Unit Conversion and Derived Units — SAT Math

Introduction

Unit conversion and derived-unit problems appear throughout the Problem-Solving and Data Analysis domain, which accounts for roughly 15% of SAT Math questions. These questions test whether you can set up conversion factors correctly, chain multiple conversions together, handle squared or cubed units (area and volume), and work with derived units built from products (kilowatt-hours) or quotients (population density, speed, mass density). Getting one conversion factor flipped or forgetting to square the linear conversion factor is exactly where test-makers hide distractors.

Core Concept

Dimensional Analysis: the Method

Dimensional analysis treats unit labels as algebraic quantities that cancel when the same unit appears in a numerator and a denominator. You multiply by conversion fractions — fractions that equal 1 — to swap one unit for another without changing the physical quantity.

General form:

\text{(given quantity)} \times \frac{\text{desired unit}}{\text{current unit}} = \text{result in desired unit}

Arrange each conversion factor so the unit you want to eliminate is on the opposite side from where it currently appears.

One-step example: Convert 45 miles per hour to miles per minute.

45 \; \frac{\text{mi}}{\text{hr}} \times \frac{1 \; \text{hr}}{60 \; \text{min}} = 0.75 \; \frac{\text{mi}}{\text{min}}

The "hr" label in the numerator of the original rate cancels with the "hr" in the denominator of the conversion fraction.

Multistep Chains

String conversion fractions together — always check that intermediate units cancel:

72 \; \frac{\text{km}}{\text{h}} \times \frac{1{,}000 \; \text{m}}{1 \; \text{km}} \times \frac{1 \; \text{h}}{3{,}600 \; \text{s}} = \frac{72{,}000}{3{,}600} \; \frac{\text{m}}{\text{s}} = 20 \; \frac{\text{m}}{\text{s}}

Square and Cubic Unit Conversions

When converting area or volume, the linear conversion factor must be raised to the matching power.

Conversion	Area	Volume
1 m = 100 cm	1 m² = $(100)^2 = 10{,}000$ cm²	1 m³ = $(100)^3 = 1{,}000{,}000$ cm³
1 ft = 12 in	1 ft² = $(12)^2 = 144$ in²	1 ft³ = $(12)^3 = 1{,}728$ in³

Key idea

Key rule: Never apply a linear conversion factor directly to an area or volume — always raise it to the 2nd or 3rd power first.

Derived Units

From a quotient: Population density, speed, and mass density are all formed by dividing one unit by another.

\text{population density} = \frac{\text{number of people}}{\text{area}} \qquad \text{speed} = \frac{\text{distance}}{\text{time}} \qquad \text{mass density} = \frac{\text{mass}}{\text{volume}}

From a product: Energy in kilowatt-hours (kWh) is power multiplied by time:

\text{Energy (kWh)} = \text{Power (kW)} \times \text{Time (h)}

To convert a derived unit, convert each component unit separately — and remember the exponent rule for squared/cubed units.

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