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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 a recommened method 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

Overview

Write the problem as a chain of factors where unwanted units cancel diagonally, leaving only the desired unit. You may use one or multiple factors, depending on the conversion.

\[\text{Given} \times \text{Conversion Factor 1} \times \text{Conversion Factor 1} = \text{Result in Desired Unit}\]

Unit Cancellation

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

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

Step-by-Step Process

  1. Identify what you are given (a value and unit)
  2. Identify the unit you want to end with
  3. Set up conversion factor(s) as fraction(s) to cancel unwanted units
  4. Multiply all numbers in the numerators together
  5. Multiply all numbers in the denominators together
  6. Divide the numerator result by the denominator result
  7. Apply rounding rules and proper abbreviation
  8. Check that only the desired unit remains

Don't forget that a you can fill in the number 1 if needed.

Single Step Example

Convert 250 mg to g.

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

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

Best Practices

Always write out every unit in every step. Skipping units to save time can produce hard-to-find errors.

Moving Forward

Dimensional analysis will be used in every remaining module. Start with single and multi-step unit conversions until you're confident before trying more complicated calculations.