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SYLLABUSMATHEMATICS (CLASS–VIII)

NUMBER SYSTEM (50 hrs)

(i) Rational Numbers: • Propertiesofrationalnumbers(includingidentities).Usinggeneralformofexpressiontodescribeproperties.

• Consolidationofoperationsonrationalnumbers.

• Representationofrationalnumbersonthenumberline.

• Betweenanytworationalnumbersthereliesanotherrationalnumber(Makingchildrenseethatifwetaketworationalnumbersthenunlikeforwholenumbers,inthiscaseyoucankeepfindingmoreandmorenumbersthatliebetweenthem.)

• Worldproblem(higherlogic,twooperations,includingideaslikearea)

(ii) Powers: • Integersasexponents.

• Lawsofexponentswithintegralpowers.

(iii) Squares and Square roots, Cubes and Cube roots: • SquaresandSquareroots.

• Squarerootsusingfactormethodanddivisionmethodfornumberscontaining(a)nomorethantotal4digitsand(b)nomorethan2decimalplaces.

• Cubesandcuberoots(onlyfactormethodfornumberscontainingatmost3digits.)

• Estimatingsquarerootsandcuberoots.Learningtheprocessofmovingnearertotherequirednumber.

(iv) Playing with Numbers: • Writingandunderstandinga2and3digitnumberingeneralizedform(100a+10b + c,wherea,b,ccanbeonlydigit

0–9)andengagingwithvariouspuzzlesconcerningthis.(Likefindingthemissingnumeralsrepresentedbyalphabetsinsumsinvolvinganyofthefouroperations.)

• Childrentosolveandcreateproblemandpuzzles.

• Numberpuzzlesandgames.

• Deducingthedivisibilitytestrulesof2,3,5,9,10foratwoorthree-digitnumberexpressedinthegeneralform.

ALGEBRA (20 hrs)

Algebraic Expressions: • Multiplicationanddivisionofalgebraicexp.(Coefficientshouldbeintegers).

• Somecommonerrors(e.g.2+x ≠2x,7x + y ≠7xy).

• Identities(a ± b)2 = a2±2ab + b2,a2–b2=(a + b)(a–b).

Factorisation(simplecasesonly)asexamplesthefollowingtypesa(x + y),(x ± y)2,a2–b2,(x + a).(x + b)

• Solving linear equations in one variable in contextual problems involvingmultiplication anddivision (wordproblems)(avoidcomplexcoefficientintheequations).

RATIO AND PROPORTION (25 hrs)

• Slightlyadvancedproblemsinvolvingapplicationsonpercentages,profit&loss,overheadexpenses,Discount,tax.

• Differencebetweensimpleandcompoundinterest(compoundedyearlyupto3yearsorhalf-yearlyupto3stepsonly),arrivingattheformulaforcompoundinterestthroughpatternsandusingitforsimpleproblems.

Prelims_VIII.INDD 3 11/25/2015 11:33:43 AM

• Directvariation–Simpleanddirectwordproblems.

• Inversevariation–Simpleanddirectwordproblems.

• Timeandworkproblems:Simpleanddirectwordproblems.

GEOMETRY (60 hrs)

(i) Understanding Shapes: • Propertiesofquadrilaterals–Sumofanglesofaquadrilateralisequalto360°.(Byverification)

• Propertiesofparallelogram(Byverification) (i) Oppositesidesofaparallelogramareequal,

(ii) Oppositeanglesofaparallelogramareequal,

(iii) Diagonalsofaparallelogrambisecteachother.

[Why(iv),(v)and(vi)followfrom(ii)]

(iv) Diagonalsofarectangleareequalandbisecteachother.

(v) Diagonalsofarhombusbisecteachotheratrightangles.

(vi) Diagonalsofsquareareequalandbisecteachotheratrightangles.

(ii) Representing 3D in 2D: • IdentifyandMatchpictureswithobjects[morecomplicatede.g.nested,joint2Dand3Dshapes

(notmorethan2]

• Drawing2-Drepresentationof3-Dobjects(Continuedandextended.)

• Countingvertices,edges&faces&verifyingEuler’srelationfor3-Dfigureswithflatfaces(cubes,cuboids,tetrahedrons,prismsandpyramids.)

(iii) Construction : ConstructionofQuadrilaterals:

• Givenfoursidesandonediagonal.

• Threesidesandtwodiagonals.

