“Dilution is the solution to pollution.”
--A misguided statement
from the EPA in the 70s.
In a previous chapter, the concepts of dilutions, alligations, and various mixtures were introduced along
with problem solving methods with simple names like the dilution formula, the alligation method and
ratio-proportions. Previously you were either given problems that could have been solved with any of
the aforementioned ways, or if they could not, you were instructed as to which method to use. This
chapter is going to review those concepts, but also help you identify which ways each problem could be
solved. So, we will be reviewing and/or learning the following concepts in this chapter:




the dilution formula,
the alligation method,
a series of ratio-proportions, and
determining which method(s) to use.
Let's begin with some simple definitions for dilution, alligation, and mixture with respect to healthcare.
dilution – Mixing a substance (such as a medication) with a diluent/solvent to decrease its
concentration is considered a dilution. As this method is intended only for when mixing 2 items, one
with a particular ingredient, and the other without the same ingredient, this can always be solved with
either the dilution formula or as a series of ratio-proportions. If the starting concentration of the
ingredient is known as well as its final (or desired) concentration, then you can use the alligation
method also because the diluent/solvent has a concentration of 0%.
alligation – An alligation is when two items with different concentrations of the same substance are
being mixed together, and the final (or desired) concentration (which must be in-between the other two
concentrations) is known. The alligation method may be used for this kind of problem. If one of the
starting ingredients is actually a diluent/solvent with a concentration of 0%, then you may also solve
this with either the dilution formula or as a series of ratio-proportions.
mixture – A mixture involves mixing 2 or more ingredients of varying concentrations together. If the
mixture only contains one starting ingredient with active substance in it, it may be treated as a dilution.
If it has two initial components containing active ingredient, and you know what the final (or desired)
concentration is supposed to be, you may treat it as an alligation. If it has two or more components
containing the measured ingredient, and you need to solve for the final concentration, then you would
use a series of ratio-proportions.
Now, we should take a moment and look at several practice problems and initially determine which
methods can be utilized to solve each one. After we have successfully identified problem solving
methods, we will review the actual processes used to solve these problems.
Practice Problems
When looking at the problems below, only determine which method(s) (alligation method, dilution
formula, and a series of ratio-proportions) would function for solving them. You do not need to solve
them.......yet.
1)
You have 50 mL of a 15% solution. You dilute the solution to 150 mL. What is the final
percentage strength of this solution?
2)
Available in stock is a 70% dextrose solution and a 40% dextrose solution. How many mL of
each stock solution would be needed to prepare a liter of 45% dextrose?
3)
On hand in the pharmacy are three different percentage strength solutions; 200 mL of 80%
solution, 100 mL of 40% solution, and 50 mL of 20% solution. If all three solutions are mixed together,
what would be the final volume and the final concentration?
4)
Determine how many mL of a 14.6% sodium chloride solution and how many mL of SWFI
need to be mixed together to make 500 mL of ¼NS (0.225% sodium chloride).
Identifying problem solving methods is something that improves with proper practice. Now let's solve
each of these problems using the various methodologies discussed. This will provide a good
opportunity to reacquaint yourself with the actual problem solving methods themselves.
Examples
1)
You have 50 mL of a 15% solution. You dilute the solution to 150 mL. What is the final
percentage strength of this solution?
First, let's demonstrate this using a series of ratio-proportions:
15g
N
=
100mL 50mL
N = 7.5g
7.5g
N
=
150mL 100mL
N = 5%
Now let's solve this with the dilution formula:
1) This problem
2) This problem
3) This problem
4) This problem
The dilution formula is C1Q1 = C2Q2
C1 = stock concentration is 15%
Q1 = stock quantity is 50 mL
C2 = final concentration is ???
Q2 = final quantity is 150 mL
(15%)(50mL) = (C2 )(150mL)
(15%)(50mL) (C2 )(150mL)
=
150mL
150mL
C2 = 5%
2)
Available in stock is a 70% dextrose solution and a 40% dextrose solution. How many mL of
each stock solution would be needed to prepare a liter of 45% dextrose?
We will solve this problem using the alligation method.
high concentration = 70%
desired concentration = 45%
low concentration = 40%
final quantity = 1000 mL
5
70%
× 1000mL = 167mLof70%solution.
5
30
25
25
× 1000mL = 833mLof40%solution.
30
45%
40%
70%
3)
On hand in the pharmacy are three different percentage strength solutions; 200 mL of 80%
solution, 100 mL of 40% solution, and 50 mL of 20% solution. If all three solutions are mixed together,
what would be the final volume and the final concentration?
