numberline icse math book 6 - roopconventschool.com

1. Sets ....................................................................................................................................................1

2. Knowing Our Numbers ...................................................................................................................14

3. Natural Numbers and Whole Numbers ...........................................................................................37

4. Negative Numbers and Integers ......................................................................................................51

5. Playing with Numbers .....................................................................................................................65

Worksheet 1 .....................................................................................................................................90

6. Fractions ..........................................................................................................................................91

7. Decimals ....................................................................................................................................... 114

8. Ratio,ProportionandUnitaryMethod .........................................................................................136

9. Percentage .....................................................................................................................................155

Worksheet 2 ...................................................................................................................................164

Test Paper 1 ..................................................................................................................................165

10. Introduction to Algebra .................................................................................................................167

11. LinearEquations ...........................................................................................................................184

12. Basic Geometrical Ideas ................................................................................................................194

13.UnderstandingElementaryShapes ...............................................................................................212

Worksheet 3 ...................................................................................................................................232

14. Three-Dimensional Shapes ...........................................................................................................234

15.Constructions ................................................................................................................................24

16. Symmetry ......................................................................................................................................253

17. Perimeter and Area .......................................................................................................................261

18. Data Handling ...............................................................................................................................276

Worksheet 4 ...................................................................................................................................295

Test Paper 2 ..................................................................................................................................296

Answers ........................................................................................................................................298

Contents

Numberline_ICSE Math_Book 6.indb 4 05-10-2018 02:02:57 PM

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Warm up In our daily life, we often use terms—flock, pack, troop, bunch, deck, school and so on to represent a collection of different objects. Observe the following pictures carefully and fill in the blanks by using the above terms at their appropriate places:

a) b) c)

A _________ of keys. A _________ of fish. A _________ of hounds.

d) e) f)

A _________ of playing cards. A _________ of soldiers. A _________ of sheep.

In mathematics, we use the word set to denote a collection of objects.

For example,

a) a set of flowers,

b) a set of students,

c) a set of numbers and so on.

In this chapter, we will learn about sets, representation of sets, types of sets and cardinality of a set.

Sets1

• Idea of sets• Representation of sets

MY LEARNING PLAN

• Types of sets: finite, infinite and empty sets

• Cardinality of a set

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SetSA collection of well-defined distinct objects is called a set. By ‘well-defined’, we mean that it should be possible to say beyond doubt whether a particular object does or does not belong to the collection and the word ‘distinct’ means that the objects of a set must be all different.

For example,

a) the collection of all natural numbers, that is, 1, 2, 3, 4, 5, … is a set since the collection is well-defined.

b) thecollectionofallvowelsintheEnglishalphabetisasetasit contains only a, e, i, o and u.

c) the collection of hard working students in your class is not a set as we cannot say definitely who are the members of this collection as no criteria to select hard working students have been given.

elements of a SetThe objects that belong to a set are called its elements or members.

All the elements of a set are enclosed in curly brackets or braces ‘{ }’ and are separated by commas.

Let A be the set of even numbers between 1 and 10. It can be written as A = {2, 4, 6, 8}.

NotationWe usually denote sets by capital letters and their members or elementsbysmalllettersofEnglishalphabet.

If x is an element of set A, we represent it as x ∈ A which means that x belongs to A.

For example,

a) 5 ∈ {the set of odd numbers}

b) a ∈{thesetofvowelsinEnglishalphabet}

If x is not an element of set A, we represent it as x ∉ A which means that x does not belong to A.

For example,

a) 3 ∉ {the set of multiples of 2}

b) 0 ∉ {the set of natural numbers}

The objects that belong to a set are called its members or elements.

If both x and y are elements of set A, we represent it as x, y ∈ A.

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Representation of a SetA set can be represented by any of the following three forms.

i) Description form

ii) Rosterortabularform

iii)Rulemethodorsetbuilderform

Description FormIn this form, the description of the elements of a set is given. This description is enclosed in curly brackets.

For example,

a) the set of natural numbers less than 100 is written in the description form as

{natural numbers less than 100}.

b) thesetofallstudentsofClassVIofanArmySchooliswritteninthedescriptionformas

{allstudentsofClassVIofanArmySchool}.

c) the set of all integers from 1 to 6 is written in the description form as

{all integers from 1 to 6}.

[Note: Description form is also known as descriptive form.]

Roster or tabular FormIn this method, we list all the elements of a set within curly brackets and the elements are separated by commas.

For example,

a) the set of natural numbers less than 6 is written in the roster form as A = {1, 2, 3, 4, 5}.

b) the set of all days in a week is written in the roster form as D = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}.

c) the set of odd natural numbers less than 6 is written in the roster form as O = {1, 3, 5}.

Remember• Theelementsofasetcanbelistedinanyorder. For example, the set {a, b, c} can be written as {a, c, b} or {b, a, c} or {b, c, a} or {c, a, b} or {c, b, a}.• Eachelementofasetislistedonlyonce. For example, the set of letters of the word NOMINATION is written as {N, O, M, I, A, T}.• Asetdoesnotchangeevenifoneormoreelementsinasetarerepeated. For example, {1, 2, 3} is the same as {1, 1, 2, 3, 3, 2}.• Ifasetcontainsalargenumberofelements,thenthesetcanberepresentedbywriting

afewmembers,whichclearlyshowthetrendoftheelementsoftheset,followedorprecededbydotsandwiththelastelementwritten(ifitexists).

Forexample,thesetofnaturalnumbersupto50canbewrittenintherosterformas {1,2,3,…,50}.

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Rule Method or Set Builder FormIn this method, we describe a set by stating the properties satisfied by the elements and enclosing it in curly brackets.

For example, if A is a set consisting of elements x having some property P, then we write it as

A = {x | x has a property P} or {x : x has a property P}.

This set is termed as ‘the set of element x, such that x has the property P’. The symbols ‘|’ and ‘:’ stand for ‘such that’.

For example,

a) the set A of natural numbers greater than 1 but less than 6 is written in the set builder form as

A = {x : x ∈ N, 1 < x < 6}.

b) the set M of months in a year having 31 days is written in the set builder form as

M = {x | x is the month of a year having 31 days}.

c) the set S = {1, 2, 3, 4, 5, 6, 7, 8} is written in the set builder form as S = {x : x ∈ N, 1 ≤ x ≤ 8}.

Some Standard Sets• N = Set of natural numbers = {Natural numbers} (Description form)

={1,2,3,4,…} (Rosterform)

= {x : x is a counting number} or {x : x is a natural number} (Set builder form)

• W = Set of whole numbers = {Whole numbers} (Description form)


= {x : x is a whole number} (Set builder form)

• I or Z = Set of integers = {Integers} (Description form)

={…,−3,−2,−1,0,1,2,3,…} (Rosterform)

= {x : x is an integer} (Set builder form)

• E =Setofevennaturalnumbers={Evennaturalnumbers} (Descriptionform)


= {x : x is an even natural number} (Set builder form)

• O = Set of odd natural numbers = {Odd natural numbers} (Description form)


= {x : x is an odd natural number} (Set builder form)

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Example 1: State whether the given collections are sets or not:

a) The collection of easy questions of an exercise in a maths book.

b) The collection of the planets of the solar system.

c) The collection of consonants of the Englishalphabet.

d) The collection of multiples of 4 which are less than 30.

Solution:

a) The collection of easy questions of an exercise in a maths book is not a set because some questions may be easy for some students but difficult for others. Hence, the collection does not have well-defined elements.

b) The collection of the planets of the solar system is a set because there are exactly eight planets in our solar system—Mercury, Venus,Earth,Mars, Jupiter, Saturn,Uranusand Neptune. Hence, this collection has well-defined elements.

c) The collection of consonants of the Englishalphabet is a set because its elements are well-defined.

d) The collection of multiples of 4 which are less than 30 is a set because its elements are well-defined.

Example 2: Write the following sets in roster and set builder forms:

a) The set of all 2-digit numbers whose sum is 9.

b) The set of integers greater than −4 but less than 6.

c) The set of composite numbers less than 13.

d) {natural numbers divisible by 3 and less than 22}.

Solution:

The above sets can be written in roster and set builder forms as below:

a) Rosterform:

A = {18, 27, 36, 45, 54, 63, 72, 81, 90}

Set builder form:

A = {x : x is a 2-digit number and the sum of its digits is 9}

b) Rosterform:

B = {−3, −2, −1, 0, 1, 2, 3, 4, 5}

Set builder form:

B = {x : x ∈ Z, −4 < x < 6}

c) Rosterform:

C={4,6,8,9,10,12}

Set builder form:

C={x : x is a composite number and x < 13}

d) Rosterform:

D = {3, 6, 9, 12, 15, 18, 21}

Set builder form:

D = {x : x is divisible by 3 and x < 22}

Example 3: Let S = {2, 4, 6, 8, 10, 12, 14} and T = {3, 6, 9, 12, 15, 18}. Fill in the blanks using the symbols ∈ or ∉.

a) 1 __ S b) 2 __ S c) 3 __ T

d) 10 __ S e) 0.1 __ T f ) 18 __ T

Solution:

a) 1 ∉ S b) 2 ∈ S c) 3 ∈ T

d) 10 ∈ S e) 0.1 ∉ T f ) 18 ∈ T

Solved examples

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1. Which of the following phrases describes a well-defined set?

a) The people in your immediate family (father, mother, sisters and brothers).

b) All science students in your school.

c) All rich cricket players in your country.

d) All odd natural numbers less than 40.

e) All honest students in your class.

f) Collectionofevennaturalnumberslessthan 20.

g) Collectionofusefulplantsinyourarea.

h) Collectionofmonthsofayearhavingfour Sundays.

i) Collectionofinterestingstoriesbyabook club.

2. Let O = Set of odd natural numbers = {1, 3,5,7,9,…}andE=Setofevennaturalnumbers = {2, 4, 6, 8, 10, …}.

Fill in the blanks using symbol ∈ or ∉.

a) 1__O b) 2__E

c) −1__E d) 3__E

e) 0.1__O f ) 28__E

3. Considerthefollowingsets.

E={x : x = 2n for n ∈ N}, O = {x : x = 2n − 1 for n ∈ N}.

Write the above sets in descriptive form.

4. Write the following sets in roster form.

a) Set of natural numbers less than 6.

b) SetofvowelsintheEnglishalphabet.

c) Set of two-digit numbers, whose sum of the digits is 7.

d) {x | x is a letter in the word ‘GEOMETRY’}.

e) {x | x is a vowel in the word ‘UMBRELLA’}.

f) Set of all months in a year.

g) Set of all months in a year having at least 30 days.

h) Set of even prime numbers.

i) {x | x ∈ N, x < 15}.

j) {m | m is a month of a year and m has 28 days}.

k) {m | m is a month of a year and m starts withtheletterJ}.

5. Rewritethefollowingsetsusingsetbuildermethod.

a) A = {2, 4, 6, 8, 10}.

b) B = {1, 3, 5, 7, 9}.

c) Set of multiples of 7.

d) Set of multiples of 2.

e) Set of vowels in the word ‘FILL’.

Exercise 1 .1

types of SetsFinite Set: A set that contains limited (countable) number of elements is called a finite set.

For example,

a) set of students in a school.

b) set of towns in India.

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c) set of people of a particular community.

d) set of natural numbers less than four.

e) A = {1, 2, 3, 4, 5, …, 10000}.

Infinite Set: A set that has unlimited (uncountable) number of elements is called an infinite set.

For example,

a) the set of all integers.

b) the set of all natural numbers.

c) the set of stars in the sky.

Empty Set: A set containing no elements is called an empty set. It is also known as a void set or a null set. It is denoted by the Greek letter ‘ϕ’ read as ‘phi’ or ‘{ }’.

For example,

a) A = {multiples of 10 but less than 10}.

