Category: Mathematics

The frequency distribution is a topic in statistics related to data handling. The term frequency is how often something occurs and distribution means dividing and sharing something. Get the examples, the definition of frequency distribution, and learn about the frequency distribution table. Find the solved examples that help you to understand the concept clearly.

Frequency Distribution Definition

The frequency distribution describes how frequencies are distributed over the data. It is a representation either in tabular or graphical format, that displays the number of observations within the given interval.

The terms of the frequency distribution are listed here:

• Presentation of Data: After collecting the data, shortly in the tabular form in order to study its features, such arrangement is known as the data presentation.
• Observation: Each entry collected as a numerical number in the given data is called the observation.
• Frequency: The number of times a particular observation repeat is known as the frequency.
• Frequency Distribution Table: It is one form of representation of the data. It contains the observations and how many times those observations occur.

Steps to Draw a Frequency Distribution Table

• Arrange the given data in ascending order.
• And find the frequency of each data value.
• Take the data value and frequency as the two columns in the table.
• Write each data value frequency in the table.

Get more useful information regarding the data handling such as the types of data, definition, and terms used in the data handling.

Example of Frequency Distribution

Example 1.

Suppose the runs scored by the 11 players of the Indian cricket team in a match are given as follows:

25, 65, 03,12, 35, 46, 67, 56, 00, 31, 17

State the frequency of each player?

Solution:

Given that,

Runs scored by the 11 players of the cricket team in a match are 25, 65, 03,12, 35, 46, 67, 56, 00, 31, 17

Arranging the data in ascending order, we get the observations as

00, 03, 12, 17, 25, 31, 35, 46, 56, 65, 67

We find that Each value has occurred only once.

We may represent the above data in a tabular form, showing the frequency of each observation. This representation is called a frequency distribution.

Frequency Distribution Table

Player	Runs Scored By Each Player (Frequency)
Player 1	00
Player 2	03
Player 3	12
Player 4	17
Player 5	25
Player 6	31
Player 7	35
Player 8	46
Player 9	56
Player 10	65
Player 11	67
Overall score in the match	357

Example 2.

In a quiz, the marks obtained by 20 students out of 30 are given as:

12, 15, 15, 29, 30, 21, 30, 30, 15, 17, 19, 15, 20, 20, 16, 21, 23, 24, 23, 21

State the frequency of each student.

Solution:

Given that,

Marks obtained by 20 students in quiz out of 30 are 12, 15, 15, 29, 30, 21, 30, 30, 15, 17, 19, 15, 20, 20, 16, 21, 23, 24, 23, 21

Arranging the data in ascending order, we get the marks list as

12, 15, 15, 15, 15, 16, 17, 19, 20, 20, 21, 21, 21, 23, 23, 24, 29, 30, 30, 30

The highest marks obtained student is 30, the lowest marks are 12, the range is 30 – 12 = 18

From the data, we find that

12 marks scored by one student,

15 marks scored by 4 students,

16, 17, 19 marks scored by one student,

20 marks scored by 2 students,

21 marks scored by 3 students,

23 marks scored by 2 students,

24, 29 marks scored by one student,

30 marks scored by 3 students,

We may represent the above data in a tabular form, showing the frequency of each observation. The number of times data occurs in a data set is known as the frequency of data

This tabular form of representation is called a frequency distribution.

Marks obtained in the quiz	Number of students(Frequency)
12	1
15	4
16	1
17	1
19	1
20	2
21	3
23	2
24	1
29	1
30	3
Total	20

Example 3.

The numbers of newspapers sold at a local shop over the 10 days are:

22, 20, 18, 23, 20, 25, 22, 20, 18, 20

State the frequency.

Solution:

Given that,

The total number of newspapers sold at the local shop past 10 days are 22, 20, 18, 23, 20, 25, 22, 20, 18, 20

By arranging the newspapers numbers in ascending order is 18, 18, 20, 20, 20, 20, 22, 22, 23, 25

The frequency table for the papers sold is given here:

Paper Sold	Frequency
18	2
20	4
22	2
23	1
25	1
Total	10

Relations and Mapping are important topics in Algebra. Relations & Mapping are two different words and have different meanings mathematically. Let’s get deep into the article to know all about the Relations, Mapping, or Functions like Definitions, Types of Relations, Solved Examples, etc.

• Ordered Pair
• Cartesian Product of Two Sets
• Relation in Math
• Domain and Range of a Relation
• Practice Test on Math Relation
• Functions or Mapping
• Domain Co-domain and Range of Function
• Math Practice Test on Functions
• Relation and Mapping Worksheets

What is a Relation?

A relation is a collection of ordered pairs. Relation in general defines the relationship between two different sets of information. An ordered pair is a point that has both x and y coordinates. Let us consider two sets x and y and set x has a relation with y. The values of set x are called Domain and the values of set y are called Range.

Relations can be represented using three different notations i.e. in the form of a table, graph, mapping diagram.

Example: (2, -5) is an ordered pair.

What is Mapping?

Mapping denotes the relation from Set A to Set B. Relation from A to B is the Subset of AxB. Mapping the oval on the left-hand side denotes the values of Domain and the oval on the right-hand side denotes the values of Range.

From the above diagram, we can say the ordered pairs are (1,c) (2, n) (5, a) (7, n).

Set{ 1, 2, 5, 7} represents the domain.

{a, c, n} is the range.

A function is a relation that derives the output for a given input.

Remember that all functions are relations but not all relations are functions.

Types of Relations

There are 8 different types of Relations and we have mentioned each of them in the following modules along with Examples.

• Empty Relation
• Universal Relation
• Identity Relation
• Inverse Relation
• Reflexive Relation
• Symmetric Relation
• Transitive Relation
• Equivalence Relation

Empty Relation: 

Empty Relation is the one in which there will be no relation between elements of a set. It is also called a Void Relation. For instance, Set A = {3, 4, 5} then void relation can be R = {x, y} where | x- y| = 7.

