Category: Mathematics

Set Operations are performed on two or more sets to obtain the combination of elements based on the Operation. There are three major types of Operations performed on Sets like Intersection, Union, Difference in Set Theory. Check out Representation of Intersection of Sets using Venn Diagram, Properties of Intersection of Sets, Solved Examples in the later modules.

Intersection of Sets Definition

Intersection of Sets A and B is the Set that includes all the Elements that are Common to Sets A and B. Intersection is represented using the symbol ‘∩’. All the elements that belong to both A and B denote the Intersection of A and B.

A ∩ B = {x : x ∈ A and x ∈ B}

If you have n sets i.e. A₁, A₂, A₃…..A_n all are Subsets of Universal Set U the intersection is the set of elements that are in common to n sets.

Intersection of Sets Venn Diagram

Consider Two Sets A and B and their Intersection is depicted pictorially using the following Venn Diagram. A, B are subsets in the Universal Set. Intersection of Sets is all those elements that belong to both the Sets A and B. Shaded Portion denotes the Intersection of Sets A and B. Intersection of Sets A and B is represented as A ∩ B and is read as A Intersection B or Intersection of A and B.

A ∩ B = {x : x ∈ A and x ∈ B}.

Clearly, x ∈ A ∩ B

⇒ x ∈ A and x ∈ B

Thus, from the Definition of Intersection, we can conclude that A ∩ B ⊆ A, A ∩ B ⊆ B.

Properties of Intersection of Sets

(i) A ∩ A = A (Idempotent theorem)

(ii)  A ∩ B = B ∩ A (Commutative theorem)

(iii) A ∩ U = A (Theorem of Union)

(iv) A ∩ ϕ = ϕ (Theorem of ϕ)

(v) A ∩ A’ = ϕ (Theorem of ϕ)

(vi) If A ⊆ B, then A ∩ B = A.

Solved Examples on Intersection of Sets using Venn Diagram

1.  If A = {a, b, d, e, g, h} B = {b, c, e, f, h, i, j}. Find A ∩ B using the Venn Diagram?

Solution:

Given Sets are A = {a, b, d, e, g, h} B = {b, c, e, f, h, i, j}

Draw the Venn Diagram for the given Sets and then find the Intersection of Sets.

Intersection is nothing but the elements that are common in both Sets A and B.

A ∩ B = { b, e, h}

2. If C = { 3, 5, 7} D = { 7, 9, 11}. Find C ∩ D using Venn Diagram?

Solution:

Given Sets are C = { 3, 5, 7} D = { 7, 9, 11}

Let us represent the given sets in the diagrammatic representation of sets

Intersection is nothing but the elements that are common in both Sets C and D.

C ∩ D = {7}

There are certain basic operations that can be performed on Sets. Similar to Addition, Subtraction, Multiplication Operations which we come across in Maths we have Union, Intersection, Difference in Set Theory. These operations are performed on two or more sets to result in a New Set depending on the Operation Performed. In the case of Union of Sets, all the elements are included in the result whereas in Intersection only common elements between the Sets are included.

At times you might get confused between Union of Sets and Universal Set. There is a slight difference between Union and Universal Set. Union of two or more sets is an Operation performed on them and includes the elements that are in both sets. On the other hand, Universal Set itself is a Set and has all the elements of other sets along with its set.

Union of Sets Definition

Union of Two Sets A and B is the set of elements that are present in Set A, Set B, or in both Sets A and B. The Union of Sets Operation can be denoted as such

A ∪ B = {a: a ∈ A or a ∈ B}

For instance Let A = { 3, 4, 5} B = { 4, 5, 6, 7} then A U B = { 3, 4, 5, 6, 7}

Go through the further sections to know How to represent the Union of Sets using Venn Diagrams.

Union of Sets Venn Diagram

Let us dive deep into the article to know about the representation of Union of Sets using the Venn Diagram. You can visualize the Operation of Sets with the Diagrammatic Representation.

