Chapter Contents

Ratio, Proportion and Continued Proportion

Chapter Contents

Introduction
Ratio and its example
Proportion and its example
What is Continued Proportion?
Questions
Examples
Worksheet 1
Worksheet 2

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 $b = \frac{a}{b}$ , 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.

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 $\frac{a}{b} = \frac{c}{d}$

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.

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, 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.

Example

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.

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 $= \frac{35}{180}$

$= \frac{7}{36}$

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

∴ Ratio of length to width = 15:x

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

3:4 = 15:x

$3 \times x = 4 \times 15$

3x = 60

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

x = 20 cm

6) Express $4 \frac{1}{2} : 3 \frac{1}{3}$ as the simplest ratio.

$\frac{9}{2} : \frac{10}{3}$
Ist method:
$\frac{9}{2} \times \frac{3}{10}$
$\frac{27}{20}$
⇒ 27:20
IInd method:
Take the LCM of denominators.
i.e. 2 and 3
∴ LCM = 6
$\frac{9}{2} \times 6 = 27$
$\frac{10}{3} \times 6 = 20$
⇒ 27:20

7) Express $\frac{2}{3} : \frac{3}{2} : \frac{1}{5}$ as the simplest ratio.

Take the LCM of 3, 2 and 5.
LCM = 30
$\frac{2}{3} \times 30 = 10$
$\frac{3}{2} \times 30 = 45$
$\frac{1}{5} \times 30 = 6$
⇒ 10:45:6

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

Since, 2 + 3 = 5
Ist part = $\frac{2}{5} \times 100 = 40$
IInd part = $\frac{3}{5} \times 100 = 60$

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 = $\frac{2}{10} \times 1200 = 240$
IInd person's share = $\frac{3}{10} \times 1200 = 360$
IIIrd person's share = $\frac{5}{10} \times 1200 = 600$

10) Divide $2900 among two persons in the ratio of $2 \frac{1}{2} : 2 \frac{1}{3}$ .

$= \frac{5}{2} : \frac{7}{3}$
Take the LCM of 2 and 3
LCM = 6
$\frac{5}{2} \times 6 = 15$
$\frac{7}{3} \times 6 = 14$
⇒ 15:14
Since, 15 + 14 = 29
Ist person's share = $\frac{15}{29} \times 2900 = $1500$
IInd person's share = $\frac{14}{29} \times 2900 = $1400$

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

Let x, 2x and 3x be sides of the tringle.
∴ perimeter = x + 2x + 3x = 120
6x = 120 $x = \frac{120}{6}$
x = 20
First side = x = 20cm
Second side = 2x = 2 × 20 = 40cm
Third side = 3x = 3 × 20 = 60cm

Worksheet 1

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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 25 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.

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unit

200

183

antecedent

Worksheet 2

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Multiple choice questions

1) A ratio equivalent to 3:4 is

a) 15 : 20

b) 15 : 4

c) 4 : 3

d) 12 : 20

2) Length and width of a playground are in ratio 4 : 3. If width is 24 cm, its length will be

a) 24 cm

b) 32 cm

c) 4 cm

d) 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

a) 8

b) 14

c) 12

d) 21

4) Mean proportional between 4 and 16 is

a) 16

b) 32

c) 8

d) 64

5) 200 mg 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?

a) 5 : 2

b) 1 : 2

c) 5 : 1

d) 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?

a) 2 : 3

b) 3 : 2

c) 2 : 5

d) 3 : 5

7) If a, b, c, d are in proportion, then

a) a² = bc

b) b² = ac

c) ad = bc

d) ac = bd

8) If a, b, c are in continued proportion, then

a) b² = ac

b) c² = ab

c) a² = bc

d) 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

a) 40 cm and 3 cm

b) 3 cm and 12 cm

c) 12 cm and 9 cm

d) 9 cm and 4 cm

10) Two parts of hydrogen and one part of oxygen form water. The ratio of hydrogen to water is

a) 1 : 2

b) 2 : 1

c) 1 : 1

d) 2 : 2

MCQ Answer Key Hide Show

Last updated on: 13-08-2024

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Maths units list

Ratio, Proportion and Continued Proportion

Introduction

Ratio and its example

Proportion and its example

What is Continued Proportion?

Frequently Asked Questions

1) What is ratio?

2) What is antecedent in ratio?

3) What is consequent in ratio?

4) What is proportion?

Solved Examples

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

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

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.

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

6) Express 4 1 2 : 3 1 3 as the simplest ratio.

7) Express 2 3 : 3 2 : 1 5 as the simplest ratio.

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

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

10) Divide $2900 among two persons in the ratio of 2 1 2 : 2 1 3 .

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

Worksheet 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 25 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.

Worksheet 2

Multiple choice questions

1) A ratio equivalent to 3:4 is

2) Length and width of a playground are in ratio 4 : 3. If width is 24 cm, its length will be

3) If the first, third and fourth terms of a proportion are 8, 14 and 21 respectively. Then the second term will be

4) Mean proportional between 4 and 16 is

5) 200 mg 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?

6) In a class test, 40 students have passed out of 100 students. What is the ratio of passed students to failed students?

7) If a, b, c, d are in proportion, then

8) If a, b, c are in continued proportion, then

9) A line segment of 21 is divided into two parts in the ratio of 4:3. The length of each part is

10) Two parts of hydrogen and one part of oxygen form water. The ratio of hydrogen to water is

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6) Express $4 \frac{1}{2} : 3 \frac{1}{3}$ as the simplest ratio.

7) Express $\frac{2}{3} : \frac{3}{2} : \frac{1}{5}$ as the simplest ratio.

10) Divide $2900 among two persons in the ratio of $2 \frac{1}{2} : 2 \frac{1}{3}$ .