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You can use the alligation method to determine how many parts of the same product, with different strengths, you will need to create the final strength.
Typology: Lecture notes
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Alligations are used when mixing two products with different percent strengths of the same active ingredient. The strength of the final product will fall between the strengths of each original product.
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Solving Alligations
You can use the alligation method to determine how many parts of the same product, with different strengths, you will need to create the final strength requested. Further, you can calculate exactly how many milliliters or grams you need of each beginning product.
The Alligation Grid
% Higher Strength
% Desired
Number/Parts of Higher Strength
Number/Parts of Lower Strength
% Lower Strength
Alligation Tips
Let’s look at the first diagonal, which contains 2.5 and 2. The difference of these two numbers will go in the bottom right box.
2
1
2
1 0.
2
1 0.
2.5 1
2
1 0.
The difference between 2.5 and 2 is 0.5, so 0.5 goes in the bottom right box.
2.5 - 2 = 0.
Now, we can work on the other diagonal, which contains 1 and 2. The difference of these two numbers will go into the top right box.
The difference between 1 and 2 is 1, so 1 goes in the top right box.
Only positive numbers can go into the alligation grid, so - 1 is changed to 1.
The numbers on the right-hand side of the alligation grid represent the number of parts per ingredient, when read straight across. This means that there should be 1 part of the 2.5% ointment and 0.5 parts of the 1% ointment to make the 2% ointment.
2
1
1 0.
2
1
1 0.
Now, by adding the numbers in the right column, we can determine the total number of parts necessary. In other words,
1 part (2.5%) + 0.5 part (1%) = 1.5 parts total
Now that we have determined the number of parts needed for each ingredient and the total number of parts to be used, we can set these up as proportions.
2.5% ointment
or
1% ointment
or
Finally, we can take the proportion of parts needed of each ointment and multiply it by the desired quantity to determine the quantity of each ingredi- ent needed.
0.5 part 1.5 parts
1 part 1.5 parts
parts needed total parts
Next, we will calculate the numbers that should go into the top right and bottom right boxes.
Higher………..0.
Desired……….0.
Lower…...0.
0.025 0.
This is done by working diagonally and taking the difference between the two numbers already in place. Let’s look at the first diagonal, which contains 0.1 and 0.05. The differ- ence of these two numbers will go in the bottom right box.
The difference between 0.1 and 0.05 is 0.5, so 0.05 goes in the bottom right box.
0.1 - 0.05 = 0.
Now, we can work on the other diagonal, which contains 0.025 and 0.05. The difference of these two numbers will go into the top right box.
0.025 0.
0.
The difference between 0.025 and 0.05 is 0.025, so 0.025 goes in the top right box.
Remember, only positive numbers can go into the alligation grid, so
The numbers on the right-hand side of the alligation grid represent the number of parts per ingredient, when read straight across. This means that there should be 0.025 part of the 0.1% cream and 0.05 part of the 0.025% cream to make the 0.05% cream.
Now, fill in the alligation grid with the information that has been provided in the problem.
Higher………..2.
Desired……….
Lower…...……..0.
Higher………..2.
Desired……….
Lower…...……..0.
1
Next, we will calculate the numbers that should go into the top right and bot- tom right boxes.
This is done by working diagonally and taking the difference between the two numbers already in place. Let’s look at the first diagonal, which contains 2.5 and 1. The difference of these two numbers will go in the bottom right box.
First, draw the alligation grid.
The difference between 2.5 and 1 is 1.5, so 1.5 goes in the bottom right box. 2.5 - 1 = 1.
1
0.5 1.
1
0.5 1.
1
0.5 1.
0.
1
0.5 1.
Now, we can work on the other diagonal, which contains 0.5 and 1. The difference of these two numbers will go into the top right box.
The difference between 0.5 and 1 is 0.5, so 0.5 goes in the top right box.
Remember, only positive numbers can go into the alligation grid, so - 0. is changed to 0.5.
The numbers on the right-hand side of the alligation grid represent the number of parts per ingredient, when read straight across. This means that there should be 0.5 part of the 2.5% cream and 1.5 parts of the 0.5% cream to make the 1% cream.
EXAMPLE 7.4 Rx—Prepare 500 mL of a 7.5% dextrose solution using SWFI and D10W. Let’s look at the information that has been provided. 10% Higher Strength 0% Lower Strength 7.5% Desired Strength 500 mL Desired Quantity
D10W stands for dextrose 10% in water, thus making it a 10% strength. Don’t forget that bases, such as SWFI are 0% strength, since they contain no ac- tive ingredient. Now, fill in the alligation grid with the information that has been provided in the problem.
Higher………..
Desired……….7.
Lower…............
10
0
Next, we will calculate the numbers that should go into the top right and bottom right boxes. This is done by working diagonally and taking the difference between the two numbers already in place. Let’s look at the first diagonal, which contains 10 and 7.5. The difference of these two numbers will go in the bottom right box.
The difference between 10 and 7.5 is 2.5, so 2.5 goes in the bottom right box. 10 - 7.5 = 2.
10
(^0) 2.
10
0 2.
10
0 2.
7.
