Free Alligation Calculator

Solution Mixing Through the Alligation Method

When dealing with solutions, the ability to adjust concentration precisely is essential. An alligation calculator (also referred to as a solution mixing calculator or concentration ratio calculator) provides a straightforward way to determine how two solutions of different concentrations must be combined to achieve a desired intermediate concentration. This technique, known as the alligation method, is widely used in pharmacy, chemistry, and other fields where exact mixture proportions matter.

Alligation vs. Dilution

It is important to distinguish alligation from simple dilution. Dilution involves lowering the concentration of a solution by adding a diluent (usually water). The well‑known dilution formula $M_1 V_1 = M_2 V_2$ handles such cases. For example, to obtain $75\ \text{ml}$ of a $9\ \text{M}$ solution from a $16\ \text{M}$ stock, one would need $42.19\ \text{ml}$ of the stock and $32.81\ \text{ml}$ of water.

Alligation, on the other hand, does not use a pure diluent. Instead, it mixes two solutions of known concentrations (both different from the target) to produce an intermediate strength. No solvent is added; the final concentration is obtained purely by blending the two given solutions. This is why alligation is sometimes called a “mixture calculator” or “concentration blending tool.”

The Alligation Formula

Let

• $C_h$ = concentration of the higher‑strength solution,
• $C_l$ = concentration of the lower‑strength solution,
• $C_r$ = required (target) concentration, with $C_l < C_r < C_h$.

The alligation method gives the ratio of the two solutions to be mixed as:

$$\text{Ratio (higher : lower)} = (C_r - C_l) : (C_h - C_r)$$

In other words, for every $(C_r - C_l)$ parts of the higher concentration, you need $(C_h - C_r)$ parts of the lower concentration.

Example:
To obtain a $9\ \text{M}$ solution from a $12\ \text{M}$ and a $5\ \text{M}$ solution, set $C_h = 12$, $C_l = 5$, $C_r = 9$.

$$\text{Higher parts} = 9 - 5 = 4, \quad \text{Lower parts} = 12 - 9 = 3$$

Thus the alligation ratio is $4:3$ (higher:lower). This means every 4 units of the 12 M solution must be mixed with 3 units of the 5 M solution to obtain the required 9 M mixture.

Calculating Volumes with the Alligation Ratio

Once the ratio is known, actual volumes can be computed if the total desired volume or any one component volume is specified.

If the required total volume is $V_r$, the volume of the higher‑concentration solution $V_h$ and the lower‑concentration solution $V_l$ are:

$$V_h = \dfrac{C_r - C_l}{C_h - C_l} \times V_r, \qquad V_l = \dfrac{C_h - C_r}{C_h - C_l} \times V_r$$

Using the previous example, suppose you want to make $612.5\ \text{ml}$ of $9\ \text{M}$ solution. Then

$$V_h = \frac{4}{7} \times 612.5 = 350\ \text{ml}, \quad V_l = \frac{3}{7} \times 612.5 = 262.5\ \text{ml}$$

If instead you know only one volume—say you have $350\ \text{ml}$ of the 12 M solution—the calculator can determine the needed volume of the 5 M solution and the final blend volume. This flexibility makes the alligation method particularly convenient for compounding.

Application in Pharmacy

Precision in concentration is critical in pharmaceutical preparations. A typical pharmacy alligation problem reads: “In what proportion should a pharmacist mix 20 % and 5 % zinc oxide ointments to prepare a 10 % zinc oxide ointment?”

Here $C_h = 20\%$, $C_l = 5\%$, $C_r = 10\%$.

$$\text{Higher parts} = 10 - 5 = 5, \quad \text{Lower parts} = 20 - 10 = 10$$

Ratio = $5:10 = 1:2$. So one part of the 20 % ointment must be blended with two parts of the 5 % ointment to obtain the desired 10 % product. The same logic applies regardless of whether the concentrations are expressed in molarity, percentage, or any other consistent unit.

How the Calculator Works

Using a solution mixing calculator (alligation calculator) is straightforward:

1. Enter the higher concentration.
2. Enter the lower concentration.
3. Enter the required (target) concentration.

The tool instantly displays the alligation ratio (e.g., 4:3).

If you wish to calculate the corresponding volumes, simply provide the volume of any one of the three solutions (either the higher, the lower, or the total). The calculator automatically computes the missing volumes based on the ratio.

This process eliminates manual arithmetic and reduces the chance of error, whether you are diluting a stock solution or blending two active ingredients. The alligation method remains a reliable technique for achieving accurate concentrations in fields ranging from chemistry to healthcare.