--- title: "Lesson 2 — Dimensional Analysis | AlabamaPharmTech.com" description: "Learn dimensional analysis for pharmacy technicians: set up equations with clean unit cancellation. Interactive examples, practice, and quizzes." lang: en json-ld: | { "@context": "https://schema.org", "@type": "LearningResource", "name": "Dimensional Analysis", "description": "Learn dimensional analysis for pharmacy technicians: set up equations with clean unit cancellation. Interactive examples, practice, and quizzes.", "url": "https://www.alabamapharmtech.com/lessons/lesson2-dimensional-analysis.html", "inLanguage": "en", "learningResourceType": "Interactive lesson", "educationalLevel": "Beginner", "isPartOf": { "@type": "Course", "name": "Pharmacy Math Mastery", "url": "https://www.alabamapharmtech.com/pharmacy-math-practice" }, "provider": { "@type": "Organization", "name": "AlabamaPharmTech", "url": "https://www.alabamapharmtech.com" } } --- AlabamaPharmTech .com An Allied Health Academy company Pharmacy Math Mastery ALBOP-Approved · PTCB & NHA Prep Overall progress 29% Lessons 1 Math Foundations 25 min · Complete ✓ 2 Dimensional Analysis 30 min · In progress 3 Day Supply Calculations 40 min 4 Dosage Conversions 35 min 5 Ratio, Proportion & Percent 35 min 6 Sig Interpretation 30 min Final exam ★ Bringing It All Together Cumulative · All 6 lessons AlabamaPharmTech.com › Pharmacy Math Mastery › Lesson 2 of 6 Core # Lesson 2: Dimensional Analysis The one method that handles every pharmacy calculation — single-step doses, weight-based pediatric dosing, and complex IV rate conversions. [Back to Course](/pharmacy-math-practice) [Back to Dashboard](/modules) Overview Examples Practice 36 Quiz The dimensional analysis (DA) setup — use this for every calculation Answer = What you have × (Conversion factor) × (Another if needed) Write units as fractions. Cancel the ones that appear on top and bottom. The unit left over = the unit of your answer. If the unit is wrong, you set up the fraction upside down. 1-step DA (dose to volume) **Formula:** dose ordered × (volume/dose on hand) Example: 500 mg ordered, have 250 mg/5 mL **500 mg × (5 mL / 250 mg) = 10 mL** Check: mg cancels, mL stays. ✓ 2-step DA (weight-based) **Step 1:** lb → kg (always divide by 2.2) **Step 2:** kg × dose/kg → total dose **Step 3:** dose × (vol/concentration) → mL _Never skip the lb→kg conversion._ 3-step DA (weight + divided dose) **Step 1:** lb → kg **Step 2:** kg × mg/kg/day = daily dose **Step 3:** daily dose ÷ doses per day = per dose **Step 4:** per-dose mg → mL using concentration _Used for TID/BID/q8h pediatric sigs._ Rate conversions (IV drips) **mcg/kg/min → mL/hr:** 1\. mcg/kg/min × weight = mcg/min 2\. mcg/min × 60 = mcg/hr 3\. mcg/hr ÷ (mcg per mL concentration) = mL/hr _Watch units: mcg vs mg matters._ **★ PTCB tip:** DA problems on the exam always have a weight in the scenario. Write your fractions out fully — it prevents errors. The PTCB won't ask you to do DA explicitly, but every multi-step dosage problem requires it. Expect 3–5 problems in Order Entry & Processing that need DA. **Top DA mistakes:** (1) Forgetting to convert lb to kg before