Alligation Calculator

What Is Alligation, and Why Pharmacists Call It "Tic-Tac-Toe Math"

Alligation alternate is a centuries-old method for working out what ratio of two ingredients, each a different strength, produces a specific target strength when mixed. According to Wikipedia's entry on alligation, the technique predates modern algebra and is still taught in pharmacy compounding curricula today because it's fast to run by hand and doesn't require setting up a full equation.

The nickname comes from how the method is laid out on paper: high strength top-left, low strength bottom-left, desired strength in the middle, with the two differences cross-subtracted diagonally into the boxes on the right, a layout pharmacy technician study guides nickname "tic-tac-toe math" because the grid resembles the game board. This calculator runs that same grid, then converts the resulting ratio into actual quantities for a specific batch size.

How to Solve Alligation Problems Step by Step (With the Grid)

Working an alligation problem by hand follows a fixed sequence: write the high strength in the top-left corner, the low strength in the bottom-left corner, and the desired strength in the center. Then subtract diagonally, desired minus low goes in the top-right box as the parts of high strength needed, and high minus desired goes in the bottom-right box as the parts of low strength needed. Adding those two numbers gives the total parts the finished mixture is built from.

Every field in the grid above updates the same way, so entering your own three strengths reproduces exactly this process instead of asking you to work the cross-subtraction out separately before typing in a final answer, the same sequence RxCalculations' alligation walkthrough uses when teaching the grid to pharmacy students.

Alligation Formula: Parts of High and Low Strength Explained

Written as formulas rather than a grid: Parts of High = |Desired − Low|, and Parts of Low = |High − Desired|. Total Parts is simply the sum of both. The absolute value bars matter because the grid works the same way regardless of whether the high strength is listed first or second, the arithmetic self-corrects as long as the desired strength actually sits between the other two.

If the desired strength falls outside the range set by the high and low strengths, the two ingredients genuinely cannot combine to reach it, no ratio of a 20% and a 40% solution will ever produce 60%, a boundary condition PTCB alligation practice problems use to test whether students actually understand the formula rather than just pattern-matching numbers into it, which is why this calculator flags that case directly instead of returning a technically-computed but meaningless negative ratio.

Worked Example: Mixing a 70% and 20% Solution to Get 50%

Take a 70% solution and a 20% solution, target 50%. Parts of the 70% solution: |50 − 20| = 30. Parts of the 20% solution: |70 − 50| = 20. That's a 30:20 ratio, which simplifies to 3:2, five total parts.

For a 1,000 mL batch, that works out to 1,000 × 30/50 = 600 mL of the 70% solution and 1,000 × 20/50 = 400 mL of the 20% solution. Checking the math confirms it: (600 × 0.70 + 400 × 0.20) ÷ 1,000 = 50%, exactly the target, the same style of worked check Omni Calculator's alligation explainer recommends running before trusting any alligation result.

Diluting With Water (0% Strength): The Special Case of Alligation

The most common real-world alligation problem isn't mixing two active solutions at all, it's diluting one active solution down with plain water or an inert base, which is just alligation with the low strength set to zero. Several dilution-specific tools, including the Sigma-Aldrich solution dilution calculator, solve exactly this case using the equivalent C1V1 = C2V2 formula without framing it as alligation at all, even though the underlying math is identical.

The Diluting With Water toggle on this calculator locks the low-strength field at 0%, so the same grid handles both the two-active-ingredient case and the plain-dilution case without switching tools or reworking the formula by hand.

Converting Parts to Actual Volumes for Your Target Batch Size

A ratio alone, 3:2 for example, doesn't tell a compounder how much of each ingredient to actually measure out. Multiplying the batch size by each ingredient's share of the total parts, Volume of High = Batch × Parts High ÷ Total Parts, gives real, usable quantities. Most alligation tools stop at the ratio and leave this last conversion step to the user, which is exactly the gap this calculator's batch-size field closes.

Entering a target batch in any unit, milliliters, grams, or liters, produces the two actual quantities directly, ready to measure rather than a ratio that still needs one more calculation, closing the gap general dilution calculators leave open by stopping at a ratio or a single C1V1 answer instead of a full two-ingredient batch breakdown.

Alligation vs. Simple Dilution: What's the Difference?

Alligation and simple dilution solve overlapping problems but aren't identical tools. Simple dilution, C1V1 = C2V2, only works when one component is pure diluent with no active ingredient. Alligation handles that case too, but also handles mixing two active solutions of different strengths, something C1V1 = C2V2 alone can't do. A Journal of Chemical Education analysis of the alligation method notes that alligation also has a real limitation dilution math doesn't: with three or more ingredients instead of two, alligation stops producing a single unique answer, since multiple valid ratios can hit the same target strength.

If your batch also needs a specific molar concentration rather than a percentage strength, our Molarity Calculator and, for water-based mixes, our Alkalinity Calculator cover the concentration math alligation's percentage-based grid doesn't.