Since there is no multiple of 10 which is less than 10, A = ϕ.

b) B = {even number between 22 and 24}.

Since there is no even number lying between 22 and 24, B = ϕ.

c) C={primenumberbetween14and16}.

Sincethereisnoprimenumberexistingbetween14and16,C=ϕ.

Non-empty Set: A set that has at least one element is called a non-empty set.

For example, A = {ϕ}, is a non-empty set.

[Note: Non-empty sets are sometimes also called non-void sets.]

Singleton Set: A non-empty set containing a single element is called a singleton set.

For example,

a) if A = {0}, then A is a singleton set.

b) if B = {odd numbers between 98 and 100}, then B = {99}.

Therefore, B is a singleton set.

c) if A = {x : x is an odd prime number and x < 5}, then A = {3}.

Therefore, A is a singleton set.

d) if A = {m : m is a month whose name begins with D}, then A = {December}.

Therefore, A is a singleton set.

Cardinal Number of a SetThe number of elements in a finite set is called the cardinal number of the set. The cardinal number is used to denote the size of a set. If A is any finite set, then its cardinal number is denoted by n(A).

Rememberϕisanemptyset.But{ϕ}or{0}arenotemptysetsaseachofthese sets contains one element.

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Let us consider the following sets:

A = {b, e, g, o, s, r, t} and B = {m, a, i, n, k, j, x, y}

The number of elements in set A is 7, therefore the cardinal number of set A is 7 or n(A) = 7.

The number of elements in set B is 8, therefore the cardinal number of set B is 8 or n(B) = 8.

[Note: The cardinal number of an empty set is zero, that is, n (ϕ) = 0.]

Equal Sets: Two sets A and B are said to be equal, if they have exactly the same elements, that is, every element of set A is an element of set B and every element of set B is an element of set A.

Let us consider two sets A and B, where A = {b, e, g, o, s, r, t} and B = {e, t, o, r, s, g, b}.

Since A and B have exactly the same elements, they are called equal sets.

Equivalent Sets: Two sets are said to be equivalent if the cardinal number of both the sets is the same. If A and B are two equivalent sets, we write it as A ↔ B.

For example,

a) If A = {a, b, c, d} and B = {p, q, r, s}, then

n(A) = 4 and n(B) = 4. Therefore, A ↔ B.

b) IfC={1,2,3,4}andD={2,c, d}, then n(C)=4andn(D) = 3.

Therefore,CisnotequivalenttoD.

Solved examples Example 1: State whether the following sets are finite or infinite:

a) A = {2, 3, 5, 7, 11, 13, …}

b) B={consonantsoftheEnglishalphabet}

c) C={multiplesof2}

d) D = {even prime numbers}

Solution:

a) Set A is infinite as it contains unlimited number of prime natural numbers.

b) Set B is finite because consonants of the Englishalphabetarelimitedandcountable.

c) Set C is infinite as it contains unlimited number of elements.

d) Set D is finite because it contains only one element, that is, 2.

Example 2: State whether the following sets are empty or not:

a) Set of odd prime natural numbers less than 3.

b) Set of even natural numbers less than 5.

c) Set of whole numbers divisible by 3 and less than 10.

d) Set of natural numbers less than 1.

Solution:

a) There are only two natural numbers less than 3, that is, 1 and 2.

Now, 1 is neither prime nor composite and 2 is an even prime number.

Therefore, the set of odd prime natural numbers less than 3 is an empty set.

b) Set of even natural numbers less than 5 = {2, 4} which is a non-empty set.

c) Set of whole numbers divisible by 3 and less than 10 = {3, 6, 9} which is a non-empty set.

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d) 1 is the smallest natural number. There is no natural number less than 1.

Therefore, the set of natural numbers less than 1 is an empty set.

Example 3: Write the cardinal number of each of the following sets:

a) A = {violet, indigo, blue, green, yellow, orange, red}.

b) B = {India, Sri Lanka, Maldives, Bangladesh, Pakistan, Nepal}.

c) C={Kolkata,Karnataka,Bombay}.

d) D = {x : x is a 2-digit number and the sum of its digits is 6}.

Solution:

a) A = {violet, indigo, blue, green, yellow, orange, red}.

Number of elements in set A is 7. Therefore the cardinal number of set A is 7 or n(A) = 7.

b) B = {India, Sri Lanka, Maldives, Bangladesh, Pakistan, Nepal}.

Number of elements in set B is 6. Therefore, the cardinal number of set B is 6 or n(B) = 6.

c) C={Kolkata,Karnataka,Bombay}.

NumberofelementsinsetCis3.Therefore,thecardinalnumberofsetCis3orn(C)=3.

d) Since D = {x : x is a 2-digit number and the sum of its digits is 6},

D = {15, 24, 33, 42, 51, 60}.

Number of elements in set D is 6. Therefore, the cardinal number of set D is 6 or n(D) = 6.

Example 4: Find whether the following sets are equal or not.

a) A = {a, 1, 3, 3, b, b, b} and B = {3, 1, 1, a, b}.

b) A = {1, b} and B = {1, a}.

Solution:

a) A = {a, 1, 3, 3, b, b, b} = {a, 1, 3, b} and

B = {3, 1, 1, a, b} = {a, 1, 3, b}

Since A and B have the same elements, they are equal.

b) Since b ∈ A but b ∉ B, the two sets A and B are not equal.

Example 5: Which of the following sets are equivalent sets?

A = {a, f, j, q},B={1,2,3,5,8},C={x, y, z, w}, D = {8, 1, 3, 5, 2}

Solution:

a) Since n(A) = 4 and n(C) = 4, A and C areequivalent sets.

b) Since n(B) = 5 and n(D) = 5, B and D are equivalent sets.

Exercise 1 .21. Classifythefollowingsetsasfiniteor

infinite.

a) Set of four seasons in a year.

b) Set of even numbers.

c) Set of multiples of 9 more than 8 and less than 46.

d) Setoflettersin‘BUSINESSMEN’.

e) Setofmembersof‘SAARC’association.

f) Set of multiples of 2.

2. Classifythefollowingsetsasemptyorsingleton sets.

a) Setofvowelsintheword‘THRILL’.

b) Set of multiples of 4 more than 4 and less than 12.

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c) Setofvowelsintheword‘CALL’.

d) Set of multiples of 7 more than 3 and less than 4.

e) Set of multiples of 6 more than 18 and less than 30.

f) Set of multiples of 3 more than 4 and less than 5.

3. Write the cardinal number of the following sets.

a) B = {u, i, o, p, a, s, m}

b) D = {6, 7, 8, 9, 10, 11, 12, 46}

c) A = {s, d, f, g, h, j, k, l, z, x, c, m}

d) C={0}

e) E={e, r, m}

f) C={a, s, d, f, g, h, j, k, l, z, n}

4. Which of the following sets are equivalent, non-equivalent or equal sets?

a) A = Set of letters in the word ‘BOTTOM’, B = Set of letters in the word‘CATCH’.

b) A = Set of vowels in the word ‘MOLECULE’,B=Setofvowelsintheword‘MOMENTUM’.

c) A = {b, c, d, y} and B = { 2, 3, 4, 5, 25}

d) A = Set of letters in the word ‘CHEMOTHERAPY’,B=Setoflettersintheword‘CHEMOSURGERY’.

e) A = Set of multiples of 8 more than 23 and less than 30, B = Set of multiples of 2 more than 23 and less than 30.

f) A = Set of multiples of 2 greater than 3 and less than 20, B = {1, 3, 5, 7, 9, 11, 13, 17}.

• A set is a collection of well-defi ned distinct objects.

• The objects in a set are known as its elements or members.

• If x is an element of a set A, then we write it as x ∈ A and read as ‘x belongs to set A’.

• If x is not an element of a set A, then we write it as x ∉ A and read as ‘x does not belong to set A’.

• A set containing no elements is called an empty set or a null set. It is denoted by { } or ϕ.

• A set which contains at least one element is called a non-empty set.

• A non-empty set containing only one element is called a singleton set.

• A set whose elements can be counted is known as a fi nite set.

• A set having unlimited number of elements is called an infi nite set.

• The number of elements in a set is called its cardinal number. The cardinal number of a set A is denoted by n(A).

• Sets A and B are said to be equal if they have exactly the same elements and we write them as A = B.

• Sets A and B are said to be equivalent if they have the same cardinal number, that is, n(A) = n(B) and we write them as A ↔ B.

• All equal sets are equivalent but all equivalent sets may or may not be equal.

Recap

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Mental Maths Fill in the blanks.

1. A collection of well-defined distinct objects is called a ___________.

2. Objects used to form a set are called its ___________ or ___________.

3. A set is said to be ___________ if it has an unlimited number of elements.

4. Asetthathasnoelementiscalleda/an___________set.

5. If two given sets have the same cardinal number, they are called ___________ sets.

Review time Multiple Choice Questions1. Which of the following collections represents

a set?

a) a collection of good books

b) a collection of good movies

c) acollectionofvowelsintheEnglishalphabet

d) a collection of beautiful necklaces

2. Which of the following sets is written in the set builder form?

a) Q = {1, 2, 3, 4, 5}

b) A = {x : x is a counting number greater than 9}

c) P = {The last four months of a year}

d) None of these

3. Asetissaidtobea/an___________setifit has a limited number of elements.

a) equal b) finite

c) infinite d) equivalent

4. A set which contains only one element is calleda/an____________set.

a) empty b) disjoint

c) singleton d) equal

5. Which of the following pairs of sets are equal?

a) P = {0, 1, 2, 3, 4}, Q = {x : x is a whole number less than 5}

b) A = {a, b, c, d}, B = {p, q, r, s}

c) M = {days of the week}, N = {months of the year}

d) None of these

Subjective Questions1. State whether the following collections are

sets or not.

a) Collectionofsmartkidsinaschool.

b) Collectionofoddprimenumberslessthan 50.

c) CollectionofdifficultchaptersinaMaths book.

d) Collectionofoldpeopleinatown.

2. Write the following sets in roster form:

a) A = {x : x is a month of the year having 31 days}

b) B = {x : x ∈ N and x ≤ 5}

c) C={lettersinthewordCERTIFICATE}

d) D = {days of the week}

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3. Write the following sets in set builder form.

a) A = {5, 10, 15, 20, 25, 30}

b) B = {consonants in the word KNOWLEDGE}

c) C={2,3,5,7,11,13,17,19,23}

d) D = {−3, −2, −1, 0, 1, 2, 3}

4. LetA={2,3,5,7,11,13}.Classifythefollowing statements as true or false.

a) 2 ∈ A b) 10 ∈ A

c) 6 ∉ A d) 13 ∉ A

5. Classifythefollowingsetsasempty,finiteorinfinite.

a) Set of months in a year having more than 31 days.

b) Set of multiples of 5.

c) Set of all even prime numbers > 2.

d) Set of even prime numbers.

e) Set of odd natural numbers whose square is less than 50.

6. Classifythefollowingsetsassingletonorempty.

a) A = {m : m is a month of the year and m has 29 days in a leap year}

b) B = {x : x is a prime natural number < 2}

7. State whether the following pairs of sets are equivalent sets or not:

a) P = {1, 2, 3, 4} and Q = {a, b, c, d}

b) R={w, x, y, z} and S = {a, e, i, o, u}

8. Write the cardinal number of the following sets:

a) A = {x : x is a letter in the word IMPORTANT}

b) B = {x : x = 2n, n ∈ W, n ≤ 5}

think and answer 1. Write the following sets in roster form and description form.

a) {x : x = 2n, n is a natural number}

b) {x : x = 2n − 1, n is a natural number}

2. State whether the following sets are equal or not.

a) C={x | x is a multiple of 2 and x < 10} and D = {2, 4, 6, 8}

b) E={x : x is a prime number less than 10} and F = {1, 3, 5, 7, 9}

Real-life Connect Yogesh’steacheraskedhimtomakeachartshowingthefooditemshetookinthelastten days. Following is the list he made:

Noodles,Burger,Apple,Pizza,Banana,MasalaDosa,Yogurt,Almonds,Pomegranate,Samosa,MomosandFriedRice.

