Alligation method explained, with examples

Alligation is the shortcut for one specific pharmacy problem: you have a higher strength and a lower strength of the same drug, and you need a strength in between. How much of each do you mix? PTCB's 2026 PTCE outline doesn't use the word "alligation," but area 4.1 lists concentration and dilution calculations, and alligation is a standard way to solve this kind of problem. Once you've drawn the grid a few times it takes under a minute.

The grid

Draw a tic-tac-toe style layout:

• Top left: the higher strength.
• Bottom left: the lower strength.
• Center: the desired strength.
• Then subtract diagonally, ignoring signs. The number you get on the right of each row is the number of parts of the strength on the left of that row.

In formula form:

• Parts of higher strength = desired - lower
• Parts of lower strength = higher - desired

The desired strength has to be between the other two. If it isn't, no mix of those two products can make it.

Example 1: finding the ratio

You have 10% and 2% solutions and need 5%.

• Parts of 10% = 5 - 2 = 3
• Parts of 2% = 10 - 5 = 5

So you mix 3 parts of 10% with 5 parts of 2%, 8 parts in total.

Check: (3 x 10% + 5 x 2%) / 8 parts = (30 + 10) / 8 = 5%. The check works because the final strength is a weighted average of the two starting strengths.

Example 2: turning parts into amounts

Now you need 400 mL of that 5% solution.

• Total parts = 3 + 5 = 8
• Each part = 400 mL / 8 = 50 mL
• 10% solution: 3 x 50 = 150 mL
• 2% solution: 5 x 50 = 250 mL

Check: 150 mL of 10% contains 15 g of drug, and 250 mL of 2% contains 5 g. That's 20 g in 400 mL, which is 5 g per 100 mL, or 5%.

Example 3: diluting with water

Plain diluent (sterile water, for example) counts as 0%. That lets alligation handle simple dilutions too.

You need 240 mL of a 5% solution from a 20% stock and water.

• Parts of 20% = 5 - 0 = 5
• Parts of water = 20 - 5 = 15
• Total = 20 parts, so each part = 240 / 20 = 12 mL
• 20% stock: 5 x 12 = 60 mL; water: 15 x 12 = 180 mL

Check with C1V1 = C2V2: 20% x 60 mL = 1,200, and 5% x 240 mL = 1,200. The two methods agree, which is what you'd expect: when one ingredient is 0%, alligation and the dilution formula are solving the same problem.

Example 4: bigger numbers

You have 70% and 10% dextrose and need 1,000 mL of 25%.

• Parts of 70% = 25 - 10 = 15
• Parts of 10% = 70 - 25 = 45
• 15 : 45 simplifies to 1 : 3, 4 parts in total, so each part = 1,000 / 4 = 250 mL
• 70% dextrose: 250 mL; 10% dextrose: 750 mL

Check: 250 mL of 70% is 175 g, and 750 mL of 10% is 75 g. That's 250 g in 1,000 mL, or 25%.

Common mistakes

• Putting the parts on the wrong row. The difference you read across from the higher strength belongs to the higher strength. If your answer uses more of the stronger product than the weaker one when the target is closer to the weaker one, you've swapped them.
• Forgetting to add the parts. The 3 and 5 in Example 1 are a ratio, not mL. Divide the total amount by the total parts first.
• Using alligation for a single-product dilution without the 0%. It still works, but only if water goes in as 0%.
• Skipping the check. Multiplying each amount by its strength and adding takes ten seconds and catches almost every error.

Practice

The alligation calculator solves the grid for any two strengths and total amount, which makes it an easy way to check practice problems. For the other calculation types on the exam, see pharmacy technician math on the PTCE, and for the next step in solution math, IV flow rate and drip rate formulas.

Encodr's PTCB course is being built now, including a calculations unit with dilution and alligation practice problems. It isn't live yet. The PTCB exam page will have it at launch.