Consider a Universal Set U in which you have the Subsets A and B. Union of Two Sets A and B is nothing but the elements in Set A and B or both the elements in A and B together. The Union of Sets is denoted by the symbol U.

A U B is the Union of Sets A and B. It is read as A Union B or Union of A and B. The Notation representing the Union of Sets A and B is given as follows

A U B = {x : x ∈ A or x ∈ B}.

It is evident that, x ∈ A U B

⇒ x ∈ A or x ∈ B

In the same way, if x ∉ A U B

⇒ x ∉ A or x ∉ B

Thus, from the definition of Union of Sets, we can say that A ⊆ A U B, B ⊆ A U B.

From the above Venn diagram, we can infer certain theorems

(i) A ∪ A = A (Idempotent theorem)

(ii) If A ⊆ B, then A ⋃ B = B

(iii) A ⋃ U = U (Theorem of ⋃) U is the universal set.

(iv) A ⋃ A’ = U (Theorem of ⋃) U is the universal set.

(v) A ∪ ϕ = A (Theorem of identity element, is the identity of ∪)

(vi) A ∪ B = B ∪ A (Commutative theorem)

Union of Sets Venn Diagram Examples with Solutions

1. If A = { 2, 5, 9, 15, 19} B = { 8, 9, 10, 13, 15, 17}. Find A U B using Venn Diagram?

Solution:

Given Sets are A = { 2, 5, 9, 15, 19} B = { 8, 9, 10, 13, 15, 17}

Let us draw Venn Diagram for the given questions considering the given sets.

A U  B is clearly {2, 5, 8, 9, 10, 13, 15, 17, 19} all the elements in both the Sets.

A U B = {2, 5, 8, 9, 10, 13, 15, 17, 19}

2. If P = { 3, 6, 9, 12, 15, 18} Q = { 2, 4, 6, 8, 10, 12, 14, 16, 18}, Find P U Q and the represent the same using Venn Diagram?

Solution:

Given Sets are P = { 3, 6, 9, 12, 15, 18} Q = { 2, 4, 6, 8, 10, 12, 14, 16, 18}

On Drawing the Venn Diagram for the given sets we have the P U Q as under

It is clearly evident from the Venn Diagram that P U Q = {2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18}

Thus, P U Q = {2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18}

Set Theory is a branch of mathematics and is a collection of objects known as numbers or elements of the set. Set theory is a vital topic and lays stronger basics for the rest of the Mathematics. You can learn about the axioms that are essential for learning the concepts of mathematics that are built with it. For instance, Element a belongs to Set A can be denoted by a ∈ A and a ∉ A represents the element a doesn’t belong to Set A.

{ 3, 4, 5} is an Example of Set. In this article of Introduction to Set Theory, you will find Representation of Sets in different forms such as Statement Form, Roster Form, and Set Builder Form, Types of Sets, Cardinal Number of a Set, Subsets, Operations on Sets, etc.

• Representation of a Set
• Types of Sets
• Finite Sets and Infinite Sets
• Power Set
• Problems on Union of Sets
• Problems on Intersection of Sets
• Difference of two Sets
• Complement of a Set
• Problems on Complement of a Set
• Problems on Operation on Sets
• Word Problems on Sets
• Venn Diagrams in Different Situations
• Relationship in Sets using Venn Diagram
• Union of Sets using Venn Diagram
• Intersection of Sets using Venn Diagram
• Disjoint of Sets using Venn Diagram
• Difference of Sets using Venn Diagram
• Examples on Venn Diagram

Set Definition

Set can be defined as a collection of elements enclosed within curly brackets. In other words, we can describe the Set as a Collection of Distinct Objects or Elements. These Elements of the Set can be organized into smaller sets and they are called the Subsets. Order isn’t that important in Sets and { 1, 2, 4} is the same as { 4,2, 1}.

