# Alligation Calculator

Created by Komal Rafay
Reviewed by Dominik Czernia, PhD candidate and Jack Bowater
Last updated: Mar 11, 2022

One of the most important concepts while studying solutions is their concentration, and that is where the alligation calculator comes into play.

You might be interested in making alligation calculations in pharmacy studies, where most of the applications for the alligation calculator lie, but we will also attempt to explain what the alligation method is and how to calculate the volume of an alligation ratio.

You will also learn about the alligation formula, along with some practical alligation calculation examples.

## What is alligation method?

The alligation definition says it's the process of calculating the proportions of two solutions to be mixed in order to produce a required solution with a specific concentration.

The number it outputs is the ratio that two solutions of the same type but of different concentrations must be mixed in to create a third solution of an intermediate concentration. That is what makes it different from dilution.

The process of dilution of a solution requires you to prepare a solution of lower concentration from a solution of higher concentration. Dilution is so simple that you can quickly obtain it by mixing a solvent into the higher concentration solution to bring it to the lower concentration.

For instance, say you have a 16 molar brine solution (the molar unit shows the concentration), and you want to obtain 75 ml of 9 molar solution. To find the result, place these values in the dilution formula:

$M_1 × V_1 = M_2 × V_2$

where:

• $M_1$ - Initial molarity;

• $V_1$ - Initial volume;

• $M_2$ - Final molarity; and

• $V_2$ - Final volume.

After rearranging the formula to calculate the required volume, it looks like this:
$V_1 = (M_2 × V_2) / M_1$

$V_1 = 9 × 75 / 16$

$V_1 = 42.19\text{ ml}$

This means that you should mix 42.19 ml of the higher concentration with 32.81 ml of water to dilute the solution enough to obtain 75 ml of the diluted solution. This is a simple method of dilution.

On the other hand, we use the alligation method formula when you have two different concentrations and are looking to obtain an intermediate concentration. The two concentrations can be of the same or different solutions, but they must be measured in the same units.

A similar "How to calculate alligation?" example might look like this:

How many parts of 12 molar and 5 molar solutions should you mix to obtain a 9 molar solution?

## Alligation ratio definition

Alligation is important when you have to change the concentration of a solution, whether it is in liquid or solid form, to make a 3rd concentration. We require the appropriate volumes of solutions with specific concentrations to make a balanced solution.

The alligation calculator estimates the alligation ratio, which tells you what ratio of the two concentrations should you use to obtain the intermediate concentration. The calculator also calculates the volumes of these different ratios.

## Alligation formula

We can represent the alligation method formula in the following form using different concentrations (conc.):

$H = \text{required conc. - lower conc.}$

$L = \text{higher conc. - required conc.}$

where:

• $H$ - Higher ratio; and

• $L$ - Lower ratio.

We usually use the alligation ratio format $H:L$ to show the final result. Isn't the alligation method formula a straightforward way to find the mixture ratio? Using our alligation calculator is even faster!

## Alligation calculation in pharmacy

Indeed, by now, you are aware of what the alligation definition is. Let's now focus on alligation and its practical calculation in pharmacy. In pharmaceuticals and medicine, the amount and concentration of solutions and their components need to be extremely precise and accurate.

An example of a common and simple question of alligation would be "In what proportion should a pharmacist mix 20% and 5% zinc oxide ointments to prepare a 10% zinc oxide ointment?" You need to use the alligation mixture formula to find the answer!

In this scenario, the proportion of ointments is a percentage, but since all the concentrations are also in percentages, it will not affect the ratio. This means that the concentrations can be in any required unit, but they all need to be the same.
So, in this example, $20%$ is the higher, $5%$ is the lower, and $10%$ is the required concentration.

• The ratio of the higher concentration is obtained by subtracting the lower from the required concentration.
• The ratio of the lower concentration is obtained by subtracting the required from the higher concentration.

So, that gives us:
$H = 10 - 5 = 5$
$L = 20 - 10 = 10$

The ratio is $5:10$, but, if you further simplify it, then the alligation ratio is $1:2$.

It means to get 10% of the ointment, mix one part of the higher concentration ointment (20%) to two parts of the lower concentration ointment (5%).

## Alligation calculator: How to calculate the volume of an alligation ratio

Using our alligation calculator is very simple. You need to know the solution concentrations you are calculating the alligation ratios for, and the calculator will do the rest for you.

To calculate the alligation ratio, you need to provide three values:

1. Higher concentration of the solution;

2. Lower concentration of the solution; and

3. Required concentration of the solution.

Once you input these three values, the calculator will use the alligation mixture formula to find the alligation ratio for you.

After that, you have the option to determine the volume of your solution if you need in order to calculate the concentrations in a per volume manner. To do that, simply input the volume of any one of the concentrations. The calculator automatically finds the other two volumes required to produce that particular amount.

Let's take an alligation calculation example to clarify things in a better way and get back to the question we left unanswered earlier.

How many parts of 12 molar and 5 molar solution should be mixed in order to obtain a 9 molar solution?

Remember, the units for all three concentrations should be the same.
Then follow the steps to obtain the ratio using the alligation calculator:

1. Input 12 as the higher concentration.
2. Input 5 as the lower concentration.
3. Input 9 as the required concentration.
4. The result is the alligation ratio, 4:3.

If you want to find the volume of these concentrations, enter the volume of your desired concentration. Suppose you want to add the volume of the higher concentration, let's say 350 ml. The calculator will tell you that you should mix 350 ml of the 12 molar solution with 262.5 ml of the 5 molar solution. This will produce 612.5 ml of 9 molar solution.

Be sure to check our other calculators, which every chemist will find very handy:

## FAQ

### How many parts of 22% and 15% solution should be mixed to produce 19% of the solution?

Using the alligation method formula you find that the alligation ratio is 4:3.

The alligation ratio suggests that to obtain a solution with concentration a of 19%, you need to mix 4 parts of the 22% solution of with 3 parts of the 15% solution.

### Is alligation the same as dilution?

No, alligation and dilution are not the same. Both methods provide two distinct solutions to the problem of obtaining specific concentrations.

Dilution is a simple method in which you simply reduce the solution's concentration by adding water or any required diluent. Alligation, however, does not use any diluent, but instead gives the ratio of two solutions to be mixed in different concentrations to produce an intermediate concentration.

### How many milliliters of 13 M and 4 M solutions do you need create 120 ml of a 8 M solution?

The alligation ratio is 4:5. It means that you should mix four parts of the 13 molar solution with five parts of the 4 molar solution to obtain the 8 molar solution.

The volume of your required concentration is 120 ml, which means you need to mix:

• Volume of higher concentration = 53.33 ml, and
• Volume of lower concentration = 66.67 ml.

### How can I calculate the alligation volumes of the alligation ratio?

Once you find the alligation ratio, the next step is to calculate the volumes of the higher and lower concentrations of the solution using volume or required concentration Vᵣ:

1. You can calculate the volume of any two solutions as long as you know at least one of the volumes and the higher H and lower L concentration alligation ratios.

2. The formula to calculate the volume of the higher concentration Vₕ is:

Vₕ = H / (H + L) × Vᵣ

3. The formula to calculate the volume of the lower concentration Vₗ is:

Vₗ = L / (H + L) × Vᵣ

Komal Rafay
The concentrations can have any unit, but they all must be same: % , w/v , mg/ml, etc.
Higher concentration
Lower concentration
Required concentration
Find the corresponding volumes
Higher concentration volume
ml
Lower concentration volume
ml
Required concentration volume
ml
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