Pharmacy Math: Concentration and Dilution Problems Solved

Concentration and Dilution Fundamentals

Concentration and dilution problems are among the most challenging calculations on the PTCE, but they follow predictable formulas. Once you understand the methods, these problems become straightforward. Let us work through the key techniques.

The Dilution Equation: C1V1 = C2V2

This equation solves problems where you need to dilute a stock solution to a lower concentration:

C1 (initial concentration) x V1 (initial volume) = C2 (final concentration) x V2 (final volume)

Example 1: Simple Dilution

How much 70% isopropyl alcohol and sterile water do you need to prepare 500mL of 30% isopropyl alcohol?

C1V1 = C2V2 gives us: 70% x V1 = 30% x 500mL. V1 = (30 x 500) / 70 = 214.3mL of 70% IPA. Add sterile water to reach 500mL total (285.7mL water).

Alligation Method

Alligation is used when mixing two different concentrations to achieve a target concentration. Draw a tic-tac-toe grid:

• Place the higher concentration in the upper left
• Place the lower concentration (or diluent at 0%) in the lower left
• Place the desired concentration in the center
• Subtract diagonally (always subtract smaller from larger)
• The results give the ratio of parts needed

Example 2: Alligation

Mix 20% and 5% dextrose solutions to make 1 liter of 12% dextrose. Higher = 20, Lower = 5, Desired = 12. Upper right: 12 - 5 = 7 parts of 20%. Lower right: 20 - 12 = 8 parts of 5%. Total parts = 15. Volume of 20%: (7/15) x 1000mL = 466.7mL. Volume of 5%: (8/15) x 1000mL = 533.3mL.

Percent Strength Expressions

• w/v (weight in volume) - grams per 100mL. A 5% w/v solution has 5g per 100mL.
• v/v (volume in volume) - mL per 100mL
• w/w (weight in weight) - grams per 100g

Practice these calculation types daily using PTCB practice problems to build speed and accuracy for exam day.