We will solve this problem using a series of ratio-proportions.
B
40g
A
80g
C
20g
= 100mL 100mL = 100mL 50mL = 100mL
200mL
A = 160g
B = 40g
C = 10g
160g + 40g + 10g = 210g
200mL + 100mL + 50mL = 350mL
210g
N
=
350mL 100mL
N = 60%
Practice Problem
The following problem we've already determined can be solved using all three methods. Attempt to
solve it using the alligation method, the dilution formula, and as a series of ratio-proportions.
Determine how many mL of a 14.6% sodium chloride solution and how many mL of SWFI need to be
mixed together to make 500 mL of ¼NS (0.225% sodium chloride).
40%
Worksheet 25-1
Name:
Date:
Determine which method(s) (alligation method, dilution formula, and a series of ratio-proportions)
would function for solving each scenario and then choose one of those methods to answer each
question.
1)
If 250 mL of a 12% solution is diluted to 1 L, what will be the percentage strength?
2)
If 15 g of a 5% topical ointment is mixed with 135 g of ointment base, what will be the final
concentration? (Hint: Don't forget to add the weights of both ointments together when determining the
final concentration.)
3)
If 500 mL of a 20% solution is diluted to 2 L, what will be the final percentage strength?
4)
If 100 mL of a 1:1500 methylbenzethonium chloride lotion is diluted with an equal volume of
water, what will be the ratio strength of this dilution? (Hint: Although you could convert your ratio
strengths to percentages, it is not necessary for solving this problem.)
5)
The pharmacy has on hand a 100 mL container of 23.4% concentrated sodium chloride solution.
How many mL of half-normal saline could be made and how many mL of diluent would be required?
(Hint: Remember that normal saline is 0.9% sodium chloride solution, so half-normal saline would be
0.45% sodium chloride solution.)
6)
A request is made to the pharmacy for a pint of 30% isopropyl alcohol solution to soak sponges
in. How many mL of 70% isopropyl alcohol and how many mL of water will be required to prepare this
dilution?
7)
A physician prescribes an otic suspension to contain 100 mg of cortisone acetate in a final
volume of 8 mL. On hand in the pharmacy is a 2.5% cortisone acetate suspension. How many mL of
the stock solution are required to make this and how many mL of NS will you use as a diluent? (Hint:
Normal saline does not contain any of the active ingredient in it.)
8)
What is the percentage strength of alcohol in a mixture containing 150 mL of witch hazel (14%
alcohol), 200 mL glycerin, and 500 mL of 50% alcohol? (Hint: Glycerin does not contain any ethyl
alcohol.)
9)
A technician is to compound a hyperalimentation with 500 mL of 5.2% amino acid, 250 mL of
70% dextrose, 72.4 mL of various micronutrients, and 177.6 mL of sterile water. What will the final
concentration of the amino acid and the dextrose be? (Hint: The amino acids and the dextrose are
different active ingredients.)
10)
You are to prepare 1 L of a 1:50 solution from 1:10 and 1:1000 stock solutions. How many mL
of each of the stock solutions do you need? (Hint: You may want to convert the ratio strengths into
percentages to simplify your work.)
11)
Available in the pharmacy is 946 mL of a 10% povidone-iodine wash concentrate. The
physician orders a diluted wash. If a pint of 1% povidone-iodine wash is ordered, how many mL of the
10% solution will be needed?
12)
If the pharmacy technician is to prepare 2 liters of a 6% betadine solution, what volume of a
10% solution would be used if it is mixed with distilled water?
13)
On hand in the pharmacy are three different percentage strength solutions; 60 mL of 75%
solution, 26 mL of 15% solution, and 14 mL of 12.5% solution. If all three solutions are mixed
together, what would be the final volume and the final concentration?
14)
An order for a parenteral nutrition requests 60 mL of micronutrients, a final concentration of
4% amino acids, a final concentration of 5% dextrose, qs with sterile water to 1000 mL. If your stock
solutions are 8.5% amino acid and 70% dextrose, how many mL of each ingredient (micronutrients,
amino acids stock solution, dextrose stock solution, and sterile water) will you need?
15)
What is the percentage strength of alcohol in a preparation if it contains 520 mL of a 95%
solution, 1 L of a 70 % solution, and a pint of a 50% solution?
Worksheet 25-2
Name:
Date:
Answer the following problems appropriately.
