Module 1 of 14 · 12 min
Dimensional analysis & ratio-proportion
Every calculation on the exam — a dose, a volume, a rate, a days' supply — is a chain of unit conversions. Most wrong answers come from one dropped unit (lb vs kg, mg vs mcg), not from hard maths.
🧠 Units are the map: if the unit you want isn't the only one left standing, the setup is wrong — don't press a calculator key yet.
📌 Memorize these
- $1\,\text{kg}$
- $2.2\,\text{lb}$
- $1\,\text{in}$
- $2.54\,\text{cm}$
- $1\,\text{tsp}$
- $5\,\text{mL}$
- $1\,\text{tbsp}$
- $15\,\text{mL}$
- $1\,\text{fl oz}$
- $\approx 30\,\text{mL}$29.57 mL exactly
- $1\,\text{g}$
- $1000\,\text{mg}$
- $1\,\text{mg}$
- $1000\,\text{mcg}$
Treat every unit as part of a fraction. A conversion factor such as $\frac{1\,\text{kg}}{2.2\,\text{lb}}$ equals 1, so multiplying by it changes the unit but not the quantity. Arrange each factor so the unit you want to get rid of sits on the opposite side (top vs bottom) and cancels.
Ratio-proportion is the same idea in two lines: if $250\,\text{mg}$ is in $5\,\text{mL}$, then $x\,\text{mg}$ is in $15\,\text{mL}$, so $\frac{250}{5} = \frac{x}{15}$. It works whenever the relationship is a straight line through zero (twice the volume, twice the drug).
Rounding: keep full precision through every intermediate step and round only the final answer — to what can actually be measured (e.g. 0.1 mL on an oral syringe) or to the number of significant figures the question implies. Write $0.5\,\text{mg}$ (leading zero), never $.5\,\text{mg}$ or $5.0\,\text{mg}$ (trailing zero).
🧭 The method
- 1Write down the unit the answer must have.
- 2Start the chain with the given quantity that matches the question (usually the patient's weight or the prescribed dose).
- 3Multiply by conversion factors oriented so each unwanted unit cancels (top against bottom).
- 4Check that only the wanted unit is left, then do the arithmetic once.
- 5Sanity-check the size (kg is a smaller number than lb; mcg is a bigger number than mg) and round at the very end.
Before each reveal, predict the next step — then check.
A child weighs 44 lb. Amoxicillin is ordered at 45 mg/kg/day divided every 12 hours. The suspension is 400 mg/5 mL. How many mL per dose? (1 kg = 2.2 lb)
- 1
Convert weight to kg — the dose is per kg, so lb must cancel.
$$44\,\cancel{\text{lb}}\times\frac{1\,\text{kg}}{2.2\,\cancel{\text{lb}}} = 20\,\text{kg}$$ Daily dose: kg cancels, mg/day is left.
$$20\,\cancel{\text{kg}}\times\frac{45\,\text{mg}}{\cancel{\text{kg}}\cdot\text{day}} = 900\,\text{mg/day}$$Every 12 hours = 2 doses a day.
$$\frac{900\,\text{mg}}{\cancel{\text{day}}}\times\frac{1\,\cancel{\text{day}}}{2\,\text{doses}} = 450\,\text{mg/dose}$$Convert mg to mL with the product strength, flipped so mg cancels.
$$450\,\cancel{\text{mg}}\times\frac{5\,\text{mL}}{400\,\cancel{\text{mg}}} = 5.625\,\text{mL}$$The whole thing as one chain — the same answer, one calculation.
$$44\,\text{lb}\times\frac{1\,\text{kg}}{2.2\,\text{lb}}\times\frac{45\,\text{mg}}{\text{kg}\cdot\text{day}}\times\frac{1\,\text{day}}{2\,\text{doses}}\times\frac{5\,\text{mL}}{400\,\text{mg}} = 5.6\,\text{mL/dose}$$
Same method — the last steps are yours. Work on paper, then reveal.
A 165 lb patient is ordered enoxaparin 1 mg/kg every 12 hours. Prefilled syringes contain 100 mg/mL. What volume is given per dose? (1 kg = 2.2 lb)
- 1
Convert weight to kg.
$$165\,\cancel{\text{lb}}\times\frac{1\,\text{kg}}{2.2\,\cancel{\text{lb}}} = 75\,\text{kg}$$ - 2
Dose in mg.
$$75\,\cancel{\text{kg}}\times\frac{1\,\text{mg}}{\cancel{\text{kg}}} = 75\,\text{mg}$$ - 3
Your turn
Now you: convert 75 mg to mL using 100 mg/mL (set it up so mg cancels).
