“This looks quite delicious. You will start cooking for four, and then there are a total of eight people waiting to have some of this delicious meal. If the ingredient list says it serves four, how could one scale up the ingredients to ensure the additional four visitors get some of this delicious oxtail and beans as well? Think about it, do the math, and let’s see how we could adjust it throughout the content of this page.”
Jamaican Oxtail and Beans Recipe
Ingredients for: 4 servings.
1 oxtail about 2-2.5 lb. Cut up
4 tbsp. cornflour
2 tbsp. cooking oil
Salt and Black pepper
4 rashers (slices) Bacon (Sugar cured, rindless)
2 medium onions sliced
1 clove garlic crushed
4 carrots pared and sliced
1 cup peeled chopped tomatoes
1 pint (16 fl. Oz) hot water
2 whole scotch bonnet peppers
2 stalks green onions finely sliced
1 spring thyme
1 can butter beans (Lima beans)
… and one large Dutch pot
By the end of this page, you should be able to:
perform calculations to reduce a formula for pharmaceutical preparations as stated in parts with 100% accuracy and without assistance.
perform calculations to reduce the concentration of one or more ingredients in a formula whilst maintaining the proportion of the base with 100% accuracy and without assistance.
perform calculations to enlarge the concentration of one or more active ingredients in a formula whilst maintaining the proportion of the base with 100% accuracy and without assistance.
solve application problems involving reducing and enlarging formulas with 100% accuracy and without assistance.
Reducing the Formula
In cooking, adjustments to ingredient quantities are made to accommodate varying numbers of people being served. Pharmaceutical compounding and manufacturing follow a similar principle. Compounders often need to prepare medications from recipes or Master formulas listed in larger quantities than necessary. For instance, official formulas in drug references like the British Pharmacopoeia (BP) or the United States Pharmacopoeia (USP) are typically specified in masses of 100 g or volumes of 1000 mL. However, the required medication quantity may be less than 100 g or 1000 mL.
To obtain the correct amounts of each ingredient for the desired product quantity, compounders must scale up or down the formulation. This process of reducing or enlarging formulas enables compounders to prepare medications in quantities different from the original formula’s specifications. Let us explore reducing the formula using the example below. Click on each to view each step.
Task #1
Using the master formula provided, determine the quantity of each ingredient that is required to produce 30 g of the ointment. Provide a new formula to make the product.
The term “parts” in the master formula is interchangeable with the basic units of measure for mass (grams) and volume (milliliters). When the total number of parts in the master formula equals 100, this unit of expression can be interchanged with percentage. In this context, 100% represents the whole, allowing each ingredient in the formula to be expressed as a ratio with respect to the whole. The image below provides additional explanation.
Determine what information you have
The total number of parts in the compounding formula = 1 part + 8 parts + 91 parts = 100 parts or 100 g
Therefore, Salicylic acid 1 part means 1 part in 100 parts
Precipitated Sulphur 8 parts means 8 parts in 100 parts
and Soft Paraffin 91 parts means 91 parts in 100 parts
Determine what you want
Salicylic acid + precipitated Sulphur Ointment * 30 g
Click on the next tab for the next step.
We have established that the formula needs to be adjusted because the master formula is for 100 g, whereas the requested quantity is 30 g. This reduction can be achieved in two ways: either by utilizing a scaling factor or by employing ratio and proportion to determine the appropriate quantity of ingredients. Let us take at look at both methods:
Now that you have successfully reduced the formula, click on the next tab to see the conclusion.
The new formula provides explicit instructions for compounding the product. It is crucial to ensure that the information provided is as accurate as possible.
New Formula
Salicylic Acid and precipitated Sulphur Ointment * 30 g
Salicylic Acid 0.3 g
precipitated Sulphur 2.4 g
Soft Paraffin 27.3 g
Let us look at another example!
Compounding Task #2
Using the master formula provided, if 30 mL is to be prepared, calculate the quantity of each ingredient and provide a new formula.
We can apply the same steps as before and use the factor or ratio and proportion to respond to this task. We can calculate the factor = 30 mL/100 mL = 0.3 and find the product of the factor and each formula quantity. However, we will use ratio and proportion to respond to this task.