For an Empty Relation R = φ ⊂ A × A

Universal Relation:

In Universal Relation every element of a set is related to each other. Consider a set A = {a, b, c}. Universal Relations will be R = {x, y} where, |x – y| ≥ 0.

For Universal Relation R = A × A

Identity Relation:

Every element of a set is related to itself in an Identity Relation. Consider a Set A = {a, b, c} then Identity Relation is given by I = { a,a}, {b,b}, {c, c}

I = {(a, a), a ∈ A}

Inverse Relation:

In Inverse Relation when a set has elements that are inverse pairs of another set. If Set A = {(a,b), (c,d)} then inverse relation will be R^-1 = {(b, a), (d, c)}

R^-1 = {(b, a): (a, b) ∈ R}

Reflexive Relation:

Every element maps to itself in Reflexive Relation. Consider Set A = { 3, 4} then Reflexive Relation R = {(3, 3), (4, 4), (3, 4), (4, 3)}. Reflexive Relation is given by (a, a) ∈ R

Symmetric Relation:

In the case of Symmetric Relation if a = b, then b = a is also true. Relation R is symmetric only if (b, a) ∈ R is true when (a,b) ∈ R.

aRb ⇒ bRa, ∀ a, b ∈ A

Transitive Relation:

In case of a transitive relation if (x, y) ∈ R, (y, z) ∈ R then (x, z) ∈ R

aRb and bRc ⇒ aRc ∀ a, b, c ∈ A

Equivalence Relation:

A Relation symmetric, reflexive, transitive at the same time is called an Equivalence Relation.

How to Convert a Relation to Function?

A relation that follows the rule that every X- Value associated with only one Y-Value is called a Function.

Example

Is A = {(1, 4), (2, 5), (3, -8)}?

Solution:

Since the set has no duplicates or repetitions in the X- Value, the relation is a function.

Mapping Diagrams

Mapping Diagram consists of two columns in which one denotes the domain of a function f whereas the other column denotes the Range. Usually, Arrows or Lines are drawn between domain and range to denote the relation between two elements.

One-to-One Mapping

Each element of the range is paired with exactly one element of the domain. The function represented below denotes the One-to-One Mapping.

Many to One Mapping

If one element in the range is associated with more than one element in the domain is called many to one mapping. In the below diagram you can see the second number II is associated with more than one element in the domain.

One to Many Mapping

If one element in the domain is mapped with more than elements in the range then it is called One to Many Mapping. In the below diagram the first element in the domain is mapped to many elements in the range therefore it is called One to Many Mapping.

Hope you got a complete idea on Relations and Mapping Concept. If you need any other help you can ask us through the comment section and we will get back to you at the earliest possibility.

Table of Cube Roots helps you to solve the cube of a number in an easy way. The approximate value of a cube root can be found easily with the help of the Table of Cube Roots.

If the given number is not a perfect cube, then one can find the number that has a cube which is nearly equal to the given number. The below table will give a clear idea of the approximate value of the cube roots of different numbers. Also, get complete knowledge on Cube and Cube Roots Concept by referring to our concepts.

Number	Cube(a3)	Cube root ∛a
1	1	1.000
2	8	1.260
3	27	1.442
4	64	1.587
5	125	1.710
6	216	1.817
7	343	1.913
8	512	2.000
9	729	2.080
10	1000	2.154
11	1331	2.224
12	1728	2.289
13	2197	2.351
14	2744	2.410
15	3375	2.466

Learn the fast and easy Method for Finding the Cube of a Two-Digit Number that makes the Cube and Cube Root calculation more simple. The easy way to find the Cube of a Two-Digit Number is by using (a + b)³ = a³ + 3a²b + 3ab² + b³.

Solving Cube of a Two-Digit Number is easier after you identify the step by step process. Have a look at the simplest way to find the cube of all two-digit numbers. Also, learn more about Cube and Cube Roots of a Number with more tips and tricks.

What is the Short Cut Method for Finding The Cubes Of a Two-Digit Number?

The cube of a two-digit number can be found with the help of tens digit = a and also with the units digit b. We can make four columns using a³, (3a² × b), (3a × b²), and b³.

The remaining process is the same as the squaring a number by the column method. a³, 3a², 3ab², and b³ are simplified as

a² × a = a³;
a² × 3b = 3a²b;
b² × 3a = 3ab²;
b² × b = b³;

Cubes Of a Two-Digit Number Solved Examples

1. Find the value of (29)³ by the short-cut method?

Answer:
Let a = 2 and b = 9
Now, find the values of a³, 3a², 3ab², and b³.

Therefore, Cube of 29 = (29)³ = 24389

Cube of 29 is 24389

2. Find the value of (71)³ by the short-cut method?

Answer:

Let a = 7 and b = 1
Now, find the values of a³, 3a², 3ab², and b³.

Therefore, Cube of 71 = (71)³ = 357911

Find the easy and short cut method to learn the cube of a two-digit number. Begin your practice now with the help of the solved examples. Cube and Cube Root of a Numbers are clearly explained here.

Cube Root of a number can be obtained by doing the inverse operation of calculating cube. In general terms, the cube root of a number is identified by a number that multiplied by itself thrice gives you the cube root of that number. The cube root of any number is denoted with the symbol ∛. Look at the Cube and Cube Roots solved examples and their explanations to learn them easily.

For example, the cube root of a number x is represented as ∛x.

How to Find the Cube Root of a Number?

Simply, Note down the product of primes a number. Then, form the groups in triplets using the product of primes a number. After that take one number from each triplet. The selected single number is the required cube root of the given number.

Note: If you find a group of prime factors that cannot form a group in triplets they remain the same and their cube root cannot be found.

Cube Root of a Number Solved Examples

(i) Find the Cube Root of a number 64?

