Properties of Proportions

An equality of two ratios is called a proportion.

Four quantities a, b, c, d are said to be in proportion if a : b = c : d (also written as a : b :: c : d) That is, if a/b = c/d.

Addendo :

If a : b = c : d = e : f =.........., then each of these ratios is equal (a + c + e + ........) : (b + d + f + ........)

Subtrahendo :

If a : b = c : d = e : f =.........., then each of these ratios is equal (a - c - e - ........) : (b - d - f - ........)

Componendo :

If a : b = c : d, then (a+b) : b = (c+d) : d

Dividendo :

If a : b = c : d, then (a-b) : b = (c-d) : d

Alternendo

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

Invertendo

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

Mean Proportional - Middle Term

If a, b, c are in continuous proportion, then the middle term b is called the mean proportional between a and c, a is the first proportional and c is the third proportional. Thus, if b is mean proportional between a and c, then

Direct and Indirect Proprotions

In a proportion a : b = c : d, all the four quantities need not be of the same type. The first two quantities should be of the same kind and last two quantities should be of the same kind.

Cross Products

In a proportion, product of extremes = product of means. If a : b = c : d are in proportion, ad = bc. This is called cross product rule.

Means-Extremes Property

The quantities a, b, c, d are called terms of the proportion; a, b, c and d are called its first, second, third and fourth terms respectively. First and fourth terms are called extremes (or extreme terms). Second and third terms are called means (or middle terms).

Continuous Proportion

Three quantities a, b, c of the same kind (in same units) are said to be in continuous proportion. Now, we can write a, b, c in proportion as given below. a : b = b : c Using cross product rule, we have b2 = ac