Differentiate the above food items in the sets of junk foods and healthy foods. Why should we not include such food items in our diet?

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Lab activity Objective: To give the idea of sets by sorting the objects of the same kind.

Materials Required:Cutoutsofplaneshapesliketriangles,rectangles,squares,circlesandovals.

Steps:

1. Bring some cutouts of the plane shapes in the class and keep them in five different bowls.

2. Divide the class into five groups.

3. Ask each group to come one by one and pick any one bowl from the table.

4. Now, ask them to sort and write the number of the cutout of each shape in their bowls in a table as shown below.

Name of the Shape Number

Triangle

Square

Rectangle

Circle

Oval

5. Tell students that each bowl represents a set.

6. Ask each group to find the cardinality of each set and also identify which sets are equivalent (if any).

7. Repeattheactivitywithdifferentobjects.

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Warm up 1. Write the following in words.

a) 52,789 b) 15,234 c) 86,742 d) 45,365

e) 57,579 f) 82,455 g) 38,222 h) 36,467

2. Write the following in figures.

a) Ninety-six thousand eight hundred fifty-four

b) Fifty-three thousand one hundred seven

c) Sixty-two thousand seven hundred nine

3. Write the place value and the face value of underlined digits in the following numbers.

a) 62,431 b) 19,654 c) 25,791 d) 15,607

e) 25,580 f) 66,769 g) 74,314 h) 61,483

4. Find the sum of the place values of the

a) two 5s in 45,425. b) two 7s in 73,721.

5. Write the following numbers in expanded form.

a) 54,036 b) 26,919 c) 30,123 d) 41,178

Knowing Our Numbers

2

• Read and write the numbers up to nine digits

• Represent numbers according to the Indian System of numeration

• Find the place value and the face value of various digits in a given number

• Write numbers in expanded form and short form

• Write the successor and the predecessor of a given number

• Compare and order large numbers

MY LEARNING PLAN

• Form greatest and smallest possible numbers using the given digits

• Represent numbers according to the International System of numeration

• Use of large numbers in measurements• Perform operations (+, −, × and ÷) on

large numbers• Round off numbers to the nearest tens,

hundreds, thousands, ten thousands and so on

• Estimate the sum, difference, product and quotient

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6. Write the following in short form.

a) 40,000 + 6,000 + 400 + 20 + 8 b) 90,000 + 2,000 + 400 + 70 + 3

c) 10,000 + 8,000 + 100 + 80 + 7 d) 70,000 + 4,000 + 200 + 50 + 4

7. Write the successor and the predecessor of the following numbers.

a) 32,368 b) 26,735 c) 73,289 d) 54,201

e) 78,789 f) 87,687 g) 56,787 h) 11,365

8. Comparethefollowingnumbersusing‘<’,‘>’or‘=’sign.

a) 10578 10758 b) 34157 34075

c) 79165 71965 d) 40897 40897

9. Write the following numbers in ascending and descending order.

a) 88,257; 12,254; 46,580; 53,053 b) 86,593; 23,957; 40,370; 59,368

c) 65,710; 87,106; 86,789; 98,215 d) 68,516; 91,240; 67,839; 54,121

10. Write the greatest and the smallest 5-digit numbers using the given digits only once.

a) 1, 7, 6, 8, 2 b) 2, 6, 5, 3, 7

c) 1, 6, 4, 7, 2 d) 8, 3, 5, 7, 6

Puzzle Solve the alphametics (identify the value of each letter with the given clue):

a) S L O W S L O W

+ O L DO W L S

S = 2L = 1

b)

LaRge NuMBeRS Numbers are one of the most important tools in our daily life, whether it is your parents’ phone number, address, number of students in your class and so on, numbers are used everywhere.

We have learnt to read and write numbers up to five digits and we know that

• 10,000 is the smallest 5-digit number. It is read as ten thousand.

• 99,999 is the greatest 5-digit number. It is read as ninety-nine thousand nine hundred ninety-nine.

Do you know? What happens when we add 1 to the greatest 5-digit number?

By adding 1 to the greatest 5-digit number, we get the smallest 6-digit number, that is,

99,999 + 1 = 1,00,000. It is read as one lakh.

A B+C4 5

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The greatest 6-digit number is 9,99,999 and it is read as nine lakh ninety-nine thousand nine hundred ninety-nine.

Similarly,

• 9,99,999+1=10,00,000(smallest7-digitnumber).

The greatest 7-digit number is 99,99,999.

• 99,99,999+1=1,00,00,000(smallest8-digitnumber).

The greatest 8-digit number is 9,99,99,999.

• 9,99,99,999+1=10,00,00,000(smallest9-digitnumber).

The greatest 9-digit number is 99,99,99,999.

There are two systems of numeration to read and write large numbers:

i) Indian System of Numeration and

ii) International System of Numeration.

INDIaN SySteM oF NuMeRatIoN Numeration system is a way of representing numbers. The system that we use in our country is known as Indian System of Numeration or Hindu-Arabic System of Numeration. It was originally developed in India over hundreds of years ago and later adopted by the Arab mathematicians.

According to this system, we use 10 symbols: 0, 1, 2, 3, 4,5,6,7,8and9torepresentanynumber.Eachofthesesymbols is called a digit or a figure. Let us have a look at the Indian place value chart.

Indian Place Value ChartPeriods Crores Lakhs Thousands Ones

Places

Ten crores Crores Ten

lakhs

Lakhs Ten

thousands

Thousands Hundreds Tens Ones

TC C TL L TTh Th H T O

Numbers 10,00,00,000 1,00,00,000 10,00,000 1,00,000 10,000 1,000 100 10 1

In the place value chart, we make the following observations:

• Theones,tensandhundredsplacesmakeones period.

• Thethousandsandtenthousandsplacesmakethousands period.

• Thelakhsandtenlakhsplacesmakelakhs period.

• Thecroresandtencroresplacesmakecrores period.

Maths Tips• Whenwewriteanumberusing

digits, it is called numeral.• Whenwewriteanumberin

words, it is called numeration.

RememberZeroisbelievedtobeinventedinIndia. It was called ShunyabytheIndians and SifrbytheArabs.

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Reading and Writing Large NumbersTo make large numbers easy to read, we place comma after each period.

For example, 73569 and 6408843 can conveniently be written as 73,569 and 64,08,843, respectively.

Now to read these numbers, read the digit in each period with the name of the period.

For example, we read 73,569 as seventy-three thousand five hundred sixty-nine.

Similarly, we read 64,08,843 as sixty-four lakh eight thousand eight hundred forty-three.

Place Value and Face Value Place ValueThe place value of a digit in a number depends on its position.

For example, in the numbers 356 and 536, 6 is at the ones place but the places where 3 and 5 occur are different.

In 356, the place value of 3 is 300 and that of 5 is 50.

In 536, the place value of 5 is 500 and that of 3 is 30.

Thus, the place value of 5 at tens place is 50 and at hundreds place is 500.

Similarly, the place value of 3 at tens place is 30 and at hundreds place is 300.

Face ValueThe face value of a digit in a number is the value of the digit itself, regardless of the place it occupies in the number.

For example, in both 356 and 536, the face value of 5 is 5, that of 3 is 3 and of 6 is 6.

exPaNDeD FoRM aND ShoRt FoRMAny numeral represented as the sum of the place values of its digits is said to be in expanded form.

Let us write 73,569 in the expanded form.

For this, we first find out the place values of all its digits.

In 73,569, the place values of 7, 3, 5, 6 and 9 are

70000, 3000, 500, 60 and 9, respectively.

Thus, the expanded form of 73,569 is 70,000 + 3,000 + 500 + 60 + 9.

Similarly, the expanded form of 64,08,843 is 60,00,000 + 4,00,000 + 0 + 8,000 + 800 + 40 + 3.

RememberTheplacevalueofthedigit0remains0regardlessofitspositioninthenumber.

Maths TipWecanfindtheplacevalueofeachdigitofanumberbymultiplyingthedigit’sfacevaluewithitsposition.

Thenumeralwritteninitsoriginal form is called in standard form or in short form.

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Example 1: Write the following numbers accordingtotheIndianPlaceValueChart.Insertcommas to separate the periods. Write number name also.

a) 42645306 b) 230830016

Solution:

a) C TL L TTh Th H T O

4 2 6 4 5 3 0 6

Number: 4,26,45,306

Number Name: Four crore twenty-six lakh forty-five thousand three hundred six

b) TC C TL L TTh Th H T O

2 3 0 8 3 0 0 1 6

Number: 23,08,30,016

Number Name: Twenty-three crore eight lakh thirty thousand sixteen

Example 2: Find the face value and the place value of each digit of the following numerals.

a) 8627572 b) 65907532

Solution:

a) 86,27,572

Digit Face Value

Place Value

8 8 8 × 10,00,000 = 80,00,000

6 6 6 × 1,00,000 = 6,00,000

2 2 2 × 10,000 = 20,000

7 7 7 × 1,000 = 7,000

5 5 5 × 100 = 500

7 7 7 × 10 = 70

2 2 2 × 1 = 2

b) 6,59,07,532

Digit Face Value

Place Value

6 6 6 × 1,00,00,000 = 6,00,00,000

5 5 5 × 10,00,000 = 50,00,000

9 9 9 × 1,00,000 = 9,00,000

0 0 0 × 10,000 = 0

7 7 7 × 1,000 = 7,000

5 5 5 × 100 = 500

3 3 3 × 10 = 30

2 2 2 × 1 = 2

Example 3: Find the sum of the place values of the two 5s in the number 25,46,758.

Solution:

The place value of 5 at the tens place = 50

The place value of 5 at the lakhs place = 5,00,000

So, the sum of the two place values

= 5,00,000 + 50 = 5,00,050.

Example 4: Find the difference between the place values of the two 1s in the number 3,17,86,751. Also, find the difference between the face values of the two 1s.

Solution:

The place value of 1 at the ones place = 1

The place value of 1 at the ten lakhs place

= 10,00,000

So, the difference between the two place values

= 10,00,000 − 1 = 9,99,999.

The face value of the two 1s will be 1.

So, the difference between the two face values= 1 − 1 = 0.

Solved examples

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1. Write the following numbers according the IndianPlaceValueChart.Insertcommastoseparate the periods.

a) 863152 b) 392689 c) 925064

d) 56903275 e) 19427632 f) 652346093

2. Write the following in words.

a) 34,566 b) 9,02,341 c) 66,89,904

3. Write the following in figures.

a) Twenty lakh seventy-three thousand four hundred one

b) Four crore fifty-eight lakh one thousand two hundred twelve

c) Ninety-nine crore sixty-two lakh forty-four thousand six hundred eighty-seven

4. Write the place value of underlined digit in the following numerals.

a) 76,59,820 b) 2,15,634 c) 2,90,67,881

Example 5: Write the following numbers in expanded form.

a) 3,00,087 b) 28,56,382

Solution:

a) 3,00,087

The place values of each digit of the given number are 7; 80; 0; 0; 0 and 3,00,000.

Thus, the expanded form of 3,00,087

= 3,00,000 + 80 + 7

b) 28,56,382

The place values of each digit of the given number are 2; 80; 300; 6,000; 50,000; 8,00,000 and 20,00,000.

