Why Does the Alligation Alternate Method Work?

Are you a pharmacy student seeking to master the concept of the alligation alternate method? Have you ever wondered why this method works and how it can be used in order to accurately solve dilution and concentration calculations? Well, look no further! In this post, we'll explore why the alligation alternate method works by taking a comprehensive look at its underlying principles and uses. With easy-to-follow explanations, guided examples, and helpful illustrations, you will leave with an understanding of how to use this valuable tool in your pharmaceutical calculations.

Related link: The Alligation Method Made Easy

Related link: 1 Super Tip on When to Use the Alligation Method to Solve Concentration Calculations Questions

Why Does the Alligation Alternate Method Work?

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So one of our subscribers asked me this question, "why does the Alligation Alternate Method work?"

I'm going to answer this question. Now, before I tell you why the Alligation alternate method works, I'm going to give a brief overview of three important things that you need to know about the alligation alternate method.

But when you talk about the Alligation Alternate method, it actually is a method by which you may calculate the number of parts of two or more components when they are to be mixed to prepare a mixture of a desired strength. So strength here actually means concentration.

So the number of parts of the components that you calculated gives you a final proportion, and that proportion permits or allows you to translate those relative parts into any specific denomination (e.g., ml, gram, etc.).

So what that means is once you know the parts of each component and you know the total quantity of preparation that you need to make, you can actually calculate the exact amount that you need to combine.

So the next thing that we want to be aware of is that the strength of the mixture must be between the strength of its components.

So what that means is that the concentration or strength of your desired mixture must be stronger than the component with the lowest concentration and should be weaker than the component with the highest concentration. Otherwise, this method doesn't work.

So now we are going to look at one example in terms of how the Alligation Alternate method works, and then I'll tell you why it actually works the way it does.

Example Calculation Solved by the Alligation Alternate Method

I'm going to start off by solving an example using the alligation method on the left-hand side of the screen. And when that's completed on the right-hand side, I'm going to demonstrate the mathematical validity of the Alligation Alternate method. And that will tell us why this method actually works the way it does.

So in this example, it says in what proportion should a 70% sorbitol solution and 40% sorbitol solution be mixed to prepare a 50% sorbitol solution. 

We can start out by setting up a grid, and in the grid we need to put in the concentrations that we have.

We have a 70% sorbitol solution, which is the higher concentration. So that goes to the top left.

And then we have a 40% sorbitol solution which is the lower concentration and that goes to the bottom left.

In the middle, you put your desired concentration, which would be the 50%.

And then we do a quick check. It's 50 between 70 and 40? Yes, it is. So we can use this method.

Now, the next thing that we do is we take the 40, which is the lower concentration and subtract that from the 50, which is your desired concentration, and we get 10. So 50 minus 40 is 10, and we put that on the top right hand side.

So this 10 actually refers to the number of parts of the 70% sorbitol solution.

The next thing that we do is we take the 50, which is the concentration of the desired mixture, we subtract that from the 70, which is the higher concentration of sorbitol solution that we are using.

So 70 minus 50 gives us 20 that goes into the bottom right and the 20 is the number of parts of the 40% solution.

So using the number of parts of the 70% solution, which is 10 and the number of parts of the 40% solution, which is 20, we can determine the proportion in which the 70% sorbitol solution and the 40 % sorbitol solution should be mixed.

And so what we will say is we will have parts of 70% sorbital solution, divided by the parts of the 40% sorbitol solution. That's going to be equal to 10 divided by 20. So the zeros can cancel out and you actually end up with 1 over 2. The proportion of 70 % to 40 % is going to be 1 is to 2.

And so the proportion of 70 % to 40 % is going to be 1 is to 2.

So now let's find out why the allegation alternate method works. And we can do that by using this general example. So here the question is saying, in what proportion should A % solution and B % solution be mixed to prepare C % solution?

So let's start off by setting up a grid. So we have three columns, essentially, to the very left would be the percentage given. In the middle column actually will be the desired concentration or the desired strength, and on the right-hand side will be the parts of each solution.

So in the top left we'll have A for A%. And then in the bottom left, we'll have B for B% which is the lowest strength solution, and in the middle column right here in the middle you have C.

But let's say we wanted to know the parts of A to take which we don't know yet so we call that X.

And then we want to find out the parts of B which we need and we'll call that Y so we have two variables.

Then what it means is we could actually say algebraically that A times X, plus B times Y, should be equal to the desired, which is C times X, plus Y.

So you can go ahead and say that your AX, plus BY, you distribute the C over the X plus Y. So that would give you CX plus CY.

You could rearrange the equation and have AX minus CX equals CY minus BY.

And then you could factor out the X on the left-hand side. So X would be A minus C. And then you can factor out the Y on the right-hand side, which will be Y equals C, minus B. And then you can divide both sides by Y. And then you can divide both sides by A minus C. And so this will cancel out and that will cancel out.

So you end up with your proportion of ratio of X over Y, being equal to C minus B, divided by A minus C.

And so there you have it. Any time you want to find the proportion in which two solutions with different concentrations should be mixed to give you the concentration of your desired mixture, you simply subtract the concentration with the lowest strength from the desired, and then you divide the value by the difference between the higher concentration and that of your desired. It always works.

So I hope you found this tutorial useful. Thank you so much, and I will see you in the next blog.

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