Examples of Sets

• Odd Numbers less than 20, i.e., 1, 3, 5, 7, 9, 11, 13, 15, 17, 19
• Prime Factors of 15 are 3, 5
• Types of Triangles depending on Sides: Equilateral, Isosceles, Scalene
• Top two surgeons in India
• 10 Famous Engineers of the Society.

Among the Examples listed the first three are well-defined collections of elements whereas the rest aren’t.

Important Sets used in Mathematics

N: Set of all natural numbers = {1, 2, 3, 4, …..}

Q: Set of all rational numbers

R: Set of all real numbers

W: Set of all whole numbers

Z: Set of all integers = {….., -3, -2, -1, 0, 1, 2, 3, …..}

Z+: Set of all positive integers

Representation of a Set

Set can be denoted using three common forms. They are given along the following lines by taking enough examples

• Statement Form
• Roaster Form or Tabular Form
• Set Builder Form

Statement Form: In this representation, elements of the set are given with a well-defined description. You can see the following examples for an idea

Example:

Consonants of the Alphabet

Set of Natural Numbers less than 20 and more than 5.

Roaster Form or Tabular Form: In Roaster Form, elements of the set are enclosed within a pair of brackets and separated by commas.

Example:

N is a set of Natural Numbers less than 7 { 1, 2, 3, 4, 5, 6}

Set of Vowels in Alphabet = { a, e, i, o, u}

Set Builder Form: In this representation, Set is given by a Property that the members need to satisfy.

{x: x is an odd number divisible by 3 and less than 10}

{x: x is a whole number less than 5}

Size of a Set

At times, we are curious to know the number of elements in the set. This is called cardinality or size of the set. In general, the Cardinality of the Set A is given by |A| and can be either finite or infinite.

Types of Sets

Sets are classified into many kinds. Some of them Finite Set, Infinite Set,  Subset, Proper Set, Universal Set, Empty Set, Singleton Set, etc.

Finite Set: A Set containing a finite number of elements is called Finite Set. Empty Sets come under the Category of Finite Sets. If at all the Finite Set is Non-Empty then they are called Non- Empty Finite Sets.

Example: A = {x: x is the first month in a year}; Set A will have 31 elements.

Infinite Set: In Contrast to the finite set if the set has infinite elements then it is called Infinite Set.

Example: A = {x : x is an integer}; There are infinite integers. Hence, A is an infinite set.

Power Set: Power Set of A is the set that contains all the subsets of Set A. It is represented as P(A).

Example:  If set A = {-5,7,6}, then power set of A will be:

P(A)={ϕ, {-5}, {7}, {6}, {-5,7}, {7,6}, {6,-5}, {-5,7,6}}

Sub Set: If Set A contains the elements that are in Set B as well then Set A is said to be the Subset of Set B.

Example:

If set A = {-5,7,6}, then Sub Set of A will be:

P(A)={ϕ, {-5}, {7}, {6}, {-5,7}, {7,6}, {6,-5}, {-5,7,6}}

Universal Set:

This is the base for all the other sets formed. Based on the Context universal set is decided and it can be either finite or infinite. All the other Sets are Subsets of Universal Set and is given by U.

Example: Set of Natural Numbers is a Universal Set of Integers, Real Numbers.

Empty Set: 

There will be no elements in the set and is represented by the symbol ϕ or {}. The other names of Empty Set are Null Set or Void Set.

Example: S = { x | x ∈ N and 9 < x < 10 } = ∅

Singleton Set:

If a Set contains only one element then it is called a Singleton Set.

Example: A = {x : x is an odd prime number}

Operations on Sets

Consider Two different sets A and B, they are several operations that are frequently used

Union: Union Operation is given by the symbol U. Set A U B denotes the union between Sets A and B. It is read as A union B or Union of A and B. It is defined as the Set that contains all the elements belonging to either of the Sets.

Intersection: Intersection Operation is represented by the symbol ∩. Set A ∩ B is read as A Intersection B or Intersection of A and B. A ∩ B is defined in general as a set that contains all the elements that belong to both A and B.