Answer:
Write the product of primes of a given number 64 those form groups in triplets.
Cube Root of 64 = ∛64 = ∛(4 × 4 × 4)
Take one number from a group of triplets to find the cube root of 64.
Therefore, 4 is the cube root of a given number 64.

4 is the cube root of a given number 64.

(ii) Find the Cube Root of a number 8?

Answer:
Write the product of primes of a given number 8 those form groups in triplets.
Cube Root of 8= ∛8= ∛(2 × 2 × 2)
Take one number from a group of triplets to find the cube root of 8.
Therefore, 2 is the cube root of a given number 8.

2 is the cube root of a given number 8.

(iii) Find the Cube Root of a number 125?

Answer:
Write the product of primes of a given number 125 those form groups in triplets.
Cube Root of 125= ∛125= ∛(5 × 5 × 5)
Take one number from a group of triplets to find the cube root of 125.
Therefore, 5 is the cube root of a given number 125.

5 is the cube root of a given number 125.

(iv) Find the Cube Root of a number 27?

Answer:
Write the product of primes of a given number 27 those form groups in triplets.
Cube Root of 27 = ∛27 = ∛(3 × 3 × 3)
Take one number from a group of triplets to find the cube root of 27.
Therefore, 3 is the cube root of a given number 27.

3 is the cube root of a given number 27.

(iv) Find the Cube Root of a number 216?

Answer:
Write the product of primes of a given number 216 those form groups in triplets.
Cube Root of 216 = ∛216 = ∛(6 × 6 × 6)
Take one number from a group of triplets to find the cube root of 216.
Therefore, 6 is the cube root of a given number 216.

6 is the cube root of a given number 216.

Finding Cube Root by Prime Factorisation Method

Find the cube root of a number using the Prime Factorisation Method with the help of the below steps.

Step 1: Firstly, take the given number.
Step 2: Find the prime factors of the given number.
Step 3: Group the prime factors into each triplet.
Step 4: Collect each one factor from each group.
Step 5: Finally, find the product of each one factor from each group.
Step 6: The resultant is the cube root of a given number.

Cube Root of a Number by Prime Factorisation Method Solved Examples

(i) Find the Cube Root of 216 by Prime Factorisation Method?

Answer:
Firstly, find the prime factors of the given number.
216 = 2 × 2 × 2 × 3 × 3 × 3
Group the prime factors into each triplet.
216 = (2 × 2 × 2) × (3 × 3 × 3)
Collect each one factor from each group.
2 and 3
Finally, find the product of each one factor from each group.
∛216 = 2 × 3 = 6
6 is the cube root of 216.

(ii) Find the Cube Root of 343 by Prime Factorisation Method?

Answer: 
Firstly, find the prime factors of the given number.
343 = 7 × 7 × 7
Group the prime factors into each triplet.
343 = (7 × 7 × 7)
Collect each one factor from each group.
7
Finally, find the product of each one factor from each group.
∛343 = 7
7 is the cube root of 343.

(iii) Find the Cube Root of 2744 by Prime Factorisation Method?

Answer:
Firstly, find the prime factors of the given number.
2744 = 2 × 2 × 2 × 7 × 7 × 7
Group the prime factors into each triplet.
2744 = (2 × 2 × 2) × (7 × 7 × 7).
Collect each one factor from each group.
2 and 7
Finally, find the product of each one factor from each group.
∛2744 = 2 × 7 = 14
14 is the cube root of 2744.

Cube Roots of Negative Numbers

Cube Root of a negative number is always negative. If -m be a negative number. Then, (-m)³ = -m³.
Therefore, ∛-m³ = -m.
cube root of (-m³) = -(cube root of m³).
∛-m = – ∛m

Solved Examples of Cube Root of a Negative Numbers

(i) Find the Cube Root of (-1000)

Answer:
Firstly, find the prime factors of the number 1000.
1000 = 2 × 2 × 2 × 5 × 5 × 5
Group the prime factors into each triplet.
1000 = (2 × 2 × 2) × (5 × 5 × 5).
Collect each one factor from each group.
2 and 5
Finally, find the product of each one factor from each group.
∛1000 = 2 × 5 = 10
∛-m = – ∛m
∛-1000 = – ∛1000 = -10
-10 is the cube root of (-1000).

(ii) Find the Cube Root of (-216)

Answer:
Firstly, find the prime factors of the number 216.
216 = 2 × 2 × 2 × 3 × 3 × 3
Group the prime factors into each triplet.
216 = (2 × 2 × 2) × (3 × 3 × 3)
Collect each one factor from each group.
2 and 3
Finally, find the product of each one factor from each group.
∛216 = 2 × 3 = 6
∛-216 = – ∛216
∛-216= – ∛216= -6
-6 is the cube root of -216.

How to Find Cube Root of Product of Integers?

Cube Root of Product of Integers can be solved by using ∛ab = (∛a × ∛b)

Solved Examples:

(i) Find ∛(125 × 64)?

Answer:
Firstly, apply the cube root to both integers.
∛ab = (∛a × ∛b)
∛(125 × 64) = ∛125 × ∛64
Then, find the prime factors for each integer separately.
[∛{5 × 5 × 5}] × [∛{4 × 4 × 4}]
Take each integer from the group in triplets and multiply them to get the cube root of a given number.
(5 × 4) = 20
20 is the cube root of ∛(125 × 64).

(ii) Find ∛(27 × 64)?

Answer:
Firstly, apply the cube root to both integers.
∛ab = (∛a × ∛b)
∛(27 × 64) = ∛27 × ∛64
Then, find the prime factors for each integer separately.
[∛{3 × 3 × 3}] × [∛{4 × 4 × 4}]
Take each integer from the group in triplets and multiply them to get the cube root of a given number.
(3 × 4) = 12
12 is the cube root of ∛(27 × 64).

(iii) Find ∛[216 × (-343)]?

