Fractions - VAGA Study – Free PDFsvaga.study/pdf/nc/fema/ncfemata07.pdfThe fractions whose numerator and denoninator add upto 10 are; (d) We can find an in-finite number of fractions

/E]Hth{r{U

Fractions

. trYaction as a part of whole.

. Representation of a fraction on number line.

" Proper, impmper and mixed fractions.. Comparison of fractions.. Addition a-nd subtraction offractioDs.. Equivalent fractions.. Simplest form of a fraction.. Like and unlike fraction.. Ascending and descending order of a fraction.

Conceptual Facts

o A frastion is a part of a whole number having numerator antl denominator. For example |, where O is

Understanding the Lesson

numerator and 7 is tlxe denoninator.

o Representation of a fraction on a nu:nber line.

for enmnte: f,

as

-2 -1 01c Prop€r ftactions: Numerator is less than the denominator.

2 6 _1For example: g, g *d

b

" Improp€r ftactlons: Numerator is bigger than the donominator.

For exanple: ;,1,+ "".:. Dtixed fractioas: It is ropresented by Quotient

For example: rl,e21 ""a +l

o Equivalent ftaotions: Two or more fracbions are said to be equivalent fractions, if they represent the

sa.me quantity.

2

6415'10

8

RemainderDiri""t

For examnle: f , and20

't09

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' Slnplest form of a fraction: A fraction is said to be sinple if numerator and the denominator have nocommon factor except 1.

For example: gimFlest form of # is :u Llke ftactioass Tbo or more fractions having sane denominators are called like fractions.

2946.tbr example: ;,;, _-,=DDDD

' Unlike fractlons: T.wo ot more frachions having different denominatorr are galled tnlike fractisns.8

(,

(u)

aaaaaaaaaa(vii)

66atTr'I 6

For eranple: I

Ql. Write the friction representing the shadedportion.

(iii) Total uumber of parts = 8

Number of shaded parts = 44

... lramon = E(iu) Total n,r'"ber ofparts = 4

Number of shaded parts = I1

;. Flaction = 7(u) Total number ofparts = 7

Ntmber of shaded parts = 3

3.'. F ractron = -

(ui) Totgl nrlmber of parts = 12

Number of shaded parts = 3

.'. Fractioo = at2(uii) Total number ofparts = 10

Number of shaded parts = 10

.'. Fraction = 410(uiii) Total number of parts = I

N"-ber of shaded parts = 4L

... Fraction = ,_(rr) Total number of parts = 8

Ntmber of shaded parts = 4

4... trachon =

E_

(*) Total number ofparts = 2

Number ofshaded part = I.'. Fractioo = 12

10

(r,

(iii) (iu)

(ui)

(viii)

(ir) (r)

Sol. (i) Total number ofparts = 4Number ofshaded parts = 2

.'. Fractioo = ?41;;1 1'o1a1 1r'mbor of parts = 9

Number of shaded parts = 8

.'. F*ctioo = 9I

ssee:a@@@

l' :': ,l'84888

110 MAT}IEMANCS-VI

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Q2. Colour the part accoriling to the given fraction.

... 1(r) -'6 (ii t4

,.,.. 1\ut,t

e,. . 3\tu ) i

b)!I

Sol. (r) *.... 1\IL) Z

(iiil L3

Atu) ,

3(iu)

7

(b)

Q3. Identi-fr the enor, if anY.

(a)

This iE12

(c)

I'his iB q4

Sol. (o) Since the shaded part is not half.I

.'. This is not ; .

(b) Since, the Parts are not equal.

.'. Shaded Part is not 1'4'

**r+

FRACNONS111

(c) Since, the part are not equal.

.'. Shaded nart is not | .

Q4. What fraction of a day is 8 hours?Sol. Since, a tlay hcn 2l hours autl ws have 8 hours,

8.'. Re+ired fraction = 7

Q6. What fraction qf s [sur is l[0 rninutog!