Complement: Usually, the Complement of Set A is represented as A^c or A^‘ or ~A. The Complement of Set A contains all the elements that are not in Set A.

Power Set: The power set is the set of all possible subsets of S. It is denoted by P(S). Remember that Empty Set and the Set itself also comes under the Power Set. The Cardinality of the Power Set is 2ⁿ in which n is the number of elements of the set.

Cartesian Product: Consider A and B to be Two Sets. The Cartesian Product of the two sets is given by AxB i.e. the set containing all the ordered pairs (a, b) where a belong to Set A, b belongs to Set B.

Representation of Cartesian Product A × B = {(a, b) | a ∈ A ∧ b ∈ B}.

The cardinality of AxB is N*M where N is the cardinality of A and M is the Cardinality of B. Remember that AxB is not the same as BxA.

If you want to Add or Subtract Rational Numbers in an Expression check out the further modules. We have listed the procedure to solve Rational Expressions involving Addition and Subtraction. Check out solved examples of Adding or Subtracting Rational Numbers explaining everything in detail.

How to Solve Rational Expressions Involving Addition and Subtraction?

To Add or Subtract Rational Numbers with the Same Denominator you just need to add/subtract numerators from each other. If the Denominators aren’t the same in the Expressions you need to find a Common Denominator. The Simplest way is to multiply the Denominators with each other. However, this might not have the simplest computations and needs further simplifying afterward.

One way of making computations easier is to find the LCD i.e. common multiple of two or more numbers present.

Solved Examples

1.  Simplify the Rational Expression x/(x+1)+2/(x+1)?

Solution:

Since the Rational Expression has common denominators we can simply add the numerators of each other while keeping the denominators unchanged.

= x/(x+1)+2/(x+1)

= (x+2)/(x+1)

Therefore, x/(x+1)+2/(x+1) on simplifcation will result in (x+2)/(x+1).

2. Simplify Rational Expression 3/(x+1)+2/(x-1)?

Solution:

Since both the denominators aren’t equal we need to find out the LCD. Simply multiply the denominators.

= 3/(x+1)+2/(x-1)

= (3*(x-1)+2(x+1))/(x+1)(x-1)

= ((3x-3)+(2x+2))/(x+1)(x-1)

= (3x-3+2x+2)/(x+1)(x-1)

= (5x-1)/(x+1)(x-1)

Therefore, 3/(x+1)+2/(x-1) on simplifying gives (5x-1)/(x+1)(x-1).

There are a few properties that are applicable while dealing with the Subtraction of Rational Numbers. Check out the Closure, Commutative, Distributive and Associative Properties of Rational Numbers under Subtraction Operation. To help you understand each and every property we have taken enough examples and explained all of them step by step.

Closure Property of Subtraction of Rational Numbers

The Difference between any Two Rational Numbers always results in a Rational Number. Let a/b, c/d be two Rational Numbers then (a/b -c/d) will also result in a Rational Number.

Example

Consider two rational numbers 5/9 and 3/9 then

Subtraction of 5/9-3/9

= 2/9

Therefore, 5/9-3/9 = 2/9 is also a Rational Number.

Commutative Property of Subtraction of Rational Numbers

Subtraction of Two Rational Numbers doesn’t obey Commutative Property. Let us consider a/b, c/d be two rational numbers then (a/b)-(c/d)≠(c/d)-(a/b). Have a look at the Example stated below and verify whether the commutative property is applicable or not.

Example

Consider the rational Numbers 5/8 and 2/8 then

= 5/8 – 2/8

= 3/8

2/8 – 5/8

= -3/8

5/8-2/8≠ 2/8-5/8

Therefore, Commutative Property isn’t applicable for Subtraction.

Associative Property of Subtraction of Rational Numbers

Subtraction of Rational Numbers is not Associative. Let us consider three Rational Numbers a/b, c/d, e/f then (a/b-(c/d -e/f)) ≠ (a/b – c/d) – e/f.

