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Ratio, Proportion and Continued Proportion

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Introduction

Ratio is a term used to compare two quantities. A Ratio of two numbers tells 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 b can be written as ab , where b ≠ 0
It is denoted by : symbol
Ratio of and b is written as a : b and read as a ratio b
a and b are called terms of ratio.

In ratio a : b, the first term a is called antecedent.
The later term b is called consequent.

Example of ratio

What is the ratio of pears in two bags with weights 15 kg and 20 kg?
The ratio of their weight = 1520
which is further = 34     (∵ 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 also.
It means the equality of two ratios is called proportion.
The symbol of proportion is ::


To understand it, 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 ab = cd
∴ the proportion of above ratios can be written as a : b :: c : d

In a : b :: c : d, the first term a and fourth term d are called extremes.
The second term b and third term c are called means.

Note

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

Example of proportion

Is this a correct 15 : 45 :: 40 : 120 proportion?
Here, 15 and 120 are called extremes and 45 and 40 are called means.
The product of extremes = 15 × 120 = 1800
The product of means = 45 × 40 = 1800
i.e. The product of extremes and means both are equal to 1800
the numbers 15, 45, 40 and 120 in proportion.

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 quantity a and third quantity c.
Third quantity, here c, is called third proportional to first quantity a and second quantity b.

Example of continued proportion

Are the numbers 4,6 and 9 are in continued proportion?
Reduce 46, which can be written as 23
Reduce 69, which can be written as 23
∴ both ratios 46 and 69 are equal to 23
∴ 46 = 69
or 4 : 6 = 6 : 9
Hence, 4, 6 and 9 are in continued proportion
Also, 6 is mean proportional between 4 and 9.
9 is third proportional to 4 and 6.

Frequently Asked Questions

1) 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.

2) What is antecedent in ratio?

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

3) What is consequent in ratio?

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

4) 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 × 60
= 180 mins
Ratio of 35 minutes to 180 mins = 35180
= 736

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 = 450300
= 32

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 = 6040
= 64
= 32

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

3:6 = 36 = 12
5:10 = 510 = 12
∴ 36 = 510
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
∴ Ratio of length to width = 15:x
Also, ratio of length to width = 3:4 (given)
3:4 = 15:x
34 = 15x
3 × x = 4 × 15
3x = 60
603
x = 20 cm

6) Express 412 : 313 as the simplest ratio.

92 : 103
Ist method:
Cross multiply 9 by 3 and 2 by 10
9 × 32 × 10
= 2720
⇒ 27:20
IInd method:
Take the LCM of denominators.
i.e. 2 and 3
∴ LCM = 6
92 × 6 = 27
103 × 6 = 20
⇒ 27:20

7) Express 23 : 32 : 15 as the simplest ratio.

Take the LCM of 3, 2 and 5.
LCM = 30
23 × 30 = 10
32 × 30 = 45
15 × 30 = 6
⇒ 10 : 45 : 6

8) Divide 100 into two parts in the ratio 2:3.

Since, 2 + 3 = 5
Ist part = 25 × 100 = 40
IInd part = 35 × 100 = 60

9) Divide $1200 among three persons in the ratio of 2:3:5. Find the share of each person.

Since, 2 + 3 + 5 = 10
Ist person's share = 210 × 1200 = 240
IInd person's share = 310 × 1200 = 360
IIIrd person's share = 510 × 1200 = 600

10) Divide $2900 among two persons in the ratio of 212 : 213.

212 : 213
= 52 : 73
Take the LCM of 2 and 3
LCM = 6
52 × 6 = 15
73 × 6 = 14
⇒ 15:14
Since, 15 + 14 = 29
Ist person's share = 1529 × 2900 = $1500
IInd person's share = 1429 × 2900 = $1400

11) If the sides of a triangle are in the ratio 1:2:3 and its perimeter is 120cm. Find the length of each side.

Let x, 2x and 3x be sides of the triangle.
∴ perimeter = x + 2x + 3x = 120
6x = 120
x = 1206
x = 20
First side = x = 20cm
Second side = 2x = 2 × 20 = 40cm
Third side = 3x = 3 × 20 = 60cm

Worksheet 1

Download PDF 1

Fill in the blanks

  1. A ratio is a comparison of one quantity to another quantity of the same kind and same ___________.
  2. The first term of the ratio is called ___________.
  3. If A's age is 15 years and the ratio of ages of A:B = 5:4. Then the age of B is ___________ years.
  4. In a leap year, there are 140 days of holidays. Then, the ratio of days of holidays and total number of days in the year is 70 : ___________.
  5. The third proportional to 5 and 10 is ___________.
  6. The ratio of boys to girls in a class is 2:3. If the total number of girls are 300, then the number of boys in the class are ___________.
  7. The angles of a triangle are in the ratio of 1:2:3. The measure of the smallest angle is ___________.
  8. 4kg : 500g = 8 : x, then the value of x is ___________.
  9. 4 : 5 :: x : 20, then the value of x is ___________.
  10. The weight of 5 boxes is 150kg. Then the weight of 2 boxes will be ___________kg.
Help iconHelp box
12
20
60
30°
unit
1
200
183
antecedent
16

Worksheet 2

Download PDF 2

Multiple choice questions

1) A ratio equivalent to 3:4 is
  1. 15 : 20
  2. 15 : 4
  3. 4 : 3
  4. 12 : 20
2) Length and width of a playground are in ratio 4 : 3. If width is 24 cm, its length will be
  1. 24 cm
  2. 32 cm
  3. 4 cm
  4. 3 cm
3) If the first, third and fourth terms of a proportion are 8, 14 and 21 respectively. Then the second term will be
  1. 8
  2. 14
  3. 12
  4. 21
4) Mean proportional between 4 and 16 is
  1. 16
  2. 32
  3. 8
  4. 64
5) 200 g yogurt and half kg of mango pulp are used to make a smoothie. What is the ratio of yogurt and mango pulp in a smoothie?
  1. 5 : 2
  2. 1 : 2
  3. 5 : 1
  4. 2 : 5
6) In a class test, 40 students have passed out of 100 students. What is the ratio of passed students to failed students?
  1. 2 : 3
  2. 3 : 2
  3. 2 : 5
  4. 3 : 5
7) If a, b, c, d are in proportion, then
  1. a2 = bc
  2. b2 = ac
  3. ad = bc
  4. ac = bd
8) If a, b, c are in continued proportion, then
  1. b2 = ac
  2. c2 = ab
  3. a2 = bc
  4. a2 = b2
9) A line segment of 21 is divided into two parts in the ratio of 4:3. The length of each part is
  1. 40 cm and 3 cm
  2. 3 cm and 12 cm
  3. 12 cm and 9 cm
  4. 9 cm and 4 cm
10) Two parts of hydrogen and one part of oxygen form water. The ratio of hydrogen to water is
  1. 1 : 2
  2. 2 : 1
  3. 1 : 1
  4. 2 : 2
MCQs Answer Key chevron-right icon
Last updated on: 31-03-2025

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