multiplying by mg/kg. (2) Flipping the fraction — putting mL on the bottom instead of top means you get mg²/mL, not mL. (3) Skipping the daily dose ÷ frequency step for divided doses. (4) Converting mcg to mg or vice versa mid-calculation without tracking the decimal. Six worked examples with full DA setups showing unit cancellation at every step. Write out your fractions — this is how you catch errors. Easy Retail Rx: amoxicillin 500 mg. Available: 250 mg/5 mL suspension. How many mL per dose? (1-step DA) Set up the fraction so mg cancels: 500 mg × (5 mL / 250 mg) = 500 × 5 / 250 = 10 mL ✓ 10 mL per dose Medium Hospital Patient: 154 lb. Gentamicin 5 mg/kg IV. Available: 40 mg/mL. How many mL to draw up? (2-step DA) Step 1 — lb to kg: 154 lb × (1 kg / 2.2 lb) = 70 kg Step 2 — kg to dose: 70 kg × (5 mg / 1 kg) = 350 mg Step 3 — mg to mL: 350 mg × (1 mL / 40 mg) = 8.75 mL ✓ Draw up 8.75 mL Medium Patient scenario Child: 44 lb. Amoxicillin 40 mg/kg/day in 3 divided doses. Available: 250 mg/5 mL. How many mL per individual dose? (3-step DA) Step 1 — weight: 44 ÷ 2.2 = 20 kg Step 2 — daily dose: 20 kg × 40 mg/kg = 800 mg/day Step 3 — per dose: 800 ÷ 3 = 266.7 mg/dose Step 4 — convert to mL: 266.7 × (5/250) = 5.33 → 5.3 mL ✓ 5.3 mL per dose, three times daily Hard Hospital Nitroglycerin: 50 mg in 250 mL D5W. Order: 20 mcg/min. What is the pump rate in mL/hr? (Rate conversion) Step 1 — find concentration in mcg/mL: 50 mg × 1,000 = 50,000 mcg / 250 mL = 200 mcg/mL Step 2 — convert rate to per hour: 20 mcg/min × 60 = 1,200 mcg/hr Step 3 — divide by concentration: 1,200 mcg/hr ÷ 200 mcg/mL = 6 mL/hr ✓ Set pump to 6 mL/hr Hard Hospital Norepinephrine: 8 mg in 250 mL D5W. Order: 0.1 mcg/kg/min for 70 kg patient. Pump rate in mL/hr? (Full rate DA chain) Step 1 — weight-based rate: 0.1 mcg/kg/min × 70 kg = 7 mcg/min Step 2 — per hour: 7 × 60 = 420 mcg/hr Step 3 — to mg/hr: 420 ÷ 1,000 = 0.42 mg/hr Step 4 — concentration: 8 mg / 250 mL = 0.032 mg/mL Step 5 — pump rate: 0.42 ÷ 0.032 = 13.1 mL/hr ✓ Set pump to 13.1 mL/hr Hard Patient scenario Child in status epilepticus: 55 lb. Lorazepam 0.1 mg/kg IV. Available: 2 mg/mL. How many mL IV — quickly and accurately? Step 1: 55 ÷ 2.2 = 25 kg Step 2: 25 × 0.1 = 2.5 mg Step 3: 2.5 ÷ 2 = 1.25 mL ✓ Draw up 1.25 mL — give slowly IV over 2 minutes **Emergency speed:** In a seizure emergency you still do all 3 steps. A 10x error here (drawing 12.5 mL) causes respiratory arrest. Practice until the chain is automatic. Difficulty Easy Medium Hard Context Retail Hospital Patient Easy Retail Single-step DA Rx calls for amoxicillin 500 mg. Available: 250 mg/5 mL suspension. How many mL per dose? Check Show solution **Answer: 10 mL** Set up: 500 mg × (5 mL / 250 mg) = **10 mL** . Units cancel: mg/mg = 1, leaving mL. Easy Retail Single-step DA Dose ordered: 200 mg ibuprofen. Available: 100 mg/5 mL. How many mL per dose? Check Show solution **Answer: 10 mL** 200 mg × (5 mL / 100 mg) = **10 mL** . Easy Retail Single-step DA Dose: 400 mg amoxicillin-clavulanate. Available: 400 mg/5 mL. Volume to dispense per dose? Check Show solution **Answer: 5 mL** 400 mg × (5 mL / 400 mg) = **5 mL** . When dose = concentration dose, volume = amount per concentration. Easy Hospital Single-step DA Order: vancomycin 750 mg IV. Available: 500 mg/10 mL vial. How many mL to draw up? Check Show solution **Answer: 15 mL** 750 mg × (10 mL / 500 mg) = **15 mL** . Easy Hospital Single-step DA Morphine 5 mg IV ordered. Available: 10 mg/mL vial. How many mL to draw up? Check Show solution **Answer: 0.5 mL** 5 mg × (1 mL / 10 mg) = **0.5 mL** . Easy Retail Single-step DA Patient needs 7.5 mg hydrocodone. Available as 7.5 mg per 15 mL elixir. How many mL per dose? Check Show solution **Answer: 15 mL** 7.5 mg × (15 mL / 7.5 mg) = **15 mL** . Easy Patient scenario Single-step DA Child needs 5 mg cetirizine. Available: 5 mg/5 mL syrup. How many mL? Check Show solution **Answer: 5 mL** 5 mg × (5 mL / 5 mg) = **5 mL** . Easy Retail Single-step DA Day 1 loading dose: 400 mg azithromycin. Available: 200 mg/5 mL. How many mL for the loading dose? Check Show solution **Answer: 10 mL** 400 mg × (5 mL / 200 mg) = **10 mL** . Med Hospital 2-step (weight-based) Patient: 154 lb. Gentamicin 5 mg/kg IV. Available: 40 mg/mL. How many mL to draw up? Check Show solution **Answer: 8.75 mL** 154 lb ÷ 2.2 = 70 kg . 70 kg × 5 mg/kg = 350 mg . 350 mg × (1 mL / 40 mg) = **8.75 mL** . Med Hospital 2-step (weight-based) Child: 33 lb. Amoxicillin 40 mg/kg/day. Available: 400 mg/5 mL. Total daily volume? How many mL per day? Check Show solution **Answer: 7.5 mL** 33 ÷ 2.2 = 15 kg . 15 × 40 = 600 mg/day . 600 × (5/400) = **7.5 mL/day** . Med Hospital 2-step (weight-based) Child: 44 lb. Morphine 0.1 mg/kg IV. Available: 2 mg/mL. How many mL per dose? Check Show solution **Answer: 1 mL** 44 ÷ 2.2 = 20 kg . 20 × 0.1 = 2 mg . 2 × (1/2) = **1 mL** . Med Retail 2-step (weight-based) Child: 22 lb. Prednisolone 1 mg/kg/day. Available: 15 mg/5 mL. How many mL per day? Check Show solution **Answer: 3.3 mL** 22 ÷ 2.2 = 10 kg . 10 × 1 = 10 mg . 10 × (5/15) = 3.33 → **3.3 mL** . Med Hospital 2-step (weight-based) DVT treatment. Enoxaparin 1 mg/kg subcut. Patient: 165 lb. Available: 100 mg/mL. How many mL to inject? Check Show solution **Answer: 0.75 mL** 165 ÷ 2.2 = 75 kg . 75 × 1 = 75 mg . 75 × (1/100) = **0.75 mL** . Med Hospital 2-step (weight-based) Child: 44 lb. Acetaminophen 15 mg/kg q6h PRN. Available: 160 mg/5 mL. How many mL per dose? Check Show solution **Answer: 9.4 mL** 44 ÷ 2.2 = 20 kg . 20 × 15 = 300 mg . 300 × (5/160) = 9.375 → **9.4 mL** . Med Hospital 2-step (weight-based) Patient: 66 lb. Cefazolin 25 mg/kg IV q8h. Available: 500 mg/10 mL. How many mL per dose? Check Show solution **Answer: 15 mL** 66 ÷ 2.2 = 30 kg . 30 × 25 = 750 mg . 750 × (10/500) = **15 mL** . Med Patient scenario 2-step (weight-based) Child: 55 lb. Ibuprofen 10 mg/kg q6h PRN. Available: 100 mg/5 mL. How many mL per dose? Check Show solution **Answer: 12.5 mL** 55 ÷ 2.2 = 25 kg . 25 × 10 = 250 mg . 250 × (5/100) = **12.5 mL** . Med Hospital 2-step (weight-based) Patient: 176 lb. Heparin bolus 80 units/kg. Available: 1,000 units/mL. How many mL to administer? Check Show solution **Answer: 6.4 mL** 176 ÷ 2.2 = 80 kg . 80 × 80 = 6,400 units . 