Registration exams: Dilution and compounding calculations demystified

Calculations in pharmacy practice are critical to patient safety and therefore central to you becoming a competent pharmacist. It is no surprise therefore that most pharmacist registration exams around the world place particular emphasis on testing mathematical competencies. In the UK, the GPhC registration exam paper 1 requires you to complete 40 calculations in 120 minutes. This comes down to three minutes per question, a very tight schedule to read the question and calculate your answer accurately with no room for error or confusion.

So, to succeed in this paper, like everything else, you need to work smart and practise, practise, practise. Make sure you practise with a wide range of pharmaceutical calculation types as listed on the GPhC framework and build up your confidence. Use good reference resources to learn and practise calculations, examples include: [Introduction to Pharmaceutical Calculations' by Judith A Rees, Ian Smith and Brian Smith, and ‘Pharmaceutical Calculations’ by Howard C. Ansel and Shelly J. Stockton.

When confronted with a calculation question, it is important to understand what is required and the essence of the mathematical approach to take. To avoid becoming frustrated and losing time, quickly scan through the question and decide whether to tackle it straight away or move on, with the hope of coming back to it later. Time management is of the essence.

Below, we show you six worked examples of dilutions and compounding calculations that we hope will help you meet some of your learning outcomes in relation to pharmaceutical calculations. Please feel free to use any calculation method that works well for you.

Question 1: Manipulating the dilution of antibiotic granules to get a desired strength

You are working as a pharmacist in charge of the off-license manufacture of specials medicines. A doctor has requested that you prepare clarithromycin suspension 202.5mg/5mL by manipulating the dilution of clarithromycin dry granules for suspension, usually diluted to 250mg/5mL. The package for clarithromycin dry granules directs the pharmacist to add 39.9mL of water to 47.8g of granules to prepare 70mL of clarithromycin 250mg/5mL. What is the volume of water that should be added to the same 47.8g granules to prepare the ordered strength of 202.5mg/5mL? Give your answer to one decimal place.

Click here for the comments, working out and answer

To get the volume to be added, you need to know three important parameters:

1. The total volume needed for the strength to be 202.5 mg/5mL.
2. The volume occupied by the granules.
3. The total weight of clarithromycin in 47.8 g of granules.

Note that the 47.8g granules is a mixture of clarithromycin plus excipients, it is NOT all clarithromycin!

Total clarithromycin in granules = 250mg/5mL x 70mL = 3,500mg. The volume occupied by 47.8 g of granules = 70mL – 39.9mL = 30.1mL. If 202.5mg = 5mL then 3,500mg = X mL. Total volume = 3,500 x 5/(202.5mL) = 86.4mL. The volume of water to add = Total Volume - Volume of granules = 86.4mL – 30.1mL = 56.3mL (reported to one decimal place).

Answer: 56.3mL

Question 2: Dilution of a liquid product before use by the patient

A 40-year-old man has been advised to take Cipramil (citalopram) 40mg/mL liquid as follows; eight drops to be mixed with 20mL of water and taken once daily. You have the following additional information on the package: one drop of Cipramil 40mg/mL liquid = 2mg of Cipramil. What is the concentration (in mg/5mL) in the final mixture after mixing the eight drops Cipramil 40mg/mL with 20mL of water? Give your answer to two decimal places.

Click here for the comments, working out and answer

You need two pieces of information to work this calculation out:

1. The amount of Cipramil in eight drops.
2. The final volume after addition of eight drops.

Please note that the final volume is NOT 20mL, as you will have added eight.

If 1 drop = 2mg, then 8 drops = 16mg. If 40mg = 1mL then 2mg = 0.05mL; if 2mg = one drop and 2mg = 0.05mL, it follows that one drop = 0.05mL. So eight drops added = 8 x 0.05mL = 0.4mL. By adding eight drops, you will have added 0.4mL to the final volume.

Final concentration = 16mg/20.4mL = 3.92mg/5mL (reported to two decimal places).

Answer: 3.92mg/5mL

Question 3. Reconstitution of antibiotic powder for injection

You are working at a hospital paediatric ward. You have been asked to reconstitute a 250mg vial of ceftriaxone with 1mL of 1% lidocaine. The displacement value for ceftriaxone powder for injection is 0.774mL/1g. What volume in (mL) of the reconstituted solution would deliver a dose of 300mg? Give your answer to two decimal places.

Click here for the comments, working out and answer

You need two pieces of information to help with your calculation:

1. The total volume after reconstitution.
2. The weight (in mg) of the ceftriaxone powder.

To get the total volume after reconstitution, you need to work out the volume occupied by 250mg ceftriaxone and then add it to 1mL.

If 1g ceftriaxone displaces 0.774mL, then 250mg displaces 0.774/4 = 0.194mL. The total volume = 0.194mL + 1mL = 1.194mL. Concentration = 250mg/1.194mL. The required volume to give 300mg = 300mg x 1.194mL/250mg = 1.43mL (reported to two decimal places).

