How to differentiate reducing and enlarging formula from dilution and concentration?

Question

By comparing and contrasting when it comes on solving problems.

Answer 1

Hi Jenson!

The quick answer to your question:

1. Enlarging and reducing formulas have to do with making a different quantity of a product. The concentration stays the same, but the amount of product changes. So, for example: If you have a formula for 50 mL of 5% Lidocaine, but you need to make 25 mL of 5% Lidocaine, then you would need to reduce the formula. If you are asked to prepare 500 mL of 5% Lidocaine, then you would need to enlarge the formula. In both cases, the strength of the lidocaine stays the same (5%), but the amount of 5% Lidocaine changes.
2. In dilution and concentration questions, you are changing the actual concentration/strength of the product. In dilution problems, you are reducing the concentration of the product. For example, you might need 5% dextrose, but you only have 50% dextrose in stock. In other words, you would need to dilute the 50% dextrose with enough diluent to reduce its concentration to 5%.
3. In concentration questions, you are increasing the concentration of the product. For example, let's say you have 1% Drug A solution, but you receive an order for 3% Drug A solution. You would need to add more active ingredient to the solution of 1% Drug A to increase its strength to 3%.

Long answer - How to tackle these problems:

In pharmacy, we reduce and enlarge formulas when you have a recipe/order for a certain quantity of a product but need to prepare a different amount of the same product.

So, for example, let's say we have the following recipe to prepare 100 mL of Magical Elixir:

Magical Elixir

Pixie Dust 10 g

Fairy Water 20 mL

Sterile water qs 100 mL

Now let's say the pharmacy receives an order for 50 mL of Magical Elixir. In this case we would have to reduce the formula since we need to prepare less than 100 mL. One way to do this is to find the factor we are reducing by:

Factor = (Quantity of formula desired) / (Quantity of formula given)

In this case, the quantity desired is 50 and the quantity given is 100. Thus the factor = 50/100 = 0.5.

Next, we multiply each ingredient by the factor in order to generate a new formula for the quantity desired:

Pixie Dust 10 g x 0.5 = 5 g

Fairy Water 20 mL x 0.5 = 10 mL

Sterile water qs 100 mL x 0.5 = 50 mL

Thus, to prepare 50 mL of Magical Elixir, we need 5 g of Pixie Dust and 10 mL of Fairy Water, and then we add sterile water until we have 50 mL of elixir.

On the other hand, let's say we want to make 400 mL of Magical Elixir. In this case, we would be enlarging the formula. We actually tackle this situation the exact same way we tackled the reduction: Finding the factor!

In this case, our quantity desired is 400 mL and our quantity given is 100 mL. Thus, our factor = 400/100 = 4. Now we multiply each ingredient by 4:

Pixie Dust 10 g x 4 = 40 g

Fairy Water 20 mL x 4 = 80 mL

Sterile water qs 100 mL x 4 = 400 mL

Thus, to produce 400 mL of Magical Elixir, we need 40 g of Pixie Dust, 80 mL of Fairy Water and then we qs with sterile water until we have 400 mL of elixir.

In terms of dilution and concentration:

1. For dilution, use the dilution formula, which is C1Q1 = C2Q2, where
2. C1 is your initial concentration and C2 is your desired concentration (both must be in the same unit)
3. Q1 is your initial quantity and Q2 is your desired quantity (both must be in the same unit)
4. Note, you may see other variations of this formula as C1V1=C2V2 or M1V1 = M2V2. They are the same thing.

Example: How many mL of 50% dextrose are needed to prepare 500 mL of 5% dextrose?

Strategy

We can tell this is a dilution problem because its concentration is being reduced.

Givens

C1 = 50%

C2 = 5%

Q1 = ?

Q2 = 500 mL

Calculations

C1Q1 = C2Q2

(50%)(Q1) = (5%)(500 mL)

Q1 = [(5%)(500 mL)] / (50%)

Q1 = 50 mL

1. Thus, we would need 50 mL of 50% dextrose to prepare 500 mL of 5% dextrose.
2. To find the amount of diluent you need to add, subtract Q2 - Q1. Thus, we would need to add 450 mL of diluent to 50 mL of 50% dextrose to prepare 500 mL of 5% dextrose.

1. For concentrating a formula, there are a lot of options in terms of how you should tackle this type of problem. I like to use alligations.

Example: How many milligrams of Drug B should be mixed with 2500 mg of a 8% w/w Drug B solution to increase it in strength to 20%?

Strategy

We can tell this is a concentration problem because the strength is being increased.

Set up:

Assuming you know how to perform alligations, the work looks as follows:

100% 20 - 8 = 12 parts of 100% Drug B

20%

8% 100 - 20 = 80 parts of 8% Drug B

Create a proportion:

x mg of 100% Drug B = 12 parts 100% Drug B

2500 mg of 8% Drug B 80 parts 8% Drug B

Now cross multiply:

(2500 mg)(12) = 80x

x = 375 mg