PTCB Quiz Prep

How to Calculate Alligation

Updated

Alligation alternate is the tic-tac-toe method for working out how much of a stronger product and a weaker product to mix to land on a strength in between. It looks intimidating until you see the pattern, then it becomes one of the quickest points on the exam. Drill it alongside the other calculations on the PTCB math practice test once you have worked the examples below.

parts of high = desired - low

parts of low = high - desired

The tic-tac-toe grid

Draw a three by three grid. Place the higher strength in the top-left corner, the lower strength in the bottom-left corner, and the desired strength in the center. Then subtract across the diagonals. The number you write in the top-right (the parts of the higher strength) is the desired strength minus the lower strength. The number in the bottom-right (the parts of the lower strength) is the higher strength minus the desired strength. Each product takes the difference measured across from the other one.

The result is a ratio of parts. To turn parts into millilitres or grams, add the parts together, divide the final quantity by that total, and scale each product up.

Worked example: mixing two liquids

You need a 5% solution, and the pharmacy stocks a 10% solution and a 2.5% solution. How much of each makes 300 mL?

A 10% and a 2.5% to make a 5%

Prepare 300 mL of a 5% solution from a 10% stock and a 2.5% stock.

  1. Step 1. Grid it: 10% top-left, 2.5% bottom-left, 5% in the center.
  2. Step 2. Parts of 10% (high) = desired - low = 5 - 2.5 = 2.5 parts.
  3. Step 3. Parts of 2.5% (low) = high - desired = 10 - 5 = 5 parts.
  4. Step 4. Total parts = 2.5 + 5 = 7.5 parts. One part = 300 mL ÷ 7.5 = 40 mL.
  5. Step 5. 10% needed = 2.5 × 40 = 100 mL. 2.5% needed = 5 × 40 = 200 mL. Check: 100 + 200 = 300 mL.

Answer: 100 mL of the 10% and 200 mL of the 2.5%

The trap: Assigning the parts to the wrong strength. The parts written across from the 10% belong to the 10%, not to the 2.5%.

Worked example: mixing two ointments

The parts method is identical for solids. Here the units are grams instead of millilitres.

A 10% and a 2% to make a 5% ointment

Prepare 120 g of a 5% ointment from a 10% ointment and a 2% ointment.

  1. Step 1. Grid it: 10% top-left, 2% bottom-left, 5% in the center.
  2. Step 2. Parts of 10% (high) = 5 - 2 = 3 parts.
  3. Step 3. Parts of 2% (low) = 10 - 5 = 5 parts.
  4. Step 4. Total parts = 3 + 5 = 8 parts. One part = 120 g ÷ 8 = 15 g.
  5. Step 5. 10% needed = 3 × 15 = 45 g. 2% needed = 5 × 15 = 75 g. Check: 45 + 75 = 120 g.

Answer: 45 g of the 10% and 75 g of the 2%

The trap: Reporting 75 g as the 10% amount. The larger part belongs to the strength that is farther from the target, which here is the 2%.

Worked example: using water as 0%

When you dilute a concentrate, the diluent is simply 0%. Put it in the low position and the method does not change.

Diluting 90% alcohol with water

Prepare 900 mL of 70% isopropyl alcohol from 90% alcohol and purified water (0%).

  1. Step 1. Grid it: 90% top-left, 0% (water) bottom-left, 70% in the center.
  2. Step 2. Parts of 90% (high) = 70 - 0 = 70 parts.
  3. Step 3. Parts of water (low) = 90 - 70 = 20 parts.
  4. Step 4. Total parts = 70 + 20 = 90 parts. One part = 900 mL ÷ 90 = 10 mL.
  5. Step 5. 90% alcohol = 70 × 10 = 700 mL. Water = 20 × 10 = 200 mL. Check: 700 + 200 = 900 mL.

Answer: 700 mL of 90% alcohol and 200 mL of water

The trap: Forgetting that water is 0%, not 100%. The diluent adds volume but no active ingredient.

The most common mistake

The number one error is swapping which product gets which parts. The parts of the higher strength are found from the lower strength side of the grid (desired minus low), and they belong to the higher strength. Read the grid across the diagonal, and always sanity-check by mixing the active back: the total active from both products divided by the total quantity has to equal the desired percentage.

Try it

  1. Using alligation, in what ratio do you mix a 20% cream and a 5% cream to make a 10% cream?
  2. You need 240 g of a 3% ointment from a 5% ointment and a 1% ointment. How many grams of each?
  3. Prepare 500 mL of a 40% dextrose solution from a 70% dextrose and a 20% dextrose. How much of each?
Show answers
  1. Parts of 20% = 10 - 5 = 5; parts of 5% = 20 - 10 = 10. Ratio is 5 to 10, which simplifies to 1 part of 20% to 2 parts of 5%.
  2. Parts of 5% = 3 - 1 = 2; parts of 1% = 5 - 3 = 2. Total 4 parts, so 240 ÷ 4 = 60 g per part: 120 g of the 5% and 120 g of the 1%.
  3. Parts of 70% = 40 - 20 = 20; parts of 20% = 70 - 40 = 30. Total 50 parts, so 500 ÷ 50 = 10 mL per part: 200 mL of the 70% and 300 mL of the 20%.

Frequently asked questions

What is alligation used for?

Alligation alternate finds how much of a stronger product and a weaker product to combine to reach a strength in between. It is the go-to method when the desired strength sits between two available strengths and neither one alone will do.

How do you set up the alligation grid?

Draw a tic-tac-toe grid. Put the higher strength top-left, the lower strength bottom-left, and the desired strength in the center. The parts of each product are the differences taken across the diagonals: parts of the higher strength equal desired minus lower, and parts of the lower strength equal higher minus desired.

How do you convert parts to actual quantities?

Add the two parts to get the total number of parts. Divide the final quantity you need by the total parts to find the size of one part, then multiply each product's parts by that value. The two amounts must add back up to the final quantity.

Can alligation use water as one of the strengths?

Yes. Purified water or any inactive diluent counts as 0%. Put 0 in the low position and the method works the same way, which is how you dilute a concentrated stock down to a target percentage.

Does the desired strength have to be between the two strengths?

Yes. Alligation only works when the target strength is greater than the lower strength and less than the higher strength. You cannot mix two products to reach a strength stronger than the strongest one you have.

Practise this

Alligation pairs naturally with concentrations and dilutions, since both turn on percentage strength. Work a mixed set on the pharmacy math practice test, then head back to the pharmacy math hub for the other formulas.

PTCB Quiz Prep is an independent study resource, not affiliated with the Pharmacy Technician Certification Board.