MATHS QUERY

# Ratio and Proportion

## Introduction

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.

## What is Ratio?

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 $$3:4$$

## What is Proportion?

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.

Note

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.

$${15} \times {120} = {45} \times {40}$$

$$\implies {1800} = {1800}$$

## 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.

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.

### Q) What is ratio?

When two quantities of same kind and same unit of measurement are divided then the value we get is called ratio. Ratio is represented by symbol :. Ratio of two quantities a and b is written as a:b and read as a ratio b, where a and b are called as terms.

### Q) What is antecedent in ratio?

Antecedent is the first term of a ratio. For example a is the antecedent in ratio a:b.

### Q) What is consequent in ratio?

Consequent is the second term of a ratio. For example b is the consequent in ratio a:b.

### Q) What is proportion?

Proportion means that the ratios are equal. Proportion is represented by symbol ::. If two ratio a:b and c:d are equal, i.e. a ÷ b = c ÷ d then we can say a,b,c and d are in proportion.

## Solved Examples

### 1) Find the ratio of 35 minutes to 3 hours.

1 hour = 60 min

3 hrs = 3 x 60

= 180 mins

Ratio of 35 minutes to 180 mins = $$\frac{35}{180}$$

= $$\frac{7}{36}$$

### 2) The number of girls and boys in a school are 300 and 450 respectively. Express the ratio of boys to girls.

Number of boys = 450

Number of girls = 300

Ratio of boys to girls = $$\frac{450}{300}$$

= $$\frac{3}{2}$$

### 3) There are 100 workers in a factory. Out of 100 workers, 60 are male workers. Find the ratio of male workers to female workers.

Total number of workers = 100

Number of male workers = 60

Number of female workers = 100 - 60 = 40

Ratio of male workers to female workers = $$\frac{60}{40}$$

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

= $$\frac{3}{2}$$

### 4) Does ratio 3:6 and 5:10 form a proportion?

3:6 = $$\frac{3}{6} = \frac{1}{2}$$

5:10 = $$\frac{5}{10} = \frac{1}{2}$$

$$\frac{3}{6} = \frac{5}{10}$$

Hence, 3:6 and 5:10 are in proportion.

### 5) The ratio of length and width of a sheet is 3:4. Find the width of sheet if its length is 15 cm.

Let width of sheet = x cm

$$\therefore$$ Ratio of length to width = 15:x

Also, ratio of length to width = 3:4 (given)

3:4 = 15:x

$$3 \times x$$ = 4 x 15

3x = 60

x = $$\frac{60}{3}$$

x = 20 cm

a) 15:20

b) 15:4

c) 4:3

d) 12:20

a) 24 cm

b) 32 cm

c) 4 cm

d) 3 cm

a) 8

b) 14

c) 12

d) 21

a) 16

b) 32

c) 8

d) 64