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Worked Examples and Case Studies

This page gives fully solved problems drawn from across the pharmacy curriculum. Each one states the problem, works through the solution step by step, and ends with a short note on the limitation or common mistake. Work each example on paper first, then check your method against the solution.

Example 1 — Alligation (Mixing Two Strengths)​

Problem. You need 500 g of a 5% (w/w) coal tar ointment. The pharmacy stocks a 2% ointment and a 10% ointment. How much of each do you mix?

Solution. Use alligation. Write the desired strength in the middle and take differences on the diagonals.

  • Higher (10%) minus desired (5%) = 5 parts of the 2% ointment.
  • Desired (5%) minus lower (2%) = 3 parts of the 10% ointment.
  • Total = 5 + 3 = 8 parts for 500 g, so 1 part = 62.5 g.

Therefore: 2% ointment = 5 × 62.5 = 312.5 g, and 10% ointment = 3 × 62.5 = 187.5 g.

Check. Coal tar delivered = (0.02 × 312.5) + (0.10 × 187.5) = 6.25 + 18.75 = 25 g. And 5% of 500 g = 25 g. Correct.

Limitation. Alligation only works between two strengths that bracket the target. You cannot make a 5% product from two stocks that are both above or both below 5%.

Example 2 — Isotonicity by Sodium Chloride Equivalent​

Problem. Prepare 30 mL of a 1% (w/v) solution of a drug whose sodium chloride equivalent (E value) is 0.20. How much NaCl must be added to make the solution isotonic with tears?

Solution.

  1. NaCl needed to make 30 mL isotonic on its own: an isotonic saline is 0.9% w/v, so 0.9 g per 100 mL → 0.009 × 30 = 0.27 g.
  2. Tonicity already contributed by the drug: drug quantity = 1% of 30 mL = 0.30 g. Its NaCl-equivalent tonicity = 0.30 × 0.20 = 0.06 g of NaCl.
  3. NaCl to add = 0.27 − 0.06 = 0.21 g.

Limitation. The E-value method assumes the drug is stable and does not react with NaCl; for some salts you use dextrose or another agent instead.

Example 3 — Creatinine Clearance and Renal Dose Adjustment​

Problem. A 70-year-old man weighs 80 kg and has a serum creatinine of 1.5 mg/dL. Estimate his creatinine clearance using the Cockcroft–Gault equation.

Solution. Cockcroft–Gault for men:

CrCl = [(140 − age) × weight (kg)] / [72 × serum creatinine (mg/dL)]

CrCl = [(140 − 70) × 80] / [72 × 1.5] = (70 × 80) / 108 = 5600 / 108 ≈ 51.9 mL/min.

For a woman the result is multiplied by 0.85.

Interpretation. A CrCl around 52 mL/min indicates mild-to-moderate renal impairment, so drugs cleared renally (for example, many aminoglycosides and some direct oral anticoagulants) may need dose reduction per their product information.

Limitation. Cockcroft–Gault estimates from a single creatinine and is unreliable in unstable renal function, extremes of body weight, or amputees.

Example 4 — Loading Dose from Volume of Distribution​

Problem. A drug has a volume of distribution (Vd) of 40 L and a target plasma concentration of 15 mg/L. Assuming complete bioavailability (IV), what loading dose achieves the target?

Solution.

Loading dose = Vd × target concentration / bioavailability (F)

Loading dose = (40 L × 15 mg/L) / 1 = 600 mg.

If the same drug were given orally with F = 0.75, the loading dose = 600 / 0.75 = 800 mg.

Limitation. This assumes a one-compartment model and that the drug distributes instantly; for two-compartment drugs the true loading dose is often given as a slow infusion or divided doses.

Example 5 — First-Order Half-Life and Steady State​

Problem. A drug is eliminated by first-order kinetics with an elimination rate constant (k) of 0.116 h⁻¹. What is its half-life, and roughly how long until steady state on repeated dosing?

Solution.

  • Half-life t½ = 0.693 / k = 0.693 / 0.116 ≈ 6 hours.
  • Steady state is practically reached after about 4–5 half-lives → 24–30 hours.

Limitation. The "5 half-lives" rule assumes linear (first-order) kinetics. Drugs with saturable (zero-order) elimination, such as phenytoin at therapeutic doses, do not follow it.

Example 6 — Pharmacology Case: Warfarin Interaction​

Case. A patient stable on warfarin is started on a course of an antibiotic and returns two weeks later with an elevated INR and minor bruising.

Reasoning.

  1. Warfarin is a vitamin-K-antagonist anticoagulant with a narrow therapeutic index, monitored by INR.
  2. Many antibiotics raise INR by inhibiting warfarin metabolism (enzyme inhibition) and by suppressing gut flora that synthesise vitamin K.
  3. The combined effect increases anticoagulation, raising bleeding risk.

Action. Recheck INR, adjust the warfarin dose per protocol, and counsel the patient on bleeding signs. Reinforce that patients should report any new medicine, including over-the-counter and herbal products.

Limitation. The direction and size of an interaction depend on the specific antibiotic; a few agents have little effect, so always check an interaction reference rather than assuming.

Example 7 — Pharmaceutical Analysis: Percentage Purity by Titration​

Problem. 0.500 g of a sample of a monoprotic acid (molar mass 122 g/mol) is titrated and requires 38.0 mL of 0.100 M NaOH to reach the endpoint. Calculate the percentage purity.

Solution.

  1. Moles of NaOH = 0.100 mol/L × 0.0380 L = 0.00380 mol.
  2. The acid is monoprotic, so moles of acid = moles of NaOH = 0.00380 mol.
  3. Mass of pure acid = 0.00380 mol × 122 g/mol = 0.4636 g.
  4. Percentage purity = (0.4636 / 0.500) × 100 = 92.7%.

Limitation. Assumes the only titratable species is the acid of interest; acidic or basic impurities distort the result.

Example 8 — Biostatistics Case: Reading a Clinical Trial Result​

Case. A trial reports that a new antihypertensive lowered the risk of stroke with a relative risk (RR) of 0.75 and a 95% confidence interval of 0.60 to 0.94.

Interpretation.

  1. RR = 0.75 means a 25% relative reduction in stroke risk versus control.
  2. The 95% confidence interval (0.60–0.94) does not cross 1.0, so the result is statistically significant at the 5% level.
  3. Relative reduction is not the same as absolute benefit; if baseline stroke risk is low, the number needed to treat may still be large.

Limitation. Statistical significance does not prove clinical importance; always look at absolute risk reduction, trial size, and how closely trial patients match your patient.

How to Build Your Own​

For any topic in the curriculum, follow the same skeleton used above: state a realistic scenario, list the numbered steps or reasoning, give a numeric or clinical answer, then write one limitation or common error. Practising this way trains you to apply concepts under exam conditions rather than only recognising definitions.