Solve Alligation Calculations

Alligation is solving calculations involving mixtures of the same product with different strengths. Alligation works for liquids, creams, gels, solutions, etc. This is useful for a compounding recipe that calls for an ingredient with a certain strength, but you have other strengths in stock, one of which is less than the desired strength and the other is stronger.

Let’s say a prescription asks for 300mL of a 40% solution. You have a 60% and a 30% in stock. Set up a tic-tac-toe-style grid with your “want” in the middle and your “haves” on the left side with the higher percentage written in the upper-left corner and the lower percentage in the lower-left corner. Notice that you won’t use all the boxes. Our example has the unused boxes shaded out.

Next, find the difference between top “have” and “want” as well as bottom “have” and “want”. You will work this diagonally.

After finding the differences, sum the two answers. These two differences and their sum give us the fraction we need to determine how many milliliters of each strength of the solution should be combined to make the desired strength.

Let’s start with the top row. Our graph shows ten parts of the 60% strength will be needed. The difference over the sum is ¹⁰⁄₃₀, which can be simplified to ⅓. Now do the same with the bottom row. The 30% strength requires twenty parts. Twenty parts placed over the sum is ²⁰⁄₃₀, which simplified is ⅔.

If you are unsure how to simplify (reduce) fractions, visit Pass the Math Review’s Reduce Lesson.

Alligation Conclusion

Last step! We are ready to calculate how many milliliters of each solution we need. Originally, the prescription called for 300 mL so we will use this amount and multiply it against the two fractions, ⅓ and ⅔.

60% solution –> ⅓ * 300mL = 100mL

30% solution –> ⅔ * 300mL = 200mL

Conclusion: Mix 100mL of the 60% solution with 200mL of the 30% solution. This mixture will give you 300mL of a 40% solution.