1. Danielle DelVillano, Pharm.D.
Pharmaceutical Calculations: Altering Product Strength, Use of Stock Solutions, and Alligation
Danielle DelVillano, Pharm.D.

2. Objectives
Perform calculations for altering product strength by dilution, concentration, or fortification
Perform calculations for the preparation and use of stock solutions
Apply alligation medial and alligation alternate in problem solving

3. Keep in mind Solution = solute + solvent Changing x% to fraction
If w/v – solute in g / solution in mL
If v/v – solute in mL / solute+solvent in mL
If w/w – solute in g / solute+solvent in g
Changing x% to fraction
X g / 100 mL
Changing 1:x ratio strength to fraction
1 g / x mL

4. Altering Product Strength
Concentrated
Addition of active ingredient
Diluted
Addition of diluent
Fortification
Addition of a calculated quantity of additional therapeutic agent

5. Relationship between Strength and Total Quantity
If mixture is diluted to twice its original quantity, then strength will be reduced by one half
Start with 100 mL of 5% solution (5 g drug)
Add 100 mL, now 200 mL of 2.5% solution (5 g drug)
If a mixture is concentrated by evaporation to one half its original quantity, the strength will be doubled
Evaporate, now 50 mL of 10% solution (5 g drug)

6. Several Methods for Solving
Inverse proportions
Q1C1 = Q2C2
Determine the quantity of active solute needed and then calculate the quantity of available solution to provide needed amount
“Box method”

7. Box method Always keep it the same When setting it up
Total amount of solution
Strength (% or ratio)
Amount of solute
Always keep it the same
When setting it up
Fill in everything from the problem
Circle what you are trying to solve for
Keep your units

8. Problem 1
If 500 mL of a 15% v/v solution are diluted to 1500 mL, what will be the percentage strength (v/v)?
500 mL solution
15%
75 g drug
Dilute with 1000 mL water 
1500 mL solution 5%
75 g drug
This stays the same 
15 g = x g
100 mL 500 mL
x = 75 g drug
x g = 75 g
100 mL mL
x = 5 %

9. Problem 2
If a syrup containing 65% w/v of sucrose is evaporated to 85% of its volume, what percentage (w/v) of sucrose will it contain?
(No volume given, chose your own) 100 mL
65%
65 g drug
Evaporate to 0.85 orioinal vol
(0.85 * 100 mL =) 85 mL
76.5%
65 g drug
This stays the same 
65 g = x g
100 mL 100 mL
x = 65 g drug
x g = 65 g
100 mL 85 mL
x = 76.5%

10. Problem 3
How many milliliters of a 1:5000 w/v solution of the preservative laurakonium chloride can be made from 125 mL of a 0.2% solution?
125 mL
0.2%
0.25 g drug
1250 mL
1:5000
0.25 g drug
This stays the same 
0.2 g = x g
100 mL 125 mL
x = 0.25 g drug
1 g = 0.25 g
5000 mL x mL
x = 1250 mL

11. Problem 4
If a cough syrup contains in each teaspoonful, 1 mg of chlorpheniramine maleate and if a pharmacist desired to double the strength, how many milligrams of that ingredient would be needed to be added to a 60 mL container of the syrup? Assume no increase in volume.
60 mL
1 mg / 5 mL
60 mL
12 mg needed
Want to double this amnt = + x mg

12. Problem 5
How many milliliters of a 1:400 w/v stock solution should be used in preparing 1 gallon of a 1:2000 w/v solution?
1:400
Add x amount of diluent
(1 gallon) 3785 mL
1:2000
757 mL
This stays the same 

13. Problem 6
How many milliliters of a 1:16 solution of sodium hypochlorite should be used in preparing 5000 mL of a 0.5% solution of sodium hypochlorite for irrigation?
400 mL

14. Problem 7
How much drug should be used in preparing 50 mL of a solution such that 5 mL diluted to 500 mL will yield a 1:1000 solution?
5 g

15. Problem 8
How many milliliters of a 17% concentrate of benzalkonium chloride should be used in preparing 300 mL of a stock solution such that 15 mL diluted to 1 liter will yield a 1:5000 solution?
23.5 mL

16. Problem 9
How many milliliters of water should be added to a pint of 5% w/v solution to make a 2% w/v solution?
709.5 mL

17. Problem 10
How many milliliters of 85% w/w phosphoric acid having a specific gravity of 1.71 should be used in preparing 1 gallon of ¼% w/v phosphoric acid solution to be used for bladder irrigation?
6.5 mL

18. Problem 11
How many grams of 20% benzocaine ointment and how many grams of ointment base (diluent) should be used in preparing 5 lbs of 2.5% benzocaine ointment?
274 g benzocaine and 1986 g diluent

19. Problem 12
How many grams of coal tar should be added to 3200 g of 5% coal tar ointment to prepare an ointment containing 20% coal tar?
600 g

20. Alligation Medial
“Weighted average” percentage strength of a mixture of two or more substances can be calculated
Percentage strength of compound is multiplied by its corresponding quantity
Sum is divided by total quantity of the mixture to give percentage strength of the mixture

21. Problem 13
What is the percentage of zinc oxide in an ointment prepared by mixing 200 g of 10% ointment, 50 g of 20% ointment, and 100 g of 5% ointment?
0.10 g * 200 g = 20 g x g = 35 g
0.20 * 50 g = 10 g g 350 g
0.05 * 100 g = 5 g x = 10%
TOTALS 350 g 35 g

22. Problem 14
What is the percentage strength of alcohol in a mixture of 500 mL of a solution containing 40% v/v alcohol, 400 mL of a second solution containing 21% v/v alcohol, and a sufficient quantity of a nonalcoholic third solution to make a total of 1000 mL?
28.4%

23. Alligation Alternate

24. Problem 15
In what proportion should 20% benzocaine ointment be mixed with an ointment base to produce a 2.5% benzocaine ointment?
20%
2.5 parts
The 20% benzocaine and ointment base shoule be mixed at a 2.5:17.5 ratio, or 1:7.
2.5% 0%
17.5 parts
20 total parts

25. Problem 16
A hospital pharmacist wants to use three lots of zinc oxide containing, respectively, 50%, 20% and 5% zinc oxide. In what proportion should they be mixed to prepare a 10% zinc oxide ointment?
50%
5 parts
The 50%, 20% and 5% lots should be mixed in a 5:5:50 proportion (or 1:1:10)
20%
5 parts
10% 5%
10 parts + 40 parts = 50 parts
60 parts total ointment

26. Problem 17
How many milliliters of 50% w/v dextrose solution and how many milliliters of 5% w/v dextrose solution are required to prepare 4500 mL of a 10% w/v solution?
500 mLof 50%
4000 mL of 5%

27. Problem 18
In what proportion must glycerin with a specific gravity of 1.25 and water be mixed to prepare a liquid having a specific gravity of 1.10?
0.10:0.15 or 2:3

28. Questions

29. Reference
Ansel, H. C. (2009) Phamaceutical Calculations (13th Ed.). Philadelphia:Lippincott Williams & Wilkins, and Wolters Kluwer Publishers