Sol. Since t houre = 60 rninutes

.'. Fraction of 40 minutee = I60

Q6. Arya, Abhimanyu and Yivek shared lutrch.Arya has brought two sa[dwiches, one -ade ofvegetable and one of Jam. Ihe other two boysforgot to bring their lune,h, Arya agxeod to sharehia sandwiches so that each psrson will havean equal shere ofeach sandwich.(o) How can Arya tlivide his eandwichsB so that

each person hes an equal share?(b) What part of a satrdwich will each boy

receive?Sol. (a) Arya has divided his sandwie,h into three

equal parts,So, each of then will gpt oue Part'

(b) Each one of them will receive t nart.

.'. Required fractiou = 13Q7. Kanchan dyes tlresses. She had to dye 30

dlpsges. Sh6 hac so fgl. ffnished 20 &Esses. Whatftaction of dreeses hps sl6 finisfusdf

Sol. Total number of dresses to be dyeil = 80

Q8.

Sol.

Number of dresses fnished = 20202

.. Required fractioo = E0 = EWrite tJre naturat Dumb€rs froE 2 to 12. Whatfraction of them ar€ prine uumbers?Natural numbers betwesn 2 and 12 are;

2,3,4,6,6,7,8,9, 10, 11, 12

Number of given natural numbsrs = 11

Number of prime numbers = 66

.'. Required fraction = 11Write the natural ounbers froo 102 to 113.

What fraction of theu are pri:ae uumbors?Natural numbers from 102 to 113 are;

102, 103, 104, 106, 106, 107, 108, 109, 110' 11r,Ltz, L].BTotal number of givsn Datutal Euab€r€ = 12

Prime numbers are 109, 107, 109, 113

.'. Number of prime tlrunbon = 4l1

.'. Required &actio". L =;

Qe.

Sol.

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Q10. What fraction of thexe circles have Xs in them?

aaaaTotal number ofcircles = 8Number of circles having X's in them = 4

,. Required fracti r" = t = +Kristin received a CD player for her birthtlay.She bought 3 CDs and received 5 others as gifts.What fraction ofher total CDs did she buy andwhat fraction tlid she receive as gifts?Number of CDs bo.Eht by her from tb.e market

Number of CD's received as gifts = 5.'. Total number of CDs = 3 + 5 = 8

.'. Fraction ofCD (bought) = ! anrt the fraction

of cDs Gifted) = !Tnv THese (Paee 137)

Ql. Show I on a number line.

Sol.

Q11.

Sol.

Sol.P

-1

ABCDEFGHIJK

Divide the number line from 0 to I into E equalparts.

The point represents ! .b

aad ff, on a number line.

o7zg41EU E 6

105lo' 10' 10

ShowQ2.

Sol.

Q3.

Sol.

0 1 2 3 4 6 6 7 8 91bo L2 B 4 6 67 891010 10 10 r0 10 10 10 10 10 r0 10

Divide the number Iine from 0 to 1 into 10 equalparts.

10.'. B represents ft, A renresents fr,-5to!'represents

1O and K represent^s fr.

Can you show any other fraction between 0 and1? Writs five more fractions that you can showand depict them on the number line.Yes, we caa show any number of fractionsbetween 0 and 1.

112 MATHEMATICS-VI

Five more fractioDs between 0 and 1 that canbe shown on nurtber line are;

d,D

Qa, How msny fractions lie between 0 aDd 1? thinhdiscuss and write your answer?

g6l. [a inffnif6 number of frastions can be foundbetween 0 and 1.

Tnv Tsese (Paae 138)

Q1. Give a proper fraction:(a) whose numerator is 5 and denominator is 7.(b) whose deno-ir'"tor is 9 and nnnerator is E.(c) whose nulerator a-nd denominator add up

to 10. How ma.ny fractions of this kind canyou make?

(d) whose denominator is 4 more than thenumerator.(Give any five. How many more catr youmake?)