Calculating for Calamine powder
If there are, 8 g in 100 mL
Then, x g in 30 mL x = 2.4 g
Calculating for Zinc Oxide Powder
If there are 8 g in 100 mL
then, x g in 30 mL x = 2.4 g
Calculating for Glycerol
If there are, 2 g in 100 mL
then, x g in 30 mL x = 0.6 g
Calculating for Bentonite Magma
If there are, 25 mL in 100 mL
then, x mL is in 30 mL x = 7.5 mL
Final Step- Generating the New Formula
Calamine Lotion
Calamine Powder 2.4 g
Zinc Oxide Powder 2.4 g
Glycerol 0.6 g
Bentonite Magma 7.5 mL
Calcium Hydroxide q.s. 30 mL
Enlarging the Formula
Depending on the specific task, you might need to scale up a formula, similar to our oxtail recipe for four, to serve 8 people while preserving its delicious flavor. To achieve this, all the ingredients in the recipe must be proportionally increased. It’s essential to note that when enlarging or reducing a formula, the concentration (strength or potency) of the formulation/product, particularly concerning the active ingredients, remains constant due to the proportional adjustment of all the components in the formula.
Let us take a look another this example!
Compounding Task #3
Using the master formula provided, if one is to prepare 250 mL of Calamine Lotion, calculate the quantity of each ingredient needed and provide a new formula.
We may use ratio and proportion or the scaling factor. In this example, we will use the scaling factor to calculate the quantities.
Recall Scaling Factor = Product Total/Formula Total
Therefore scaling factor = 250 mL/100 mL = 2.5
Now let us determine the quantities of each Ingredient!
Calamine Powder 8 g * 2.5 = 20 g
Zinc Oxide Powder 8 g * 2.5 = 20 g
Glycerol 2 g * 2.5 = 5 g
Bentonite Magma 25 mL * 2.5 = 62.5 mL
Since we are combining liquids and solids to create this compound, we will not calculate the volume of Calcium Hydroxide in this case. Different substances have different densities and will therefore occupy varying volumes; this implies that the volume occupied by 8 g of Calamine powder and 8 g of Zinc Oxide powder will not be identical. The master formula suggests making up to the final volume; hence, we will add a sufficient quantity to reach a final product volume of 250 mL by using the appropriate quantity of Calcium Hydroxide.
New Formula- Calamine Lotion
Calamine Powder 20 g
Zinc Oxide Powder 20 g
Glycerol 5 g
Bentonite Magma 62.5 mL
Calcium Hydroxide q.s. 250 mL
Enlarging or Reducing the Formula while Altering Concentration
There are occasions when it becomes necessary to either expand or reduce a formula and modify the product’s concentration. This is especially common in formulations involving semi-solids like ointments and creams. For instance, a physician might request an adjustment in concentration, especially for babies, if the formula is deemed excessively potent or too concentrated for the specific age-group.
When adjusting the concentration of a formula, it involves modifying the quantities of active ingredients by either increasing or decreasing them. Consequently, the base or diluent of the product, which carries the active pharmaceutical ingredient and other excipients, is adjusted to accommodate the altered quantities of the active ingredient. This ensures a balanced and effective formulation in pharmaceutical products.
Compounding Task #4
You are provided with the master formula below and asked to prepare 30 g of Double Strength Whitefield’s Ointment:
First, What you Need to Know
Single Strength is an expression of concentration with respect to the active ingredients in this master formula.
You must be able to correctly identify which of the ingredients are active pharmaceutical ingredients (API), and which are base(s) or excipients.
If the API in the formula increases the base must decrease proportionally to accommodate the change.
What we have
Formula for Whitefield’s Ointment BP Single Strength
Formula total = 91 g + 6 g + 3 g = 100 g or 100%
What we want
Formula for Whitefield’s Ointment BP Double Strength
Product total = 30 g
Step 2- Reduce or Enlarge the formula based on the Quantity Requested
As per the request, we realize that we need to reduce the formula since the quantity requested is less than the master formula total. We can use the scaling factor or ratio and proportion and reduce the formula. Let us use the scaling factor for this example:
Scaling factor = 30 g/100 g = 0.3
Emulsifying Ointment = 91 g * 0.3 = 27.3 g
Benzoic Acid = 6 g * 0.3 = 1.8 g
Salicylic Acid = 3 g * 0.3 = 0.9 g
Step 3 – Alter the concentration of the formula to the match the concentration in the compounding request.
We established that the concentration of the master formula is single strength with reference to the quantities of the APIs and concentration of the product is double strength with reference to the APIs. We will therefore use this information to find the scaling factor to alter the concentrations.
Recall, the scaling factor is normally applied to all ingredients in the formula to reduce or enlarge the formula. However, since this scaling factor [conc.] is calculated on the change in concentration, we will only apply it to the APIs in the formula.
