Definition of Ratio
Ratio is the simplest form of comparison between two or More quantities.
If $A = 5000$ and $B = 8000$ then $A:B = 5 : 8$
Definition of Proportion
If two ratios are equal then we say both are in 'Proportion'.
If two ratios are in proportion then "product of extremes is equal to product of means".
$a:b \equiv c:d$ $ \Rightarrow ad = bc$
Proportion in Fractional format
$a:b = c:d \Rightarrow \dfrac{a}{b}=\dfrac{c}{d}$
Fourth Proportional
$a:b \equiv c:\boxed{d}$ then $d$ is fourth proportional.
Third Proportion 1
$a:b \equiv \boxed{c}:{d}$ then $c$ is third proportional.
Third Proportion 2
Third proportional to $a:b$ is $a:b \equiv b:\boxed{c}$
Mean Proportion
Mean proportional to $a:b$ = $\sqrt {ab} $
Continued Proportion
If $a,\;b,\;c,\;d,\;...$ are in continued proportion if they are in $G.P$
$\dfrac{a}{b} = \dfrac{b}{c} = \dfrac{c}{d}$ etc., then a, b, c, d are in Geometric Progression.
1 Duplicate ratio of $a : b$ $= a^2 : b^2$
2 Sub Duplicate ratio of $a : b$ $ = \sqrt a :\sqrt b $
3 Triplicate ratio of $a : b$ $= a^3 : b^3$
4 Sub Tripliate ratio of $a : b$ $ = {a^{1/3}}:{b^{1/3}}$
5 If $\dfrac{a}{b} = \dfrac{c}{d} = \dfrac{e}{f}$ $= \dfrac{{a + c + e}}{{b + d + f}}$
6 If $\dfrac{a}{b} = \dfrac{c}{d}$, then $\dfrac{b}{a} = \dfrac{d}{c}$ (Invertendo)
7 If $\dfrac{a}{b} = \dfrac{c}{d}$, then $\dfrac{a}{c} = \dfrac{b}{d}$ (Alternendo)
8 If $\dfrac{a}{b} = \dfrac{c}{d}$, then $\dfrac{{a + b}}{b} = \dfrac{{c + d}}{d}$ (Componendo)
9 If $\dfrac{a}{b} = \dfrac{c}{d}$, then $\dfrac{{a - b}}{b} = \dfrac{{c - d}}{d}$ (Dividendo)
10 If $\dfrac{a}{b} = \dfrac{c}{d}$, then $\dfrac{{a + b}}{{a - b}} = \dfrac{{c + d}}{{c - d}}$ (Componendo & dividendo)
Definition of Proportion
Proportion in Fractional format
Fourth Proportional
Third Proportion 1
Third Proportion 2
Mean Proportion
Continued Proportion
1 Duplicate ratio of $a : b$ $= a^2 : b^2$
2 Sub Duplicate ratio of $a : b$ $ = \sqrt a :\sqrt b $
3 Triplicate ratio of $a : b$ $= a^3 : b^3$
4 Sub Tripliate ratio of $a : b$ $ = {a^{1/3}}:{b^{1/3}}$
5 If $\dfrac{a}{b} = \dfrac{c}{d} = \dfrac{e}{f}$ $= \dfrac{{a + c + e}}{{b + d + f}}$
6 If $\dfrac{a}{b} = \dfrac{c}{d}$, then $\dfrac{b}{a} = \dfrac{d}{c}$ (Invertendo)
7 If $\dfrac{a}{b} = \dfrac{c}{d}$, then $\dfrac{a}{c} = \dfrac{b}{d}$ (Alternendo)
8 If $\dfrac{a}{b} = \dfrac{c}{d}$, then $\dfrac{{a + b}}{b} = \dfrac{{c + d}}{d}$ (Componendo)
9 If $\dfrac{a}{b} = \dfrac{c}{d}$, then $\dfrac{{a - b}}{b} = \dfrac{{c - d}}{d}$ (Dividendo)
10 If $\dfrac{a}{b} = \dfrac{c}{d}$, then $\dfrac{{a + b}}{{a - b}} = \dfrac{{c + d}}{{c - d}}$ (Componendo & dividendo)
