Equality of two ratios.

### Terminology

- A proportion written as \(\dfrac {a} {b} =\dfrac {c} {d}\), where a : b :: c : d is read as “
*a*is to*b*as*c*is to*d*” and is written :*ad*=*bc*. - The elements
*a*and*d*are called**extremes**(or*extreme terms*) and the elements*b*and*c*are called the**means**(or*mean terms*). - In a proportion with three terms \(\dfrac {a}{b} =\dfrac {b} {c}\) , the second term
*b*is called a**mean proportional**.

The following operations can be carried out based on the proportion \(\dfrac {a} {b} =\dfrac {c} {d}\) :

\(\dfrac{a + b}{b}\) = \(\dfrac{c + d}{d}\) and \(\dfrac{a\space –\space b}{b}\) = \(\dfrac{c\space –\space d}{d}\)

\(\dfrac{a}{a + b}\) = \(\dfrac{c}{c + d}\) and \(\dfrac{a}{a\space –\space b}\) = \(\dfrac{c}{c\space –\space d}\)

\(\dfrac{a}{b}\) = \(\dfrac{c}{d}\) = \(\dfrac{a + c}{b + d}\) = \(\dfrac{a\space –\space c}{b\space –\space d}\)

### Examples

To calculate 15% of 200, we start by writing the following proportion : \(\dfrac{15}{100}\) = \(\dfrac{x}{200}\).

The rules to solve a first-degree equation in one unknown are then applied, and we obtain :

*x* = 30.

### Educational note

In the case of a proportion with three terms, the mean proportional (second term) is the geometric mean of the first and third terms.