, the other decreases so that their product remains constant.

Dimensional Analysis

A method for converting units by multiplying by conversion factors written as fractions equal to 1. Units cancel algebraically, leaving the desired unit in the answer.

Scale Factor

The constant multiplier that relates corresponding measurements in two similar figures or proportional situations. If the scale factor is $k$, then every length in the original is multiplied by $k$ in the scaled version.

Constant of Proportionality

The fixed ratio $k$ in a proportional relationship $y = kx$. It represents the unit rate and can be found by dividing any $y$-value by its corresponding $x$-value.

Part-to-Whole vs. Part-to-Part Ratios

A part-to-part ratio compares two parts of a whole (e.g., boys to girls), while a part-to-whole ratio compares one part to the total. Converting between them requires knowing all the parts.

Cross-Multiplication

A technique for solving proportions. Given $\frac{a}{b} = \frac{c}{d}$, cross-multiplying gives $ad = bc$. This eliminates fractions and produces a solvable linear equation.

Key Terms at a Glance

Ratio:A comparison of two quantities by division, written as $a:b$ or $\frac{a}{b}$.

Rate:A ratio that compares two quantities measured in different units, such as miles per hour or cost per item.

Proportion:An equation stating that two ratios are equal, such as $\frac{a}{b} = \frac{c}{d}$.

Unit Rate:A rate expressed per one unit of the denominator quantity, making comparison and scaling straightforward.

Cross-Multiplication:A method for solving proportions: if $\frac{a}{b} = \frac{c}{d}$, then $ad = bc$.

Direct Variation:A relationship in which $y = kx$ for a positive constant $k$; the ratio $\frac{y}{x}$ remains constant.

Inverse Variation:A relationship in which $xy = k$; as one variable increases, the other decreases proportionally.

Dimensional Analysis:A systematic method for converting between units by multiplying by conversion factors expressed as fractions equal to 1.

Scale Factor:The constant ratio between corresponding lengths in similar figures or between a model and the real object.

Constant of Proportionality:The unchanging value $k$ in a proportional relationship $y = kx$, equal to the unit rate.

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