Fractions

What Is a Fraction?

A fraction represents a relationship between two quantities. It has two parts:

• Numerator — the top number
• Denominator — the bottom number

\[\frac{\text{numerator}}{\text{denominator}}\]

In dimensional analysis, every conversion factor is a fraction. The numerator and denominator represent the same quantity expressed in two different units. Fractions also express rates Which unit you place on top determines which unit will survive after cancellation.

\[\frac{1\text{ g}}{1000\text{ mg}} \qquad \frac{1000\text{ mg}}{1\text{ g}}\]

These are the same relationship written two ways. You choose the orientation based on which unit you need to cancel.

Multiplying Fractions

Multiply numerators together and denominators together.

\[\frac{1}{2} \times \frac{3}{4} = \frac{1 \times 3}{2 \times 4} = \frac{3}{8}\]

This is the only operation you need for every calculation in this course. Dimensional analysis chains are a series of fractions multiplied together — nothing more.

Simplifying

Dividing both numerator and denominator by the same number does not change the value of a fraction.

\[\frac{6}{8} = \frac{6 \div 2}{8 \div 2} = \frac{3}{4}\]

In a calculation chain, simplifying before multiplying reduces the size of the numbers you work with and makes errors easier to spot. It is not required — the answer is the same either way.

In this course

You will only multiply fractions. Dimensional analysis handles every conversion and every dosage calculation through multiplication alone. The orientation of each factor — which unit is on top — determines the result. You will not need to divide, add, or subtract fractions anywhere in this course.

Self-Check