• Threesidesandtwoincludedangles.

• Twoadjacentsidesandthreeangles.

MENSURATION (15 hrs)

• Areaofatrapeziumandapolygon.

• Conceptofvolume,measurementofvolumeusingabasicunit,volumeofacube,cuboidandcylinder.

• Volumeandcapacity(measurementofcapacity).

• Surfaceareaofcube,cuboid,cylinder.

DATA HANDLING (15 hrs)

(i) Readingbar-graphs,ungroupeddata,arrangingitintogroups,representationofgroupeddatathroughbar-graphs,con-structingandinterpretingbar-graphs.

(ii) SimplePiechartswithreasonabledatanumbers.

(iii) Consolidatingandgeneralisingthenotionofchanceineventsliketossingcoins,diceetc.Relatingittochanceinlifeevents.Visualrepresentationoffrequencyoutcomesofrepeatedthrowsofthesamekindofcoinsordice.

(iv) Throwingalargenumberofidenticaldice/coinstogetherandaggregatingtheresultofthethrowstogetlargenumberofindividualevents.Observingtheaggregatingnumbersoveralargenumberofrepeatedevents.Comparingwithdataforacoin.Observingstringsofthrows,notionofrandomness.


INTRODUCTION TO GRAPHS (15 hrs)

Preliminaries:

(i) Axes(Sameunits),CartesianPlane

(ii) Plottingpointsfordifferentkindofsituations(perimetervslengthforsquares,areaasafunctionofsideofasquare,plottingofmultiplesofdifferentnumbers,simpleinterestvsnumberofyearsetc.)

(iii) Readingofffromthegraphs.

• Readingoflineargraphs.

• Readingofdistancevstimegraph.


1 CONCEPTS

Rational numbers and their properties

Representation of rational numbers on the

number line

Rational numbers between two rational

numbers

RATIONAL NUMBERS AND THEIR PROPERTIES

The numbers of the form pq

, where p and q are integers and (q ≠ 0), are called rational numbers.

For example, 34

53

29

611, , ,− etc.

Standard Form of a Rational Number: A rational number

pq is said to be in standard form if p and q are integers

having no common divisor other than 1 and p is positive.

Notes:

(i) Every positive rational number is greater than 0.

(ii) Every negative rational number is less than 0.

(iii) Rational numbers are closed under addition, subtraction, multiplication and division (provided divisor

is not zero).

(iv) Commutativity of addition is true for natural numbers, whole numbers and integers. It is also true

for rational numbers.

(v) Associativity of addition is true for natural numbers, whole numbers and integers. It is also true for

rational numbers.

Properties of Rational Numbers

Additive Identity Element: Zero is the identity element for addition and subtraction of natural numbers,

whole numbers, integers and rational numbers.

For examples,

(i) 3 + 0 = 0 + 3 = 3

(ii) 0 + 5 = 5 + 0 = 5

(iii) 34

+ 0 = 0 + 34

= 34

etc.

Multiplicative Identity Element: One is the multiplicative identity for natural numbers, whole numbers,

integers and rational numbers.

Rational Numbers

CONCEPT IN A NUTSHELL

1

MBD_SUPR_RFR_MATH_G8_C01.indd 1

11/10/2015 1:17:42 PM

Super RefresherAll chapters as per NCERTSyllabus and Textbook

Every chapter divided into Sub-topics

Concept in a Nutshellprovides a complete and comprehensive summary of the concept

Highlights essential information which must be remembered

Rational Numbers 3

Numbers Associative for

Addition Subtraction

Multiplication Division

Rational Numbers Yes No Yes No

Integers Yes No Yes No

Whole Numbers Yes No Yes No

Natural Numbers Yes Yes Yes No

Try These [Textbook Page 13] Q. 1. Find using distributivity

(i) 75

× 312

+ 75

× 512

−

(ii) 916

× 412

+ 916

× 39

−

Sol. (i) 75

312

75

512

×−

+ ×

= 75

312

512

×−

+

[By distributivity property]

= 75

3 512

×− +

= 75

212

1460

730

× = =

(ii) 9

164

12916

39

×

+ ×

−

= 9

164

123

9× +

−

= 9

164

1239

× −

= 9

1612 12

36×

−

= 9

16036

× = 0576

= 0

TEXTBOOK EXERCISE 1.1 Q. 1. Usingappropriatepropertiesfind:

(i) −−

23

× 35

+ 52

35

× 16

(ii) 25

× 37

16

× 32

+ 114

× 25

−

−

Sol. (i) We have: − × + − ×2

335

52

35

16

Try These [Textbook Page 6]

Q. 1. Complete the following table:

Numbers Commutative for



Rational Numbers Yes … … …

Integers … No … …

Whole Numbers … … Yes …

Natural Numbers … … … No

Sol.







