## Introduction

Ratio is a term used to compare two quantities. In that comparison, we are able to see how many times one quantity is to another quantity.

## Ratio and its example

When two quantities, which are of the same kind and have the same units of measurement, are compared by dividing one quantity to another quantity, it is called ratio.

∴ ratio of two quantities a and
$\text{b}=\frac{\text{a}}{\text{b}}$,
where b ≠ 0

It is denoted by :

Ratio of and b is written as a:b and read as **a ratio b**

a and b are called **terms** of ratio.

The first term **a** is called **antecedent** and later term b is called **consequent**.

Let’s learn by an example.

There are two bags of pears with weights 15 kg and 20 kg.

The ratio of their weight
$=\frac{15}{20}$

which is further
$=\frac{3}{4}$
(∵ the ratio must be expressed in its lowest terms.)

So, the ratio is 3:4

## Proportion and its example

When two ratios are equal, that implies the two ratios are in proportion.

It means the equality of two ratios is called proportion.

To understand it better, consider four quantities a, b, c and d.

If the ratio of first two quantities a and b is equal to ratio of last two quantities c and d , then four
quantities a, b, c and d are said to be in proportion.

It is written as
$\frac{\text{a}}{\text{b}}=\frac{\text{c}}{\text{d}}$

The symbol of proportion is ::

∴ the above proportion can be written as a : b :: c : d

The **first term** and **fourth term** in proportion are called **extremes**.

The **second term** and **third term** in proportion are called **means**.

If four terms are in proportion then:
**Product of extremes = Product of means**

Let’s take an example of four numbers **15, 45, 40, 120** which are in proportion.
i.e.
$\frac{15}{45}=\frac{40}{120}$

How?

When we reduce
$\frac{15}{45}$
to its lowest term, it is equal to
$\frac{1}{3}$

and
$\frac{40}{120}$
to its lowest term, it is equal to
$\frac{1}{3}$.

Here, 15 and 120 are called **extremes** and 45 and 40 are called **means**.

The interesting fact is, the product of extremes is equal to the product of means.

15 × 120 = 45 × 40

i.e. 1800 = 1800

## What is Continued Proportion?

Let there are three quantities a, b and c. If the ratio between first and second quantity is equal to ratio between second and third quantity. It implies that the three quantities are in continued proportion.

It is written as following:

a : b = b : c
**Second quantity**, here b, is called **mean proportional** between first and third quantities.

**Third quantity**, here c, is called **third proportional** to first and second quantities.

4,6 and 9 are in continued proportion.

Why?

∵ 4 : 6 = 6 : 9

How?

$\frac{4}{6}=\frac{6}{9}$

i.e.
$\frac{2}{3}=\frac{2}{3}$

So, here, 6 is **mean proportional** between 4 and 9.

Further, 9 is **third proportional** to 4 and 6.