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.