# Percentages Formulas

Percent means for every hundred.  $x\% = \displaystyle\frac{x}{{100}}$
Percentage conversions: *
$\displaystyle\frac{1}{2}$ $=50\%$;
$\displaystyle\frac{1}{3}$ $=33\frac{1}{3}\%$ or $33.33\%$;
$\displaystyle\frac{1}{4}$ $=25\%$; $\displaystyle\frac{3}{4} = 75\%$ ;
$\displaystyle\frac{1}{5}$ $=20\%$; $\dfrac{2}{5}$ $= 40%$; $\dfrac{3}{5}$ $= 60\%$; $\dfrac{4}{5}$ $= 80\%$;
$\displaystyle\frac{1}{6}$ $=16\dfrac{2}{3}\%$ (or) $16.66\%$;
$\displaystyle\frac{1}{7}$ $=14\dfrac{2}{7}\%$ (or) $14.28\%$;
$\displaystyle\frac{1}{8}$ $=12\dfrac{1}{2}\%$ (or) $12.5\%$;
$\displaystyle\frac{1}{9}$ $= 11\dfrac{1}{9}\%$ (or) $11.11\%$ ;
$\displaystyle\frac{1}{{10}}$ $=10\%$;
$\displaystyle\frac{1}{{11}}$ $=9\dfrac{1}{{11}}\%$  or  $9.09\%$;
$\displaystyle\frac{1}{{12}}$ $=8\dfrac{1}{3}\%$

Formula 1:
A is what percentage of B?
$\Rightarrow \displaystyle\frac{{\rm{A}}}{{\rm{B}}}{\rm{ \times 100}}$

Formula 2:
A is how much percent greater than B?
$\Rightarrow \displaystyle\frac{{{\rm{A - B}}}}{{\rm{B}}}{\rm{ \times 100}}$

Formula 3:
A is howmuch percent less than B?
$\Rightarrow \displaystyle\frac{{{\rm{B - A}}}}{{\rm{B}}}{\rm{ \times 100}}$

Formula 4:
If A is increased by K% then the new number is
$\begin{array}{cl} \text{A is increased by K%} &\Rightarrow A\left( {1 + \dfrac{K}{{100}}} \right) \\[4px] &\Rightarrow A\left( {\dfrac{{100 + K}}{{100}}} \right) \\[4px] & \Rightarrow A + K\% (A) \end{array}$

Example: 600 is increased by 20% then the new number is
$\begin{array}{cl} Method\;1: &\Rightarrow 600\left( {1 + \dfrac{20}{{100}}} \right) \\&= 600\left( {1 + \color{red}{\dfrac{1}{5}}} \right) \\&= 600\left( {\dfrac{{6}}{{5}}} \right) \\& = 720 \\\\[4px] Method\;2:&\Rightarrow 600\left( {\dfrac{{100 + 20}}{{100}}} \right) \\& = 600\left( {\dfrac{{120}}{{100}}} \right) \\&= 720 \\\\[4px] Method\;3:& \Rightarrow 600 + 20\% (600) \\&= 600+120 \\&= 720 \end{array}$

Formula 5:
If A is decreased by K% then the new number is:
$\begin{array}{cl} \text{A is decreased by K%} &\Rightarrow A\left( {1 - \dfrac{K}{{100}}} \right) \\[4px] &\Rightarrow A\left( {\dfrac{{100 - K}}{{100}}} \right) \\[4px] & \Rightarrow A - K\% (A) \end{array}$

Example: 500 is increased by 25% then the new number is
$\begin{array}{cl} Method\;1: &\Rightarrow 500\left( {1 - \dfrac{25}{{100}}} \right) \\&= 500\left( {1 - \color{red}{\dfrac{1}{4}}} \right) \\&= 500\left( {\dfrac{{3}}{{4}}} \right) \\&= 375 \\\\[4px] Method\;2:&\Rightarrow 500\left( {\dfrac{{100 - 25}}{{100}}} \right) \\&= 500\left( {\dfrac{{75}}{{100}}} \right) \\&= 375 \\\\[4px] Method\;3:& \Rightarrow 500 - 25\% (500) \\&= 500-125 \\&= 375 \end{array}$

Formula 6:
$A\% (B) = B\% (A)$

Formula 7:
If several percentages are acting on the same number then we can add all the percentages.
${x_1}\% (K) + {x_2}\% (K) + {x_3}\% (K)...$$= ({x_1} + {x_2} + {x_{3...}})\% (K)$

Formula 8:
If a number K got increased by A% and B% successively then the final percentage is given by $\left( {A + B + \displaystyle\frac{{AB}}{{100}}} \right)\%$
Note1: If decreased then substitute $+A%$ with $-A%$
Note2: Any two dimensional diagram like square, rectangle, rhombus, triangle, circle, parrellogram, sides got increased or decreased by certain percentages, then the percentage change in the area can be calculated by the above formula