Natural Numbers Yes No Yes No

Think, Discuss and Write [Textbook Page 11] Q. 1. If a property holds for rational numbers, will

it also hold for integers? For whole numbers? Which will? Which will not?

Sol. Try yourself.






Rational Numbers … … … No

Integers … … Yes …

Whole Numbers Yes … … …

Natural Numbers … Yes … …

MBD_SUPR_RFR_MATH_G8_C01.indd 3 11/10/2015 1:17:45 PM

Rational Numbers

5

Q. 10. Write:

(i) The rational number that does not have a

reciprocal.

(ii) The rational numbers that are equal to

their reciprocals.

(iii) The rational numbers that is equal to their

negative.

Sol. (i) 01

(ii) 1 and (–1). (iii) Zero.

Q. 11. Fill in the blanks:

(i) Zero has reciprocal.

(ii) The numbers and are their

own reciprocals.

(iii) The reciprocal of –5 is .

(iv) Reciprocal of 1 ,x

where x≠0is.

(v) The product of two rational numbers is

always a .

(vi) The reciprocal of a positive rational number

is .

Sol. (i) no (ii) 1 and –1

(iii) −1

5

(iv) x

(v) rational number (vi) positive

SELF PRACTICE 1.1

1. Find using distributivity:

(i) −

− −

3

4×

23

+34

×56

(ii) −

−

2

3×

56

+23

×72

2. Using appropriate properties find:

23

×37

114

37

×35− − −

3. Find the additive inverse of each of the

following:

(i) 13

(ii) 239

(iii) −311

(iv) −−87

4. Verify that: –(–x) = x for

(i) x = 1317

(ii) x = − 21

31

5. Find the multiplicative inverse of the following:

(i) 12 (ii) –8 (iii) 516

(iv) −1417

(vi) −1

\ Multiplicative inverse of −1 is 1

1−,

i.e., −1

1 = −1

Q. 5. Name the property under multiplication

used in each of the following:

(i) −4

5 × 1 = 1 ×

−45

= −45

(ii) − −13

17×

27

= − −27

×1317

(iii) −

−1929

×29

19 = 1

Sol. (i) Multiplicative identity.

(ii) Commutative property of multiplication.

(iii) Multiplicative inverse.

Q. 6. Multiply 613

by the reciprocal of −716

.

Sol. 613

716

×

−

multiplication inverse of

= 613

167×−

= 96

91

9691

−= −

Q. 7. Tell what property allows you to compute:

13

× 6×43

as

13

×6 ×43

.

Sol. Associativity of multiplication.

Q. 8. Is 89

the multiplicative inverse of –1 18

. Why

or why not?

Sol. −118

= −98

\ Multiplicative inverse of −9

8 is

89−

. i.e., −89

.

\89

is not the multiplicative inverse of −118

.

Q. 9. Is 0.3 the multiplicative inverse of 313

. Why

or why not?

Sol. 313

= 10

3

\ The multiplicative inverse of 10

3 is

310

.

i.e., 0.3.

\ Yes, 0.3 is the multiplicative inverse of 313

.


11/10/2015 1:17:57 PM

Important Questions fromexamination point of viewto ensure passing marks

Self Practice questions forconsolidation of each concept

Try These and Do Thiswith page numbersfully solved to helpthe learners

NCERT Textbook Exerciseswith detailed solution


Time – 2 Hours Class VII Max. Marks – 50

General Instructions:

● All questions are compulsory.

● Section A comprises of 5 questions carrying 1 mark each.

● Section B comprises of 5 questions carrying 2 marks each.

● Section C comprises of 5 questions carrying 3 marks each.

● Section D comprises of 5 questions carrying 4 marks each.

SecTion A

1. How many angles are formed when 2 lines intersect?

2. Evaluate: (20)2 + (31)0 + 40

3. If the circumference of a circular sheet is 154 m, find its radius.

4. Find third angle of the triangle which have two angles as 30° and 80°.

5. Find the whole quantity if 10% of its is 7.

SecTion B

6. Raju has solved 24

part of an exercise while Sameer solved 12

part of it. Who has solved more?