Thus, the expanded form of 28,56,382

= 20,00,000 + 8,00,000 + 50,000 + 6,000 + 300 + 80 + 2

Example 6: Write the following numbers in short form.

a) 40,00,000 + 30,000 + 4,000 + 300 + 3

b) 8,00,00,000 + 10,00,000 + 2,00,000 + 30,000 + 2,000 + 300 + 80 + 5

Solution:

a) For 40,00,000 + 30,000 + 4,000 + 300 + 3,

the corresponding numeral is 40,34,303.

b) For 8,00,00,000 + 10,00,000 + 2,00,000 + 30,000 + 2,000 + 300 + 80 + 5,

the corresponding numeral is 8,12,32,385.

5. Write the face value of underlined digits in the following numerals.

a) 84,537 b) 78,24,580 c) 3,56,909

6. Write the following numbers in expanded form.

a) 29,48,028 b) 3,84,64,739c) 2,83,80,074

7. Write the following numbers in short form.

a) 5000000 + 60000 + 3000 + 100 + 50 + 6

b) 800000 + 40000 + 2000 + 800 + 30 + 8

c) 100000000 + 3000000 + 50000 + 400 + 70 + 5

8. Find the sum of the place values of the two 6s in the number 1,68,97,634.

9. Find the difference between the place values of the two 9s in the number 5,69,26,590.

10. Find the product of the face value of the digit at the tens place and the place value of 5 in the number 7,65,14,208.

Exercise 2.1

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SuCCeSSoR aND PReDeCeSSoRThe successor of a number is the number that comes just after it. To find the successor of a given number, just add 1 to it.

For example, the successor of 44,598 = 44,598 + 1 = 44,599.

The predecessor of a number is the number that comes just before it. To find the predecessor of a given number, just subtract 1 from it.

For example, the predecessor of 44,598 = 44,598 − 1 = 44,597.

CoMPaRISoN oF LaRge NuMBeRSThe following two cases may arise while comparing two given numbers:

i) When the numbers have different number of digits.

ii) When the numbers have the same number of digits.

Case 1: When the numbers have different number of digits

If the two numbers have different number of digits, then the number with more digits is greater.

For example, in 3,62,783 and 64,738, the first number has 6 digits and the second number has 5 digits.

So, the first number is greater than the second number, that is, 3,62,783 > 64,738.

Case 2: When the numbers have the same number of digits

If the given numbers have the same number of digits, then we start comparing them from the highest place. The number with the greater digit at the highest place is greater. If the digits at the highest place are the same, then we compare the digits at the next place and so on.

Let us compare 73,463 and 73,946.

As both the numbers have 5 digits, we compare the digits at the highest place, that is, ten thousands place.

In both the numbers, the digit at the ten thousands place is 7, so we need to compare the thousands place. The digit at the thousands place in both the numbers is also the same, that is, 3.

So, we compare the next place. In the first number, the digit at the hundreds place is 4 and in the second number, it is 9.

Since 4 < 9; 73,463 < 73,946.

aSCeNDINg aND DeSCeNDINg oRDeRSThe pattern of writing numbers from the smallest to the greatest is known as ascending order and the pattern of writing numbers from the greatest to the smallest is known as descending order.

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Let us arrange the following numbers in ascending and descending orders:

86,257; 7,43,827; 6,38,658; 21,653

Ascending order: 21,653; 86,257; 6,38,658; 7,43,827

Descending order: 7,43,827; 6,38,658; 86,257; 21,653

FoRMINg gReateSt aND SMaLLeSt NuMBeRSWhen Digits Cannot Be RepeatedTo form the greatest number, we arrange the digits in descending order.

For example, the greatest 4-digit number that can be formed by using the digits 9, 8, 2 and 7 is 9872.

To form the smallest number, we arrange the digits in ascending order.

For example, the smallest 4-digit number that can be formed by using the digits 9, 8, 2 and 7 is 2789.

In the case when one of the digits is zero, we place 0 at the second place from the left and arrange the remaining digits in ascending order to form the smallest number.

For example, the smallest 4-digit number that can be formed by using the digits 9, 0, 4 and 2 is 2049.

When Digits Can Be RepeatedFirst form the smallest or greatest number using the given digits each only once without repetition. Then, in the number formed, replace the smallest or greatest digit by repeating it the number of times allowed.

For example, the smallest 3-digit number formed by using the digits 3 and 6, repeating 3 (the smallest digit) twice, is 336 while the greatest 3-digit number formed by using the same digits and repeating 6 (the greatest digit) two times is 663.

Similarly, the smallest 4-digit number formed by using the digits 3, 5 and 0, repeating 0 twice, is 3005 while the greatest 4-digit number formed by using the same digits and repeating 5 two times is 5530.

Solved examples Example 1: Find the successor of the following numbers.

a) 3,67,828 b) 86,18,699

Solution:

a) The successor of 3,67,828 is

3,67,828 + 1 = 3,67,829

b) The successor of 86,18,699 is

86,18,699 + 1 = 86,18,700

Example 2: Find the predecessor of the following numbers.

a) 7,35,261 b) 1,24,530

Solution:

a) The predecessor of 7,35,261 is

7,35,261 − 1 = 7,35,260

b) The predecessor of 1,24,530 is

1,24,530 − 1 = 1,24,529

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Example 3: Find the numbers whose predecessor and successor is 7,56,351.

Solution:

7,56,351 is the predecessor of its successor, that is,

7,56,351 + 1 = 7,56,352.

7,56,351 is the successor of its predecessor, that is,

7,56,351 − 1 = 7,56,350.

Thus, the numbers are 7,56,350 and 7,56,352.

Example 4: Comparethefollowingnumbers.

a) 3,46,289 and 34,628

b) 53,628 and 52,781

c) 6,87,467 and 6,88,754

Solution:

a) The first number has 6 digits and the second number has 5 digits.

Hence, 3,46,289 > 34,628.

b) Both the numbers have 5 digits.

Both the numbers also have 5 at the highest place, that is, at the ten thousands place.

The digits at the next place, that is, at the thousands place are 3 and 2, respectively.

As 3 > 2; 53,628 > 52,781.

c) Both the numbers have 6 digits.

The digit at the highest place, that is, at the lakhs place, is 6 in both the numbers.

The digit at the next place, that is, at the ten thousands place, is 8 in both the numbers.

The numbers at the next place, that is, at the thousands place are 7 and 8, respectively.

As 7 < 8; 6,87,467 < 6,88,754.

Example 5: Write the following numbers in ascending and descending orders.

2,67,049; 12,65,824; 5,987; 8,95,313

Solution:

Ascending order:

5,987 < 2,67,049 < 8,95,313 < 12,65,824

Descending order:

12,65,824 > 8,95,313 > 2,67,049 > 5,987

Example 6: Form the greatest and the smallest 6-digit numbers using the digits 0, 7, 2, 5, 9 and 3 only once.

Solution:

To form the greatest 6-digit number, arrange the digits in descending order.

We observe that 9 > 7 > 5 > 3 > 2 > 0.

So, the greatest 6-digit number is 9,75,320.

To form the smallest 6-digit number, arrange the digits in ascending order.

We observe that 0 < 2 < 3 < 5 < 7 < 9.

As we cannot have 0 at the extreme left, exchange it with the digit on its immediate right.

Thus, the smallest 6-digit number is 2,03,579.

Example 7: Write the greatest and the smallest 5-digit numbers using the digits 3, 8, 4 and 0 and repeating any one of them.

Solution:

To form the greatest 5-digit number, arrange the digits in descending order and repeat the greatest digit twice. We observe that 8 < 4 < 3 < 0.

Thus, the greatest 5-digit number is 88,430.

To form the smallest 5-digit number, we repeat the smallest digit twice.

Thus, the smallest 5-digit number is 30,048.

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Exercise 2.2

Example 8: Find the greatest and the smallest 6-digit numbers that can be formed using all different digits.

Solution:

To form the greatest 6-digit number, we use six greatest digits, which are 9, 8, 7, 6, 5 and 4. Arranging these digits in decreasing order, we get 9,87,654 as the greatest 6-digit number with all different digits.

To form the smallest 6-digit number, we use the six smallest digits, which are 0, 1, 2, 3, 4 and 5. Arranging them in increasing order, we get

0 1 2 3 4 5

We cannot put 0 at the highest place as it will result in 012345, which is a 5-digit number.

Therefore, we place 1 at the highest place and 0 at the second highest place. Thus, the smallest 6-digit number with all different digits is 1,02,345.

1. Find the successor and the predecessor of the following numbers.

a) 24,32,499 b) 20,00,000 c) 30,00,900

2. Find the numbers whose successors are the following numbers.

a) 3,67,879 b) 23,90,189 c) 9,13,47,000

3. Find the numbers whose predecessors are the following numbers.

a) 3,78,908 b) 41,099 c) 81,30,000

4. Comparethefollowingnumbersusing<or>sign.

a) 27,984 64,378

b) 46,52,798 46,20,089

c) 83,78,395 83,78,095

5. Write the following numbers in ascending order.

a) 3,87,018; 45,372; 47,980; 1,23,457

b) 9,038; 92,34,517; 9,074; 9,526

c) 3,56,492; 92,781; 23,468; 69,805

6. Write the following numbers in descending order.

a) 54,723; 7,65,902; 32,415; 8,79,605

b) 3,479; 75,689; 21,345; 6,584

c) 48,672; 75,896; 4,32,687; 658

7. Form the smallest and the greatest 7-digit numbers using the digits 7, 0, 5, 4, 1, 9 and 3 without repetition.

8. Form the smallest and the greatest 6-digit numbers using the digits 6, 0, 2, 3, 1, 9 and 4 without repetition.

9. Write the greatest and the smallest 6-digit numbers using the digits 5, 7, 1, 4 and 0 repeating any one of them.

10. Write the smallest 5-digit number that can be formed with the smallest digits without repeating any.

11. Write the greatest 6-digit number that can be formed using all different digits.

12. Write the smallest 6-digit number that can be formed if only one digit is allowed to appear twice.

13. Write the smallest 4-digit number that can be formed with all the digits same.

14. Write the greatest 4-digit number that can be formed with all the digits same.

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INteRNatIoNaL SySteM oF NuMeRatIoNSo far, we have learnt about the Indian System of Numeration. The system of numeration using worldwide is known as International System of Numeration.

LetushavealookattheInternationalPlaceValueChart.

International Place Value Chart

Periods Millions Thousands Ones

Places

Hundred millions

Ten millions

Millions Hundred thousands

Ten thousands

Thousands Hundreds Tens Ones

HM TM M HTh TTh Th H T O

Numbers 100,000,000 10,000,000 1,000,000 100,000 10,000 1,000 100 10 1

In the place value chart, we make the following observations:

• Theones period has three place values—ones, tens and hundreds.

• Thethousands period has three place values—thousands, ten thousands and hundred thousands.

• Themillions period has three place values—millions, ten millions and hundred millions.

Like the Indian System, we separate periods using commas in the International System.

In the International System of Numeration, we place comma after every 3 digits starting from the right.

Some numbers in the International System of Numeration are shown below:

Numbers Names in International System of Numeration

631,725 Six hundred thirty-one thousand seven hundred twenty-five

934,566,045 Nine hundred thirty-four million five hundred sixty-six thousand forty-five

Comparing the two SystemsLet us read the number 760500831 in both the systems.

Indian System International System

Written as: 76,05,00,831 Written as: 760,500,831

Read as: Seventy-six crore five lakh eight hundred thirty-one

Read as: Seven hundred sixty million five hundred thousand eight hundred thirty-one

Remember• 1lakh=100thousands • 1million=10lakhs• 10millions=1crore • 100millions=10crore

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Solved examples Example 1: For the numbers given below, rearrange the commas as per the International PlaceValueChart.

a) 22,41,36,001 b) 1,19,57,642

Solution:

a) 22,41,36,001 = 224,136,001

b) 1,19,57,642 = 11,957,642

Example 2: Write the following numbers accordingtotheInternationalPlaceValueChart.Insert commas to separate the periods. Write number name also.

a) 71285364 b) 160532692

Solution:

Millions Thousands Ones

HM TM M HTh TTh Th H T O

a) 7 1 2 8 5 3 6 4

b) 1 6 0 5 3 2 6 9 2

a) Number: 71,285,364

Number Name: Seventy-one million two hundred eighty-five thousand three hundred sixty-four

b) Number: 160,532,692

Number Name: One hundred sixty million five hundred thirty-two thousand six hundred ninety-two

Example 3: Write the following in figures.

a) Nine hundred seventy-nine million eight hundred forty-six thousand four hundred thirty-three

b) Four hundred five million two hundred nine thousand twenty-five

Solution:

a) 979,846,433 b) 405,209,025

Example 4: Express4567329inboththeIndianand International Systems of Numeration.

