Alligation is a mathematical technique used to determine the quantity of 2 concentrations needed to create a desired concentration. The desired concentration will have a value somewhere in the range of the 2 starting concentrations.
There are 2 ways to perform alligation calculations:
- Alligation Tables
- Intuitive Calculation
This unit focuses on the use of intuitive equations. Take a look at both strategies, decide which one works best for your style of learning.
Problem with Alligation Tables
In the unit on alligation tables there are step-by-step instructions that show how the complex mathematical concept is distilled into a standardized orientation of concentrations on a grid.
That unit also discusses the disadvantages of using alligation tables.
Most notably, the heavy reliance on memorization of where each value has to be placed on the grid.
Once you have the correct orientation you must subtract diagonally, add vertically then work horizontally.
Any misstep and your calculation will be incorrect.
This much memorization isn’t an issue if you are studying for an exam in the near future or if you use alligation tables all the time. With consistent use it becomes second nature.
If alligation is not part of your usual practice you need a more intuitive way to perform alligation calculations so that you are prepared whenever the need arises. For those purposes I recommend alligation by intuitive equations. Performing alligation a skill that is not used often but when it is needed it is of critical importance. If you work with neonatal or pediatric patients, you should have a strong understanding of alligation.
Even if you are just studying for an exam, I will always advocate for approaches that rely on deriving answers from foundational concepts rather than memorization. There is no need to memorize every new concept. That will not serve you. You will not retain the skill of alligation for much longer than the duration an exam.
Terminology
Alligation calculations can be broken down into three concentrations. Two starting concentrations and one desired concentration. There are also 3 quantities. A desired quantity of a desired concentration and the quantity needed of each of the initial products. These 6 elements of alligation are presented below along with the symbols that will used to represent them throughout this unit.
Intuitive Calculation
The desired concentration will have a value between those of the 2 starting concentrations:
C1 > C3 > C2
The desired quantity will be the sum of the quantities needed for each of the starting concentrations:
Q1 + Q2 = Q3
This above equation is based on the fundamental property of mass.
Mass is Fundamental
Consider 3 jars of beads. The first jar contains 8 red beads, the second contains 5 blue beds, the 3rd jar is empty. If we add the contents of the first 2 jars into the 3rd, we would have a jar of 13 beads.
These beads represent the fundamental property of mass.
Mass a measure of the amount of matter in an object or substance. Mass is fundamental. Is does not depend on nor is it influences by any other properties. The only way to change mass is to add or remove from it. You can change weight without any change to the structure of the substance by changing the pull of gravity, therefore weight is not fundamental. No matter where an object is, its mass is the same.
This means that when 2 masses are added together the resultant mass will always be the sum of the individual masses. How does this relate to alligation?
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Concentration|Volume|Mass
Concentration is the mass per unit of quantity of a substance. If we multiple concentration by a given quantity we are left with the fundamental property of mass. Remember mass cannot change.
If mass 1 + mass 2 = mass 3 is also means that:
This is the basis of our alligation equation: the mass of the 2 solutions combined must equal the mass of the final combined solution.
Alligation Equation
Let’s put this equation to the test with some example calculations.
Example Calculations
We will use the same examples from the unit Alligation Tables to show that we can derive the same answer without the use of alligation tables.
We can arrive at the same values by leveraging the fundamental property of mass that tells us total mass must equal the sum of the masses. Since concentration multiplied by volume will give us mass, we equate the sum of the products of our given concentrations and volumes to derive a given volume of our desired concentration.
Personally, this is my preferred method of alligation. It is more intuitive. It avoids having to memorize the various orientations required for alligation tables. We all learn different so check out the unit on Alligation Tables for an alternative method for alligation to find your preference!
GlobalRph provides an alligation calculator but this is best used as a double check for your own calculations. No calculator will relieve you of liability in medical compounding so I highly recommend understanding how to calculate versus the “plug and chug” use of calculators.
If this unit has been helpful I would love to hear from you! Leave a question or comment below.
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