Example

2/8-(4/8-1/8) = 2/8-(3/8)

= -1/8

(2/8-4/8)-1/8 = (-2/8)-1/8

= -3/8

Therefore, 2/8-(4/8-1/8) ≠(2/8-4/8)-1/8.

Distributive Property of Subtraction of Rational Numbers

Multiplication of Rational Numbers is Distributive Over Subtraction. Consider three Rational Numbers then a/b*(c/d-e/f) = a/b*c/d-a/b*e/f.

Example

Consider three Rational Numbers 1/2, 2/3, 4/5 then

1/2(2/3-4/5) = (1/2*2/3 – 1/2*4/5)

= (2/6-4/10)

= (2*5/6*5 – 4*3/10*3)

= (10/30 -12/30)

= -2/30

= 1/2*2/3 – 1/2*4/5

= 2/6-4/10

= -2/30

Therefore, 1/2*(2/3-4/5) = 1/2*2/3 – 1/2*4/5

If you need assistance with the Subtraction of Rational Numbers then take the help of this article. Check out the steps you need to follow while Subtracting Rational Numbers be them with the Same or Different Denominator. Let us consider a/b, c/d to be two rational numbers then subtracting c/d from a/b is adding the additive inverse of c/d to the rational number a/b.

We can denote the Subtraction of c/d from a/b as a/b – c/d.

Thus we have, a/b – c/d = a/b+(-c/d)[since -c/d is the additive inverse of c/d]

How to Subtract Rational Numbers?

The Procedure to Subtract Rational Numbers having Same and Different Denominators is better explained in the coming modules by considering a few examples. Have a glance at them and solve the Subtraction of Rational Numbers Problems you encounter easily.

Examples

1. Subtract 2/3 from 4/5?

Solution:

Subtracting 4/5 from 2/3

= 4/5-2/3

= 4/5+(-2/3)

= 4*3/5*3+(-2*5/3*5)

= 12/15+(-10/15)

= (12-10)/15

= 2/15

Therefore, 4/5 -2/3 = 2/15.

2. Subtract 3/9 from 2/5?

Solution:

Subtracting 2/5 from 3/9 we get

= 3/9 -2/5

= 3/9+(-2/5)

= 3*5/9*5+(-2*9/5*9)

= 15/45+(-18/45)

= -3/45

= -1/15

Therefore, 2/5-3/9 = -1/15

3. Simplify -6/7-5/8?

Solution:

Given Expression is -6/7-5/8

= -6/7+(-5/8)

= -6*8/7*8+(-5*7/8*7)

= -48/56+(-35/56)

= (-48-35)/56

= -83/56

Therefore, -6/7-5/8 = -83/56

4. What should be subtracted from 4/5 to get the Rational Number 6/15?

Solution:

Let the number to be subtracted as x

From the given data we can write the equation as 4/5-x = 6/15

Rearranging the equation to get the variable x aside we have

4/5-6/15=x

4*3/5*3-6*1/15*1 = x

x= 12/15-6/15

=6/15

The Number to be Subtracted from 4/5 to obtain 6/15 is 6/15.

5. What is the Rational Number to be Added to -7/10 to get 4/7?

Solution:

Let us consider the number to be added as x which results in 4/7

-7/10+x= 4/7

-7/10-4/7 =x

x =-7/10-4/7

= -7*7/10*7-4*10/7*10

= -49/70-40/70

= (-49-40)/70

= -89/70.

Therefore -89/70 when added to -7/10 results in 4/7

6. Sum of Two Rational Numbers is 5/3 if One of the Numbers is 9/20 find the Other?

Solution:

Sum of two rational numbers = One Number + Other Number

5/3 = 9/20+Other Number

Other Number = 5/3-9/20

= 5*20/3*20 – 9*3/20*3

= 100/60-27/60

= 73/60

Therefore, the Other Number to be added to 9/20 to get the sum 5/3 is 73/60.

