Alligation Method for Questions With a Diluent or a Pure Compound

In this tutorial, I'm going to show you how to solve Alligation Calculation Questions when one component is either a diluent or a pure compound, and we are starting right now.

Related link: The Alligation Method Made Easy

Related link: Alligation Pharmacy Calculations for 3 Components

Alligation Method for Questions With a Diluent or a Pure Compound

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So let's get right to it by reviewing an example question which illustrates the form in which you normally will see questions where you want to use the Alligation method.

Alligation Calculation Example 1 

So in this example, the question says how many milliliters of a 23.4% sodium chloride solution will be required to prepare one liter of a 10% sodium chloride solution for irrigation using a 0.9% sodium chloride solution and a 23.4% sodium chloride solution.

So for a typical scenario, you will normally have three different concentrations. In this question, we have the 23.4%, we have the 10% and we have the 0.9%. Now, all of these percentages are referring to the amount of sodium chloride in the solution.

And so the way you handle this is you typically set up the Alligation grid where you have your two vertical lines and the two horizontal lines.

And so here the higher concentration will be the 23.4 %. And then the lower concentration goes to the bottom left. The lower concentration is the 0.9. And then your desired is 10, which goes in the middle.

So as a reminder, the desired concentration, which in this example is 10, must always be between the higher and the lower concentration.

So now what you do is you take your desired, you subtract the lower concentration, so that would be 10 minus 0.9.

So that gives us 9.1 and then 9.1 goes in the top right. So you have 9.1 which signifies the number of parts of the 23.4%.

Then we do a similar thing by taking the desire, which is 10, subtracting that from the higher concentration, which is 23.4, that gives us 13.4.

So now, because we are required to prepare a 1 L solution, that is the total volume we are making. And so we also need the total parts.

And so the total parts, you obtain that by adding the parts of the 23.4, which is 9.1 to the 13.4, which is the parts of the 0.9 % sodium chloride solution. That gives us 22.5 parts, and that represents the total quantity in your preparation.

So the question is asking us how many milliliters of the 23.4% sodium chloride solution do we need? But we are making one liter, so we want to make sure that the liters is in milliliters.

And so we take one liter, multiply that by the conversion factor, one liter is 1,000 milliliters. So the litters cancel out and that should be equal to 1,000 milliliters.

And so to find the actual quantity in milliliters of the 23.4 % sodium chloride solution, we will take the parts of the 23.4% sodium chloride solution, which is 9.1, divide that by the total part, which is 22.5,

and set up a proportion saying that that should be equal to some quantity in milliliters, divide it by the total volume we are making, and we are making 1,000 mL.

So now we can go ahead and solve for X, which is our unknown. So X is going to be equal to 9.1 divided by 22.5, times 1,000. And that is going to be equal to 404.4 millileters.

Now, the thing about this type of question is you have three different concentrations all giving you. The numbers are given to you explicitly.

Alligation Calculation Example 2

But now let's look at an example where you are not given all the three numbers, but rather you're given some additional information which you're supposed to use to fill in the third concentration which you do not know. So let's go through this question.

So here the question says, how many milligrams of diluent should be combined with 10 grams of a 15% boric acid ointment to prepare a 12% boric acid ointment.

So here we have only two concentrations given us directly. We have the 15% and the 12%. So what happened to the third concentration? Now in the question, you have enough information to determine what the third concentration is.

So let's first start off by setting up our grid. We have two vertical lines and two horizontal lines.

So now let's analyze the question and determine whether we can deduce what the third concentration actually is.

So this question is saying how many milligrams of diluent? The key word here is “diluent.” Now, what it tells us is we are going to take the 15 % boric acid ointment and we are going to lower the concentration, which will mean that the concentration of the additional components should be lower than 12% for sure.

Now, because it says “diluent,” it means your component has 0% of the boric acid in there, 0%. Now, all these concentrations, the 12% and the 15%, refer to the amount of boric acid in a given amount of ointment.

So when it says diluent here, it implies that you have 0% of the boric acid in that component. And so your third concentration here is 0.

And so what that means is your highest concentration will be 15, and that goes to the top left, your lowest concentration would be 0. And that comes from the keyword diluent. And then your desired would be the 12. So that goes in the middle.