1)
Define the term dilution with respect to pharmacy math.
2)
Define the term alligation with respect to pharmacy math.
3)
Define the term mixture with respect to pharmacy math.
4)
A 500 mL bag of 3% sodium chloride is requested.
a)
How many milliliters of a 23.4% stock solution of sodium chloride would be required to
prepare this?
b)
How many mL of sterile water for injection (SWFI) will be required to make this bag?
5)
How many mL of a 2% stock solution should be used to prepare 500 mL of a 0.025% solution?
6)
How many mL of 95% ethyl alcohol and how many mL of NS will be needed to prepare 3 L of
50% ethyl alcohol?
7)
The OR requests 3.9 L of 0.1% quaternary ammonium compound to cleanse their surgical
instruments. If you have on hand a 6.5% concentrated quaternary ammonium compound, how many
mL of this and how many mL of sterile water will you need to prepare the requested solution?
8)
On hand in the pharmacy are several different percentage strength solutions; 50 mL of 75%
solution, 115 mL of a 50% solution, 25 mL of 15% solution, and 10 mL of 12.5% solution.
a)
If all of these solutions are mixed together what would be the final volume?
b)
What will be the final concentration of this mixture?
9)
An old pharmacy recipe is supposed to make a pint of a 117 alcohol proof spirit (2 proof = 1%).
If 95% alcohol is used to prepare this, how many mL of 95% alcohol will be required?
10)
An order is for 60 g of a 0.5% ointment. On hand is 454 g of a 10% ointment. The 10%
ointment must be diluted with petrolatum. How many grams of each ingredient must be used?
11)
One mL of a 1:5000 isoproterenol HCl solution is diluted to 10 mL with NS.
a)
What is the percentage strength of the final product?
b)
What is its ratio strength?
c)
What is its concentration in mcg/mL?
12)
A compounded medication requires that two syrups be mixed together in equal parts dissolving
a medication that has negligible powder volume. The final suspension contains 2 ounces of a syrup that
has an 85% w/v sucrose content, and 2 ounces of another syrup that has a 60% w/v sucrose content.
a)
What is the final percentage strength of sucrose in this suspension?
b)
How many grams of sucrose would be in each tablespoon of this suspension?
13)
Respiratory requests 3 mL of a 3% hypertonic sodium chloride solution intended for inhalation.
If you decide to use a 14.6% sodium chloride stock solution and normal saline, how many mL of each
will you need to prepare the requested solution?
14)
A physician orders 250 mL of a 1:200 solution of potassium permanganate (KmnO4) to be used
as a foot wash on a patient. In stock the pharmacy has a 5% KMnO4 solution. How many mL of stock
solution and how many mL of diluent (water) will be needed to prepare the requested order?
15)
You want to make 120 mL of 50% IPA solution, and you have on hand 70% IPA and purified
water. How many mL of each do you need?
16)
You receive a request for 6 mL of a 0.2% tobramycin ophthalmic solution. The pharmacist asks
you to prepare it using 0.3% tobramycin ophthalmic solution and NS. How many mL of each do you
need?
17)
You receive another request for a tobramycin ophthalmic solution, but this time it needs to be
fortified. The order wants 6 mL of a 1.5% tobramycin ophthalmic solution. The pharmacist asks you to
use a 0.3% tobramycin ophthalmic solution and 80 mg/2 mL tobramycin injection to make this
preparation. How many mL of each do you need?
18)
A physician orders a vial that will be used for IM injections with 10 mL of 200 mg/mL edetate
and 3.3 mL of 1% tetracaine.
a)
What is the final concentration of edetate in mg/mL?
b)
What is the final percentage strength of tetracaine?
19)
A nurse wants half-strength Dakin's solution (0.25% sodium hypochlorite) for dressing changes
on a patient. The pharmacy agrees to send a liter of it to the unit for the nurse to use. On hand in the
pharmacy is a 3% sodium hypochlorite solution and sterile water for irrigation. How many mL of each
will you need to prepare this?
20)
A large volume parenteral ended up leaking in the refrigerator. To try and prevent excessive
microbial growth, you are asked to prepare 250 mL of a 0.3% sodium hypochlorite solution that the
inside of the refrigerator can be wiped down with. On hand you have a 3% sodium hypochlorite
solution and sterile water for irrigation. How many mL of each will you need to prepare this?