6,400 × (1/1000) = **6.4 mL** . Med Hospital 2-step (weight-based) Patient: 132 lb. Vancomycin 15 mg/kg IV q12h. Available: 500 mg/10 mL. How many mL to draw up? Check Show solution **Answer: 18 mL** 132 ÷ 2.2 = 60 kg . 60 × 15 = 900 mg . 900 × (10/500) = **18 mL** . Hard Hospital 3-step (weight + divided) Child: 44 lb. Amoxicillin 40 mg/kg/day in 3 divided doses. Available: 250 mg/5 mL. How many mL per individual dose? Check Show solution **Answer: 5.3 mL** 44 ÷ 2.2 = 20 kg . 20 × 40 = 800 mg/day . 800 ÷ 3 = 266.7 mg/dose . 266.7 × (5/250) = 5.33 → **5.3 mL** . Hard Hospital 3-step (weight + divided) Child: 22 lb. Gentamicin 2.5 mg/kg q8h IV. Available: 10 mg/mL. How many mL per dose? Check Show solution **Answer: 2.5 mL** 22 ÷ 2.2 = 10 kg . 10 × 2.5 = 25 mg/dose . 25 × (1/10) = **2.5 mL** . Hard Hospital 3-step (weight + divided) Patient: 154 lb. Morphine PCA bolus: 0.05 mg/kg. Available: 1 mg/mL. How many mL per PCA bolus? Check Show solution **Answer: 3.5 mL** 154 ÷ 2.2 = 70 kg . 70 × 0.05 = 3.5 mg . 3.5 × (1/1) = **3.5 mL** . Hard Hospital 3-step (weight + divided) Patient: 165 lb. Tobramycin 3 mg/kg loading dose. Available: 40 mg/mL. How many mL to draw up? Check Show solution **Answer: 5.6 mL** 165 ÷ 2.2 = 75 kg . 75 × 3 = 225 mg . 225 ÷ 40 = 5.625 → **5.6 mL** . Hard Hospital 3-step (weight + divided) Child: 66 lb. IV acetaminophen (Ofirmev) 15 mg/kg. Available: 10 mg/mL (100 mL bag). How many mL to infuse? Check Show solution **Answer: 45 mL** 66 ÷ 2.2 = 30 kg . 30 × 15 = 450 mg . 450 ÷ 10 = **45 mL** . Hard Hospital 3-step (weight + divided) Child: 33 lb. Dexamethasone 0.6 mg/kg IM for croup. Available: 4 mg/mL. How many mL per dose? Check Show solution **Answer: 2.25 mL** 33 ÷ 2.2 = 15 kg . 15 × 0.6 = 9 mg . 9 ÷ 4 = 2.25 mL = **2.25 mL**. Hard Hospital 3-step (weight + divided) Patient: 176 lb. Metronidazole 15 mg/kg IV. Bags available: 500 mg each. How many full bags needed? How many 500 mg bags are required? A. 1 bag (insufficient)B. 3 bags (1,500 mg — sufficient)C. 4 bags — requiredD. 2 bags (insufficient) Show solution **Answer: 3 bags (1,500 mg — sufficient)** 176 ÷ 2.2 = 80 kg . 80 × 15 = 1,200 mg . 1,200 mg ÷ 500 mg/bag = 2.4 → need **3 bags (1,500 mg)** to cover the full dose. Hard Patient scenario 3-step (weight + divided) Child: 55 lb in status epilepticus. Lorazepam 0.1 mg/kg IV. Available: 2 mg/mL. How many mL IV? Check Show solution **Answer: 1.25 mL** 55 ÷ 2.2 = 25 kg . 25 × 0.1 = 2.5 mg . 2.5 ÷ 2 = **1.25 mL** . Med Hospital Rate conversions Dopamine ordered at 5 mcg/kg/min. Patient weighs 70 kg. What is the infusion rate in mcg/min? Check Show solution **Answer: 350 mcg/min** 5 mcg/kg/min × 70 kg = **350 mcg/min** . This is then used to calculate the mL/hr pump rate based on the drug concentration. Med Hospital Rate conversions Heparin infusion at 18 units/kg/hr. Patient weighs 80 kg. What is the infusion rate in units/hr? Check Show solution **Answer: 1,440 units/hr** 18 units/kg/hr × 80 kg = **1,440 units/hr** . Hard Hospital Rate conversions Nitroglycerin: 50 mg in 250 mL D5W. Order: 20 mcg/min. What is the pump rate in mL/hr? Check Show solution **Answer: 6 mL/hr** Concentration: 50 mg/250 mL = 0.2 mg/mL = 200 