Work it out on paper, then reveal.
$$75\,\cancel{\text{mg}}\times\frac{1\,\text{mL}}{100\,\cancel{\text{mg}}} = 0.75\,\text{mL}$$
Type your number. Your first try decides when this module comes back for review.
A patient takes 2 teaspoonfuls of a cough syrup three times daily for 10 days. What volume should be dispensed? (1 tsp = 5 mL)
mL per dose × doses per day × days.
= 300 mL
Amoxicillin 250 mg/5 mL: how many mg are in one tablespoonful? (1 tbsp = 15 mL)
Set up 250 mg / 5 mL = x mg / 15 mL.
= 750 mg
A child weighs 33 lb. Ibuprofen 10 mg/kg per dose is ordered; the suspension is 100 mg/5 mL. How many mL per dose? (1 kg = 2.2 lb)
The dose is per kg — convert lb first. Did you get 16.5 mL? That is the lb-as-kg trap.
= 7.5 mL
Digoxin oral solution is 50 mcg/mL. The dose is 0.125 mg. How many mL?
Units must match before you divide: convert mg to mcg.
= 2.5 mL
A patient is 5 ft 4 in tall. What is the height in cm? (1 in = 2.54 cm, 1 ft = 12 in)
5 ft 4 in is NOT 5.4 ft. Convert everything to inches first.
= 162.56 cm
A 26.4 lb child is ordered a drug at 2.5 mg/kg/day divided three times daily for 7 days. The oral liquid is 25 mg/mL. What total volume is needed for the full course? (1 kg = 2.2 lb)
Total course = daily dose × days. Dividing into 3 doses doesn't change the daily total.
= 8.4 mL
⚠️ Tricky
A 4-year-old child (15 kg, 102 cm, temperature 38.4 °C) is prescribed a drug at 40 mcg/kg once daily. The oral solution contains 0.5 mg/mL. How many mL per dose?
Age, height and temperature are not needed. Put the dose and the concentration in the same mass unit before dividing.
= 1.2 mL
⚠️ The trap: Dividing 600 mcg by 0.5 mg/mL without converting gives 1200 mL — a thousand-fold error.
⚠️ Tricky
A child weighing 48.4 lb is prescribed cephalexin 12.5 mg/kg/dose q6h for 10 days. The suspension is 250 mg/5 mL. What total volume must be dispensed for the full course? (1 kg = 2.2 lb)
q6h means a dose every 6 hours — how many doses is that in 24 hours?
= 220 mL
⚠️ The trap: Reading q6h as three times daily gives 3 doses/day and only 165 mL — the bottle runs out on day 8.
⚠️ Tricky
A 132 lb adolescent is prescribed azithromycin 10 mg/kg on day 1 (maximum 500 mg per dose). The suspension is 200 mg/5 mL. How many mL should be given on day 1? (1 kg = 2.2 lb)
Calculate the weight-based dose, then compare it with the maximum before converting to mL.
= 12.5 mL
⚠️ The trap: Ignoring the 500 mg cap and giving the full 600 mg weight-based dose gives 15 mL.
⚠️ Tricky
Eye drops: instil 1 drop into each eye twice daily. The bottle contains 5 mL and the dropper delivers 20 drops/mL. How many full days will one bottle last?
Count drops per day for BOTH eyes.
= 25 days
⚠️ The trap: Counting only one eye (2 drops/day) gives 50 days, double the real supply.
⚠️ Tricky
A patient is prescribed 5 mg every other day for 8 weeks, starting today. Only 2.5 mg tablets are available. How many tablets should be dispensed?
8 weeks = 56 days, but a dose is taken only on alternate days.
= 56 tablets
⚠️ The trap: Treating 'every other day' as daily gives 112 tablets — twice what is needed.
🕵️ Spot the mistake
A 55 lb child needs a drug at 8 mg/kg/day divided twice daily. What is each dose? (1 kg = 2.2 lb)
A student wrote
⚠️ Traps examiners set
- Using the weight in lb as if it were kg (or multiplying by 2.2 instead of dividing).
- Mixing mg and mcg in the same division (0.125 mg ÷ 50 mcg/mL).
- Reading 5 ft 4 in as 5.4 ft.
- Confusing a per-dose amount with a per-day amount (or forgetting to multiply by the number of days).
- Rounding an intermediate result and carrying the error into the final answer.
💬 Explain it in your own words
Why can you multiply any quantity by a fraction like 1 kg / 2.2 lb without changing how much of something you have — and how does writing the units tell you whether to multiply or divide?