Lesson 1.3

Dimensional Analysis

Units matter. A number without a label is just a number — but attach "feet" or "seconds" and it becomes a measurement. Learn how to convert between units by multiplying by clever forms of the number 1.

Introduction

Dimensional analysis (also called the factor-label method) is a technique for converting units by multiplying by fractions equal to . Since , multiplying by this fraction changes the unit without changing the value.

Past Knowledge

You know how to multiply fractions and simplify common factors.

Today's Goal

Convert between units (feet↔inches, hours↔minutes↔seconds) using conversion factors.

Future Success

Science courses and rate problems in later units all depend on unit conversions.

Key Concepts

1. The Conversion Factor

A conversion factor is a fraction where the top and bottom are equal amounts in different units. For example:

2. The Process

Write down what you're starting with as a fraction (put it over 1).

Multiply by a conversion factor so the unwanted unit cancels.

Simplify — the old unit disappears, leaving only the new unit.

3. Common Conversion Facts

Measurement	Equivalence
Length
Length
Time
Time

Worked Examples

Example 1: Feet to Inches

Basic

Convert feet to inches.

Set Up

Cancel & Multiply

"ft" cancels, leaving inches.

Example 2: Minutes to Hours

Intermediate

Convert minutes to hours.

Set Up (flip the factor!)

Cancel & Divide

Example 3: Two-Step Conversion

Advanced

Convert hours to seconds.

Chain Two Factors

Multiply Through

Common Pitfalls

Flipping the Fraction the Wrong Way

If you want feet → inches, inches must be on top. If the unit you want ends up on the bottom, you'll get the reciprocal answer.

Forgetting to Cancel Units

Always draw a line through matching units. If units don't cancel, something is set up incorrectly.

Real-Life Applications

Nurses convert between milligrams and micrograms every day when calculating medication dosages. A mistake in the conversion factor could mean giving a patient 10× too much — dimensional analysis literally saves lives in the medical field.