Answer:
Firstly, apply the cube root to both integers.
∛ab = (∛a × ∛b)
∛[216 × (-343)] = ∛216 × ∛-343
Then, find the prime factors for each integer separately.
[∛{6 × 6 × 6}] × [∛{(-7) × (-7) × (-7)}]
Take each integer from the group in triplets and multiply them to get the cube root of a given number.
[6 × (-7)] = -42
-42 is the cube root of ∛[216 × (-343)].

Cube Root of a Rational Number

The Cube Root of a Rational Number can be calculated with the help of ∛(a/b) = (∛a)/(∛b). Apply the Cube Root separately to each integer available on the numerator and the denominator to find the cube root of a rational number.

Solved Examples of Cube Root of a Rational Number

(i) Find ∛(216/2197)

Answer:
Firstly, apply the cube root to both integers.
∛(a/b) = (∛a)/(∛b)
∛(216/2197) = ∛216/∛2197
Then, find the prime factors for each integer separately.
[∛(6 × 6 × 6)]/[ ∛(13 × 13 × 13)]
Take each integer from the group in triplets to get the cube root of a given number.
6/13
6/13 is the cube root of ∛(216/2197).

(ii) Find ∛(27/8)

Answer:
Firstly, apply the cube root to both integers.
∛(a/b) = (∛a)/(∛b)
∛(27/8) = ∛27/∛8.
Then, find the prime factors for each integer separately.
[∛(3 × 3 × 3)]/[ ∛(2 × 2 × 2)]
Take each integer from the group in triplets to get the cube root of a given number.
3/2
3/2 is the cube root of ∛(27/8).

How to Find the Cube Root of Decimals?

The Cube Root of Decimals can easily be solved by converting them into fractions. After converting the decimal number into a fraction apply the cube root to the numerator and denominator separately. Then, convert the resultant value to decimal.

Cube Root of Decimals Solved Examples

(i) Find the cube root of 5.832.

Answer:
Conver the given decimal 5.832 into a fraction.
5.832 = 5832/1000
Now, apply the cube root to the fraction.
∛5832/1000
Apply the cube root to both integers.
∛(a/b) = (∛a)/(∛b)
∛5832/1000 = ∛5832/∛1000.
Then, find the prime factors for each integer separately.
∛(2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3)/∛(2 × 2 × 2 × 5 × 5 × 5)
Take each integer from the group in triplets to get the cube root of a given number.
(2 × 3 × 3)/(2 × 5) = 18/10
Convert the fraction into a decimal
18/10 = 1.8
1.8 is the cube root of 5.832.

A perfect cube is the prime factors of a number that are arranged in triplets of equal factors. Check more about the perfect cubes and How to find if the given number is a perfect cube or not from the Cube and Cube Roots Chapter.

How to Find if the Given Number is a Perfect Cube?

To Find if the Given Number is a Perfect Cube, we need to find the prime factors of that number. If the prime factors are grouped together with the triplets, then the number is considered as the perfect cube. If not, the number is not a perfect cube. If you see the prime factor is a single factor or it is a double factor, then that particular number is not treated as a perfect cube.

Let’s find whether the given numbers are perfect cubes or not.

1. Find the below Numbers are Perfect Cubes?
(i) 18 (ii) 27,000 (iii) 120

Answer:

(i) 18
First, find the prime factors of the given number.
The prime factors of the given number 18 are 2, 3, 3
Therefore, 18 = 2 × 3 × 3
The prime factors 2 and 3 are single and double factors. There is no triple factor available in the prime factors of the given number. So, the given number is not a perfect cube.

The given number 18 is not a perfect cube.

(ii) 27,000

First, find the prime factors of the given number.
The prime factors of the given number 27,000 are 30, 30, 30
Therefore, 27,000 = 30 × 30 × 30
The prime factor 30 is grouped in triples. There is no other prime factor is left as single and double factors. So, the given number is a perfect cube.

The given number 27,000 is a perfect cube.

(i) 120
First, find the prime factors of the given number.
The prime factors of the given number 120 are 2, 2, 2, 3, 5
Therefore, 120 = 2 × 2 × 2 × 3 × 5
The prime factors 3 and 5 are single factors. There is one prime factor grouped in triples. As the single prime factors available in the product of prime factors of the given number, it is is not a perfect cube.

The given number 120 is not a perfect cube.

Smallest Multiple that is a Perfect Cube

The smallest multiple that make a number perfect cube can be calculated by following the step by step method product of prime factors of a number. A clear explanation is given To Find if the Given Number is a Perfect Cube and what is the smallest multiple that needs to make it a perfect cube. Check the below examples for a better understanding.

2. Find the smallest number by which 1372 must be multiplied so that the product is a perfect cube?

To find the smallest number by which 1372 must be multiplied so that the product is a perfect cube, you need to find the prime factors of the given number.
The prime factors of the given number 1372 are 7, 7, 7, 2, 2
1372 = 7 × 7 × 7 × 2 × 2
1372 = 7³ × 2 × 2
From above, by multiplying 2 the number 1372 becomes the perfect cube.
Therefore, the smallest number is 2.

3. Find the smallest number by which 2000 must be multiplied so that the product is a perfect cube?

To find the smallest number by which 2000 must be multiplied so that the product is a perfect cube, you need to find the prime factors of the given number.
The prime factors of the given number 2000 are 5, 5, 5, 4, 4
2000 = 5 × 5 × 5 × 4 × 4
2000 = 5³ × 4 × 4
From above, by multiplying 4 the number 2000 becomes the perfect cube.
Therefore, the smallest number is 4.

Generally, a cube is defined as a solid figure that has all sides equal. In other words, a Cube is a solid three-dimensional figure that consists of 6 square faces, 8 vertices, and 12 edges. When six identical square face together along with their edges form a cube. Also, the three edges of a cube join at each corner to form a vertex. The cube in a Cube and Cube Roots chapter is the first and basic concept that will let you know about the perfect cubes or cube numbers.