Learn about the Subtraction of Rational Numbers with Different Denominators from here. Get to know the detailed procedure to follow while solving Problems on Subtraction of Rational Numbers having Different Denominators. Check out a few examples explaining step by step and get a good hold of the concept and learn to solve problems on your own.

How do you Subtract Rational Numbers with Different Denominator?

Follow the guidelines listed below to Subtract Rational Numbers with Different Denominator. They are in the following fashion

Step 1: Check out whether the Denominator is Positive or not. If either of the Denominators is negative rearrange them so that denominators become positive.

Step 2: Obtain the Denominators of Rational Numbers in the earlier step.

Step 3: Determine the Least Common Multiple of the Denominators of the given Rational Numbers.

Step 4: Express the Rational Numbers in terms of a Common Denominator using the LCM Obtained.

Step 5: Write a Rational Number whose numerator is the difference of numerators of rational numbers obtained in the earlier step and denominator is the LCM obtained in the earlier steps.

Step 6: Rational Number obtained in step 5 is the required rational number. Simplify if it is required.

Solved Examples on Subtracting Rational Numbers with Different Denominator

1. Subtract 9/5 from 5?

Solution:

Given Rational Numbers are 5/1 and 9/5

Since the denominators aren’t the same find the LCM of Denominators.

LCM(1, 5) = 5

Express the Rational Numbers given using the LCM Obtained.

5/1 = 5*5/1*5 = 25/5

9/5 = 9*1/5*1 = 9/5

Subtracting we get 25/5 -9/5

= 16/5

Therefore, 5-9/5 = 16/5

2. Find the Difference of 4/3 and 6/5?

Solution:

Find the LCM of the Denominators 3, 5

LCM(3, 5) = 15

Express the Rational Numbers with Common Denominator using the LCM obtained.

4/3 = 4*5/3*5 = 20/15

6/5 = 6*3/5*3 = 18/15

difference = 20/15 – 18/15

= 2/15

Therefore, Difference of 4/3 and 6/5 is 2/15.

3. Simplify 4/12-5/-6?

Solution:

Since one of the denominators isn’t positive change it to positive by rearranging it.

5/-6 =5*(-1)/-6*(-1)

= -5/6

Find the LCM of the Denominators 12, 6

LCM(12,6) = 12

Express the Rational Numbers in terms of a Common Denominator using the LCM obtained.

4/12 = 4*1/12*1 = 4/12

-5/6 = -5*2/-6*2 = -10/12

Subtracting the Rational Numbers Obtained we get

= 4/12-(-10/12)

= 14/12

= 7/6

If you are worried about How to Subtract Rational Numbers having the Same Denominator you have come the right way. To help you out we have given the Step by Step Process to follow while solving problems on Subtraction of Rational Numbers having the Common Denominator. Have a glance at the solved examples and better understand the concept.

How do you Subtract Rational Numbers with the Same Denominator?

Go through the easy guidelines listed to Subtract Rational Numbers with the Same Denominator. Following these steps, you will arrive at the solution easily. They are along the lines

Step 1: Firstly, obtain the numerators of given rational numbers along with the common denominator.

Step 2: Subtract the First Numerator from the Second One.

Step 3: Note down the Rational Number whose numerator is the difference of two given Rational Numbers in the earlier step and retain the common Denominator unchanged. Simplify the Rational Number Obtained if Required.

From the above-provided information, we can say that for two rational a/b and c/b having the common denominator b, a/b – c/b is equal to (a-c)/b.

Solved Examples on Subtraction of Rational Numbers with Common Denominator

1. Find the Difference Between 4/7 and 12/7?

Solution:

Given Rational Numbers are 4/7 and 12/7

Difference between Rational numbers = 12/7 -4/7

= (12-4)/7

= 8/7

Therefore 12/7-4/7 = 8/7.