7. In the figure below, DCDE ≅DQPR. What is m∠D?

8. Find the mode and median of the data:

13, 16, 12, 14, 19, 12, 14, 13, 14

9. Verify that a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c) for the values of a, b and c as a = 12, b = −4 and c = 2.

10. ABC is a triangle right-angled at C. If AB = 25 cm and AC = 7 cm, find BC.

SecTion c

11. If Meenakshi gives an interest of `45 for one year at 9% rate p.a. What is the sum she has borrowed?

12. Write the following numbers in the expanded form:

(i) 279404

(ii) 20068

(iii) 2806196

13. A picture is painted on a cardboard 8 cm long and 5 cm wide such that there is a margin of 1.5 cm along each

of its side. Find the total area of the margin.

14. Construct ΔPQR is PQ = 5 cm, m∠PQR = 105°, m∠QRP = 40°.

Sample Paper - i

338

Brain Teasers.indd 338

11/10/2015 2:36:20 AM

MBD Super Refresher Mathematics-VIII

8

i.e., 35

2020

× = 60100 and 3

425

25× = 75

100

\ Ten rational numbers between 60

100 and 75

100 can be any of these: 61100

62100

63100

64100

65100

, , , , , …, 70100 , 71

100 , 72100 ,

73100 , 74

100 .

SELF PRACTICE 1.2 1. Represent the following numbers on the

number line: (i) −13 (ii) 2

7 (iii) 72 (iv) −3

7 .

2. Find a rational number lying between 13 and 1

2 .

3. Find three rational numbers lying between 3

and 4. 4. Find three rational numbers lying between 23

and 34 . 5. Find ten rational numbers between −5

6 and 58 .

6. Find three rational numbers between 14 and 1

2 .

7. Find three rational numbers between —2 and 0.

8. Find two rational numbers between 15 and 1

2 .

9. Find seven rational numbers between 13 and

12 .

1. Which of the following statement is false?

(a) Rational numbers are not closed under

addition. (b) Whole numbers are closed under addition.

(c) Integers are closed under addition.

(d) Natural numbers are closed under addition.

2. Which of the following statement is false?

(a) Rational numbers are commutative for

addition. (b) Integers are not commutative for addition.

(c) Natural numbers are commutative for addition.

(d) Whole numbers are commutative for addition.

3. Which of the following statement is true?

(a) Integers are associative for subtraction.

(b) Natural numbers are associative for subtraction.

(c) Whole numbers are not associative for

subtraction. (d) Rational numbers are associative for

subtraction. 4. Which of the following statement is true?

(a) Rational numbers are not associative for

multiplication. (b) Integers are associative for multiplication.

(c) Whole numbers are not associative for

multiplication. (d) Natural numbers are not associative for

multiplication.

MULTIPLE CHOICE QUESTIONS (MCQs)

In each of the following questions four options are given. Choose the correct answer. 5. Which of the following statement is false?

(a) Rational numbers are closed under

subtraction. (b) Integers are closed under subtraction.

(c) Natural numbers are closed under subtraction.

(d) Whole numbers are not closed under

subtraction. 6. Which of the following statement is true?

(a) Rational numbers are not commutative for

subtraction. (b) Natural numbers are commutative for

subtraction. (c) Whole numbers are commutative for

subtraction. (d) Integers are commutative for subtraction.

7. Which of the following statement is true?

(a) Whole numbers are not closed under

multiplication. (b) Integers are not closed under multiplication.

(c) Rational numbers are not closed under

multiplication. (d) Natural numbers are closed under

multiplication. 8. Which of the following statement is false?

(a) Integers are not commutative for

multiplication? (b) Rational numbers are commutative for multi-

plication.MBD_SUPR_RFR_MATH_G8_C01.indd 8

11/10/2015 1:18:12 PM

MBD Super Refresher Mathematics-VIII

10

Q. 8. The rational number 10.11 in the form pq is

_____________.

Q. 9. The two rational numbers lying between −2

and −5 with denominator as 1 are ________

and ________.

ANSWERS

6. Positive rational number 7. Opposite

8. 1011100

9. –3, –4

True/FalseIn questions 10 to 13, state whether the given statements

are true (T) or false (F).