In mathematical terms, a cube of a number is explained as the multiplication of a number by three times. The cube of a number represents as a superscript 3 or 3 is written a little up to the right of the numbers.

Representation of a cube of a number: m³, where 3 is the power of m and read as “m cubed”.

Cube of a Number

A number is multiplied by itself 3 times to find the cube of that number.
cube of m = m × m × m
cube of m = m³

Examples:

(i) Find the cube of a number 3?

Solution:
cube of 3 = 3³
The cube of 3 = 3³ = 3 × 3 × 3
cube of 3 = 3³ = 27

The cube of 3 is 27

(ii) Find the cube of a number 2?

Solution:
Cube of 2 = 2³
The cube of 2 = 2³ = 2 × 2 × 2
cube of 2 = 2³ = 8

The cube of 2 is 8

(iii) Find the cube of a number 4?

Solution:
cube of 4 = 4³
The cube of 4 = 4³ = 4 × 4 × 4
cube of 4 = 4³ = 64

The cube of 4 is 64

(iii) Find the cube of a number 5?

Solution:
Cube of 5 = 5³
The cube of 5 = 5³ = 5 × 5 × 5
cube of 5 = 5³ = 125

The cube of 5 is 125

(iv) Find the cube of a number 6?

Solution:
cube of 6 = 6³
The cube of 6 = 6³ = 6 × 6 × 6
cube of 6 = 6³ = 216

The cube of 6 is 216

(v) Find the cube of a number 7?

Solution:
cube of 7 = 7³
The cube of 7 = 7³ = 7 × 7 × 7
cube of 7 = 7³ = 343

The cube of 7 is 343

Perfect Cubes and Cube Roots

A perfect cube is defined as a number that is the cube of an integer or the cube of some natural number.

Examples:
Cube of 1 = 1³ = 1 × 1 × 1 = 1
Cube of 2 = 2³ = 2 × 2 × 2 = 8
The Cube of 3 = 3³ = 3 × 3 × 3 = 27
Cube of 4 = 4³ = 4 × 4 × 4 = 64
The Cube of 5 = 5³ = 5 × 5 × 5 = 125

Cube of Negative Numbers

Cubing the number is nothing but raising the number to its 3rd power. The Cube of a negative number is always a negative number. If -m is a number, then the cube of -m is (-m)³ = -m × -m × -m

Examples:

(i) Find the cube of -1?

Answer:
cube of -1 = (-1)³ = -1 × -1 × -1 = -1

The cube of -1 is -1

(ii) Find the cube of -2?

Answer:
cube of -2 = (-2)³ = -2 × -2 × -2 = -8

The cube of -2 is -8

(iii) Find the cube of -3?

Answer:
cube of -3 = (-3)³ = -3 × -3 × -3 = -27

The cube of -3 is -27

(iv) Find the cube of -4?

Answer:
cube of -4 = (-4)³ = -4 × -4 × -4 = -64

The cube of -4 is -64

(v) Find the cube of -5?

Answer:
cube of -5 = (-5)³ = -5 × -5 × -5 = -125

The cube of -5 is -125

Cube of a Rational Number

Finding the cube of a rational number is represents as the (a/b)³ that is also eqaul to the a/b × a/b × a/b = (a × a × a)/(b × b × b) = a³/b³
Therefore, (a/b)³ = a³/b³

Examples:

(i) Find the cube of (1/2)³?

Answer:

cube of 1/2 = (1/2)³
The cube of 1/2 = (1/2)³ = (1/2) × (1/2) × (1/2)
cube of 1/2 = 1³/2³
cube of 1/2 = (1/2)³ = 1³/2³ = (1 × 1 × 1)/(2 × 2 × 2)
The cube of 1/2 = (1/2)³ = 1³/2³ = 1/8

The cube of (1/2) is 1/8

(i) Find the cube of (-4/3)³?

Answer:
cube of (-4/3) = (-4/3)³
The cube of (-4/3) = (-4/3)³ = (-4/3) × (-4/3) × (-4/3)
cube of (-4/3) = (-4)³/3³
cube of (-4/3) = (-4/3)³ = (-4)³/3³ = (-4 × -4 × -4)/(3 × 3 × 3)
The cube of (-4/3) = (-4/3)³ = (-4)³/3³ = -64/27

The cube of (-4/3) is -64/27

Cube number Properties:

(i) The cube of even integers is always even.
(ii) The cube of odd integers is always odd.

Perfect Cube Solved examples

1. Is 189 a perfect cube?

Answer:
Separate 189 into different prime factors
The prime factors for 189 are 3, 3, 3, 7
189 = 3 × 3 × 3 × 7
189 = 3 × 7
Divide 189 with 7 to make it a perfect cube.
So, 189 is not a perfect cube.

2. Find the number 216 is a perfect cube?

Answer:

Firstly, find the prime factors of 216
The prime factors of 216 are 2, 2, 2, 3, 3, 3
So, 216 = 2 × 2 × 2 × 3 × 3 × 3
216 = (2 × 3) × (2 × 3) × (2 × 3)
216 = 6 × 6 × 6
The 216 = 6³ = cube of 6
216 is a perfect cube as it is the cube of 6.

216 is a perfect cube

3. Find the smallest number that makes the 3087 a perfect cube?

Answer:
To know the smallest number that makes the 3087 a perfect cube, first, we need to find the prime factors of 3087.
The 3087 prime factors are 3, 3, 7, 7, 7
3087 = 3 × 3 × 7 × 7 × 7
If the product of prime factors is multiplied by the number 3, then 3087 becomes a perfect cube.

The required number is 3

4. Which number needs to divide from 392 to make it a perfect cube?

Answer:
To find the number that makes 392 a perfect cube, we need to find the prime factors of 392
The prime factors of 392 are 2, 2, 2, 7, 7
392 = 2 × 2 × 2 × 7 × 7
The 392 will becomes a perfect cube if 7 × 7 is divided from the product of prime factors.

