Alligation

Updated

Alligation is the tic-tac-toe method for mixing two strengths of the same drug into a target strength between them: write the strengths on a grid, subtract across the diagonals, and the differences become the parts of each stock to mix.

Removed from the 2026 PTCE outlineDropped effective 2026-01-06 (official outline PDF) — still taught in coursework and useful on the job, so the tool stays.

Alligation calculator

Mix 200 mL of the 70% with 300 mL of the 20%.

Ratio 20 : 30 (high : low) · 50 total parts across 500 mL.

70high %
20parts of 70%
40desired %
20low %
30parts of 20%
Read the diagonals: |40 − 20| = 20 parts high (top right) · |70 − 40| = 30 parts low (bottom right). Multiply each by volume ÷ total parts.

Alligation alternate — the grid

  1. High strength top-left, low strength bottom-left, target in the center.
  2. Subtract along each diagonal, writing the absolute difference on the right.
  3. Top-right = parts of the HIGH stock; bottom-right = parts of the LOW stock.
  4. Multiply each part share by (final volume ÷ total parts).

The calculator above draws this grid with your numbers as you type.

Alligation medial — the average

Medial answers the opposite question: mix 300 mL of 50% with 200 mL of 20% — what strength results? Multiply each volume by its strength, add, and divide by the total volume: (300 × 0.5 + 200 × 0.2) ÷ 500 = 38%. It’s a weighted average, and it’s the quickest way to CHECK an alternate answer.

Dilution or alligation — which one?

Use C₁V₁ = C₂V₂ when…

you dilute ONE stock with a plain diluent (water, NS) and want a volume or concentration. One product + diluent → algebra with one unknown.

Use the alligation grid when…

you mix TWO strengths of the same ingredient (a diluent counts as 0%) and need the RATIO of the two. Two components → grid. If your “low” is 0%, both methods agree — the grid just shows the parts.

Practice problems

Unlimited generated alligation problems, each with the grid worked step by step.

Alligation (legacy) · difficulty 4/5

Removed from the 2026 outline

You need 240 mL of a 40% solution, prepared by mixing 70% and 30% stock solutions. How many mL of the 30% solution are needed?

Common questions

What does alligation mean?

Alligation is an arithmetic method for mixing two strengths of the same ingredient to hit a target strength in between. It converts the strength differences into mixing parts — no algebra required — which is why pharmacy uses it for combining stock solutions and creams.

What is the law of alligation?

The amounts of each component are inversely proportional to how far its strength sits from the target: the further a stock strength is from what you need, the fewer parts of it you use. The tic-tac-toe grid is just this law drawn as diagonal subtraction.

What are the types of alligation?

Alligation medial averages known quantities into a final strength (weighted average); alligation alternate runs in reverse — given a target strength, it finds the mixing ratio. The grid on this page is alligation alternate, the version pharmacy calculations actually use.

How is alligation used in pharmacy?

Compounding: diluting concentrated stock with a weaker stock or diluent to a prescribed strength — mixing 70% alcohol with 20% to get 40%, or cutting a 23.4% sodium chloride concentrate down to 0.9%. Working techs also sanity-check compound ratios with it.