Q. 10. 56

lies between 23

and 1.

Q. 11. If xy

is the additive inverse of, cd

thenxy

cd

− = 0.

Q. 12. The negative of the negative of any rational

number is the number itself.

Q. 13. The rational number −−83

lies neither to the

right nor to the left of zero on the number line.

ANSWERS

10. True 11. False 12. True 13. False

Short Answer Type Questions

Q. 14. The cost of 194

metres of wire is `171

2. Find

the cost of one metre of the wire.

Sol. Cost of 194

m of wire = `171

2

Cost of 1 m of wire = `171

2194

÷

= 171

2419

× = 18

Cost of 1 m wire = `18

Q. 15. 711

of all the money in Hamid’s bank account

is `77,000. How much money does Hamid

have in his bank account?

Sol. Let the total amount in Hamid’s bank account

= `x

As per question, 711

of x = 77,000

NCERT EXEMPLAR QUESTIONS (SOLVED)

Multiple Choice Questions (MCQs)

In questions 1 to 5, out of the four options only one is

correct. Write the correct answer.

Q. 1. The numerical expression 38

57

+−( )

=−1956

shows that

(a) Rational numbers are closed under

addition.

(b) Rational numbers are not closed under

addition.

(c) Rational numbers are closed under

multiplication.

(d) Addition of rational numbers is not

commutative.

Q. 2. The multiplicative inverse of −1 17

is

(a) 87

(b) −87

(c) 78

(d) 78−

Q. 3. If y be the reciprocal of rational number x,

then the reciprocal of y will be

(a) x (b) y (c) xy

(d) yx

Q. 4. Between two given rational numbers, we can

find

(a) One and only one rational number.

(b) Only two rational numbers.

(c) Only ten rational numbers.

(d) Infinitely many rational numbers.

Q. 5. x y+

2 is a rational number

(a) Between x and y.

(b) Less than x and y both.

(c) Greater than x and y both.

(d) Less than x but greater than y.

ANSWERS

1. (a) 2. (d) 3. (a) 4. (d) 5. (a)

Fill in the Blanks

In questions 6 to 9, fill in the blanks to make the

statement true.

Q. 6. The reciprocal of a positive rational number is

a _______________.

Q. 7. The rational numbers 13

and −13

are on the

_________ sides of zero on the number line.


11/10/2015 1:18:16 PM

Rational Numbers

13

Now, the rational numbers lying between

them will be −991000

, −98

1000, ...,

01000

,

11000

,2

1000, ...,

991000

.

Thus, we conclude that rational numbers

lying between two given rational numbers are

uncountable.

Q. 2. Find a rational number between (a + b)–1 and

(a–1 + b–1), given that a = 13

, =27b .

Sol. a = 13

, b = 27

a + b = 13

27+ =

7 621+ =

1321

(a + b)–1 = 1321

–1

= 2113

a–1 + b–1 = 13

27

11

+

−−

...(i)

=

31

72+ =

6 72+ =

132

...(ii)

We now need a rational number between 2113

and 132

.

Mean of 2113

and 132

= 12

2113

132+

= 12

42 16926+

=

12

21126

=

21152

Hence, the required rational number is 21152

.

Q. 3. If the price of 12 tables is `360025

and the

price of 6 chairs is `300034

, find the total

price of 4 tables and 4 chairs.

Sol. Price of 12 tables = `360025

= `18002

5

\ Price of 1 table = `18002

5 ÷ 12

= `

180025

× 112

= ̀9001

30

\ Price of 4 tables = 4 ×

900130

= `18002

15d

Price of 6 chairs = ` 3000

34

= `12003

4

\ Price of 1 chair = `

120034

÷ 6

=

120034

× 16

= `4001

8

\ Price of 4 chair = `4 × 4001

8 = `

40012

Hence, total price of 4 tables and 4 chairs.

= `18002

15

40012+

= `

36004 60015

30+

= `96019

30 = `

1930

VALUE BASED QUESTIONS (VBQs)

Q. 1. Two students Shrey and Hitesh gave the

following statements, respectively.

(a) If a number is divisible by 3 it will also be

divisible by 9.

(b) If a number is divisible by 9 it will also be

divisible by 3.

Who is telling a lie? What is the importance

of truth in life?

