Averages, Ratio and Proportion

• 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)%
• Note: If there is a decrease, then take a negative sign before the percentage.

• 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 $$\frac{a-x}{b-x} < \frac{a}{b}$$

Equivalence of ratios:  $$a : b = ak : bk$$, where $$k$$ is a constant.

$$a,b,c,d$$ four terms are in proportion.

Invertendo: $$a:b=c:d$$ then $$b:a=d:c$$

Alternendo: $$a:b=c:d$$ then $$a:c=b:d$$

Componendo: $$a:b=c:d$$ then $$\dfrac{a+b}{b}=\dfrac{c+d}{d}$$

Dividendo: $$a:b=c:d$$ then $$\frac{a-b}{b}=\frac{c-d}{d}$$

Componendo-Dividendo: $$a:b=c:d$$ then $$\dfrac{a+b}{a-b}=\dfrac{c+d}{c-d}$$

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 => $$x=\dfrac{k}{y}$$