The required number is 7 × 7

5. Find the cube of each of the following?
(i) 8 (ii) (2/5) (iii) 0.2 (iv) 2 3/4 (v) -5

Solutions:

(i) 8
cube of 8 = 8³
The cube of 8 = 8³ = 8 × 8 × 8
cube of 8 = 8³ = 512

The cube of 8 is 512

(i)(2/5)
cube of (2/5) = (2/5)³
The cube of (2/5) = (2/5)³ = (2/5) × (2/5) × (2/5)
cube of (2/5) = (2/5)³ = 2³/5³ = (2 × 2 × 2)/(5 × 5 × 5) = 8/125

The cube of (2/5) is 8/125

(iii) 0.2
cube of 0.2 = 0.2³
The cube of 0.2 = 0.2³ = (0.2) × 0.2 × 0.2
cube of 0.2 = (0.2)³ = 0.008

The cube of (0.2) is 0.008

(iv) 2 3/4
2 3/4 = 11/4
cube of 11/4 = (11/4)³
The cube of (11/4) = (11/4)³ = (11/4) × (11/4) × (11/4)
cube of (11/4) = (11/4)³ = (11 × 11 × 11)/(4 × 4 × 4) = 1331/64

The cube of (2 3/4) is 1331/64

(i) -5
cube of -5 = -5³
The cube of -5 = -5³ = -5 × -5 × -5
cube of -5 = -5³ = -125

The cube of (-5) is -125

The cube of a number is clearly explained along with examples and explanations.

In this article, you will learn how to find the Complement of a Set. Check out the below sections to get an idea of how to find the Complement of a Set. Refer to Questions on Complement of a Set along with Answers explained in detail and get a grip on the concept. To learn completely about various operations on sets have a glance at the Set Theory and get a good hold of all the underlying topics.

Solved Example Questions on Complement of a Set

1. If U = {2, 4, 6, 8, 10, 12, 14} and A = {1, 3, 5}. Find the Complement of A?

Solution:

A^c or A^‘ = U – A

= {2, 3, 4, 5, 6, 8, 10, 12, 14} – {1, 3, 5}

= { 2, 4, 6, 8, 10, 12, 14}

The Complement of A is the Elements that are in the Universal Set and not in the Set A.

2. Given Sets B = {1 Orange, 1 Pineapple, 1 Guava, 1 Pomegranate} and U = {1 Guava, 1 Orange, 1 Apricot, 1 Pomegranate, 1 Custard Apple, 1 Banana, 1 Mango, 1 Apple, 1 Kiwifruit }. Find the Complement of B?

Solution:

B = {1 Orange, 1 Pineapple, 1 Guava, 1 Pomegranate}

U = {1 Guava, 1 Orange, 1 Apricot, 1 Pomegranate, 1 Custard Apple, 1 Banana, 1 Mango, 1 Apple, 1 Kiwifruit }

B^c or B^‘ = U – B

= {1 Guava, 1 Orange, 1 Apricot, 1 Pomegranate, 1 Custard Apple, 1 Banana, 1 Mango, 1 Apple, 1 Kiwifruit } – {1 Orange, 1 Pineapple, 1 Guava, 1 Pomegranate}

= { 1 Apricot, 1 Custard Apple, 1 Banana, 1 Mango, 1 Apple, 1 Kiwifruit}

3. Find the Complement of B if the Sets are B = {1, 3, 4, 5} U = {1, 2, 3, 4, 5, 6, 7, 8, 9 }?

Solution:

Given Sets are

U = {1, 2, 3, 4, 5, 6, 7, 8, 9}

B = {1, 3, 4, 5}

B^c or B^‘ = U – B

= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {1, 3, 4, 5}

= {2, 6, 7, 8, 9}

4. Find the Complement of A

A = { x / x is a number bigger than 6 and smaller than 10}

U = { x / x is a positive number smaller than 10}

Solution:

Given

A = { x / x is a number bigger than 6 and smaller than 10}

= { 7, 8, 9}

U = { x / x is a positive number smaller than 10}

= { 1, 2, 3, 4, 5, 6, 7, 8, 9 }

A^c or A^‘ = U – A

= { 1, 2, 3, 4, 5, 6, 7, 8, 9 } – { 7, 8, 9}

= {1, 2, 3, 4, 5, 6}

5. Let U = {x : x is an integer, –7 ≤ x ≤ 4}, P = {-4, -2, 0, 7, 8, 9, 6}. Find the Complement of P?

Solution:

P = {-4, -2, 0, 7, 8, 9, 6}

U = {x : x is an integer, –7 ≤ x ≤ 4}

U = {-7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4}

P^c or P^‘ = U – P

= {-7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4} – {-4, -2, 0, 7, 8, 9, 6}

= { -7, -6, -5, -3, -1, 1, 2, 3, 4}

Let’s learn about Sets before diving deep into the article Complement of a Set. Check out the Definition of Complement of a Set, its Representation, Venn Diagram of the Complement of the Set. Refer to Solved Examples for finding the Complement of a Set. To learn more about the Operations of Sets you can have a look at the Set Theory and clarify all your concerns easily.

Sets Definition

A Well Defined Collection of Objects or Elements is called Sets. Any set containing all the objects or elements related to a particular context is known as a Universal Set and is represented by U.

For any Set A that is a subset of the Universal Set U, the Complement of the Set A contains the elements that are members of Universal Set but not from Set A. The Complement of Set A is denoted by A′.

Complement of a Set

If U is a Universal Set and A is a Subset of U then the Complement of A is the set of all the elements in the Universal Set but not aren’t elements of Set A.

A′ = {x : x ∈ U and x ∉ A}

In Other Words, we can say that the Complement of A is nothing but the difference between the Universal Set and the Sub Set A.