Answer: 1.43mL

Question 4. Double dilution of chlorhexidine digluconate

You have been asked to supply a concentrated stock solution of chlorhexidine digluconate (A), such that if you dilute 35.5mL of it with 64.5mL of water you form an intermediary solution (B). If you further dilute the intermediary solution (B), 3 parts of it with 27 parts of water a final solution of 0.71% chlorhexidine digluconate (C) is obtained. What is the strength of the concentrated solution of chlorhexidine digluconate (A) in (mg/mL)? Give your answer to the nearest whole number.

Click here for the comments, working out and answer

This calculation question is a serial dilution. The question is unique in that in a single question, you are tested on your knowledge of converting units, rounding, dealing with parts and other expressions of concentrations.

1. Find the first dilution factor (DF1) from (A) to (B). 35.5mL of chlorhexidine digluconate (A) + 64.5mL of water = 100mL, therefore DF1 = 100/35.5.
2. Find the second dilution factor (DF2) from (B) to (C). 3 parts of (B) + 27 parts of water = 30 parts (C) total. Therefore this dilution factor = 30/3 = 10.
3. DF1 x DF2 = (100/35.5) x10 = 1,000/35.5
4. Concentration of concentrated stock solution (A) = 1,000 x 0.71/35.5% = 20%.
5. 20% = 20g/100mL = 20,000mg/100mL = 200mg/mL (reported to the nearest whole number).

Check your answer: Concentrated stock (A) at 200mg/mL would yield an intermediate solution (B) at 71mg/mL, and a final diluted solution (C) at 7.1mg/mL (or 0.71%).

Answer: 200mg/mL

Question 5: Compounding Drug A-B-C using lactose as a diluent

You are working in the specials department. A prescription request requires the following: Rx: Drug A-B-C containing: Drug A: 10mg; Drug B: 20mg; Drug C: 30mg; Lactose, Q.S; Mitte 200 capsules. You have the following pertinent information: 1. The weight of contents of a capsule filled with Drug A without the capsule shells = 670 mg; 2. The weight of the contents of the capsule filled with Drug B without the capsule shells = 650mg; 3. The weight of contents of the capsule filled with Drug C without the capsule shells = 645mg and 4. The weight of the contents of a capsule filled with lactose without the capsule shells = 350 mg. What is the weight of lactose in (g) that is required to fill 200 capsules of Drug A-B-C? Give your answer to the nearest whole number.

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You need to calculate the amount of lactose displaced by each of the drugs A, B, and C as follows:

1. For Drug A 10mg/X = 670mg/350mg; 10mg drug X displaces 5.2mg lactose
2. For Drug B 20mg/X = 650mg/350mg; 20mg of drug B displaces 10.8mg lactose
3. For Drug C 30mg/X = 645mg/350mg; 30mg of drug C displaces 16.3mg lactose

The amount of lactose needed per capsule = 350mg – (5.2mg + 10.8mg + 16.3mg) = 317.7mg

For 200 capsules, the quantity of lactose needed = 200 x 317.7mg = 63,540mg = 63.54g or 64g when reported to the nearest whole number.

Answer: 64g

Question 6: Dilution and infusion rates

A seven-year-old boy weighing 23kg is due to receive salbutamol infusion at a rate of 5micrograms/kg/min in glucose 5%. The infusion bag is made up by diluting 10mL of salbutamol 5mg/5mL ampoules with 40mL of glucose 5% solution. What flow rate in (micro drops per minute) would you set the infusion to deliver the prescribed dose? Give your answer to the nearest whole number (1mL = 60 micro drops).

Click here for the comments, working out and answer

Three pieces of information are important to work this out:

1. The dilution when 10mL of the salbutamol 5mg/5mL are added to 40mL of glucose 5% must be done accurately.
2. The concentration of salbutamol in the final infusion volume in (mg/mL) should be calculated.
3. The infusion rate in (mg/min) must be calculated using the information provided.

The final volume after dilution is 50mL (from 10mL salbutamol 5mg/5ml + 40mL glucose 5%). 10mL salbutamol 5mg/5mL provides 10mL x 5mg/5mL = 10mg. Final concentration is 10mg in 50mL i.e. 0.2mg in 1mL. Infusion rate for a weight of 23kg = 5micrograms/kg/min = (5micrograms/kg/min) x 23kg x (1mg/1,000micrograms) = 0.115mg/min (note conversion of micrograms to mg); From 2. If 0.2mg = 1mL; 0.115mg = X; X/0.115 = 1/0.2; X =0.115/0.2 = 0.575mL/min; if 1mL = 60 micro drops, 0.575mL = 0.575 x 60 micro drops/min = 34.5 micro drops / min = 35 micro drops/min (reported to the nearest whole number. Also note: micro drops per min = mL/hr so you could also calculate your mL/hr and yield the same answer as micro drops/min.

Answer: 35micro drops/min

The key to success with calculations is understanding the key concepts and regular practise of a wide range of questions. Make sure you understand the question that is being asked and choose the quickest way of getting to the answer. You should be able to check your answer by backward calculation. When you are preparing for the exam, make use of a timer as this will mirror the real exam conditions.

Acknowledgements: These questions were kindly provided by: Focus Pre-Reg Revision

The calculations presented here are for educational purposes only and the directions for dilution written on the Summary of Product Characteristics (SmPC) should be strictly adhered to during your professional practice as a pharmacist. The views in this article are our own and do not represent the views of any organisations we are associated with.