Sol. (a) Given that:Numerator = 5Denominator = 7

_6.. lYaction=;

(b) Given that: t

Nunerator = 5Denomiaator = 9

5.'. tsYactron = -I

63i'et? and

1

,

(c) Nume-rator

Denorninator

Sum ofEumerator

anddeaominator

FractioE

0 r0 0+10=10 0

10

I 9 1+9=10 1I

2 8 2+8=lO 2

8

3 7 3+7=10 3

i4 6 4+6=10 4

6

D 5 5+5=10 q6

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.'. The fractions whose numerator anddenoninator add upto 10 are;

(d) We can find an in-finite number of fractionswhose denominator is 4 more than thenumerator.

12349'8'7'6

367'9'

15

2L3i'e'E'

0

10and

etc.

5=b

(b) If the numerator is equal to the denominator,then the fraction is equal to 1.

246For example: r, Z' 8,.1".

Q3. F'iI up using one of these 5', '? or'=',

..1\a),

(c) 1

I

1

1

7

8

13=b4

Z

(6) 9b

,alFor example:

11'

Q2. A fraction is given.

How will you decide, by just looking at it,whether the fraction is(a) lese than 1? (b) equal to 1?

Sol. (a) If numerator is less than the denominator,then the fraction is less than 1.

For example: etc.

Ql. Draw number lines and locate the points ontJrem.

'We have divided the number line from 0 to1 into eight equal parts.

B represents

C ropresenta

D represents

So1.

2005 r--r -te) ,r* [--] r

1(d,

(c) 1

1

7

81

A BCDE1

2005(e) ,O*

and H renresents ]..2 3lc) =, =,bb

1

2378'8'8

..113 4\a) 2'2,4'A

3Z

(u) *,I82

83

8

2584=r=,=r=bbbb

(c

ltg4Sol. <d ,, n,7, n

04

1

42 4

4

4EB

8bA

We have divided the number line from 0 to1 into four equal parts.

01234667Io6866s665

From the above number line, we have

C represents

D represents

E represents

35

C DE FGH I

C represents

B represents

D represents

2

4'1

43

Z

I2 2

=b3:b

4=tand E represents t,i..., t.

1(b) ;, 8

2

8'A

1012a46678888888888

37 8and I represents

U

Q2. Express the following as '"ired fractions:..t7\c) n

8IB CDE FGH ..20\a) g

@)2:t

b

(e) (r)

11

6l96

FRACTIONS

9

113

(b)

(d)

)

l.€.

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sor. to) f;

(a)

@)

Sol. (a)

(b)

(c)

@)

(e)

v)

We have,

s IZo-( 6-18

@)+

2

10

'W'e have,

6Ir816-26

3

20 ^2- T=o5- 1lt0 -_

t

We have,

6TTTI2-10-f

. 11 _"166

..77\c) n

We have,7 tT7\2-14

28 _356

..19\e) =o

'We have,

6)-rc-( g-18-1

19 -166

q) I'We have,

e fsi-( g_27-T

<"> z9b

a8t"3 31

4 4

_6 4tb- = -77

2

3

36

. 77 _.377

36 -899

Q3. Express the following as i:aproper fractions:_3 ._. _6t -; \b) b;+l

,o; @)sx

-3 7 x4+344

-6 6x7+6h- = _ =

-5 2x6+566

169 _ lOxS+g55

^3 9x7+39-=-=77^4 8x9+4tt- =

-

=99

Tny THese (Paee 142)

er. Are j,"af 'f ""aGive reason.

22its

31

4

4t;77

?53

6

66-=

76

T

61766

353bb

eg=9q

8!=1996

114

"oa $ "eoi"ut"otf

MATHEMATICS-VI

sol. ra l and ?"3 7

WehavelxT=7and2x3=6Since 7 ;" 6

| _2.: t and 7 are not eguivalent fractions.

....2 .2(rr) E

and -We have,2x7 =74, 2 x 6 = 10

Since 14 ;c 10

2 _2.'.

E and

7 are not eluivalent fractions.

.-...2 6\LLL) 6

aarl -We have,2 x27 =64,6x9=64Since 64 = 54

26.'. d

and 7 are eguivalent fractions.