Venn Diagram for Complement of a Set

The Venn Diagram representing the Complement of a Set is given by

Solved Examples on Complement of a Set

1.  Let U be a Universal Set that consists of natural numbers greater than 10 and less than 20. A, B are Subsets of U and given by

A ={x:x ∈ U and x is a perfect square}

B = {8, 9, 11, 12, 13}

Find the complement of sets A and B?

Solution:

U = {11,12,13,14,15,16,17,18,19}

A = {x:x ∈ U and x is a perfect square}

= {16}

B = {8, 9, 11, 12, 13}

A′ = U – A

{11,12,13,14,15,16,17,18,19} – {16}

= { 11, 12, 13, 14, 15, 17, 18, 19}

B′ = U – B

= {11,12,13,14,15,16,17,18,19} – {8, 9, 11, 12, 13}

= { 12, 14, 15, 16, 17, 18, 19}

2. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = { 1, 3, 5, 7}. Find the Complement of A?

Solution:

Given U = {1, 2, 3, 4, 5, 6, 7, 8,, 9, 10}

A = {1, 3, 5, 7}

A′ = U – A

= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} – {1, 3, 5, 7}

= { 2, 4, 6, 8, 9, 10}

3. If U = {3, 5, 7, 9, 11, 13, 15, 17, 19}, A = {6, 10, 4, 16}. Find the Complement of A?

Solution:

U = {3, 5, 7, 9, 11, 13, 15, 17, 19}

A = {6, 10, 4, 16}

A′ = U – A

= {3, 5, 7, 9, 11, 13, 15, 17, 19} – {6, 10, 4, 16}

= {3, 5, 7, 9, 11, 13, 15, 17, 19}

In Set Theory we usually perform different operations on sets such as intersection, union, complement. The Difference of Sets is also a similar kind of operation which we perform on sets. You will understand the difference between intersection and difference of sets clearly after going through this article. Check out Set Theory to be clear with the concepts of Sets Operations.

How to find the Difference of Sets?

In general, the Difference between the Two Sets is the Set of elements present in A but not in B. It is represented as A – B. You can see the difference in the orange shaded region of the below Venn diagram. In the same way, the region shaded in violet indicates the difference between B and A.

Identities Involving Difference of Sets

• If Set A and B are equal then A-B = A-A = ϕ
• If you subtract an empty set from a Set then the result is the Set itself i.e. A – ϕ = A.
• In the Similar Way, if you subtract a Set from an Empty Set then the result is an Empty Set i.e. ϕ – A = ϕ
• If you subtract a Superset from Subset the result is an empty set i.e. A – B = ϕ if A ⊂ B
• If Two Sets A and B are disjoint then A – B = A, B – A = B

Solved Examples for finding Difference of Sets

1. If A = {4, 5, 6} and B = {7, 8, 9}. Find the Difference between Sets A and B and B and A?

Solution:

Given A = { 4, 5, 6}

B = {7, 8, 9}

A-B = {4, 5, 6} – { 7, 8, 9}

= { 4, 5, 6}

B-A = {7, 8, 9} – {4, 5, 6}

= {7, 8, 9}

Since two sets A, B are disjoint the difference between A and B yields A and difference between B and A gives B.

2. Let A = {c, d, e, f, g, h, i} and B = {b, d, f, g, i, h} find A-B and B-A?

Solution:

Given A = {c, d, e, f, g, h, i}

B = {b, d, f, g, i, h}

A-B = {c, d, e, f, g, h, i} – {b, d, f, g, i, h}

= { c, e}

The elements c, e belong to Set A but not B

B-A = {b, d, f, g, i, h} – {c, d, e, f, g, h, i}

= {b}

Thus, the element b belongs to Set B but not A.

3. Given Sets A = {3 , 4 , 8 , 9 , 11 , 12 } B = {3 , 4 , 8 , 9 , 11 , 12 }. Find the difference between them i.e. A-B?

Solution:

A = {3 , 4 , 8 , 9 , 11 , 12 }

B = {3 , 4 , 8 , 9 , 11 , 12 }

A-B = A- A( since both the sets are equal)

= ϕ or {}

When you subtract two equal sets the difference between them will be an Empty Set.

4. Given three sets P, Q and R such that:

P = {x : x is a natural number between 12 and 18},

Q = {y : y is a even number between 14 and 20} and

R = {5, 9, 10, 12, 18, 20}

(i) Find the difference between two sets P and Q

(ii) Find Q – R

(iii) Find R – P

(iv) Find Q – P

Solution:

Given P = {x : x is a natural number between 12 and 18}

P = { 13, 14, 15, 16, 17}

Q = {y : y is a even number between 14 and 20}

Q = { 16, 18}

R = {5, 9, 10, 12, 18, 20}

(i) Difference between two sets P and Q i.e. P-Q

P-Q = { 13, 14, 15, 16, 17} – { 16, 18}

= {13, 14, 15, 17}

Elements 13, 14, 15, 17 are there in P but not in Q.

(ii)Q – R

Q = { 16, 18}

R = {5, 9, 10, 12, 18, 20}

Q – R = { 16, 18} – {5, 9, 10, 12, 18, 20}

= {16}

16 is the element that is present in Q but not in R.

(iii) R – P

R = {5, 9, 10, 12, 18, 20}

P = {13, 14, 15, 16, 17}

R – P = {5, 9, 10, 12, 18, 20} – {13, 14, 15, 16, 17}

= { 5, 9, 10, 12, 18, 20}

(iv) Q – P

Q = { 16, 18}

P = {13, 14, 15, 16, 17}

Q – P = { 16, 18} – {13, 14, 15, 16, 17}

= { 18}

18 is the element that is present in Q but not in P.

A plane is any flat surface that can go on infinitely in both directions. Coordinate geometry or analytical geometry is the link between algebra and geometry through graphs having curves and lines. It provides geometric aspects in algebra and enables them to solve geometric problems. Get the detailed information about the coordinate graph, all four quadrants, coordinates of points, others in the following sections.