mcg/mL. Convert rate: 20 mcg/min × 60 = 1,200 mcg/hr. 1,200 ÷ 200 = **6 mL/hr** . Hard Hospital Rate conversions Insulin infusion: 100 units in 100 mL NS = 1 unit/mL. Order: 8 units/hr. What is the pump rate in mL/hr? Check Show solution **Answer: 8 mL/hr** Concentration: 1 unit/mL. 8 units/hr ÷ 1 unit/mL = **8 mL/hr** . When concentration = 1 unit/mL, units/hr = mL/hr. Hard Hospital Rate conversions Norepinephrine: 8 mg/250 mL D5W. Order: 0.1 mcg/kg/min for 70 kg patient. What is the pump rate in mL/hr? Check Show solution **Answer: 13.1 mL/hr** Drug rate: 0.1 mcg/kg/min × 70 kg = 7 mcg/min. Convert: 7 × 60 = 420 mcg/hr = 0.42 mg/hr. Concentration: 8/250 = 0.032 mg/mL. 0.42 ÷ 0.032 ≈ **13.1 mL/hr** . Hard Hospital Rate conversions Amiodarone maintenance: 450 mg in 250 mL D5W. Order: 0.5 mg/min. What is the pump rate in mL/hr? Check Show solution **Answer: 16.7 mL/hr** Rate in mg/hr: 0.5 mg/min × 60 = 30 mg/hr. Concentration: 450/250 = 1.8 mg/mL. 30 ÷ 1.8 = **16.7 mL/hr** . Med Retail Multi-step mixed Patient needs 300 mg phenytoin/day. Available: 125 mg/5 mL suspension. How many mL per day total? Check Show solution **Answer: 12 mL** 300 mg × (5 mL / 125 mg) = **12 mL/day** . Med Patient scenario Multi-step mixed Adult: 198 lb. Prednisone 1 mg/kg/day. Available: 5 mg tablets. How many tablets per day? A. 16 tabletsB. 9 tabletsC. 20 tabletsD. 18 tablets Show solution **Answer: 18 tablets** 198 ÷ 2.2 = 90 kg . 90 × 1 = 90 mg/day . 90 ÷ 5 = **18 tablets** . Hard Hospital Multi-step mixed Patient: 110 lb. Tobramycin 2 mg/kg every 8 hours. Available: 40 mg/mL. How many mL per dose? Check Show solution **Answer: 2.5 mL** 110 ÷ 2.2 = 50 kg . 50 × 2 = 100 mg/dose . 100 ÷ 40 = **2.5 mL** . Hard Patient scenario Multi-step mixed Child: 44 lb. Augmentin ES 45 mg/kg/day in 2 divided doses. Available: 400 mg/5 mL. How many mL per individual dose? Check Show solution **Answer: 5.6 mL** 44 ÷ 2.2 = 20 kg . 20 × 45 = 900 mg/day . 900 ÷ 2 = 450 mg/dose . 450 × (5/400) = 5.625 → **5.6 mL** . **How this quiz works:** Pick your question count and difficulty, then start. After answering, click _Reveal explanation_ to see the full step-by-step solution — answers never show automatically. Score and type breakdown appear at the end. ⓘ This lesson has 36 questions in the bank. A 10-question quiz draws a random subset. Run multiple sets to cycle through all questions. How many questions? 102030 Question difficulty Random mixEasy onlyMedium onlyHard only Options Randomize question order Shuffle every time you start Timed mode 45-second countdown per question Number of sets Each set draws a fresh pool − 1 + 10 Questions 1 Sets 10 Total ~10 min Est. time ▶  Start quiz — Dimensional Analysis Question 1 of 10 0 correct · 0 skipped \--% Score Quiz complete! Lesson 2 · Dimensional Analysis · AlabamaPharmTech.com \-- Correct \-- Wrong \-- Skipped \-- Time Performance by question type \--% Overall accuracy \-- Weakest area Miss rate by question type Retry missed + skipped Next set New quiz Answer review — every question this session [Home](/) [Back to Dashboard](/modules) Previous [Back to Course](/pharmacy-math-practice) Lesson 2 of 6 Next lesson