Sol. Shrey is telling a lie. It is important to speak

the truth in life as it developes faith, love and

transparency in the minds of other people and

keeps the person, who speaks the truth, calm

and relaxed.

CHAPTER ASSESSMENT

1. Choose the correct option in each of the

following:

(i) A number of the form pq

is said to be

rational number, if

(a) p and q are integers.

(b) p and q are integers and q ≠ 0.

(c) p and q are integers and p ≠ 0.

(d) p and q are integers and p ≠ 0 also q ≠ 0.

(ii) The additive inverse of 247

is

(a) 187

(b)

−718

(c) −18

7

(d) 718


11/10/2015 1:18:30 PM

Mathematics

Value-Based Questionsto apply mathematical conceptsto real life situations with stress onsocial values

Four Sample Papersof 50 marks each

Multiple Choice Questions (MCQs)for testing conceptual skills of students

NCERT Exemplar Problemswith complete solution tosupplement the NCERTsupport material

Chapter Assessment with answers at the end of each chapter


1 Rational Numbers 1–14

2 Linear Equations in One Variable 15–34

3 Understanding Quadrilaterals 35–58

4 Practical Geometry 59–68

5 Data Handling 69–88

6 Squares and Square Roots 89–112

7 Cubes and Cube Roots 113–126

8 Comparing Quantities 127–150

9 Algebraic Expressions and Identities 151–172

10 Visualising Solid Shapes 173–185

11 Mensuration 186–215

12 Exponents and Powers 216–226

13 Direct and Inverse Proportions 227–243

14 Factorisation 244–259

15 Introduction to Graphs 260–278

16 Playing with Numbers 279–292

Sample Papers (1–4) 293–300

CONTENTS


1CONCEPTS

Rational numbers and their properties Representation of rational numbers on the

number line Rational numbers between two rational

numbers

RATIONAL NUMBERS AND THEIR PROPERTIES

The numbers of the form pq , where p and q are integers and (q ≠ 0), are called rational numbers.

For example, 34

53

29

611

, , , − etc.

Standard Form of a Rational Number: A rational number pq is said to be in standard form if p and q are

integers having no common divisor other than 1 and p is positive.

Notes: (i) Every positive rational number is greater than 0. (ii) Every negative rational number is less than 0. (iii) Rational numbers are closed under addition, subtraction, multiplication and division (provided divisor

is not zero). (iv) Commutativity of addition is true for natural numbers, whole numbers and integers. It is also true

for rational numbers. (v) Associativity of addition is true for natural numbers, whole numbers and integers. It is also true for

rational numbers.

Properties of Rational Numbers Additive Identity Element: Zero is the identity element for addition and subtraction of natural numbers,

whole numbers, integers and rational numbers. For examples, (i) 3 + 0 = 0 + 3 = 3 (ii) 0 + 5 = 5 + 0 = 5

(iii) 34

+ 0 = 0 + 34

= 34

etc.

Multiplicative Identity Element: One is the multiplicative identity for natural numbers, whole numbers, integers and rational numbers.

Rational Numbers


1


MBD Super Refresher Mathematics-VIII2

For example,

(i) 6 × 1 = 1 × 6 = 6

(ii) (– 7) × 1 = 1 × (– 7) = – 7

(iii) 53

× 1 = 1 × 53

= 53

etc.

Additive inverse: For every rational number pq

, there exists a rational number pq

pq

+

−

such that;

pq

pq

+

−

= 0 and similarly,

−

+

=

pq

pq

0.

Then, −pq

is called the additive inverse of pq

.

Multiplicative inverse (Reciprocal): Every non-zero rational number pq

has its multiplicative inverse

pq . For example

pqqp×

= q

ppq×

= 1

\ qp

is called the reciprocal of pq

.

Note:

(i) Zero has no reciprocal (ii) Reciprocal of 1 is 1 (iii) Reciprocal of –1 is –1

Distributive law of multiplication over addition: For any three rational numbers abcd

, and ef

, we ave:

ab

cd

ef

× +

=

abcd

ab

ef

×

+ ×

.

NCERT TEXTBOOK EXERCISE (SOLVED)


Q. 1. Fill in the blanks in the following table:

Numbers Closed Under



Rational Numbers Yes Yes … No

Integers … Yes … No


Natural Numbers … No … …

Sol.

