# Properties of Ratio and Proportion

## Invertendo Property

For four quantities a, b, c, d, if a : b = c : d, then b : a = d : c, which says, if two ratios are equal, then their inverse ratios are also equal.

If a : b = c : d, then b : a = d : c.

The property can also be expressed in fractions.

If \dfrac{a}{b}=\dfrac{c}{d}, then \dfrac{b}{a}=\dfrac{d}{c}

## Alternendo Property

For four quantities a, b, c, d, if a : b = c : d, then a : c = b : d, which says, if two ratios are equal, if the second and third term interchange their places, then also the four terms are in proportion.

If a : b = c : d, then a : c = b : d.

In fraction form, the property is written as

If \dfrac{a}{b}=\dfrac{c}{d}, then \dfrac{a}{c}=\dfrac{b}{d}

## Componendo Property

For four quantities a, b, c, d, if a : b = c : d then (a + b) : b = (c + d) : d, which says, if two ratios are equal, the ratios of whole to the respective part are equal.

## Dividendo Property

If a : b = c : d, then (a - b) : b = (c - d) : d, which says, if two ratios are equal, the ratios of difference of parts to the respective single part are equal.

In fraction form,

If \dfrac{a}{b}=\dfrac{c}{d}, then \dfrac{a-b}{b}=\dfrac{c-d}{d}

## Convertendo Property

For four quantities a, b, c, d, if a : b = c : d then a : (a - b) = c : (c - d), which says, if two ratios are equal, the ratios of the respective single part to the difference of parts are equal.

## Componendo Dividendo Property

For four quantities a, b, c, d, if a : b = c : d then (a + b) : (a - b) = (c + d) : (c - d), which says, if two ratios are equal, the ratios of the sum to the difference of parts are equal.

## Addendo Property

For six quantities a, b, c, d,e,f, if a : b = c : d = e : f, then a : b = c : d = e : f=(a + c + e) : (b + d + f) which says, if multiple ratios are equal, the value of each ratio is equal to the ratio of the sum of numerators to the sum of denominators of each ratios.

## Equivalent ratio property

If a : b = c : d then (a ± c) : (b ± d) = a: b = c : d.