2. Find the Difference between 3/4 and -5/4?

Solution:

Given Rational Numbers are 3/4, 5/4

=3/4-(-5/4)

= 8/4

=2/1

3. Subtract 2/17 from -6/17?

Solution:

Rational Numbers are 2/17, -6/17

= -6/17-2/17

= -8/17

4. Subtract Rational Number 5/3 from 17/3?

Solution:

Given Rational Numbers are 5/3 and 17/3

= 17/3-5/3

= 12/2

= 4/1

5. Find the Difference between 4/-5 and 2/5?

Solution:

Given Rational Numbers are 4/-5 and 2/5

Changing the Rational Number with negative Denominator to Positive Denominator we get

4/-5 = 4*(-1)/-5*(-1) = -4/5

= -4/5-2/5

= -6/5

6. Subtract Rational Number -2/3 from -4/3?

Solution:

Given Rational Numbers are -2/3 and -4/3

= -4/3-(-2/3)

= (-4+2)/3

= -2/3

Refer to the complete article and get an idea of how to find the Equivalent Form of Rational Numbers i.e. expressing the given rational number in different forms. Furthermore, you can witness the equivalent form of rational numbers having the common denominator.

1. Express Rational Number -48/60 with the denominator 5?

Solution:  

In order to express the Rational Number -48/60 with denominator 5 firstly figure out the number which gives 5 when 60 is divided by it.

It is clearly evident that such a number = (60÷5) = 12

Dividing the numerator and denominator of -48/60 by 12 we get

-48/60 = (-48÷12)/(60÷12)=-4/5

Therefore, expressing the Rational Number -48/60 with the denominator 5 is -4/5.

2. Fill in the blanks with the suitable number in the numerator 4/-7 = ...../-35 = ...../-77

Solution:

4/-7 = 4*7/-5*7 = 28/-35

4/-7 = 4*11/-7*11 = 44/-77

3. Find an equivalent form of the rational numbers 3/9 and 4/6 having a common denominator?

Solution:

It is evident that the denominator is the LCM of 9, 6

LCM of 9, 6 is 18

Now, 18 ÷ 9 = 2, 18 ÷ 6 = 3

Therefore 3/9 = 3*2/9*2 = 6/18

4/6 = 4*3/6*3 = 12/18

Given rational numbers 3/9, 4/6 with common denominators are 6/18, 12/18.

4. Find an equivalent form of rational numbers 4/3, 7/5, 10/12 having a common denominator?

Solution:

We need to convert given rational numbers into equivalent rational numbers having the common denominator.

LCM of 3, 5, 12 is 60

60 ÷ 3 = 20, 60 ÷ 5 = 12, 60 ÷ 12 = 5

Therefore 4/3 = 4*20/3*20 = 80/60

7/5 = 7*12/5*12 = 84/60

10/12 = 10*5/12*5 = 50/60

Hence the given rational numbers with common denominator are 80/60, 84/60, 50/60.

In this article of ours, you will learn how to find Equivalent Rational Numbers by multiplication and division. Get to see solved examples in the coming modules.

Equivalent Rational Numbers by Multiplication

Let suppose a/b is a rational number and m is a non-zero integer then (a*m)/(b*m) is a rational number equivalent to a/b.

For instance, 16/20, 40/50, -56/-70, -96/-120 are equivalent fractions and are equal to the rational number 4/5.

On multiplying the numerator and denominator of a fraction with the same integer the fraction value doesn’t change.

Example: Fractions 4/8 and 16/32 are equivalent because the numerator and denominator can be obtained by multiplying with each of them with 4.

Also, -4/5 = -4*(-1)/5*(-1) = -4*(-2)/5*(-2) = -4*(-3)/5*(-3) and so on ……

If the denominator of a rational number is a negative integer then by using the above-mentioned property we can convert it to positive by multiplying the numerator and denominator by -1.

