# Alligation Pharmacy Calculations for 3 Components

How do you use the alligation method when you have three components? In this tutorial, I'm going to show you exactly how to do that.

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## Video Transcription

The alligation method is a really powerful method that allows you to calculate the number of parts of two or more components when they are to be mixed to prepare a mixture of a desired strength.

Now, once you've identified what the final proportions are, then you can actually translate those proportions into the actual quantities of the components that you need in any specific denomination.

When it comes to the alligation method, you commonly see scenarios where you apply the alligation method to two components.

But in this tutorial, what we are going to do is I'll start out by reviewing a scenario where you use the alligation method for two components, and then we'll go ahead and look at three strategically selected examples on how you can use the alligation method for three components.

## Example 1

Let's get right to it. This question says, In what proportion should the pharmacist mix 20% and 5% zinc oxide ointments to prepare a 10% zinc oxide ointment?

Let's start off by setting the grid. We have the alligation grid, and we can begin to put in the concentration on the grid.

Now, the way it works is that the higher concentration goes into the top left corner.

The lower concentration goes in the bottom left

And then your desired concentration goes into the middle.

In the scenario presented, the higher concentration is the 20%, so that will go to the top left. So you have 20 in the top left.

And then the lower concentration is 5%, so that goes into the bottom left.

And then your desired concentration is 10%, and that goes into the middle.

And that will give you five. So 10 minus 5 is 5. That goes into the top right corner. And the five there represent the parts of the 20% zinc oxide ointment.

You also take the 10, which is the concentration you desire. And subtract that from the 20, which is the higher concentration.

And then you have 10, so 20 minus 10 is 10 and that goes into the bottom right corner and that lets us know the number of parts of the 5% zinc oxide ointment.

And so if we were to determine the proportion, we have 5 parts of the 20% zinc oxide ointment to 10 parts of the 5% zinc oxide ointment.

Now we can break it down some more. So if you divide both the five and the 10 by 5, you end up with a 1 is to 2 ratio.

So one part of the 20% zinc oxide ointment. You mix that with two parts of the 5% zinc oxide ointment, and that should give you your 10% zinc oxide ointment.

### Example 2

So now that we reviewed how you could use the alligation method for two components, let's go ahead and take a look at the examples where you apply the alligation method for three components.

This question says, In what proportion should a pharmacist mix 70%, 50% and 20% drug Z solution to prepare a 30% drug Z solution?

Here we have three components and we can go ahead and set up the alligation grid as follows.

Now the alligation grid will be similar to how it looks like for two components, except there will be a slight difference. You have your two vertical lines. And then you have one horizontal across, a second one, and normally if you have two components, you stop right here.

But because we have three components, we will go ahead and draw another horizontal line. And so our grid looks like this.

So let's go ahead and put the concentrations on the grid. One thing we want to recall is that when you're using the alligation method, your desired concentration must be between the strengths of the component.

And so what that will mean is we have 70%, 50%, and 20%. Now the 30% will be between the 50% and the 20%. And so why that is significant is because it lets us know exactly where all the numbers should go on the grid.

Now in this example, your highest strength is 70% and that will go to the top left of the grid. So your 70 will go right here.

Now the next highest concentration is the 50% and that will go right below the 70 on the left hand side. Your 50 goes here.

Now the lowest concentration is at 20% and that goes to the bottom left. Your 20 goes here.

Now the desired strength which is 30, goes in the middle.

So one of the ways you can think about using the alligation method for three components is as though you were doing it for two components, but you are doing it twice.

What that means is that in this example, we could go ahead and do the alligation method for the 50 and 20 and the 30.

In which case you could take the 30, subtract the 20, which is the lowest strength from it, so 30 minus 20 and that gives you 10 and that will go right here.

So that will be 10 parts of the 50 % drag Z solution.

Then you can go ahead and take the 30, which is the desired Subtract that from the 50, which is the higher concentration.

So 50 minus 30, that gives you 20 and that will be 20 parts of the 20% solution.

But it don't stop here. So you need to repeat the process for the 70, the 20 and the 30.

So once again, you take the 30 which you are desired. Subtract the 20% which is your lower concentration.

So 30 minus 20, that gives you 10. And this time the 10 goes to the top right. So that means you have 10 parts of the 70% drug Z solution.

You also go ahead and take the 30 which your desired concentration, subtract that from the 70.

And that means you have 70 minus 30, and that gives you 40. And that also means that you have 40 parts, so plus 40 parts of the 20% drug Z solution.

So now what that means is you have 10 parts of the 70% drug Z solution.

10 parts of the 50% drug Z solution

And 20 plus 40, which would give us 60. So 60 parts of the 20% solution.

So one of the things you would observe is when you are using the alligation method for three components, on whichever side of the grid that you have two components. So in this example, you have the 70 and the 50 before your desired 30 and then the 20, wherever those two components are, you typically have the same number of parts of each of those components.

So here you have 10% plus 70% drug solution and 10 % plus 50 % drug solution. Just keep an eye on that. When we look at the additional examples, you'll see exactly this pattern repeats itself.

But having said that, what is our proportion? You have 10 parts of the 70% drug Z solution, which is to 10 parts of the 50% drug Z solution, which is to 60 parts of the 20% drug Z solution.

So you can break this proportion down some more by dividing every number there by 10.

And so 10 divided by 10 is 1, is to 10 divided by 10, which is 1, is to 60 divided by 10, which is 6. So the proportion will be 1:1:6.

So one part of 70% drug Z solution, one part of 50% drug Z solution, and six parts of the 20% drug Z solution.