Solution:

Indian System of Numeration:

4567329 is written as 45,67,329 and read as forty-five lakh sixty-seven thousand three hundred twenty-nine.

International System of Numeration:

4567329 is written as 4,567,329 and read as four million five hundred sixty-seven thousand three hundred twenty-nine.

Exercise 2.31. Write the following numerals according to

the International System of Numeration. Mark the periods with commas.

a) 4517301 b) 9340001

c) 70707971 d) 8579832

e) 5883279 f) 1417530

2. Write the following in figures and place commas correctly.

a) Ninety-seven lakh fifty-three thousand fifteen

b) Seventy-three crore five lakh eighty-five thousand three hundred five

c) Eighty-fivelakhtwo

d) Six million seven hundred eighty-two thousand five hundred six

e) Eighthundredeighty-eightmillioneighthundred thousand eighty-eight

f) One hundred million two hundred thousand five

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3. RearrangethecommasaccordingtotheInternationalPlaceValueChart.

a) 89,76,86,824 b) 49,00,70,100

c) 98,05,45,283 d) 25,36,87,304

e) 29,70,99,140 f) 11,06,79,402

g) 58,35,44,738 h) 67,87,36,438

4. Fill in the blanks.

a) 1 crore = _____ lakhs

b) 1 crore = _____ ten lakhs

c) _____ lakhs = 1 million

d) _____ millions = 1 crore

e) 1 lakh = _____ ten thousands

5. Fill in the blanks for the coloured digit. One has been done for you.

Number Indian system Place value International system

Place value

a) 23908748 2,39,08,748 9,00,000 23,908,748 900,000

b) 98105782

c) 87410235

d) 75693546

e) 564802894

f) 706573921

uSe oF LaRge NuMBeRS IN MeaSuReMeNtSWe frequently use large numbers while measuring length, mass and capacity.

Measuring LengthWe generally use the units millimetre (mm), centimetre (cm), metre (m) and kilometre (km) for measuring length, and we know that

• 1km=1000m

• 1m=100cm

• 1cm=10mm

So, we can write

1 km = 1000 m = 1000 × 100 cm = 1,00,000 cm = 1,00,000 × 10 mm = 10,00,000 mm.

Measuring MassWe generally use the units milligram (mg), gram (g) and kilogram (kg) for measuring mass, and we know that

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• 1kg=1000g

• 1g=1000mg

So, we can write

1 kg = 1000 g = 1000 × 1000 mg = 10,00,000 mg.

Measuring CapacityWe generally use the units litre (L), kilolitre (kL) and millilitre (mL) for measuring capacity, and we know that

• 1kL=1000L

• 1L=1000mL

So, we can write

1 kL = 1000 L = 1000 × 1000 mL = 10,00,000 mL.

Example: Comparethefollowingmeasurements:

a) 6 cm and 60 mm b) 20 L and 2000 mL c) 67 g and 6700 mg d) 1.5 m and 1500 cm

Solution:

a) 1 cm = 10 mm

6 cm = 6 × 10 mm = 60 mm

Since 60 mm = 60 mm, 6 cm = 60 mm.

b) 20 L = 20 × 1000 mL = 20,000 mL

Since 20,000 mL > 2000 mL, 20 L > 2000 mL.

c) 67 g = 67 × 1000 mg = 67,000 mg

Since 67,000 mg > 6700 mg, 67 g > 6700 mg.

d) 1.5 m = 1.5 × 100 cm = 150 cm

Since 150 cm < 1500 cm, hence, 1.5 m < 1500 cm.

Exercise 2.4Compare the following measurements and insert <, > or = symbol.

1. 4.5 m _________ 150 cm

2. 420 km _________ 42,00,000 m

3. 130 cm _________ 13 mm

4. 2800 g _________ 28 mg

5. 250 g _________ 2.5 kg

6. 3450 g _________ 3.45 kg

7. 3 L _________ 300 mL

8. 2.3 L _________ 23,000 mL

9. 9000 mL _________ 9 L

10. 12020 g __________ 12 kg 200 g

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oPeRatIoNS oN LaRge NuMBeRS Example 1: An automobile showroom sold two identical cars and a motorbike for a total of ` 26,38,450. If the cost of a car is ` 12,36,050, find the cost of the motorbike.

Solution:

The sum will be solved in the following two steps:

1 1

` 1 2 3 6 0 5 0+ ` 1 2 3 6 0 5 0

` 2 4 7 2 1 0 0

5 13

` 2 6 3 8 4 5 0− ` 2 4 7 2 1 0 0

` 1 6 6 3 5 0

Step 1: Step 2:

Thus, the cost of motorbike is ` 1,66,350.

Example 2: The difference between two numbers is 10,43,152. If the larger number is 35,84,769, find the smaller number.

Solution:

Larger number = 35,84,769

Difference between two numbers = 10,43,152

∴ Smaller number = 3 5 8 4 7 6 9− 1 0 4 3 1 5 2

2 5 4 1 6 1 7

Thus, the smaller number is 25,41,617.

Example 3: The cost of a wooden table is ` 2375. What is the cost of 236 such tables?

Solution:

Costof1table=` 2375

Costof236tables=` 2375 × 236

Thus, the cost of 236 wooden tables is ` 5,60,500.

Example 4: Anirudh earns ` 1,15,230 salary for 6 months. What is his monthly salary? Also, find out his annual income.

Solution:

a) Anirudh’s earnings in 6 months = ` 1,15,230

Monthly earning = ` 1,15,230 ÷ 6 = ` 19,205

So, Anirudh earns ` 19,205 every month.

b) Salary for 1 month = ` 19,205

Salary for 12 months = ` 19,205 × 12 = × ` 2,30,460

So, Anirudh’s annual income would have been ` 2,30,460.

L TTh Th H T O2 3 7 5

× 2 3 61 4 2 5 07 1 2 5 0

+ 4 7 5 0 0 05 6 0 5 0 0

1 9 2 056 1 1 5 2 3 0

− 65 5

−5 4

1 2 − 1 2

3 0

− 3 0

0

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Exercise 2.5Read these stories carefully and solve them.

1. There are 9,24,09,540 women; 7,65,85,372 men and 3,78,49,075 children in a city. What is the total population of the city?

2. ABCAirlinesrequiresapilottohave4,00,000 hours of flying experience before promotion.IfVinayakhascompleted3,46,928 hours, how many more hours does he need to qualify?

3. A,BandCcontestedforanelection.Thetotal number of votes were 75,95,394. Findthe votespolledforcandidateC,ifA and B got 23,95,710 and 3,21,001 votes, respectively. Who won the election and by how many votes?

4. Green Public School has two libraries. The number of books in the junior library is 48,473 and the number of books in the senior library is 73,602. What is the total number of books in both the libraries? How many more books does the senior library have than the junior library?

5. A school is preparing for its annual day functions. They plan to decorate the school

building with 15,340 strings of lights and each string of light has 112 bulbs. How many bulbs will be there for decoration?

6. At the college cafeteria, a large North-Indian thali costs ` 425 and a small thali costs ` 215. The college students purchased 110 large and 154 small thalis on a day. How much money did the college cafeteria collect for large thalis and small thalis—separately and in total?

7. A fruit company packed 2550 oranges in one carton. If the company has packed 275 cartons, how many oranges are packed in all?

8. There were 62,910 participants for the sports day which were divided into groups. There were 30 participants in each group. How many groups are there?

9. Sakshi got a notebook with 4100 pages. She has to use it for 5 subjects. How many pages can she have for each subject if she has to divide them equally?

10. A company distributes ` 24,78,900 bonus equally on diwali among its 80 employees. How much money does each employee get?

aPPRoxIMatIoN Observe the following examples:

1. Our school has approximately 1500 students.

2. About 1,00,000 people watched the cricket match.

3. This newspaper has a circulation of about 3,00,000 in UttarPradesh.

In the above examples, we have used the numbers that are close to the actual number but not the actual number. This is called approximation or estimation.

For estimation, numbers are generally rounded off to the nearest tens, hundreds, thousands and so on.

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Rounding off NumbersRounding off a Number to the Nearest TensFor rounding off a number to the nearest tens, follow these rules:

• Ifonesdigit<5,replaceitby0andkeeptheotherdigitsasitis.

• Ifonesdigit≥5,replaceitby0andincreasethetensdigitby1.

Rounding off a Number to the Nearest HundredsFor rounding off a number to the nearest hundreds, follow these rules:

• Iftensdigit<5,replacethetensandonesdigitsby0andkeeptheotherdigitsasitis.

• Iftensdigit≥5,replacethetensandonesdigitsby0andincreasethehundredsdigitby1.

Rounding off a Number to the Nearest ThousandsFor rounding off a number to the nearest thousands, follow these rules:

• Ifhundredsdigit<5,replacethehundreds,tensandonesdigitsby0andkeeptheotherdigitsthesame.

• Ifthehundredsdigit≥5,replacethehundreds,tensandonesdigitsby0andincreasethethousandsdigit by 1.

Rounding off a Number to the Nearest Ten ThousandsFor rounding off a number to the nearest ten thousands, follow these rules:

• Ifthousandsdigit<5,replacethethousands,hundreds,tensandonesdigitsby0andkeeptheotherdigits as it is.

• Ifthousandsdigit≥5,replacethethousands,hundreds,tensandonesdigitsby0andincreasetheten thousands digit by 1.

Rounding off a Number to the Nearest LakhsFor rounding off a number to the nearest lakhs, follow these rules:

• Iftenthousandsdigit<5,replacethetenthousands,thousands,hundreds,tensandonesdigitsby0and keep the other digits as it is.

• Iftenthousandsdigit≥5,replacethetenthousands,thousands,hundreds,tensandonesdigitsby0 and increase the lakhs digit by 1.

Solved examples Example 1: Roundoff4694tothenearesta) tens b) hundreds c) thousands

Solution:

a) In 4694, the ones digit is 4.

As 4 < 5, the ones digit becomes 0 and the rest of the digits remain unchanged.

Therefore, 4694 rounded off to the nearest tens as 4690.

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b) In 4694, the tens digit is 9.

As 9 > 5, the ones and tens digits become 0 and we increase the hundreds digit by 1.

Therefore, 4694 rounded off to the nearest hundreds as 4700.

c) In 4694, the hundreds digit is 6.

As 6 > 5, the ones, tens and hundreds digits become 0 and we increase the thousands digit by 1.

Therefore, 4694 rounded off to the nearest thousands as 5000.

Example 2: Roundoff86,67,405tothenearest

a) ten thousands b) lakhs

Solution:

a) In 86,67,405, the digit in the thousands place is 7.

As 7 > 5; 86,67,405 rounded off to the nearest ten thousands as 86,70,000.

b) In 86,67,405, the digit in the ten thousands place is 6.

As 6 > 5, 86,67,405 rounded off to the nearest lakhs as 87,00,000.

Estimating the Sum or DifferenceThere are many situations where we need to find a rough estimate of answers without performing actual addition or subtraction.

For estimating the sum or difference of two or more numbers, we need to round off the numbers.

Look at the following examples:

Example 1:Estimatethesum5679+12,345.