And so you take the 12, you subtract the lower concentration from that, and that gives you 12. That goes to the top right.

And this means you have 12 parts of the 15%. 

Now you also take the 12, which is the desired concentration, subtract that from the 15 and that gives you 3. That goes to the bottom right. And so you have three parts of the diluent. 

And so we take a look at the question again and we see that we have 10 grams of the 15% boric acid ointment.

So in this question, we were not given the total quantity we are making and so we do not need to calculate the total parts here.

Rather, what we are going to do is we are going to take the quantity of the 15% boric acid ointment, so you have 10 grams, and divide that by the parts of the 15 % boric acid ointment, so that is 12.

And we'll set up a proportion saying that that should be equal to some quantity in grams divided by the parts of the diluent, because that's what we are looking for now.

So we go ahead and solve for X. X equals 10 grams times 3, divided by 12. And that gives us 2.5 grams.

Now, we don't stop here because the question is asking about milligrams. And so we take the 2.5 grams and we convert that to milligrams. Using the conversion factor, 1 gram is 1,000 milligrams.

So grams cancel out and you end up with 2,500 milligrams.

So the take-home message from this particular question is, even though we're given just the two concentration numbers, especially in the question, the third concentration is implied from the word “diluent.” 

And so when you see that word diluent, it implies 0 % concentration.

Alligation Calculation Example 3

Now, let's take a look at another example. This question says, how many grams of Petrolatum should be added to 200 g of a 20% ichthammol ointment to make an 8 % ichthammol ointment?

Now, once again, in this question, we only have two numbers which represent concentration of the ointment which is given in the question explicitly. But to use the allegation method, we need three concentrations.

So let's first set up the grid. So we have our two vertical lines and then the two horizontal lines. So now let's analyze the question more closely to find out exactly how we should proceed.

Now in the question, we have 200 grams of a 20 % HTML ointment, and we want to make 80 % HTML ointment. So what it means is that the third component should have a concentration that is lower than the 80% HTML ointment for this to work.

And although the question doesn't say diluent, it tells us that we are mixing the 200 grams or the 20% HTML ointment with petrolatum. The implication here is that the petrolatum is the diluent and it has 0 % HTML in it. So that's your clue right there. The word petrolatum implies a diluent which has 0% ichthammol in it.

And so we proceed by putting the 20% concentration in the top left because that's the higher concentration. And then the lower concentration will be the 0% because there's 0% ichthammol in the Petrolatum, and our desired will be 8, and that goes in the middle.

And so next we proceed by taking the 8, which is the desired concentration, subtracting the zero from it. So 8 minus 0, that gives us 8.

And that goes to the top right. So 8 goes here, and that is the part of the 20% ichthammol ointment. 

We also take the eight, which is the desired concentration subtracted from the 20. So 20 minus eight gives us 12, that goes to the bottom right.

And the 12 represents the parts of the petrol atom. 

Now, because we've been given the quantity of the 20% ichthammol ointment, and we do not know the total quantity, there's really no need to find the total parts here.

And so what we will do here is we'll take the 200 grams of the 20% ichthammol ointment, so 200 g divided by the parts of the 20% ichthammol ointment, so that's 8 parts,

and set that equal to some quantity in grams divided by the parts of the petrolatum, which would be 12.

So we go ahead and solve for X, which is our unknown. So X equals 200 g, times 12, divided by eight, and that is equal to 300 g. So here we had two concentrations which was given to us in numbers directly in the question.

The third concentration we did use from the term petrolatum, and the petrolatum here is an ointment base. There is 0 % ichthammol in it. And so that's why the third concentration is 0 %.

So that is how you solve alligation calculation questions where you have a diluent and two other concentrations given you in the question.

Alligation Calculation Example 4

So now let's take a look at the scenario where you have a pure compound and two concentrations explicitly given to you in the question.

So let's take a look at this question which says, How many grams of pure zinc oxide should be added to 4,000 grams of 10 % zinc oxide ointment to prepare a 30 % zinc oxide ointment?

In this question, we've been given two concentrations in numbers directly, and we need a third concentration for the alligation method to work.