Example: 7/-5 = 7*(-1)/-5*(-1) = -7/5

Equivalent Rational Numbers by Division

If a/b is a rational number and m is the common divisor of a, b then (a÷m)/ (b÷m) is a rational number equivalent to a/b.

Rational Numbers -24/-30, -28/-35, 40/50, 60/75 are equivalent to the rational numbers 4/5.

24/32 = (24÷8)/(32÷8) = 3/4

Solved Examples

1. Find the Two Rational Numbers Equivalent to 4/7?

Solution:

4/7 = (4*4)/(7*4) = 16/28

4/7 = (4*7)/(7*7) = 28/49

Thus, the two rational numbers equivalent to 4/7 are 16/28 and 28/49.

2. Determine the smallest equivalent rational number of 100/125?

Solution:

100/125 = (100÷5)/(125÷5) = 20/25 = (20÷5)/(25÷5) = 4/5

Thus, the Equivalent Rational Number of 100/125 is 4/5.

3. Write down the following rational numbers with a positive denominator 4/-9, 11/-22, -17/-3?

Solution:

4/-9 = 4*(-1)/-9*(-1) = -4/9

11/-22 = 11*(-1)/-22*(-1) = -11/22

-17/-3 = -17*(-1)/-3*(-1) = 17/3

Therefore Rational Numbers 4/-9, 11/-22, -17/-3 changed with a positive denominator are -4/9, -11/22, 17/3.

4. Express -4/7 as a Rational Number with the numerator

(i) -16 (ii) 24

Solution:

(i) In order to make -4 as a rational number having the numerator -16 we first need to find a number when multiplied by results in -16.

Clearly, such number is (-16 )÷ (-4) = 4

Multiplying both the numerator and denominator with 4 we get

-4/7 = (-4*4)/(7*4) = -16/28

(ii) In order to make -4 as a rational number having the numerator 24 we first need to find a number when multiplied by results in 24.

Clearly, such number is (24 )÷ (-4) = -6

Multiplying both the numerator and denominator with -6 we get

-4/7 = (-4*-6)/(7*-6) = 24/-42

All the examples listed above are for Equivalent Rational Numbers.

Get a Complete Idea of Negative Rational Numbers from this article. You can see the conditions for Negative Rational Numbers along with a few examples.

A Rational Number is said to be negative if the numerator and denominator are of opposite sign i.e. any one of them is a positive integer and the other is a negative integer. You can also say that a Rational Number is Negative if the numerator and denominator are of opposite signs.

All the Rational Numbers -1/7, 4/-5, -25/11, 10/-19, -13/23 are negative. Rational Numbers -11/-14, 2/3, -3/-4, 1/2 are not negative.

Is every negative integer a negative rational number?

We know -1 = -1/1, -2 = -2/1, -3 = -3/1, -4 = -4/1 ……

We can express negative integer n in the form of n/1 where n is a negative integer and 1 is a positive integer.

Thus, every negative integer is a negative rational number. On the other hand, Rational Number 0 is neither positive nor negative.

Determine whether the following rational numbers are negative or not?

(i) 3/(-6)

3/(-6) is a negative rational since the denominator and numerator are having opposite signs.

(ii) (-1)/(-4)

(-1)/(-4) is not a negative rational as both the numerator and denominator are having the same sign.

(iii) 11/23

11/23 is not a negative rational since both the numerator and denominator are of the same sign.

(iv) 9/-14

9/-14 is a negative rational since both the numerator and denominator are of opposite signs.

(v) (-64)/(-8)

(-64)/(-8) is not a negative rational as both the numerator and denominator are of the same sign.

(vi) 20/24

20/24 is not a negative rational as you have both the numerator and denominator of the same sign.

(vii) (-13)/39

(-13)/39 is a negative rational since we have both the numerator and denominator of opposite signs.

(viii) (-31)/7

(-31)/7 is a negative rational since we have both the numerator and denominator of opposite signs.

Thus, from the above examples, we can say that a negative rational number is the one that has both the numerator and the denominator of the opposite sign.