Numbers Closed Under



Rational Numbers Yes Yes Yes No

Integers Yes Yes Yes No




Rational Numbers 3







Natural Numbers Yes Yes Yes No

Try These [Textbook Page 13] Q. 1. Find using distributivity

(i) 75

× 312

+ 75

× 512

−

(ii) 916

× 412

+ 916

× 39

−

Sol. (i) 75

312

75

512

×−

+ ×

= 75

312

512

×−

+

[By distributivity property]

= 75

3 512

×− +

= 75

212

1460

730

× = =

(ii) 916

412

916

39

×

+ ×

−

= 916

412

39

× +−

= 916

412

39

× −

= 916

12 1236

×−

= 916

036

× = 0576

= 0

TEXTBOOK EXERCISE 1.1 Q. 1. Usingappropriatepropertiesfind:

(i) −−

23

× 35

+ 52

35

× 16

(ii) 25

× 37

16

× 32

+ 114

× 25

−

−

Sol. (i) We have: − × + − ×2

335

52

35

16






Rational Numbers Yes … … …

Integers … No … …


Natural Numbers … … … No

Sol.








Think, Discuss and Write [Textbook Page 11] Q. 1. If a property holds for rational numbers, will

it also hold for integers? For whole numbers? Which will? Which will not?

Sol. Try yourself.






Rational Numbers … … … No

Integers … … Yes …

Whole Numbers Yes … … …

Natural Numbers … Yes … …



= −× − × +

23

35

35

16

52

(By commutativity)

= 35

23

16

52

−−

+ (By distributivity)

= 35

4 16

52

− −

+ =

35

56

52

×−

+

= − +12

52

= − +1 5

2

= 42

= 2

(ii) We have: 25

37

16

32

114

25

× −

− × + ×

= 25

37

114

25

16

32

× −

+ × − ×

(By commutativity)

= 25

37

114

14

− +

− =

25

6 114

14

− +

−

= 25

514

14

×−

−

= − − =− −

=−1

714

4 728

1128

Q. 2. Write the additive inverse of each of the following:

(i) 28

(ii) −59

(iii) −−

65

(iv) 29−

(v) 196−

Sol. (i) Additive inverse of 28

is −28

.

(ii) Additive inverse of −59

is 59

because

= −

+5

959

= − +5 5

9=

09

= 0

(iii) We may write; −−

65

= ( ) ( )( ) ( )− × −− × −

6 15 1

= 65

\ Additive inverse of 65

is −65

because

−65

+ 65

= − +6 6

5 =

05

= 0

(iv) In standard form, we write; 29−

as −29

.

\ Additive inverse of −29

is +29

because − +2

929

= − +2 29

= 09

= 0

(v) In standard form, we write; 19

6− as −

196

.

\ Additive inverse of – 196

is 196

because −

+196

196

= − +19 19

6 = 06

= 0

Q. 3. Verify that: −(−x) = x for:

(i) x = 1115

(ii) x = −1317

Sol. (i) For x = 1115

⇒ –(–x) = − −

=( )1115

1115

= x

Thus; –(–x) = x is verified.

(ii) For x = −1317

⇒ – (–x) = – −−

( )1317 = −

1317 = x

Thus –(–x) = x is verified.

Q. 4. Find the multiplicative inverse of the following:

(i) –13 (ii) −1319

(iii) 15

(iv) − −58

× 37

(v) –1 × −25

(vi) –1

Sol. (i) –13 \ Multiplicative inverse of −13 is

113−

,

i.e., −113

.

(ii) −1319


is 1913−

,

i.e., −1913

.

(iii) 15

\ Multiplicative inverse of 15

is 51

, i.e., 5.

(iv) −

×−5

83

7 =

( ) ( )− × −×

5 38 7

= 1556


is 5615

.

(v) − ×−1 25

= ( ) ( )− × −1 2

5 =

25


is 52

.


Rational Numbers 5

Q. 10. Write: (i) The rational number that does not have a

reciprocal. (ii) The rational numbers that are equal to

their reciprocals. (iii) The rational numbers that is equal to their

negative.

Sol. (i) 01

(ii) 1 and (–1). (iii) Zero.

Q. 11. Fill in the blanks: (i) Zero has reciprocal. (ii) The numbers and are their

own reciprocals. (iii) The reciprocal of –5 is .

(iv) Reciprocal of 1 ,x

where x≠0is .

(v) The product of two rational numbers is always a .

(vi) The reciprocal of a positive rational number is .

