How this instrument works
Alligation alternate is a shortcut for a problem that shows up constantly in pharmacy, brewing, and the lab bench: you have a stock that's too strong and a stock that's too weak, and you need something in between. Rather than setting up algebra, you cross-subtract on a diagonal. Put the desired concentration in the middle, the high concentration above it, the low concentration below it. Subtract diagonally — desired minus low gives the 'parts' of the high solution you need; high minus desired gives the parts of the low solution. Those two differences are the mixing ratio.
The reason the cross-subtraction works is a lever-balance argument: think of the low concentration and high concentration sitting at opposite ends of a number line, with the desired concentration somewhere between them. The distance from desired down to low, and the distance from high down to desired, set the ratio in which the two solutions balance around that middle point — exactly like a see-saw balances heavier weights closer to the fulcrum. This instrument runs that logic for you, then scales the resulting ratio to whatever total volume you asked for.
The name is old — 'alligation' comes from the Latin alligare, 'to bind together' — and the technique predates modern algebra notation by centuries; it shows up in Renaissance-era arithmetic texts for mixing metals of different purity, and pharmacists have kept using it because it's faster than solving simultaneous equations by hand at the compounding bench.
- Enter the higher concentration in the first field — the strength of your more concentrated stock solution.
- Enter the lower concentration — the strength of your weaker stock solution or diluent.
- Enter the desired (required) concentration — it must sit between the low and high values, or the instrument will flag it.
- Enter the total volume of mixture you want to end up with.
- Read the two results: the ratio (H:L) and the actual volume of each solution to combine to hit that total.
Worked example — mixing 12 M and 5 M to get 9 M
A stockroom holds a 12 M stock solution and a 5 M stock solution, and a protocol calls for 612.5 mL of 9 M. Cross-subtracting: H = 9 − 5 = 4, L = 12 − 9 = 3, so the two solutions mix in a 4:3 ratio. Scaled to 612.5 mL total, that works out to 350 mL of the 12 M solution and 262.5 mL of the 5 M solution — the exact split this instrument returns, and a textbook alligation-alternate problem reproduced on chemistry course pages and pharmacy calculation guides alike.
Check it by weighted average: (350 × 12 + 262.5 × 5) ⁄ 612.5 = (4200 + 1312.5) ⁄ 612.5 = 9 M exactly. That's the whole method in one line — alligation just finds the split that makes this weighted average come out right, without you needing to set up and solve the equation from scratch.
Questions
What is alligation in chemistry?
Alligation is a shortcut arithmetic method for figuring out how much of a higher-concentration solution and a lower-concentration solution (or diluent) to combine to hit a target concentration in between. Instead of solving a weighted-average equation algebraically, you cross-subtract the three concentrations on a diagonal grid to get a mixing ratio directly. It's taught heavily in pharmacy calculations because compounding pharmacists need a fast, reliable way to work out dilutions and admixtures by hand.
Does alligation work with percentages as well as molarity?
Yes — alligation only cares about the numbers being in the same units on both sides, so it works identically whether you're mixing molar concentrations, percent-strength ointments, percent alcohol by volume, or percent w/v solutions. Just make sure the higher, lower, and desired values are all expressed the same way (all percent, or all molarity) before you enter them; mixing units silently is the most common way to get a wrong answer.
What happens if the desired concentration isn't between the high and low values?
The method breaks down — alligation only solves for a target that sits strictly between your two stock concentrations, because it's finding a weighted average, and a weighted average of two numbers can never fall outside the range between them. This instrument checks for that and will tell you to adjust your inputs rather than return a nonsensical negative volume.
Can alligation handle mixing three or more solutions?
The classic alligation-alternate grid is built for exactly two components. With three or more stock concentrations, pharmacists typically pick two convenient ones (often the closest above and below the target) and alligate those, or fall back on setting up and solving simultaneous equations directly. Some pharmacy references show extended three-component alligation grids, but they're a specialized variant beyond this two-solution calculator's scope.
How is alligation different from a simple weighted average?
They're the same relationship solved in opposite directions. A weighted average takes two known volumes and concentrations and predicts the resulting mixture strength; alligation starts from the desired strength and works backward to find the volumes. Alligation is really just the weighted-average formula rearranged and turned into a quick mental cross-subtraction so you don't have to solve an equation from scratch every time.