Mixtures and Solutions

• Alligation Rule: This is used to find the ratio of individual components in a mixture. If two components A and B, costing Rs. X and Rs. Y individually, are mixed and the resultant mixture has an average price of Rs. Z, then the ratio of A and B in the mixture is $$\frac{Z-Y}{X-Z}$$

Just like averages, ratios, and proportions, we can use the concept of weighted averages, which is nothing but the formula version of alligation, to solve many questions.

If $$x_1$$, $$x_2$$, $$x_3$$, $$...$$, $$x_n$$ are the percentage or fractional values of the concentrations of a solute in $$n$$ solutions, which are respectively mixed in the ratio $$y_1:y_2:y_3:...:y_n$$, we get the concentration of that solute in the resultant mixture as:

$$\frac{x_1y_1 + x_2y_2 + x_3y_3 + ... +x_ny_n}{y_1+y_2+y_3+...+y_n}$$

The result will be a percentage or fractional value of concentration of the solute, depending on what we start with.

In a mixture of mixtures, two quantities of some mixtures are mixed to get a mixture of mixtures.

Let Mixture 1 have ingredients A and B in ratio a: b, and Mixture 2 have ingredients A and B in ratio x : y.

Now, the M unit of mixture 1 and N unit of mixture 2 are mixed to form a resultant mixture. Then, in the resultant mixture, the ratio of A and B is

$$\dfrac{Q_a}{Q_b}=\ \dfrac{M\left(\frac{a}{a+b}\right)+N\left(\frac{x}{x+y}\right)\ }{M\left(\frac{b}{a+b}\right)+N\left(\frac{y}{x+y}\right)}$$

In N liters mixture of a solution C, if the amount of liquid A is 'a' liters then the concentration of A in the solution is a*100/N %

If a container has 'a' liters of liquid A and if 'b' liters of solution is withdrawn and is replaced with an equal volume of another liquid B and the operation is repeated for 'n' times, then after nth operation,

The final quantity of Liquid A in the container = $$\left(\ \frac{\ a-b}{a}\right)^{^n}\times\ a$$