Solution:

Step 1: First check which is the smaller number. 5679 < 12,345.

As the smaller number has 4 digits, we will round off both the numbers to the nearest thousands.

Step 2: 12,345 rounded off to the nearest thousands = 12,000

5679 rounded off to the nearest thousands = + 6000

Estimatedsum = 18,000

Now, check whether the answer is a good estimate or not.

Actual sum = 5679 + 12,345 = 18,024

As the estimated sum is close to the actual sum, our estimate is correct.

Example 2:Estimatethedifference2,18,567−53,204.

Solution: We will round off both the numbers to the nearest ten thousands.

2,18,567 rounded off to the nearest ten thousands = 2,20,000

53,204 rounded off to the nearest ten thousands = −50,000

Estimateddifference = 1,70,000

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Now, check whether the answer is a good estimate or not.

Actual difference = 2,18,567 − 53,204 = 1,65,363

As the actual difference is close to the estimated difference, our estimate is correct.

estimating the Product or quotientWhen we estimate the product or quotient of given numbers, we round off each number to its greatest place value and then find the estimate by finding the product or quotient of rounded off numbers.

Let us elaborate the process with the help of a few examples.

Example 1:Estimate53×178.

Solution:

Roundoffthenumberstotheirhighestplaces.

In 53, the highest place is tens, so it is rounded off to 50.

In 178, the highest place is hundreds, so it is rounded off to 200.

Multiply the rounded numbers.

50 × 200 = 10,000

Actual product is 9434.

As the estimated product is close to the actual product, our estimate is correct.

Example 2:Estimate:a)62÷18 b)4968÷216

Solution:

a) First, round off the dividend and the divisor to their highest places.

62 is rounded off to 60.


Divide the rounded off numbers.

60 ÷ 20 = 3

Thus, the estimated quotient is 3.

b) First, round off the dividend and the divisor to their highest places.



To find the estimated quotient, divide 5000 by 200.

5000 ÷ 200 = 25

Thus, the estimated quotient is 25.

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Exercise 2.61. Roundoffthefollowingnumberstothe

nearest tens, hundreds and thousands.

a) Tens 39 55 48 64 13

b) Hundreds 872 104 211 394 822

c) Thousands 9722 7625 3181 1486 6101

2. Roundoffthefollowingnumberstothenearest ten thousands.

a) 77,723 b) 6,81,795

c) 8,61,392 d) 20,251

3. Roundoffthefollowingnumberstothenearest lakhs.

a) 44,69,744 b) 6,29,636

c) 54,38,361 d) 3,41,141

4. Estimatethefollowingsumsbyroundingoffthe numbers to their nearest thousands and compare the result with the actual sum.

a) 34,253 + 5524 b) 2507 + 3223

c) 19,887 + 43,501 d) 68,283 + 1843

e) 8650 + 26,181 f) 23,886 + 2899

5. Estimatethefollowingdifferencesbyrounding off the numbers to their nearest hundreds and compare the result with the actual difference.

a) 9739 − 5524 b) 12,507 − 223

c) 45,301 − 3501 d) 8283 − 543

e) 5979 − 2181 f ) 886 − 289

6. Estimatethefollowingproducts.

a) 9410 × 130 b) 2547 × 11

c) 3895 × 270 d) 2041 × 36

e) 7720 × 626 f) 8106 × 872

7. Estimatethefollowingquotients.

a) 4212 ÷ 52 b) 6496 ÷ 58

c) 3885 ÷ 15 d) 828 ÷ 18

e) 5580 ÷ 310 f) 940 ÷ 20

8. A music concert was held for four days in a city. The number of tickets sold at the counter on the first, second, third and fourth days was, respectively, 1,51,094, 81,812, 97,550 and 2,42,751. Find the approximate number of tickets sold in all four days to the nearest thousands.

9. There are 345 rows of flowers in a garden. Eachrowcontains48flowers.Findtheestimated number of flowers.

10. A factory manufactures 3,42,825 toys on Monday and 89,723 toys on Tuesday. Find, how many more toys were manufactured on Monday to the nearest hundreds.

• The place value of a digit in a number shows its place in the number.

• The face value of a digit in a number simply represents the digit’s value.

• The expanded form of a number represents the sum of the place values of all its digits.

• The successor of a number is the number that comes just after it.

• The predecessor of a number is the number that comes just before it.

• In the Indian System of Numeration, the periods are ones, thousands, lakhs, crores and so on.

Recap

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• In the International System of Numeration, the periods are ones, thousands, millions, billions and so on.

• The greatest number from given digits is formed by arranging the digits in descending order.

• The smallest number from given digits is formed by arranging the digits in ascending order. (Do not write 0 at the extreme left, that is, the highest place value).

• For estimation, numbers are generally rounded off to the nearest tens, hundreds or thousands.

Mental Maths 1. Find the numbers between 234 and 334 that remain the same when their digits are reversed.

For example, 303.

2. How many times does the digit 7 appear between 2567 and 2589?

3. Changethedigitsof89,570toformthegreatestandthesmallest5-digitnumbers.

Review time Multiple Choice Questions1. The number corresponding to three crore

thirty-five lakh two thousand one hundred twenty is

a) 3,05,02,120 b) 3,35,02,120

c) 30,52,120 d) 30,05,02,120

2. The sum of the face value and the place value of the digit 1 in 67,62,100 is

a) 1001 b) 101

c) 100 d) 1

3. The number formed by 10,00,000 + 3,00,000 + 4000 + 900 + 7 is

a) 1,34,097 b) 13,04,907

c) 13,04,097 d) 1,34,00,497

4. The successor of the number 15,638 is

a) 15,637 b) 15,640

c) 15,639 d) 15,638

5. The predecessor of the number 72,670 is

a) 72,669 b) 72,671

c) 72,672 d) 72,670

6. Which of the following numbers is the greatest?

a) 67,81,613 b) 1,68,73,683

c) 56,47,836 d) 1,35,426

7. Which of the following numbers is the smallest?

a) 2,67,81,613 b) 68,73,683

c) 6,41,806 d) 10,38,226

8. The smallest number that can be formed using the digits 3, 8, 9 and 0 is

a) 389 b) 3089

c) 3809 d) 3890

9. The greatest number that can be formed using the digits 3, 8, 9 and 0 is

a) 9830 b) 9803

c) 9876 d) 9038

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10. The sum of 6059 and 5876 can be rounded off to the nearest thousands as

a) 10,000 b) 11,000

c) 12,000 d) 13,000

Subjective Questions1. Find the sum of the place value of two 5s in

the number 6,78,62,575.

2. Write the following numbers in their expanded form as well as in words according to the Indian System of Numeration.

a) 6,02,81,546 b) 1,82,17,364

3. Find the successor and the predecessor of the following numbers.

a) 9,37,83,767 b) 24,45,676

4. Find the greatest and the smallest 4-digit numbers that can be formed using the digits

3, 5, 7 and 1 without repeating any of the digits.

5. Roundoffthenumbersasindicated.

a) 3,52,016 to the nearest hundreds

b) 1,19,930 to the nearest ten thousands

c) 25,49,063 to the nearest lakhs

d) 21,295 to the nearest thousands

6. Estimatethefollowingsumsanddifferencesby rounding off the numbers to their nearest thousands.

a) 97,318 + 3,94,032 b) 25,800 − 3,090

c) 3,775 + 4,77,834 d) 3,89,373 − 24,338

7. Estimatethefollowingproductsandquotients.

a) 1,68,205 ÷ 195 b) 8708 × 115

c) 21,295 × 45 d) 21,056 ÷ 241

think and answer 1. ATVnetworkclaimedthatabout4,50,00,000people(roundedtonearesttenlakhs)watchedthe

coverage of a special event. What could be the greatest possible and least possible number of peoplewhocouldhavewatchedtheprogrammesoastomaketheclaimofTVnetworkcorrect?

2. The first digit of a number is 8. Is it greater than a number whose first digit is 4? Think, discuss and write. A number has 7 in its thousands place and is less than 54,000. How many such numbers can you find?

3. ThetallestmountainintheUnitedStatesisMountMcKinleyinAlaska.Itis20,320feettall.ThetallestbuildingintheUnitedStatesistheSearstowerinIllinois.Itis1454feettall.ThetallestmountainintheworldisMountEverestwhichisabout28,848feettall.

a) How many times is Mount McKinley higher than Sears tower?

b) HowmanytimesisMountEveresthigherthanSearstower?

c) HowmanytimesisMountEveresthigherthanMountMcKinley?

Real-life Connect The table given below shows the prices and the quantity of some items sold by a grocer in a particular year.

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Items Price Quantity Sold

Rice ` 35 per kg 60,405 kg

Pulses ` 70 per kg 52,040 kg

Flour ` 22 per kg 80,560 kg

Sugar ` 36 per kg 3000 kg

Chips ` 25 per packet 2000 packets

Toothbrush ` 20 per piece 1500 pieces

a) Find the total weight of rice and sugar sold by the grocer.

b) Find out the amount of money the grocer earned by selling flour.

c) How much money does the grocer earn by selling sugar and chips? Does it exceed the amount he got from selling sugar and toothbrush?

d) Find out the amount of money the grocer earned by selling each of the items. By selling which item does he earn the maximum amount?

Interlinking with other Subjects Geography:

The table given below shows the population of eight major countries of the world.

Country Population (Estimated in year 2010)

Country Population (Estimated in year 2010)

Indonesia 23,41,81,400 USA 30,99,75,000

Pakistan 17,02,60,000 China 1,33,91,90,000

Bangladesh 16,44,25,000 Brazil 19,33,64,000

India 1,18,46,39,000 Nigeria 15,82,59,000

Study the table carefully and rank the countries from most populous to least populous.

Lab activity Objective: To form 6-, 7- and 8-digit numbers and write their names

Materials Required: Number cards (0 to 9)

Steps:

1. Teacher will pick up any six cards randomly from the ten available cards and arrange them to form a 6-digit number. Suppose, the number formed is 210867.

2. Teacher will then ask the students to write the number names in both the Indian and International systems for every arrangement.

3. The students are also asked to rearrange the digits and make the greatest as well as the smallest numbers (using the same digits) and then write their number names.

4. Perform this activity for numbers with 7, 8 and 9 digits.

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Warm up Make 10 number cards with digits from 0 to 9 written on them. Place them upside down. Pick any three cards and

a) form the greatest 3-digit number. Also, write its successor and predecessor.

Greatest 3-digit Number: _____; Successor: _____; Predecessor: _____

b) form the smallest 3-digit number. Also, write its successor and predecessor.

Smallest 3-digit Number: _____; Successor: _____; Predecessor: _____

Similarly, form greatest and smallest numbers with more than 3 digits.

NatuRaL NuMBeRSThe numbers 1, 2, 3, … are used for counting various things. These are called counting numbers or natural numbers.

Natural numbers are used for counting various things.

For example, the number of days, fruits in a basket, number of dolls in a shop.

[Note: 1 is the smallest natural number. As counting can go on endlessly, there is no largest natural number or we can say that there are infinite natural numbers.]

Natural Numbers and Whole Numbers

3

• Identify natural numbers and whole numbers

• Represent the whole numbers on a number line

• Perform four basic operations on the whole numbers using a number line

MY LEARNING PLAN

• Apply different properties of basic operations on the whole numbers

• Identify different patterns in whole numbers

Remember• Everynaturalnumber

except 1 has a predecessor.• Everynaturalnumberhasasuccessor.

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As has been explained in the previous chapter, when we add 1 to a natural number, we get the next natural number, which is its successor.

Similarly, when we subtract 1 from a natural number, we get its predecessor.

