measurement

Unit Conversions

Pre-Algebra20 minDifficulty

You should know: ratios and proportions, fractions and decimals

Overview

Unit conversion uses multiplication by conversion factors (fractions equal to 1) to change measurements from one unit to another without changing the underlying quantity. For example, 1 foot = 12 inches, so multiplying by (12 inches / 1 foot) converts feet to inches. Dimensional analysis tracks units through calculations.

Intuition

A conversion factor is a fraction equal to 1. Multiplying by 1 doesn't change a quantity, but it can change its units. To convert 3 feet to inches: 3 ft * (12 in / 1 ft) = 36 in. The 'ft' cancels. Chain multiple conversions: miles to inches = miles * (5280 ft/mi) * (12 in/ft).

Formal Definition

Definition

If unit A and unit B satisfy: 1 A = k B, then the conversion factor is (k B)/(1 A) = 1. To convert a measurement from A to B, multiply by this factor. Units are treated like algebraic quantities that can cancel. Dimensional analysis: every equation must balance dimensionally (same units on both sides).

x \text{ A} \times \frac{k \text{ B}}{1 \text{ A}} = xk \text{ B}

Unit conversion formula

1 \text{ mile} = 5280 \text{ ft},\quad 1 \text{ ft} = 12 \text{ in}

Length conversion factors

1 \text{ kg} = 1000 \text{ g},\quad 1 \text{ hour} = 3600 \text{ s}

Metric conversion factors

Notation

Notation	Meaning
$[Q]$	The units of quantity Q

Theorems

Theorem 1: Dimensional Homogeneity

A physically meaningful equation must have the same units on both sides. This principle (dimensional analysis) can be used to check formulas and derive relationships between quantities.

Worked Examples

1
1 hour = 60 minutes, so the conversion factor is 60 min/hr.
$2.5 \text{ hr} \times \frac{60 \text{ min}}{1 \text{ hr}} = 150 \text{ min}$

✓ Answer

150 minutes

Practice Problems

Easyfill in blank

Convert 3 kilometers to meters.

Common Mistakes

Common Mistake

Dividing instead of multiplying by the conversion factor.

Set up the conversion factor so the unit you want to eliminate cancels. If going from feet to inches, multiply by (12 in/1 ft), not divide.

Quiz

To convert from km to cm, you multiply by:

101001000100000

Historical Background

Standardized measurement systems developed gradually. Ancient civilizations used body-based units (cubit = forearm length) that varied by region. The metric system was introduced in France after the 1789 Revolution to create universal standards. The International System of Units (SI) was formalized in 1960. Today most of the world uses metric; the US uses a mix of customary and metric units.

1795
French Republic adopts the metric system
1960
SI (International System of Units) formalized

Summary

Conversion factor: a fraction equal to 1 built from a unit equality (1 ft = 12 in).
Multiply the measurement by the conversion factor; original units cancel.
Chain multiple factors for multi-step conversions (e.g., miles to cm).
Dimensional analysis: every term in a valid equation has the same units on both sides.

References

BookLarson, R. Pre-Algebra. Holt McDougal, 2010.