# Proportion

## Proportion

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.