Sol. (i) no (ii) 1 and –1

(iii) −15

(iv) x

(v) rational number (vi) positive

SELF PRACTICE 1.1 1. Find using distributivity:

(i) −

− −

34

× 23

+ 34

× 56

(ii) −

−

23

× 56

+ 23

× 72

2. Using appropriate properties find:

23

× 37

114

37

× 35

−− −

3. Find the additive inverse of each of the following:

(i) 13

(ii) 239

(iii) −311

(iv) −−

87

4. Verify that: –(–x) = x for

(i) x = 1317

(ii) x = − 2131

5. Find the multiplicative inverse of the following:

(i) 12 (ii) –8 (iii) 516

(iv) −1417

(vi) −1

\ Multiplicative inverse of −1 is 11−

,

i.e., −11

= −1

Q. 5. Name the property under multiplication used in each of the following:

(i) −45

× 1 = 1 × −45

= −45

(ii) − −1317

× 27

= − −27

× 1317

(iii) −

−19

29× 29

19 = 1

Sol. (i) Multiplicative identity. (ii) Commutative property of multiplication. (iii) Multiplicative inverse.

Q. 6. Multiply 613

by the reciprocal of −716

.

Sol. 613

716

×−

multiplication inverse of

= 613

167

×−

= 9691

9691−

= −

Q. 7. Tell what property allows you to compute:

13

× 6× 43

as

13

×6 × 43

.

Sol. Associativity of multiplication.

Q. 8. Is 89

the multiplicative inverse of –1 18

. Why

or why not?

Sol. −1 18

= −98


is 89−

. i.e., −89

.

\89

is not the multiplicative inverse of −1 18

.

Q. 9. Is 0.3 the multiplicative inverse of 3 13

. Why

or why not?

Sol. 3 13

= 103

\ The multiplicative inverse of 103

is 3

10.

i.e., 0.3.

\ Yes, 0.3 is the multiplicative inverse of 3 13

.



Sol. (i) A = 15

, B = 45

, C = 55

= 1, D = 85

, E = 95

.

(ii) J = −116

, I = −86

, H = −76

, G = −56

, F = −26


RATIONAL NUMBERS BETWEEN TWO RATIONAL NUMBERS

If x and y be two rational numbers, such that x < y,

then, 12

(x + y) is a rational number between x and y.

For example, between 13

and 12

, the required rational

number is = 12

13

12

+

=

12

2 36+

=

12

56

× = 5

12

Hence, 512

is a rational number lying between 13

and12

.

Let us see, whether, we are able to say like this in the

case of numbers like 3

10 and

710

.

You might have thought that they are 4

105

10, , and

610

.

You can also write 3

10 as

30100

and 710

as 70100

.

Now, the numbers, 31

10032100

33100

, , ,…, 68

10069100

,

all between 3

10 and

710

. The number of these

rational numbers is 39. This is called the density property of rational numbers.

TEXTBOOK EXERCISE 1.2

Q. 1. Represent these numbers on the number line:

(i) 74

(ii) −56

Sol. (i)

(ii) –12__

6

–11__6

–10__6

__

6

–9 __

6

–8 __

6

–7 __

6

–6 __

6

–5 __

6

–4 __

6

–3 __

6

–2 __

6

–1 0

–2 –1

6. Name the property under multiplication used in each of the following:

(i) −

316

× 815

= 815

× 316−

(ii) 23

× 67

× 1415

= 23

× 67

× 1415

−

−

(iii) 56

× 45

710

= 56

× 45

56

× 710

−+

−

−

+

−

7. Multiply −719

by the reciprocal of 513

.

8. Tell what property allows you to compute:

34

× 8× 25

as

34

×8 × 25


REPRESENTATION OF RATIONAL NUMBERS ON THE NUMBER LINEYou have learnt to represent natural numbers, whole numbers, integers and rational number on a number line. We shall revise them.

(i) Natural numbers: e.g.,

1 2 3 4 5 6 7

(ii) Whole numbers: e.g.,

(iii) Integers: e.g.,

(iv) Rational numbers: e.g.,

(a)

(b)

(c)

Try These [Textbook Page 17]Write the rational number for each point labelled with a letter.

(i)

(ii)


MBD CBSE Super RefresherMathematics Class 8

Publisher : MBD GroupPublishers

Author : Sudhanshu SekharSwain

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