Coordinate Geometry Definition

Coordinate geometry is one of the branches of geometry where the position of a point is defined using coordinates. Using the coordinate geometry, you can calculate the distance between two points, find coordinates of a point, plot ordered pairs, and others. The basic terms of coordinate geometry for class 8 students are listed below.

• Coordinate Geometry Definition
• Coordinates of a Coordinate Geometry
• Coordinate Plane
• Quadrants

What is meant by Coordinate and Coordinate Plane?

A coordinate plane is a two-dimensional plane created by the intersection of two axes names horizontal axis (x-axis) and the vertical axis (y-axis). These lines are perpendicular to each other and meet at the point called origin or zero. the axes divide the coordinate plane into four equal sections, and each section is known as the quadrant. The number line which is having quadrants is also known as the cartesian plane.

A set of values that represents the exact position on the coordinate plane is called coordinates. Usually, it is a pair of numbers on the graph denoted as (x, y). Here, x is called the x coordinate, y is called the y coordinate.

Quadrants: Four different quadrants and their respective signs are given below:

• Quadrant 1: In this quadrant both x and y are positive. The point is represented as (+x, +y).
• Quadrant 2: X-axis is negative and the y-axis is positive. So, the point is shown as (-x, +y).
• Quadrant 3: Here, both x and y axes are negative. The point in Q3 is represented as (-x, -y).
• Quadrant 4: In this quadrant, x is positive, and y is negative. Coordinates are (+x, -y).

How to Plot Coordinates of a Point on Graph?

Following are the simple steps to plot coordinates of a point on a graph. Have a look at them and check out how to plot a graph, represent ordered points, and identify signs of axes of a point.

• First of all, take a point which is having both x coordinate ad y coordinate.
• And know the signs of each value in the given point.
• Using those signs, identify under which quadrant the point falls.
• From the respective axes, take those numbers and put a dot on the graph.

Linear Equation

The general form of a line in the coordinate geometry is Ax + By + C = 0

Where A is the coefficient of x

B is the coefficient of y

And C is the constant value.

Intercept form of a line is y = mx + c. Where (x, y) is a point on the line and m is the slope.

Graph of Area vs. Side of a Square

To plot a graph of the square area and square side, you need to have the square area for each side length. Measure the square side length each time, find the area by performing the square of side length. Note down those values and take side length on the x-axis, area on the y-axis. As the square side is always positive, the points will automatically come in the first quadrant. Mark points such as side length as x-coordinate and area as y-coordinate for each point in the graph. Join those points to make a graph of area vs square side length.

Graph of Distance vs. Time

Drawing the graph for area vs side of a square and distance vs time is the same. Here, we are checking the distance traveled by an object in a certain amount of time, what happens when a change happens in either time or distance. Make sure that, the unit of both time and distance must be the same, if not convert them into the same unit. Take distance on the y-axis and time on the x-axis, so the x coordinate will the time, and the y coordinate will be the distance. Mark those points in the first quadrant and join them to draw a graph of distance vs time.

Example Questions

Example 1.

Plot that the points A (0, 0), B (1, 1), C (2, 2), D (3, 3) and show that these points form a line?

Solution:

Given Points are A (0, 0), B (1, 1), C (2, 2), D (3, 3)

Point A (0, 0) is the origin.

From the graph, we can say that the points form a straight line and that line passes through the origin.

Example 2.

Plot each of the following points on a graph?

a. (5, 2) b. (8, 0) c. (-5, -2) d. (9, -1)

Solution:

Given points are (5, 2), (8, 0), (-5, -2), (9, -1)

Coordinates of the point (5, 2) both abscissa and ordinate are positive so the point lies in the first quadrant. On the x-axis, take 5 units to the right of the y-axis and then on the y-axis, take 2 units above the x-axis. Therefore, we get the point (5, 2).

Coordinates of the point (8, 0) both abscissa and ordinate are positive so the point lies in the first quadrant. On the x-axis take 8 units and take 0 units on the y-axis to get the point (8, 0).

For the point (-5, -2), both abscissa and ordinate are negative so the point lies in the third quadrant. Take -5 units on the x-axis and take -2 units on the y-axis to obtain the point (-5, -2).

The point (9, -1) lies in the fourth quadrant because x-coordinate is positive and y-coordinate is negative. To plot this point, take 9 units on the x-axis and take -1 units on the y-axis.

Example 3.

Draw the graph of the linear equation y = x + 1?

Solution:

Given linear equation is y = x + 1

The given equation is in the form of y = mx + c

slope m = 1, and constant c = 1

By using the trial and error method, find the value of y for each value of x.

If x = 0, y = 0 + 1, then y = 1

If x = 1, then y = 1 + 1 = 2

If x = 2, then y = 2 + 1 = 3

x	0	1	2
y	1	2	3

Plot the graph for the points mentioned in the above table.

Mark the points (0, 1), (1, 2), (2, 3) on the graph.

Join those points to get a line equation.

FAQs on Coordinate Geometry

1. Why do we need coordinate geometry?

Coordinate geometry has various applications in real life. Some places where we use coordinate geometry is in integration, in digital devices such as mobiles, computes, in aviation to determine the position and location of airplane accurately in GPS, and to map the geographical locations using longitudes and latitudes.

2. Who is the father of Coordinate Geometry?

The father of coordinate geometry is Rene Descartes.

3. What is the name of horizontal and vertical lines that are drawn to find out the position of any point in the Cartesian plane?

The name of horizontal and vertical lines that are drawn to find out the position of any point in the Cartesian plane is determined by the x-axis and y-axis respectively.

4. What is Abscissa and Ordinates in Coordinate Geometry?

Abscissa and Ordinates are used to identify the position of a point on the graph. The horizontal value or x-axis is called the abscissa and the vertical line or the y-axis is called the ordinate. For example, in an ordered pair (1, 8), 2 is abscissa and 8 is ordinate.