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

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

\(\therefore\) ratio of two quantities a and b \(= \frac{a}{b}\), where \(b \neq 0\)

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.

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 \(= \frac{3}{4}\) (\(\because\) the ratio must be expressed in its lowest terms.)

So, the ratio is \(3:4\)

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 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 \(\frac{a}{b} = \frac{c}{d}\)

The symbol of proportion is \(::\)

\(\therefore\) 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:

\(Product \; of \; extremes \; = \; Product \; of \; means\)

Example

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, product of extremes is equal to product of means.

\({15} \times {120} = {45} \times {40}\)

\(\implies {1800} = {1800}\)

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.

Example

4,6 and 9 are in continued proportion.

Why?

\(\because\) \({4} : {6} = {6} : {9}\)

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.