But first, let's proceed by setting up the alligation grid. So we have our two vertical lines and two horizontal lines. And so let's go ahead and do a deeper analysis of the question and see what is going on.

So we have a 10 % zinc oxide ointment, and we want to increase the concentration to a 30 % zinc oxide ointment. And so to do that, it means our third component, which is whatever we are adding to the 10 % zinc oxide ointment, should have a concentration which is much, much more than the 30 % which we desire. Otherwise, there's no way you can actually end up with a 30 % zinc oxide ointment.

And so the clue in the question is the word “pure.” And so because it's pure, it means it's 100% zinc oxide. 

And so we can proceed by putting the 100% in the top left corner of the grid. Because the 100 is going to be your higher concentration, and it represents the pure zinc oxide that you're going to mix with the 10%.

Now the lower concentration is going to be 10, and our desired is going to be 30, and that goes in the middle.

And so the next thing we'll do is we'll take the 30, subtract the lower concentration from it, which is 10. 

So 30 minus 10 is 20, and that goes to the top right. So 20 represents the parts of the pure zinc oxide.

We also take the desired concentration, which is 30, subtract that from the pure concentration, which is 100. 

So 100 minus 30 is 70. That goes to the bottom right. And so you have 70 parts of the 10%.

Now in the question, we do not have a total quantity. So there is no need here to find the total parts. 

Rather, we've been given the amount of the 10% zinc oxide ointment. So we'll take that amount, which is 4,000, so 4,000 g, and we will divide that by the parts of the 10%, which is 70. 

And we will set that equal to some quantity in grams divided by the parts of the pure zinc oxide, which is 20.

So we go ahead and solve for X. So X equals 4,000 g times the 20, divided by 70, and that should be equal to 1,142.9 g.

So the key word in the question is the word “pure,” and that indicates that you have 100% zinc oxide. Everything in the zinc oxide is zinc oxide, so that's why it's 100%.

Alligation Calculation Example 5

Let's take a look at another example. This question says, how many grams of pure sulfur should be added to 150 g of 1.5% sulfur ointment in order to prepare a 2% sulfur ointment?

So in this question, we have two concentrations given to us explicitly in the question in numbers, but for the Alligation method to work, we need three concentrations.

And so let's proceed by first putting down our Alligation grid. So you have the two vertical lines, two horizontal lines, and then we can now do an analysis of the question to determine what that third number could be.

So we have a 1.5% sulfur ointment, and we want to increase the concentration to 2%. What that implies is we need to add a third component, and that component is what you're going to mix with the 1.5% sulfur ointment to get the 2%. The concentration of that ointment should be greater than the 2% ointment.

Now, the keyword in this question that points you to the concentration of that component is the word “pure.” So whenever we see pure, it implies you have 100%.

So here we have 100% sulfur. And so to proceed, we'll put 100% in the top left, which would be the higher concentration. Now in the bottom left, we'll put the 1.5, and in the middle, we'll put our desired concentration, which would be 2.

And so the next thing we'll do is we'll take the 2% and we'll subtract from it the lower concentration. So 2 minus 1.5 should give us 0.5.

And so 0.5 goes in the top right, and that represents the parts of the 100%.

And so next we will take the 2 and subtract that from the 100. 

So 100 minus 2, equals 98, and that goes in the bottom right. So 98 parts of the 1.5%.

Now in this question, we have not been given the total quantity, so there is no need to calculate the total parts. You only calculate the total parts when you have the total quantity. 

But rather what we'll do is we'll take the quantity of the 1.5 % sulfur ointment, which is 150 grams, and we'll divide that by the parts of the 1.5 % ointment, which is 98.

And we'll set that equal to some quantity in grams divided by the parts of the 100%, which is the pure sulfur. And so that will be divided by 0.5.

And so we can go ahead and solve for X. X is going to be equal to 150 g times 0.5, divided by 98.

And that gives us 0.77 g.

So that is how you solve Alligation questions where you have two concentrations given to you in terms of numbers. And then the third concentration is implied using the word "pure".

So whenever you see the word pure, it indicates you have 100%. And so with that information, you can go ahead and proceed to solve the question using the Alligation method as you normally would. 

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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