Q2. Give example offour equivalent fractions.Sol. The following pair of fractions are equivalent

... 1 .4(r) , and E

.... 3 6Qt) ,, at(I

-..... 5 . 15(zrz) - and

-''9 27...6 .L2(zDl

- and -' ' 11 22

Q3. Identi& the fractions in each. Are these fractionsequivalent?

(D

(iii)

(i) Figure represenLs

(ii) Figure represents

(iii) Figure represents

(iu) Figure represents

(rr)

(iu)

6 6+2 3

8 8+2 4

9 9+3 3

L2 L2+3 4

L2 12+ 4 316 76+4 4

15 15+5 320 20+5 4

IIrIIIIIII

IIIIIIIIII

ITIIIIII

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Since allrthe fraction feCual to

Z .

,16ano ,6 are

912A'16

3 3x6 18

2 4 6 8 r0 .L2s' 6' g' t2' 16

and IE are

6

1

b

.'. They are equivalent fractions.

Tnv Txese (Paee 1431

Ql. Find five equivalent fractions of each of thefollowing:

Gi

(iu

2

53=b

1

b

b

9)

2Sol. (i)

Qii)

(ii

22x243x2 6

2x4 I

2xg 6

3

2

3xg 9

2xE 10

3

2

3

2

t212

3x42xG

3 3x6 16

equivalentSo,

fractions.I=bI=b1

D

Lx2 2 1 1x3 3

6x2 10

1x4 4

5x3 15

rxb b

6x4 20

1x6 6

6x5 26

so,

6x6 30

2

So, tho simplest form of

an<l $ are euuivarent1

tt

34516',0'%0I

fractions.(iii), (iu) Try Yoursef.

Tnv Trese (Paee 146)

Ql. Write the simplest form of:

.. 16 .... 16 ..... 77{il * \zt) i, \tLL) il

42 ..80Gv) u lu)

2416

Sol. (i) *We have, 15=3x6

76=3x6x5.'. HCF of 15 and 76 = 3 x 6 = 16

15 15 +- 15 1

76 76+LB 5 15 1

FRACNONS

765

115

.... 16\LL) i,

(iii)

We have, L6=2x2x2x272=2x2x2xBxB

.'. HCF of 16 and 72=2x2x2=816 16+8 2

72 72+8 92

So,

L772 9

61

We have, L7 =7xL761 =3x17

.'. HCF of 17 and 61 = 17

17 \7 +L7 1" 61 51+17 3

42(iu) UWe have,42 =2xBx7

28=2x2x7.'. HCFof 42 atd,28=2x7 =14

42 I)+LA 3

16

28

42

28+L4 2

3So,

..80\D)

2a

'Wehave,80=2x2x2x2x624=2x2x2x3

.. HCF of 80 and 24 = 2 x 2 x 2 = 8

80 80+ 8 10

2

" 24 24+8 3

-80 10So,Za=T

Az. U fl in its simplest form?

Sol.49MWe have, fagtols of49 = 1, 7 and 49

Factors of 64 = 1,2,4,8,16, 32 and 64.

Common factor of 49 and 64 is 1.

LA-.- : is in its eimDlest form.

64

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Q1. Write the fractions. Are all these fractionsequivalent?

(a)

(a (ii\

(iii) (iu)

(6)

(, (,0 (iii)

ooo(iu)

Sol. (o) (i) Shaded part =

(ii) Shaded part =

(iii) Shaded part =

(iu) Shaded part =

(b) (i) Shaded part=

(di) Shaded part =

(iii) Shaded part =

(lu) Shaded part =

8+4 2. . All fractions are equivalent.

4 4+4 1

u

1

,2

Z3;o4

s

2+2 1

4+2 2

3-r 3 1

6+3 24+ 4 1

12q92

61;6

L2+4 33+ 3 I9+3 3

2+.2 1

6+2 3

(u) Shaded part 6+3 215 15+3 5

Since all the fractions in their simplestform are not equal..'. They are not equivalent fractiors.

