Reduction and Enlargement of Formulae

What is a formula?

A formula is a list of the ingredients in a preparation, with the quantity of each, for a stated total amount. It may be written in a book, a pharmacopoeia or on a prescription.

You will often need a different amount from the one the formula gives. A formula for 100 mL might have to be enlarged to 5 litres for a hospital, or reduced to 30 g to make a small supply.

The scale factor method

scale factor = quantity required ÷ quantity in the formula

Then multiply every ingredient by the scale factor.

• A factor greater than 1 enlarges the formula.
• A factor less than 1 reduces it.

Example 1: enlargement. A formula for 100 mL contains drug A 5 g, drug B 2 g and glycerin 10 mL, with purified water to 100 mL. Make 250 mL.

• Scale factor = 250 ÷ 100 = 2.5

Example 2: reduction. An ointment formula for 500 g contains A 50 g, B 150 g and C 300 g. Make 60 g.

• Scale factor = 60 ÷ 500 = 0.12

The amounts add up to 60 g, so the working is correct.

Example 3: a very large enlargement. A formula for 100 mL is enlarged to make 5 litres (5 000 mL).

• Scale factor = 5 000 ÷ 100 = 50
• Drug A: 5 g × 50 = 250 g

Example 4: reducing to a small pack. A formula for 1 kg contains A 25 g, B 75 g and C 900 g. Make a 30 g tube.

• 1 kg = 1 000 g, so scale factor = 30 ÷ 1 000 = 0.03
• A: 25 × 0.03 = 0.75 g
• B: 75 × 0.03 = 2.25 g
• C: 900 × 0.03 = 27 g
• Check: 0.75 + 2.25 + 27 = 30 g ✓

After reducing a formula, some amounts may be too small to weigh accurately. In that case use the aliquot method or a dilution.

Ingredients in parts

Some formulae give the ingredients as parts, in a ratio.

Example. A base is made from three ingredients in the ratio 1 : 2 : 7. Make 500 g.

1. Total parts = 1 + 2 + 7 = 10
2. One part = 500 ÷ 10 = 50 g
3. The ingredients are 50 g, 100 g and 350 g.
4. Check: 50 + 100 + 350 = 500 g ✓

Formulae written per dose

When a formula is written for each dose (for example "each 5 mL contains"), find the number of doses first.

Example 1. Each 5 mL contains 125 mg of drug. Make 200 mL.

1. Number of doses = 200 ÷ 5 = 40
2. Drug needed = 40 × 125 mg = 5 000 mg = 5 g

Example 2. Each 10 mL contains drug A 200 mg and drug B 50 mg. Make 150 mL.

1. Number of doses = 150 ÷ 10 = 15
2. Drug A = 15 × 200 = 3 000 mg = 3 g
3. Drug B = 15 × 50 = 750 mg = 0.75 g

"Quantity sufficient" and making up to volume

Formulae often end with a line such as purified water to 100 mL (written aqua ad 100 mL or q.s. ad).

This means: dissolve the ingredients, then add water until the total volume reaches 100 mL. The dissolved ingredients take up some space, so the water you add will be less than 100 mL.

Checking a scaled formula

• Did you multiply every ingredient by the same factor?
• Are all quantities in the same units (g with g, mL with mL)?
• Do the weights of solids add up to the total weight?
• Is the answer the right size? A factor of 2.5 should give amounts a little more than double.
• Are any amounts too small to weigh accurately?