Pharmacy calculations revision: nine worked examples

How many milligrams of Drug A are in 250 mL of a 0.05% w/v solution? Express the strength as a ratio too.

0.05% w/v means 0.05 g in 100 mL. In 250 mL: 0.05 × 250 ÷ 100 = 0.125 g = 125 mg. As a ratio, 0.05 g in 100 mL is 1 g in 2,000 mL, which is 1 in 2,000.

You need 500 mL of a 0.02% w/v solution, and you have a 5% w/v stock. How much stock do you use?

Amount of drug is unchanged, so C1 × V1 = C2 × V2. V1 = 0.02 × 500 ÷ 5 = 2 mL of stock, made up to 500 mL (498 mL of diluent).

A vial holds 1 g of Drug B as powder. The product information says the powder displaces 0.8 mL. You want a final concentration of 250 mg/mL. How much water do you add, and what volume gives a 600 mg dose?

Final volume = 1,000 mg ÷ 250 mg/mL = 4 mL. The powder already takes up 0.8 mL, so add 4 − 0.8 = 3.2 mL of water. A 600 mg dose is 600 ÷ 250 = 2.4 mL.

The trap is adding the full 4 mL, which gives a weaker solution than you meant to make.

Drug C is prescribed at 60 mg/m² once daily. The patient's body surface area is 1.72 m². Give the dose to the nearest 5 mg.

60 × 1.72 = 103.2 mg, which rounds to 105 mg. Only round at the stage the question tells you to. Rounding mid-calculation is a common way to lose a mark.

Drug D is prescribed at 5 micrograms/kg/minute for a 72 kg patient. The infusion is 200 mg in 50 mL. What rate in mL/hour?

Dose: 5 × 72 = 360 micrograms/minute. Per hour: 360 × 60 = 21,600 micrograms = 21.6 mg/hour. Concentration: 200 mg ÷ 50 mL = 4 mg/mL. Rate: 21.6 ÷ 4 = 5.4 mL/hour.

Gravity drip version: 1 litre over 8 hours through a set delivering 20 drops/mL. 1,000 × 20 ÷ (8 × 60) = 41.67, which is 42 drops/minute.

In a trial, 12% of the control group and 9% of the treatment group had the event over three years. Find the absolute risk reduction, the relative risk reduction and the number needed to treat.

ARR = 12% − 9% = 3% (0.03). RRR = 0.03 ÷ 0.12 = 25%. NNT = 1 ÷ 0.03 = 33.3, which rounds up to 34 over three years. Always round an NNT up, and quote the time period.

A new treatment costs £18,000 more than the current one per patient and adds 0.75 quality-adjusted life years (QALYs). What is the incremental cost per QALY?

£18,000 ÷ 0.75 = £24,000 per QALY. A question will give you a threshold to compare it with, if it needs one.

Drug E has a half-life of 6 hours. The plasma level is 80 mg/L. What will it be 18 hours later, assuming no further doses?

18 hours is 3 half-lives: 80 → 40 → 20 → 10 mg/L.

Creatinine clearance (Cockcroft-Gault): a man aged 70, weight 80 kg, serum creatinine 110 micromol/L. With creatinine in micromol/L, CrCl = (140 − age) × weight × 1.23 ÷ creatinine (1.04 instead of 1.23 for women). (140 − 70) × 80 × 1.23 ÷ 110 = 62.6, which is 63 mL/minute.

Where 1.23 comes from: the original 1976 equation used creatinine in mg/dL and divided by 72. Converting from mg/dL to micromol/L multiplies by 88.4, and 88.4 ÷ 72 = 1.228. For women the original multiplied by 0.85, and 0.85 × 1.228 = 1.04. The assessment gives you complex formulae when you need them, but knowing where the constant comes from stops you using the wrong one.

Drug F oral solution: 5 mL four times a day for 7 days. It comes in 100 mL bottles. How many bottles?

5 × 4 × 7 = 140 mL. One bottle holds 100 mL, so you need 2 bottles.

A reducing course: Drug G 5 mg tablets. Take 30 mg a day for 5 days, then reduce by 5 mg every 3 days until you stop. How many tablets?

First 5 days: 30 × 5 = 150 mg. Then 25, 20, 15, 10 and 5 mg a day, each for 3 days: (25 + 20 + 15 + 10 + 5) × 3 = 225 mg. Total: 375 mg ÷ 5 mg = 75 tablets over 20 days.