Q2. Write the fractions and pair up the equivalentfractions fron each row.

ooooooooooooo ooo ooo oo

oo

o

oo o

ooooooooooooooo

II116

@) (b) (c)Sol. (a) Given that:

MATHEMANCS-M

IIIIIIII@)

(,

G)

(aSol

c

(e

(iv)

1

,3 3+3 I9 9+3 3

3

Z6 6+6 1

18 18+6 3

number:

(u)

... 4 4+2 2(b) -=-=-'6 6+2 3

..2 2+2 1@) 8= 8-J=A

tfl and (;u) = 3

rat aad (u) = i

,'. 5tb) -8_10

)

(ii) (iii)

@)

(,) .... 4 4+4 1' ' 8 8+4 2

-.. 8 8+4 2\LD)

12= n+4= g..... 12 12+ 4 3ILLL|

-=_=_' 16 16+4 4

. 4 4+4 1(D) -=-=-" 16 16+4 4

The following pairs represent the equivalentfrastion6:

t.,l andbi) = *fcl ad (;) = *tel and (rrr) = i

Q3. Replace ! in each ofthe follorring by the correct

8

20

..18\e) * 4

46 15

60

..2la) - =

3(c);=D

2

7

IIITITII

IIIIIIrI

8

!=

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+2x =7x8 + z=ff=,,2

7

(b) Given that: f, =

8

28

0I + 6x =8x10

n=T=*5

8

3

5

10

(c) Giventhat: I =6=n='fo="

t2

6x =3x20

= 45x =15x60(d) Given that: ffi =

20

15

_ 16x60_2046

16

60 20

46

(e)Giventhat: j!4

+ 24x =18x4

18x4 ^24

18

Q4. Find the equivalent fraction of 3 **(o) denominator 20 (6) numerator 9(c) denoyninsfor 30 (d) numerator 27

Sol. (a)Here, we require deno"ninator 20.

Let N be the numerator ofthe fractions.N3_.. ;=; + 6xN=20x8

= rt= 2o'3=tz5

.'. The required fraction is ffi.(b) Hore, we required numerator 9.

let D be the denominator of the fraction.93D6

244

= 3xD=9x5

3

FRACTIONS

o= f =rs

117

.'. The required fractioo is ft.(c) Here, we requirgd flsaorninstol $Q.

Let N be the numerator of the fraction.

... N_g + 6xN=BxB0305

,- w= 3'30 = 186 -'

.'. The required fraction is S.(d) Here, we required numerator 27.

Let D be the denominator of the fraction.

'. 27 =9 = s yl=gx2lD6n _ 6x27 _rR

. . The required fraction is fl.45

Q6. Find the equivalent fraction of f,$

with

(a) numerator 9 (b) deno-'i"ator 4Sol. (a) Given that numerator = 9

936.. ;=48- = Dx36=9x48

:3 n= 9'48 = t.2

aro

So, the equiva.lent fraction is |.t2(b) Given that denominator = 4

N364 -8

30

= Nx48=4x36

\r_ 4x36_a

So, the equivalent fraction is | .4

Q6. Check whetherthe given fractions are equivalent:

30 -.312 _-7 6,64 (6) 10,60 (c) 13' 1r

andol.S

(a

(a

6')-'96)-'9 64We havo 6x64=270and 9 x3O =27OHere 5x64=9x30

. $ ura f| a* woi"uteot fractions.

(b) ,oa

a,,a *1?

We have 3x60=160and loxl2=120

tr-

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4A

6oL2

n4860

SoI. (o)

Here 3x50*10x123 -t2.r

16 and

_: are not eguivalent fractions.

r"r a and I'13 11

Wehave 7 x Ll =77 and 5 x 13 = 65HereTx 11 *5x 13

7x.: 1r=

ald 3 are not equivalent fractions.