Calculating APIs quantities
Benzoic Acid 1.8 g * 2.0 = 3.6 g
Salicylic Acid 0.9 g * 2.0 = 1.8 g
What is the difference in the weight of APIs added?
Product API (3.6 g + 1.8 g) – Formula API (1.8 g + 0.9 g) = 2.7 g
Since we will add 2.7 g of APIs to change the concentration to Double Strength, we must now reduce the base quantities by 2.7g.
Base Quantity in New Formula = 27.3 g – 2.7g = 24.6 g
Step 4- Generate the New Product Formula
Notice that although the active ingredients were doubled the base was not. If you merely doubled the base you will end up with the same concentration of actives in the single strength formulation.
You should also check that all quantities add up to the intended product quantity of 30 g
24.6 g + 3.6 g + 1.8 g = 30 g.
In this example, we first reduced the formula and then adjusted the concentration. You can also adjust the concentration then reduce the formula. Let us explore with this next example.
Compounding Task #5
You are provided with the master formula below and asked to prepare 30 g of a Half Strength Whitefield’s Ointment. Calculate the quantity of ingredients for compounding this ointment and provide a new formula.
Step 1- Adjust the concentration of the formula.
Scaling Factorconc. = 0.5 (Half)/1 (Single) = 1/2 or 0.5 in decimal
Benzoic Acid 6 g * 0.5 = 3 g
Salicylic Acid 3 g * 0.5 = 1.5 g
The difference in API = (3 g + 1.5 g) – (6 g + 3 g) = – 4.5 g
Since the APIs were reduced by 4.5 g, we need to adjust (add) 4.5 g to the base quantity.
Base quantity = 91 g + 4.5 g = 95.5 g
Step 2- Generate the New Formula.
Step 3- Enlarge or Reduce the Formula
Scaling Factor = 30 g/100 g = 0.3
Emulsifying Ointment = 95.5 g * 0.3 = 28.65 g
Benzoic Acid = 3 g * 0.3 = 0.9 g
Salicylic Acid = 1.5 g * 0.3 = 0.45 g
Step 4- Generate the New Product Formula
In our previous tasks, there was one base (excipient) in this formulation. Let us see what approach we would take if there are multiple bases.
Compounding Task #6
You are provided with the master formula below and asked to prepare 30 g of a double strength ointment.
This is a variation in the Whitefield’s Ointment Formula. Notice there is an added ingredient. The White Soft Paraffin is a part of the base in the formula. Note the concentration of the base has not changed (91%) it is the same as in the other formula, but in this formula the base is made up of more than one ingredient which is fairly common.
Step 1- Alter the concentration
Scaling Factorconc. = 2/1 = 2.0
Benzoic Acid = 6 g * 2.0 = 12 g
Salicylic Acid = 3 g * 2.0 = 6 g
What is the difference in APIs?
= Product (12 g + 6 g) – Formula (6 g+ 3 g) = 9 g
This means we need to adjust (reduce) the base by 9 g to account for this additional quantity of APIs.
Step 2- Adjust the Base Quantities to Accommodate the Increased Quantity of APIs.
Total Base in Master Formula = 51 g (EO- Emulsifying Ointment) + 40g (WSP- White Soft Paraffin) = 91 g of Base.
Adjust the base by 9 g due to the added APIs = 91 g – 9 g = 82 g or 82%
Note well– We now know that the total base in the adjusted formula is 82 g but we do not know the exact quantity for each ingredient. We must determine the proportional quantity of each base ingredient such that it is similar to the master formula. This is to ensure the integrity of the product we are compounding.
We will use ratio and proportion to calculate the base quantities.
Step 3- Generate the New Formula
Step 4- Reduce or Enlarge the Formula
The Scaling Factor = 0.3
EO 45.96 g * 0.3 = 13.79 g
WSP 36.04 g * 0.3 = 10.81 g
Benzoic Acid 12 g * 0.3 = 3.6 g
Salicylic Acid 6 g * 0.3 = 1.8 g
Step 5- Generate the New Product Formula.
So, when adjusting the concentration of a formula the compounder must identify the active ingredients and the base. If the concentration of the active(s) will increase then the base needs to be decreased to accommodate this increase. Likewise, if the concentration of the actives decreases the base will need to increase.
Reducing and enlarging formulas allow the compounder to prepare a medication that is listed in larger or smaller portions than the original formula. Whether scaling up or down the compounding pharmacist or pharmacy technician must be proficient in such calculations. Reducing and enlarging the formula can be done by ratio and proportion calculations or using a scaling factor.
