• The average of n terms equals $$\frac{x_1+x_2+. . .+x_n}{n}$$

• The weighted average of n terms equals $$\frac{w_1*x_1+ w_2*x_2+. . .+ w_n*x_n}{ w_1+ w_2+ . . . w_n}$$

• Percentage Change = (Final Value – Initial  Value)/Initial Value * 100
• x % of T = $$\dfrac{x}{100}\times\ T$$

• For two successive changes of of x% and y%, the total % change is (x + y + xy/100)%

• 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 $$\frac{Z-Y}{X-Z}$$

• If a, b, c, d and x are positive integers such that $$\frac{a}{b}=\frac{c}{d}$$

  1. If $$a < b$$, $$ \frac{a+x}{b+x} > \frac{a}{b}$$
  2. If $$a > b$$, $$ \frac{a+x}{b+x} < \frac{a}{b}$$
  3. If $$a > b$$, then $$\frac{a-x}{b-x} > \frac{a}{b}$$
     
  4. If $$a < b$$, then $$\frac{a-x}{b-x} < \frac{a}{b}$$
  5. $$\frac{a+c}{b+d}=\frac{a}{b}=\frac{c}{d}$$
     
  6. $$\frac{a+b}{a-b}=\frac{c+d}{c-d}$$

If a container has 'a' liters of liquid A and if 'b' liters of solution is withdrawn and is replaced with an equal volume of another liquid B and the operation is repeated for 'n' times, then after nth operation,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 = $$\left(\ \frac{\ a-b}{a}\right)^{^n}\times\ a$$

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 x is directly proportional to y => x = ky

If x is directly proportional to square of y => x = $$ky^2$$

If x is directly proportional to cube of y => x = $$ky^3$$

Here, 'k' is a proportionality constant.

When x is indirectly proportional to y => $$x=\dfrac{k}{y}$$

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

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