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Dimensional Analysis

What Is Dimensional Analysis?

Dimensional analysis (also called the factor-label method or unit-factor method) is a systematic approach to solving conversion and dosage problems by tracking units through every step of the calculation.

It is the preferred method in nursing because:

  • It works for simple and complex problems alike
  • Units cancel visually so errors are easy to spot
  • It reduces the need to memorize formulas
  • It produces a clear written trail you can check

How It Works

Write the problem as a chain of fractions where unwanted units cancel diagonally, leaving only the desired unit.

\[\text{Given} \times \frac{\text{conversion factor 1}}{1} \times \frac{\text{conversion factor 2}}{1} = \text{answer in desired unit}\]

Units that appear in both a numerator and denominator cancel out:

\[\frac{\cancel{\text{mg}}}{1} \times \frac{1}{\cancel{\text{mg}}} = 1\]

Step-by-Step Process

  1. Identify what you are given (starting value and unit)
  2. Identify what unit you want to end with
  3. Set up conversion fractions so unwanted units cancel
  4. Multiply all numerators together
  5. Multiply all denominators together
  6. Divide and apply rounding rules
  7. Check that only the desired unit remains

Single Step Example

Convert 250 mg to g.

\[250 \cancel{\text{ mg}} \times \frac{1 \text{ g}}{1000 \cancel{\text{ mg}}} = \frac{250}{1000} \text{ g} = 0.25 \text{ g}\]

Note how mg cancels leaving only g.

Multi-Step Example

Convert 3 tsp to L.

\[3 \cancel{\text{ tsp}} \times \frac{5 \cancel{\text{ mL}}}{1 \cancel{\text{ tsp}}} \times \frac{1 \text{ L}}{1000 \cancel{\text{ mL}}} = \frac{15}{1000} \text{ L} = 0.015 \text{ L}\]

Note how tsp and mL both cancel, leaving only L.

Dosage Calculation Example

Applying Dimensional Analysis to Dosing

Dimensional analysis is not just for conversions — it is the foundation of all dosage calculations in later modules.

Example: Order: 500 mg of amoxicillin Stock: 250 mg per tablet How many tablets?

\[500 \cancel{\text{ mg}} \times \frac{1 \text{ tablet}}{250 \cancel{\text{ mg}}} = \frac{500}{250} \text{ tablets} = 2 \text{ tablets}\]

Example: Order: 0.5 g of medication Stock: 250 mg per tablet How many tablets?

Step 1 — set up the full chain:

\[0.5 \cancel{\text{ g}} \times \frac{1000 \cancel{\text{ mg}}}{1 \cancel{\text{ g}}} \times \frac{1 \text{ tablet}}{250 \cancel{\text{ mg}}} = \frac{500}{250} \text{ tablets} = 2 \text{ tablets}\]

All units cancel except tablets — exactly what we want.

Weight-Based Example

Example: Order: gentamicin 5 mg/kg Patient weight: 176 lb How many mg?

\[176 \cancel{\text{ lb}} \times \frac{1 \cancel{\text{ kg}}}{2.2 \cancel{\text{ lb}}} \times \frac{5 \text{ mg}}{1 \cancel{\text{ kg}}} = \frac{880}{2.2} \text{ mg} = 400 \text{ mg}\]

Practice Problems

Problem 1

Convert 2500 mcg to g using dimensional analysis.

Answer
\[2500 \cancel{\text{ mcg}} \times \frac{1 \cancel{\text{ mg}}}{1000 \cancel{\text{ mcg}}} \times \frac{1 \text{ g}}{1000 \cancel{\text{ mg}}} = \frac{2500}{1{,}000{,}000} \text{ g} = 0.0025 \text{ g}\]

Problem 2

Order: 750 mg. Stock: 500 mg per tablet. How many tablets?

Answer
\[750 \cancel{\text{ mg}} \times \frac{1 \text{ tablet}}{500 \cancel{\text{ mg}}} = 1.5 \text{ tablets}\]

Problem 3

Order: 0.25 g. Stock: 125 mg per capsule. How many capsules?

Answer
\[0.25 \cancel{\text{ g}} \times \frac{1000 \cancel{\text{ mg}}}{1 \cancel{\text{ g}}} \times \frac{1 \text{ capsule}bundles}{125 \cancel{\text{ mg}}} = \frac{250}{125} = 2 \text{ capsules}\]

Problem 4

A patient weighs 88 lb. A drug is ordered at 2 mg/kg. What is the dose in mg?

Answer
\[88 \cancel{\text{ lb}} \times \frac{1 \cancel{\text{ kg}}}{2.2 \cancel{\text{ lb}}} \times \frac{2 \text{ mg}}{1 \cancel{\text{ kg}}} = \frac{176}{2.2} = 80 \text{ mg}\]

Problem 5

Convert 3 tbsp to L.

Answer
\[3 \cancel{\text{ tbsp}} \times \frac{15 \cancel{\text{ mL}}}{1 \cancel{\text{ tbsp}}} \times \frac{1 \text{ L}}{1000 \cancel{\text{ mL}}} = \frac{45}{1000} = 0.045 \text{ L}\]

Clinical Tip

Always write out every unit in every step. Skipping units to save time is how errors happen. The few extra seconds it takes to write units out could prevent a serious medication error.

Moving Forward

Dimensional analysis will be used in every remaining module. If any problem in this course feels confusing, come back here and set it up as a unit-cancellation chain — it will almost always clarify the path to the answer.