Now, we can work on the other diagonal, which contains 0 and 7.5. The difference of these two numbers will go into the top right box.
The difference between 0 and 7.5 is 7.5, so 7.5 goes in the top right box.
Remember, any negative number must be changed to a positive number to be used in the grid.
The numbers on the right-hand side of the alligation grid represent the number of parts per ingredient, when read straight across. This means that there should be 7.5 parts of the D10W and 2.5 parts of the SWFI to prepare a 7.5% dextrose solution.
EXAMPLE 7.5 Rx—Prepare 1 L of a 20% alcohol solution using a 90% alcohol and a 10% alcohol. Let’s look at the information that has been provided. 90% Higher Strength 10% Lower Strength 20% Desired Strength 1 L (1000 mL) Desired Quantity
Now, fill in the alligation grid with the information that has been provided in the problem.
90
20
10 70
Next, we will calculate the numbers that should go into the top right and bottom right boxes. This is done by working diagonally and taking the difference between the two numbers already in place. Let’s look at the first diagonal, which contains 90 and 20. The difference of these two numbers will go in the bottom right box.
Higher………..
Desired……….
Lower…...……..
90
20
10
The difference between 90 and 20 is 70, so 70 goes in the bottom right box. 90 - 20 = 70
Now, we can work on the other diagonal, which contains 10 and 20. The difference of these two numbers will go into the top right box.
90
20
10 70
90
20
10 70
10
90
20
10 70
10
The difference between 10 and 20 is 10, so 10 goes in the top right box.
Remember, any negative number must be changed to a positive number to be used in the grid.
The numbers on the right-hand side of the alligation grid represent the number of parts per ingredient, when read straight across. This means that there should be 10 parts of the 90% alcohol and 70 parts of the 10% alcohol to prepare a 20% alcohol solution.
Calculate the following alligations.
1. Rx silver nitrate 0.25% solution 1 L You have a gallon of silver nitrate 1% stock solution, which you will dilute with distilled water. How many milliliters of each will you need to make the final product? Note that the percent strength of water is zero. ________________ ________________ 2. Rx soaking solution 1:100 1 L You have a 1:25 stock solution and water. How many milliliters of each will you need to make the final product? ________________ ________________ 3. Rx coal tar 5% ointment 120 g You have coal tar 10% ointment and coal tar 2% ointment. How many grams of each will you use to prepare the final product? ________________ 4. You are instructed to prepare 454 g of a 15% ointment. In stock you have 5% and 30%. How much of each will you need to use to make the order? ________________ 5. Rx—Prepare 480 mL of a 1:30 solution using a 1:10 solution and a 1: solution. What quantities will be used of each stock solution to make the 1: solution? ________________ 6. You need to prepare 80 g of a 9% cream using a 20% stock cream and a cream base. How much are needed of each? ________________ 7. You are asked to prepare 1 L of a 1:300 soaking solution, using a stock 1:500 soaking solution and distilled water. How much of each will you need to use? ________________ 8. How much SWFI would need to be added to 500 mL stock normal saline (0.9% NaCL) to produce a 0.45% sodium chloride solution? ________________ 9. Rx alcohol 30% How many milliliters of 90% alcohol should you add to 25 mL of 10% alcohol to make 30% alcohol? ________________ 10. Rx hydrocortisone 2% ointment
How many grams of petrolatum should you add to 30 g of hydrocortisone 2.5% ointment to reduce its strength to 2.0%? The percent strength of petrolatum is zero. ________________
11. Rx normal saline
How many milliliters of water must you add to 500 mL of a 10% stock solution of sodium chloride to make a batch of normal saline (sodium chloride 0.9% solution)? ________________
12. Rx ichthammol 5% ointment
How many grams of ichthammol 10% ointment should you add to 20 g of ichthammol 2% ointment to make ichthammol 5% ointment?
13. Rx benzalkonium chloride 1:1000 solution
How many milliliters of water should you add to 50 mL of benzalkonium chloride 0.25% solution to prepare the order?
14. Rx zinc oxide 10% ointment 45 g
How many grams of zinc oxide 20% ointment and zinc oxide 5% ointment should you mix to prepare the order? ________________
15. Rx aluminum acetate 1:400 solution 1 gallon
How many milliliters of Burrow’s solution (aluminum acetate 5%) should you use to prepare the order? ________________
16. Rx histamine phosphate 1:10,000 solution 10 mL
How many milliliters of a histamine phosphate 1:10 solution do you need to prepare the order? ________________
17. Rx benzocaine 5% ointment 2 oz.
How many grams of benzocaine 2% ointment should you mix with 22.5 g of benzocaine 10% ointment to prepare the order? ________________
18. When using a 0.5% cream and a 2% cream to produce a 1.25% cream, how many parts of each are needed? ________________ 19. In what proportion would you add SWFI with D10W to produce D6W? ________________ 20. In what proportion should you add a 1:20 soaking solution with distilled water to create a 1:50 solution? ________________
SUMMARY
In certain situations, a pharmacy must use the alligation method to combine two varying strengths of a drug or combine a drug with a base or diluent to achieve the prescribed strength. While these calculations can be confusing at first, once you master the alligation grid you should be able to perform these calculations easily.