For example, 372 is the successor of 371 and 371 is the predecessor of 372.

the NuMBeR ZeRo Rajeshisfascinatedbythenumberzeroandaskedhismotherwhatexactlyzero was. His mother decided to explain it with an example. She said, “Suppose you have 11 marbles and you want to distribute them equally among three of your friends, how many marbles will you be left with?” Rajeshsaid,“Itissimple.Iwillgivethem3eachandtherewillbe2leftwith me.” His mother said, “What if you had 12 marbles?” He said “Oh! Eachofthemwouldget4andIwouldbeleftwithnothing.”

His mother said, “Well! This absence of marbles is represented by zero.

Youcansaythatyouwouldbeleftwithzeromarbles.”

WhoLe NuMBeRSWhen zero is included in the group of natural numbers, then this collection is called whole numbers. Thus, the whole numbers are 0, 1, 2, 3, … .

So, in whole numbers, 1 has a predecessor, that is, 0. Zero is the smallest whole number. The collection of whole numbers is also infinite, that is, there is no largest whole number.

[Note:Everynaturalnumberisawholenumberbuteverywholenumberisnotanaturalnumber.]

Remember• ThesetofnaturalnumbersisdenotedbyN={1,2,3,4,5,…}.• ThesetofwholenumbersisdenotedbyW={0,1,2,3,4,5,…}.

Representing Whole Numbers on a Number LineWe are already familiar with the number line. We also know that the distance between two consecutive points is called unit distance. Let us learn how to represent whole numbers on a number line.

Step 1: Draw a straight horizontal line.

Step 2: Mark a point on the extreme left and label it as 0.

Step 3: Mark another point to the right of 0. Label it as 1. The distance between the points 0 and 1 is one unit distance.

Step 4: Mark one more point to the right of 1 at one unit distance from 1. Label it as 2.

Step 5: Keep on marking points and labelling them as 3, 4, 5, … at equal distances.

Zero means absence of something.

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Thus, we can represent whole numbers on the number line as shown below:

0 1 2 3 4 5 6 7 8 9

Natural numbersWhole numbers

10

By observing the number line shown above, we can easily find the distance between any two points.

For example, the distance between the points 1 and 5 is 4 units.

On a number line, the numbers that are to the right of a given number are greater and that to the left are smaller.

For example, the number 4 is to the right of 1 (so 4 > 1) and 3 is to the left of 6 (so 3 < 6).

Example: Draw a number line and represent 12 and 15 on it. What is the distance between 12 and 15?

Solution:

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

The distance between 12 and 15 is 3 units.

aDDItIoN oN the NuMBeR LINeLet us add 5 and 4 using a number line.

Start from 5. As we have to add 4 to 5, make 4 jumps to the right of 5.

0 1 2 3 4 5 6 7 8 9 10

With the fourth jump, we reach at 9.

Thus, 5 + 4 = 9.

SuBtRaCtIoN oN the NuMBeR LINeLet us subtract 3 from 7 using a number line.

As we have to subtract 3 from 7, we start from 7 and then move 3 points backward.

0 1 2 3 4 5 6 7 8 9 10

With three steps backward, we reach at 4.

Thus, 7 − 3 = 4.

MuLtIPLICatIoN oN the NuMBeR LINeTo find 5 × 3 using a number line, we start from 0 and move 3 points forward at a time.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

After 5 such moves, we reach at 15.

Thus, 5 × 3 = 15.39


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DIVISIoN oN the NuMBeR LINeAs division is repeated subtraction, it can also be shown on the number line.

Let us find 14 ÷ 2.

We start from 14 and move 2 steps backward at a time till we reach 0.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Now, count the number of jumps we made to reach 0. It is 7.

Thus, 14 ÷ 2 = 7.

Exercise 3.11. Fill in the blanks.

a) The smallest whole number is ______ .

b) The smallest natural number is ______ .

c) ______ is the only whole number which is not a natural number.

d) The whole number ______ has no predecessor in whole numbers.

e) The natural number ______ has no predecessor in natural numbers.

f ) On the number line, the smaller whole number lies to the ______ of the greater whole number.

g) There are ______ whole numbers up to 1000.

2. Representthefollowingwholenumbersonanumber line.

a) 5 b) 0 c) 3 d) 11

3. In each pair of whole numbers, write the number which will come to the left side on the number line.

a) 763, 125 b) 1,01,987, 8,91,001

c) 8094, 91,000 d) 4873, 6945

e) 7651, 8198 f ) 8687, 8689

4. Solve the following using a number line.

a) 6 + 7 b) 15 − 7

c) 4 × 5 d) 12 ÷ 6

5. How many whole numbers are there between 72 and 97?

PRoPeRtIeS oF WhoLe NuMBeRSIn this section, we will learn the properties of whole numbers on basic operations, that is, addition, subtraction, multiplication and division.

Properties of additionClosure Property: When we add two whole numbers, the sum will always be a whole number.

For example, 3 + 4 = 7 and 7 is a whole number.

Thus, if x and y are any two whole numbers and x + y = z, then z is also a whole number.

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Commutative Property: When two whole numbers are added, the sum does not change even if the order in which the numbers are added is changed.

For example, 71 + 19 = 90 and 19 + 71 = 90.

⇒ 71 + 19 = 19 + 71

Thus, if x and y are any two whole numbers, then x + y = y + x.

Associative Property: When we add three or more whole numbers, the numbers can be grouped in any way, their sum remains the same.

For example,

11 + (13 + 15) = 11 + 28 = 39

(11 + 13) + 15 = 24 + 15 = 39

(11 + 15) + 13 = 26 + 13 = 39

Thus, if x, y and z are any three whole numbers, then x + (y + z) = (x + y) + z = (x + z) + y.

Additive Identity: When 0 is added to any whole number, the value of the whole number remains the same. Hence, 0 is the additive identity of whole numbers.

For example, 6 + 0 = 6 = 0 + 6.

Thus, if x is a whole number, then x + 0 = x = 0 + x.

Properties of SubtractionClosure Property: When one whole number is subtracted from another whole number, the difference is not always a whole number.

For example, 37 − 6 = 31 is a whole number but 5 − 32 = −27 is not a whole number.

Thus, if x and y are any two whole numbers and x − y = z, then z is not always a whole number. z will be a whole number only if x > y or x = y.

Commutative Property: The commutative property does not hold true for the subtraction of whole numbers.

For example, 75 − 67 = 8 and 67 − 75 = −8

⇒ 75 − 67 ≠ 67 − 75

Thus, if x and y are any two whole numbers, then x − y ≠ y − x.

Associative Property: Associative property does not hold true for the subtraction of whole numbers. Grouping affects the result of subtraction.

For example,

18 − (7 − 6) = 18 − 1 = 17

(18 − 7) − 6 = 11 − 6 = 5

⇒ 18 − (7 − 6) ≠ (18 − 7) − 6

Thus, if x, y and z are any three whole numbers, then x − (y − z) ≠ (x − y) − z.

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Property of Zero: When we subtract 0 from a whole number, the value of the whole number remains the same.

For example, 59 − 0 = 59 and 84 − 0 = 84.

Properties of MultiplicationClosure Property: When we multiply two whole numbers, the product will always be a whole number.

For example, 29 × 2 = 58 and 58 is a whole number.

Thus, if x and y are any two whole numbers and x × y = z, then z is a whole number.

Commutative Property: When two whole numbers are multiplied, the product does not change even if the order in which the numbers are multiplied is changed.

For example, 7 × 9 = 63 and 9 × 7 = 63

⇒ 7 × 9 = 9 × 7.

Thus, if x and y are any two whole numbers, then x × y = y × x.

Associative Property: When we multiply three or more whole numbers, the numbers can be grouped in any way, their product remains the same.

For example,

6 × (3 × 5) = 6 × 15 = 90

(6 × 3) × 5 = 18 × 5 = 90

(6 × 5) × 3 = 30 × 3 = 90

⇒ 6 × (3 × 5) = (6 × 3) × 5 = (6 × 5) × 3.

Thus, if x, y and z are any three whole numbers, then x × (y × z) = (x × y) × z = (x × z) × y.

Multiplicative Identity: When a whole number is multiplied by 1, the product is the number itself.

For example, 16 × 1 = 16 = 1 × 16 and 47 × 1 = 47 = 1 × 47.

Thus, if x is a whole number, then x × 1 = x = 1 × x.

Multiplicative Property of Zero: When a whole number is multiplied by 0, the product will always be 0.

For example, 83 × 0 = 0 × 83 = 0.

Thus, if x is a whole number, then x × 0 = 0 × x = 0.

Properties of DivisionClosure Property: The closure property does not hold true for division of whole numbers because the result of division of two whole numbers is not always a whole number.

For example, 77 ÷ 11 = 7 is a whole number but 3 ÷ 9 = 13

is not a whole number.

Thus, if x and y are any two whole numbers and x ÷ y = z, then z need not be a whole number.

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Commutative Property: The commutative property does not hold true for the division of whole numbers.

For example, 13 ÷ 39 = 3 is a whole number but 13 ÷ 39 = 1339

= 13

is not a whole number.

⇒ 39 ÷ 13 ≠ 13 ÷ 39.

Thus, if x and y are any two whole numbers, then x ÷ y ≠ y ÷ x.

Associative Property: Associative property does not hold true for the division of whole numbers.

For example,

24 ÷ (6 ÷ 3) = 24 ÷ 2 = 12

(24 ÷ 6) ÷ 3 = 4 ÷ 3 = 43

⇒ 24 ÷ (6 ÷ 3) ≠ (24 ÷ 6) ÷ 3.

Thus, if x, y and z are any three whole numbers, then x ÷ (y ÷ z) ≠ (x ÷ y) ÷ z.

Distributive PropertyDistributive Property of Multiplication over Addition

Observe the given example:

3 × (5 + 9) = 3 × 14 = 42 and

3 × 5 + 3 × 9 = 15 + 27 = 42

We conclude that 3 × (5 + 9) = 3 × 5 + 3 × 9.

This property is called the distributive property of multiplication over addition.

Thus, for any three whole numbers x, y and z, we have x × (y + z) = x × y + x × z.

Distributive Property of Multiplication over Subtraction

Observe the given example:

15 × (20 − 7) = 15 × 13 = 195 and

15 × 20 − 15 × 7 = 300 − 105 = 195

We conclude that

15 × (20 − 7) = 15 × 20 − 15 × 7.

This property is called the distributive property of multiplication over subtraction.

Thus, for any three whole numbers x, y and z, we have x × (y − z) = x × y − x × z.

Remember• Zeromultipliedbyanywholenumbergives0. Forexample,0×x=0.

• Zerodividedbyanywholenumbergives0. Forexample,0÷x=0,wherex ≠0.

• x÷0=notdefined.

• Anynumbermultipliedby1givesthenumberitself.

For example, x×1=x.

• Anywholenumberdividedby1givesthenumberitself.

For example, x÷1=x.

• Anynumberdividedbyitselfgives1. For example, x÷x=1.

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Solved examples Example 1: State the property represented by the following:

a) 27 × (77 + 65) = (27 × 77) + (27 × 65)

b) 44 × 1 = 44

c) 66 × 0 = 0

d) 71 + 0 = 71

e) 277 + 12 = 12 + 277

Solution:

a) Distributive property of multiplication over addition

b) Multiplicative identity

c) Multiplicative property of zero

d) Additive identity

e) Commutativepropertyofaddition

Example 2: Multiply the following by suitable rearrangements:

a) 23 × 16 b) 6 × 18 × 250

Solution:

a) 23 × 16 = 23 × (10 + 6)

= 23 × 10 + 23 × 6

= 230 + 138 = 368

b) 6 × 18 × 250 = (6 × 250) × 18

= 1500 × 18 = 27,000

Example 3: Solve the following using distributive property:

a) 15 × 9 + 17 × 9 b) 11 × 5 − 11 × 2

Solution:

a) 15 × 9 + 17 × 9 = 9 × (15 + 17) = 9 × 32 = 288

b) 11 × 5 − 11 × 2 = 11 × (5 − 2) = 11 × 3 = 33

Example 4: Simplify: 75 × 55 + 26 × 99.

