AlabamaPharmTech.com›Pharmacy Math Mastery›Lesson 2 of 6Core
Lesson 2: Dimensional Analysis
The one method that handles every pharmacy calculation — single-step doses, weight-based pediatric dosing, and complex IV rate conversions.
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. ✓
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.
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.
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.
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.
EasyRetail
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
MediumHospital
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
MediumPatient 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
HardHospital
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
HardHospital
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
HardPatient 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
Context
Easy
Retail
Single-step DA
Rx calls for amoxicillin 500 mg. Available: 250 mg/5 mL suspension.
How many mL per dose?
Answer: 10 mL
Set up: 500 mg × (5 mL / 250 mg) = 10 mL. Units cancel: mg/mg = 1, leaving 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?
Answer: 10 mL
200 mg × (5 mL / 100 mg) = 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?
Answer: 5 mL
400 mg × (5 mL / 400 mg) = 5 mL. When dose = concentration dose, volume = amount per concentration.
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?
Answer: 15 mL
750 mg × (10 mL / 500 mg) = 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?
Answer: 0.5 mL
5 mg × (1 mL / 10 mg) = 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?
Answer: 15 mL
7.5 mg × (15 mL / 7.5 mg) = 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?
Answer: 5 mL
5 mg × (5 mL / 5 mg) = 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?
Answer: 10 mL
400 mg × (5 mL / 200 mg) = 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?
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.
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?
Answer: 7.5 mL
33 ÷ 2.2 = 15 kg. 15 × 40 = 600 mg/day. 600 × (5/400) = 7.5 mL/day.
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?
Answer: 1 mL
44 ÷ 2.2 = 20 kg. 20 × 0.1 = 2 mg. 2 × (1/2) = 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?
Answer: 3.3 mL
22 ÷ 2.2 = 10 kg. 10 × 1 = 10 mg. 10 × (5/15) = 3.33 → 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?
Answer: 0.75 mL
165 ÷ 2.2 = 75 kg. 75 × 1 = 75 mg. 75 × (1/100) = 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?
Answer: 9.4 mL
44 ÷ 2.2 = 20 kg. 20 × 15 = 300 mg. 300 × (5/160) = 9.375 → 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?
Answer: 15 mL
66 ÷ 2.2 = 30 kg. 30 × 25 = 750 mg. 750 × (10/500) = 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?
Answer: 12.5 mL
55 ÷ 2.2 = 25 kg. 25 × 10 = 250 mg. 250 × (5/100) = 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?
Answer: 6.4 mL
176 ÷ 2.2 = 80 kg. 80 × 80 = 6,400 units. 6,400 × (1/1000) = 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?
Answer: 18 mL
132 ÷ 2.2 = 60 kg. 60 × 15 = 900 mg. 900 × (10/500) = 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?
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.
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?
Answer: 2.5 mL
22 ÷ 2.2 = 10 kg. 10 × 2.5 = 25 mg/dose. 25 × (1/10) = 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?
Answer: 3.5 mL
154 ÷ 2.2 = 70 kg. 70 × 0.05 = 3.5 mg. 3.5 × (1/1) = 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?
Answer: 5.6 mL
165 ÷ 2.2 = 75 kg. 75 × 3 = 225 mg. 225 ÷ 40 = 5.625 → 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?
Answer: 45 mL
66 ÷ 2.2 = 30 kg. 30 × 15 = 450 mg. 450 ÷ 10 = 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?
Answer: 2.25 mL
33 ÷ 2.2 = 15 kg. 15 × 0.6 = 9 mg. 9 ÷ 4 = 2.25 mL = 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?
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.
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?
Answer: 1.25 mL
55 ÷ 2.2 = 25 kg. 25 × 0.1 = 2.5 mg. 2.5 ÷ 2 = 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?
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.
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?
Answer: 1,440 units/hr
18 units/kg/hr × 80 kg = 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?
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.
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?
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.
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?
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.
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?
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.
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?
Answer: 12 mL
300 mg × (5 mL / 125 mg) = 12 mL/day.
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?
Answer: 18 tablets
198 ÷ 2.2 = 90 kg. 90 × 1 = 90 mg/day. 90 ÷ 5 = 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?
Answer: 2.5 mL
110 ÷ 2.2 = 50 kg. 50 × 2 = 100 mg/dose. 100 ÷ 40 = 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?
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.
44 ÷ 2.2 = 20 kg. 20 × 45 = 900 mg/day. 900 ÷ 2 = 450 mg/dose. 450 × (5/400) = 5.625 → 5.6 mL.
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ⓘ This lesson has 36 questions in the bank. A 10-question quiz draws a random subset. Run multiple sets to cycle through all questions.
ⓘ This lesson has 36 questions in the bank. A 10-question quiz draws a random subset. Run multiple sets to cycle through all questions.
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Lesson 2 · Dimensional Analysis · AlabamaPharmTech.com
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