## Example 3

Let's take a look at another example. Here, the question says a compounding pharmacist wants to mix 20%, 15% and 5% zinc oxide ointments to produce 100 grams of a 10% zinc oxide ointment. How many grams of the 15% ointment will the pharmacist use?

So here, the strategy will be to use the alligation method to first determine the number of parts of each of the components that is being used, and then go ahead and use the final proportions to determine the amount of the 15% ointment that is actually required.

So to do that, we'll go ahead and start out by setting up our alligation grid. It will look as follows. So you draw your two straight vertical lines, and then you have your first horizontal line, the second one, and then you need a third one.

So we take a look at the question. We have 20, 15, and 5, and our desired is 10.

The highest concentration will be the 20% to go to the top most left. So you have your 20 right here.

15%, which is the next highest concentration, will go to the row below it on the left hand side. So the 15 goes here.

Your lowest concentration is 5%, so that goes to the bottom left.

And your desired is 10%, so it will go on the second row from the bottom in the middle column.

So here we'll proceed with the method using the 15% and the 5%.

And the way that will work is you take the 10% which is your desired concentration, subtract the 5 from it.

So 10 minus five, that gives you five, and that will go right here. So that means you have five parts of the 15% zinc oxide ointment.

And then you go ahead and take the 10% which is your desired concentration, subtract that from the 15% so 15 minus 10 is going to be five. So you have five parts of the 5% zinc oxide ointment.

We go ahead and more or less repeat the process for the 20 and the 5.

And so we will have 10 minus 5. So 10 is your desired concentration, 5 is your lower concentration.

So 10 minus 5, that gives you 5, and the 5 goes to the top right. So that means you have five parts of the 20% zinc oxide ointment.

So you go ahead and take the 10%, which is your desired concentration. Subtract that from the 20%, which is your highest concentration.

So 20 minus 10 is equal to 10. So we have plus 10 parts of the 5% zinc oxide ointment.

So we can go ahead and review our grid. We have five parts of the 20 % zinc oxide ointment. We have five parts of the 15% zinc oxide ointment, and we have 15 (5 plus 10), so we have 15 parts of the 5% zinc oxide ointment.

So in terms of a final proportion, we have 5 parts of the 20% is to five parts of the 15% zinc oxide ointment, is to 15 parts of the 5% zinc oxide ointment.

So we can reduce this proportion into its lowest form by dividing every number there by five.

So we end up with 1 is to 1, is to 3.

So we have one part of the 20% zinc oxide ointment, one part of the 15% zinc oxide ointment, and three parts of the 5% zinc oxide ointment.

So now we’ve determined the final proportions of how you need to mix the various components. But the question says, how many grams of the 15% ointment do we need?

So first we need to find the total parts. So total parts is going to be equal to 1 plus 1, plus 3, which gives us 5 parts total.

So we can go ahead and find the amount in grams or the 15% ointment by taking the parts of the 15% ointment, dividing that by the total parts and multiplying that by the quantity desired, which would be 100 grams.

That would imply that we have one part of the 15% ointment, divided by total parts, which is 5, and we can multiply that by the desired quantity, which is 100 grams.

100 divided by five is 20.

So we would need 20 grams of the 15% ointment for this preparation.

## Example 4

Let's take a look at another example. This question says the solvent for the extraction of a vegetable drug is 70% alcohol. In what proportion may 95%, 60% and 50% alcohol be mixed to prepare a solvent of the desired concentration?

The first thing that we can do is go ahead and set up the alligation grid. We have two vertical lines and then three horizontal lines across those two vertical lines. We have the first horizontal line, we have the second one, and we need the third one because we have three components.

We start by filling the grid with the highest concentration, which will be the 95%. Because it's a component, it will go to the topmost left-hand side of the alligation grid. Your 95 % will be right here.

Now, the next thing that will go on the grid will be the 70% alcohol. That's your next highest concentration. Because that is your desired concentration, it will go on the second row in the second column. Your 70% will be right here.

The next highest concentration is 60%. That's also a component. That will go on the left hand side on the third row, so your 60 goes here.

Then the lowest concentration is 50%, and that will go below the 60 in the bottom most left corner. So 50 will be right here.

So now we proceed by working with the 95, the 60, and the 70.

We will take the 70, which is your desired centration, subtract the 60 from that.

So 70 minus 60 is 10. So you have 10 parts of the 95% alcohol solution.

Then we'll take the 70, subtract that from 95 so it will be 95 minus 70

And that will give us 25. So we have 25 parts of the 60% alcohol solution.

And then we go ahead and work with the 95, the 50 and the 70.

And so what that would look like is you have the 70 minus 50, which would be your desired concentration minus the lowest concentration.

So 70 minus 50 is 20. So you have plus 20 parts of the 95% alcohol.

And then you go ahead and take the 70, which is your desired strength, subtract that from the 95%, which is the highest strength.

So 95 minus 70, that gives you 25. So you have 25 parts of the 50% alcohol.

And so what that would mean is you have 10 plus 20, which is 30 parts of the 95% alcohol. You have 25 parts of the 60% alcohol, and you have 25 parts of the 50% alcohol.

And so your final proportion will be 30 parts of the 95% alcohol, is to 25 parts of the 60% alcohol, is to 25 parts of the 50% alcohol.

Now you can go ahead and reduce this to the lowest possible form. So if we divided all the numbers by 5

You end up with 30 divided by 5, which is 6, is to 25 divided by 5, which is 5, is to 25 divided by 5, which is once again, 5.

So your final proportion will be six parts of the 95% alcohol. Five parts of the 60% alcohol. And five parts of the 50% alcohol.

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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