Solution:

75 × 55 + 26 × 99

= 75 × (50 + 5) + 26 × (100 − 1)

= 75 × 50 + 75 × 5 + 26 × 100 − 26 × 1

= 3750 + 375 + 2600 − 26

= 6699

Example 5: In a class, there are 20 girls and 15 boys. Eachchildbrings5booksfordonatingto a charity home. Find the total number of books donated by the class.

Solution:

There are two methods of solving this problem.

Method 1:

Number of books donated by girls = 20 × 5

Number of books donated by boys = 15 × 5

Total number of books = (20 × 5) + (15 × 5)

= 100 + 75 = 175

Method 2:

Total number of students = 20 + 15

Number of books given by each student = 5

Total number of books = (20 + 15) × 5

= 35 × 5 = 175

In this example, we see that

(20 × 5) + (15 × 5) = (20 + 15) × 5 = 17 5.

Exercise 3.21. State the property represented by the

following:

a) 18 + 23 = 23 + 18

b) 25 × 31 = 31 × 25

c) 16 × (9 + 41) = 16 × 9 + 16 × 41

d) 35 + (17 + 21) = (35 + 17) + 21

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e) 16 × (41 − 29) = 16 × 41 − 16 × 29

f) 3 × (5 + 9) = 3 × 5 + 3 × 9

2. Fill in the blanks.

a) 219 × _____ = 0

b) 456 ÷ 456 = _____

c) 1971 ÷ 1 = _____

d) 314 × 18 = 314 × 10 + 314 × _____

e) _____ + 936 = 936 + 487

f ) 8769 + 2000 + 135 = 135 + _____ + 8769

g) 37 × (93 + 7) = 37 × _____

h) 251 × 100 = 251 × 94 + _____ × 6

3. State whether the following statements are true or false.

a) (32 ÷ 8) ÷ 2 = 32 ÷ (8 ÷ 2)

b) 3 × (6 + 1) = 3 × 6 + 3 × 1

c) (25 − 5) × 4 = 25 × 4 + 5 × 4

d) 0 ÷ 5 = 5

e) 8 ÷ 8 = 0

f ) 7 ÷ 7 = 1

g) (9 × 8) × 0 = 9 × (8 × 0)

4. Multiply the following by suitable rearrangements.

a) 250 × 38 × 40

b) 8 × 693 × 125

c) 439 × 5 × 60

d) 2 × 4 × 8 × 50 × 125

5. Evaluatethefollowingusingthesuitableproperty of whole numbers.

a) 24598 × 159 − 24598 × 59

b) 61725 × 92 + 61725 × 8

c) 584 × 99 + 584 × 1

6. Find the product of the greatest 4-digit number and the greatest 3-digit number using the properties of whole numbers.

7. The budget for a birthday party was ` 50,000. Out of this, ` 17,500 was spent on refreshments, ` 25,750 was given to the event management company and ` 4580 was given as tips to various workers. The balance was given for charity. Find the amount given for charity.

8. RaveenalearnsBharatnatyamatanacademyand also plays badminton in a sports complex. She pays ` 4500 per month for her dance classes and ` 1750 per month for her badminton coaching. Find her total expenditure on the two for one year.

PatteRNS IN WhoLe NuMBeRSLet us recall an interesting way of arranging numbers in a square and in a triangle using small dots.

Some numbers like 3, 6 and 10 can be arranged in a triangle. Such numbers are called triangular numbers.

3 6 10 …

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Some numbers like 4, 9 and 16 can be arranged in a square. Such numbers are called square numbers.

4 9 16 …

Now, let us look at other shapes in which we can arrange numbers.

We can arrange all numbers in a line.

1 2 3 4 5 …

We can arrange some numbers in a rectangle. 12 can be shown in two different rectangles.

Solve Problems using Number PatternsWe can also use the number patterns to simplify mathematical problems. Observe the given patterns:

a) 54 + 9 = 54 + 10 − 1

= (54 + 10) − 1

= 64 − 1 = 63

54 + 19 = 54 + 20 − 1

= (54 + 20) − 1

= 74 − 1 = 73

b) 138 + 8 = 138 + 10 − 2

= (138 + 10) − 2

= 148 − 2 = 146

138 + 18 = 138 + 20 − 2

= (138 + 20) − 2

= 158 − 2 = 156

c) 54 − 9 = 54 − 10 + 1

= (54 − 10) + 1

= 44 + 1 = 45

54 − 19 = 54 − 20 + 1

= (54 − 20) + 1

= 34 + 1 = 35

From the above patterns, we find that, it becomes simple to calculate the sum or the difference of two numbers, when one of the numbers is closer to 10, 20, 30, 40 and so on.

Now, observe the following pattern:

1 × 8 + 1 = 9

12 × 8 + 2 = 98

123 × 8 + 3 = 987

1234 × 8 + 4 = 9876

Solve 12,345 × 8 + 5 orally using the above pattern.

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Exercise 3.31. List the square, rectangular, triangular

and line numbers from the whole numbers between 1 and 50.

2. Study the patterns and fill in the blanks.

a) 6 × 2 − 5 = 7

7 × 3 − 12 = 9

8 × 4 − 21 = 11

9 × ______ − 32 = 13

10 × 6 − 45 = ______

b) 5 × 11 = 55

55 × 101 = 5555

555 × 1001 = 5,55,555

5555 × 10,001 = ______

55,555 × ______ = ______

3. Study the given pattern of triangular numbers and extend it to 3 more steps.

8 × 1 + 1 = 9 = 3 × 3

8 × 3 + 1 = 25 = 5 × 5

8 × 6 + 1 = 49 = 7 × 7

8 × 10 + 1 = 81 = 9 × 9

• Counting numbers 1, 2, 3, … are called natural numbers.

• The natural numbers along with 0 form the collection of whole numbers.

• While performing addition on the number line, we move from left to right. For performing subtraction on the number line, we move from right to left.

• Whole numbers are closed with respect to addition and multiplication.

• Addition of two whole numbers is also a whole number and multiplication of two whole numbers is also a whole number.

• Whole numbers are not closed with respect to subtraction and division.

• Addition and multiplication of whole numbers are commutative and associative.

• Multiplication of whole numbers is distributive over addition and subtraction.

Recap

Mental Maths 1. 0 ÷ 160 = _________.

2. 92 + (________ + 21) = (________ + 7) + 21

3. _________ and _________ property do not hold true for division of whole numbers.

4. Multiplication distributes over _________ but _________ never distributes over multiplication. (Fill both blanks with a common word.)

5. _________ is the only natural number which when multiplied to itself gives the number itself, while _________ is the only whole number which when added to itself gives the number itself.

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6. _________ is the smallest natural number while _________ is the smallest whole number.

7. x + y = y + x represents _________ property.

8. x − (y − z) ≠ (x − y) − z shows that _______ property does not hold true for subtraction of whole numbers.

Review time Multiple Choice Questions1. How many natural numbers are there

between 17 and 22?

a) 4 b) 6

c) 5 d) 8

2. 16 + _____ = 7 + 16

a) 5 b) 6

c) 7 d) 8

3. ______ is the multiplicative identity of whole numbers while ______ is the additive identity of whole numbers.

a) 1, 0 b) 0, 1

c) 1, 1 d) 0, 0

4. Multiplicative identity is always true for

a) whole numbers

b) natural numbers only

c) both for whole numbers and natural numbers

d) none of these

5. The predecessor of the greatest 4-digit number is

a) 10,000 b) 9998

c) 10,002 d) 9999

6. Zero multiplied by any whole number is _________ .

a) 1 b) 0

c) infinite d) 2

7. Zero divided by any whole number is _____.

a) 1 b) 0

c) not defined d) number itself

8. Which of the following is not true?

a) (13 + 5) + 12 = 13 + (5 + 12)

b) (13 × 5) × 12 = 13 × (5 × 12)

c) 13 + 5 × 12 = (13 + 5) × (13 + 12)

d) 13 × (5 + 12) = (13 × 5) + (13 × 12)

9. Which of the following is true?

a) 0 ÷ 0 = 0 b) 0 × 1 = 0

c) 0 + 1 = 0 d) 0 × 0 = 1

10. Which of the following statement is true?

a) The difference of two whole numbers is also a whole number.

b) 0 is the multiplicative identity for whole numbers.

c) Division of two whole numbers is commutative.

d) Multiplication is distributive over addition for whole numbers.

Subjective Questions1. Solve the following using number line.

a) 5 + 8 b) 16 − 5

c) 7 × 3 d) 18 ÷ 3

2. State the property represented by the following:

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a) (75 + 32) + 29 = 75 + (32 + 29)

b) 21 + 52 = 52 + 21 = 73

c) 5456 × 0 = 0

d) 7678 × 1 = 7678

e) 62 × (9 − 7) = 62 × 9 − 62 × 7

f ) (64 + 453) + 36 = 453 + (64 + 36)

3. Give three examples of each of the following properties of whole numbers.

a) Closurepropertyofaddition

b) Commutativepropertyofaddition

c) Associative property of multiplication

d) Distributive property of multiplication over subtraction

4. Solve the following by suitable rearrangements.

a) 199 × 25 × 4 b) 31 + 25 + 19

c) 456 + 933 + 144 d) 36 × 25

e) 175 + 933 + 25 f ) 6 × 50 × 4

g) 173 + 577 + 27 h) 32 × 60 × 8

5. Solve the following using the distributive property.

a) 35 × 6 + 9 × 35

b) 722 × 24 + 722 × 5 + 722 × 16

6. Fill in the blanks and state the property used:

a) 23 + 55 + 17 = 55 + ______

b) 758 × ______ = 0

c) 2 × ( ______ −3) = 2 × 5 − 2 × 3

d) 32 + (22 + 2) = (32 + ______ ) + 2

e) 7 × 674 × 25 = 674 × ______

7. How many whole numbers are there between 56 and 89?

8. Which is greater—the sum of the first 50 whole numbers or the product of the first 100 whole numbers?

9. A city had 32 mm of rain on Sunday. On Monday, it increased by 3 mm and on Tuesday, it decreased by 5 mm. A similar pattern was predicted to be followed on subsequent days. How much rainfall occurred on Saturday?

10. There were 47 students in a school bus. 23 students got off at a stop and 19 students got off at another stop. How many students are left in the bus?

think and answer 1. A tempo can carry 472 boxes of muffins weighing 16 kg each whereas a van can carry 528 boxes

each of the same weight. Find the total weight carried by both the vehicles.

2. A box contains 8 strips having 12 tablets of 750 mg medicine in each tablet. Find the total weight in grams of medicines in 32 such boxes.

Real-life Connect 1. In a class, there were 50 students. 43 of them were boys and the remaining were girls. How many

girls were there in four such similar classes?

2. There are 12 similar dinner plates. 5 sweets were placed on each plate. 3 sweets were removed from each of the 11 plates and dispatched away while 7 sweets were added to the last plate. How many total sweets are there now?

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Lab activity Objective: To verify that the multiplication of whole numbers is commutative

Materials Required: Graph paper, sheets of paper, glue stick, pair of scissors and pencil

Steps:

1. Take a graph paper and cut 3 strips of length 5 units from it as shown below.

a)

2. Paste these strips on the sheet of a paper as shown in the figure alongside.

This represents 3 × 5.

3. Take another graph paper and cut 5 strips of length 3 units from it and paste these strips on a sheet of paper as shown below.

c) d)

This represents 5 × 3.

4. Countthetotalnumberofboxesinfiguresb)andd).

We observe that both the figures b) and d) have 15 boxes.

Thus, 3 × 5 = 5 × 3 = 15.

b)

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