21)
A two liter bag of a 3-in-1 parenteral nutrition is ordered for a patient.
a)
If the stock solution of amino acid has a concentration of 8.5%, and the final concentration is to
be 4%, how many mL of stock solution is required?
b)
If the stock solution of dextrose has a concentration of 70%, and the final concentration is to be
19%, how many mL of stock solution is required?
c)
If this PN includes 250 mL of 20% lipid emulsion, what is the final concentration of lipids in
this preparation?
d)
If there are 127 mL of micronutrients being added to this bag and all the rest is sterile water,
how many mL of sterile water will be needed?
e)
If this bag is being infused over 24 hours what is the infusion rate in mL/hr?
f)
If this is being infused through standard tubing with a drop factor of 20, what is the drip rate?
Chapter 25 - Dilutions & Alligations
Worksheet 25-1
of ratio-proportions. 31.9% alcohol
1)
The problem can be solved using either the 9)
The problem can be solved using either the
dilution formula or a series of ratio-proportions. 3%dilution formula or a series of ratio-proportions.
2)
The problem can be solved using either the 2.6% amino acid; 17.5% dextrose
dilution formula or a series of ratio-proportions.
10)
The problem can be solved using the
0.5%
alligation method. 192 mL of 1:10 solution; 808
3)
The problem can be solved using either the mL of 1:1000 solution
dilution formula or a series of ratio-proportions. 5%11)
The problem can be solved using either the
4)
The problem can be solved using either the dilution formula, a series of ratio-proportions, or
dilution formula or a series of ratio-proportions.
the alligation method. 48 mL of 10% povidone1:3000
iodine
5)
The problem can be solved using either the 12)
The problem can be solved using either the
dilution formula, a series of ratio-proportions, or dilution formula, a series of ratio-proportions, or
the alligation method. 5200 mL half-normal saline; the alligation method. 1200 mL of 10% solution
5100 mL diluent
13)
The problem can be solved using a series of
6)
The problem can be solved using either the ratio-proportions. 50.65%
dilution formula, a series of ratio-proportions, or 14)
The problem can be solved using either the
the alligation method. 205.7 mL 70% isopropyl
dilution formula, a series of ratio-proportions, or
alcohol; 274.3 mL water
the alligation method. 60 mL micronutrients; 470.6
7)
The problem can be solved using either the mL of 8.5% amino acids; 71.4 mL of 70%
dilution formula, a ratio-proportion, or the
dextrose; 398 mL of sterile water.
alligation method. 4 mL 2.5% cortisone acetate
15)
The problem can be solved using a series of
suspension; 4 mL NS
ratio-proportions. 71.7% alcohol
8)
The problem can be solved using a series
Worksheet 25-2
1)
Mixing a substance (such as a medication)
with a diluent/solvent to decrease its concentration
is considered a dilution.
2)
An alligation is when two items with
different concentrations of the same substance are
being mixed together and the final (or desired)
concentration (which must be in-between the other
two concentrations) is known.
3)
A mixture involves mixing 2 or more
ingredients of varying concentrations together.
4)
a)
64.1 mL of 23.4% sodium chloride
b)
435.9 mL of SWFI
5)
6.25 mL of 2% stock solution
6)
1579 mL of 95% ethyl alcohol; 1421 mL of
normal saline
7)
60 mL of 6.5% concentrated quaternary
ammonium compound; 3840 mL of SWFI
8)
a)
200 mL
b)
50%
9)
295.6 mL of alcohol
10)
3 g of 10% ointment; 57 g of petrolatum
11)
a)
0.002%
b)
1:50,000
c)
20 mcg/mL
12)
a)
72.5%
b)
10.875 g
13)
0.46 mL of 14.6% sodium chloride; 2.54
mL of NS (0.9% sodium chloride)
14)
25 mL of 5% KMnO4 solution; 225 mL of
water
15)
85.7 mL of 70% IPA; 34.3 mL of water
16)
4 mL of 0.3% tobramycin; 2 mL of NS
17)
1.95 mL of 80 mg/2 mL of tobramycin; 4.05
mL of 0.3% tobramycin
18)
150 mg of edetate; 0.25% tetracaine
19)
83.3 mL of 3% sodium hypochlorite; 916.7
mL of sterile water for irrigation
20)
25 mL of 3% sodium hypochlorite; 225 mL
of sterile water for irrigation
21)
a)
b)
c)
d)
e)
f)
941.2 mL of 8.5% amino acid
542.9 mL of 70% dextrose
2.5% lipid emulsion
138.9 mL of sterile water
83 mL/hr
28 gtt/min