Concept of Ratio

Sometimes we have to deal with quantities that require comparison. Suppose in a class of 45 students, 15 of the students are girls, and 30 of the students are boys. We can compare the number of boys with the number of girls in two different ways.

1. There are 15 more boys than girls in the class. In this case we are comparing the number of boys and the number of girls by finding their difference.

2. The number of boys is twice the number of girls in the class. Here we are comparing the number of boys and the number of girls by finding a fraction consisting of the number of boys over the number of girls; i.e., the fraction is $\frac{{30}}{{15}}$

A ratio is a fraction of two quantities of the same kind.

In other words

A ratio is the comparison of two homogeneous quantities, or a ratio is the division of two quantities $a$ and $b$ having the same units.

It is denoted by $\frac{a}{b}$ (read as “a ratio b”) or $a \div b$

A ratio has no unit. It is a numerical quantity which indicates how many times one quantity is greater than the other.

For example: Boys : girls, i.e. 30 : 15 or 2 : 1 indicates that the number of boys is twice the number of girls.

For example: Girls : boys, i.e. 1 : 2 or $\frac{1}{2}$ indicates that there are half girls than boys.

Example:
Find the ratio of         (1) 50g to 200g                       (2) 700g to 1kg

Solution:
(1) The ratio of 50g to 200g can be found in two ways.
50 : 200                       or         50 : 200
=$\frac{{50}}{{200}}$                     or         $\frac{{50}}{{50}}:\frac{{200}}{{50}}$ (divided by 50)
=$\frac{1}{4}$                    or         1 : 4
=1 : 4

Hence, the ratio of 50g to 200g is 1 to 4.

(2) 700g and 1kg have different units. First, we express them in the same units. We know that

700g : 1kg
=700g : 1000g (since 1kg=1000g)
=700 : 1000     (divided by 100)
=7 : 10

Hence, the ratio of 700g to 1kg is 7 to 10.

Example:
Suppose three men receive their share of profit as 4000, 3000 and 1000. Find the ratio between their profits.

Solution:
The ratio of their share (profits) is
4000 : 3000 : 1000
= 4 : 3 : 1         (divided by 1000)

Example:
Simplify the ratio of 3.5 : 2.5

Solution:
We are given
3.5 : 2.5
=$\frac{{35}}{{10}}:\frac{{25}}{{10}}$
= 35 : 25          (multiply by 10)
= 7 : 5              (Dividing by 5)

Example:
Express in the form of $a & :b$ and fraction
(a) 25, 45         (b) 30min, 1hour

Solution:

1. 25, 45

$a & :b$
= 25 : 45
=5 : 9         (Divided by 5)

And in fraction form, we have $\frac{a}{b} = \frac{{25}}{{45}} = \frac{5}{9}$

1. 30min, 1hour

$\therefore$ 30min : 1hour          (since 60min=1hour)
= $\frac{1}{2}$h : 1h                      (1min=$\frac{1}{{60}}$hour)
= $\frac{1}{2}$ : 1                          (30min=$\frac{{30}}{{60}}$hour=$\frac{1}{2}$h)
= 1 : 2                          (Multiply by 2)

And in fraction form we have $\frac{a}{b} = \frac{{(\frac{1}{2}h)}}{{1h}} = \frac{1}{2}$