Q7. Reduce the following fractions to simplest form:._ . 150 84(b) 60 (c) ,,tdL"24

We have

48 = t,2,3,4,6,8, ,16,24,86O = 1,2,3,4, 5,6, 10, 15, 20, 30, 60

So' 60*72 - 6

Hence,([84606

60We have

150 = 1, 2, 3, 5, 6, LO,15,26, 50,75, 160

,60

(a

@

HCF = 12

48+12 4

150b

60 = t,2,3, 4, 5,6, LO, t2, L5,20,.'. HCF = 30

^ 150 + 30 5so' 60* 30 = ,

150 5flenc€,

To- = -..84(cl

-"98We haveM=7,2,3,98 = L,2,7,

HCF = 14

6,7,72, ,21,28,42,U49,98

4

So,

Hence,84

98

84+1498+14

6

i6

i6g'62

We have 12 = 7,2,3, ,6, L2

52 = L,2,HCF=4

118

13,26,62400+50 8

MATTIEMANCS-VI

So,12+4 3

52+ 4 13

t23E2 13

Hence,

*r*

Q8.

Sol.

WehaveT=1,28 = 1,2,4,

HCF=77 +7 I

14,28

So, 28+7 47t

Henc€, ,6 = ZRamesh had 28 pencils, Sheelu had 50 pencilsand Jamaal had 80 pencils. A.fter 4 months,Ramesh used up 10 pencils, Sheelu used up 26pencils a-nd Jernoql used up 40 pencils. Whatfraction did each use up? Check ifeach hss usedup an equal fraction ofher/his pencils.

Ramesh used up 10 pencils out of20 pencils.10 I... lraffion = Z0

= 2

Sheelu used up 25 peocils out of60 pencils.

26 26+26 1.'. -ffACtrOIr = 50 6O+28 2Jamaal used up 40 pencils out of80 pencils.

Fraction404180 82

Factorsof400=2x bxbHCF=2x6x6=60

260+60 5

Yes, eacl hsc used up an equal fractio*, i.u., |.Q9. Match the equivalent fractions aad write two

more for each.

,r::3 @"(r, # ,ur?

..... 660 . . 1(ra., ggo \c) ,

. (,u)::3 (d 3

..220 Itu) U, te)

,O

SoI. (, ffiFactors of250 =2 x 6 x 6 x 6

2x2x2

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260 5 _.._ 400

* 8 or (r.) e (d]

Two additional examples of equivalent fr actionsare..5 3 16\a) Bx g= 2a.... 180\tL)

-Factors of 180 =

Factors of 200 =

*,:,; 25

40

3x3x2xEx

HCF=2x2x5=20180+20 I2gQ+20 10

180 9 ,..,

-<-)-or{rzt(-)tet200 10Two additional exa-mples of equivalent fr actionsare

9 218 ...9 3 27(l.,) -x-=-

(D)-x-=-'-' to 2 20 10 3 30.....660ILLL) ggo

SSQ+10 66ggQ+10 99

Factorsof66=2x x 11

Factorsof99=3x x11HCF=3x11 =33

gS+33 2" 99+33 3

sq ffi*; or (i;;)+r(o)

Two additional examples of equivalent fractionsare

...2 3 6(b) -x-=-''3 3 9..2 2 4\o sxr= 6

.. . 180tru) 5*

180+10 18

360- 10 = 36

Factors of 18 =Factors of36 =

rlf,rG\,>ibfu

HCF=2xBx3=1818+18 1

tg+ 18 2

s", #*;or(;u)e(c)Two additional examples of equivalent fractionsare

..1ta) ix

2

2

5

5

3

3

2

2

FRACNONS

3336

._. 1 6 5(b) rx6= 10

119

..220tu) u*

220+LO 22

550+10 55Factorsof22=2xFactorsofS6=5x

HCF = 11

22+11 2

55+11 6

s,, 88223.-';

or (u)e (6)

Two additional examples of equivalentfractions are

..2 2 4\o ;xr= ro

Tnv Tuese (Paee 148)

Ql. You get one-fifth of a bottle of juice and yoursister geLs one.third of a bottle of juice. Whogets more?

Sol. Let us divide a rectangle into five equal partsand shade one of tb.em.

A person gBts oneffth, i.e., I ofa bottle ofjuice.

23 6to) ;x5 = *

Le.,

Tnv Txese (Paee 1491

Ql. Which is the larger fraction?

For his sister, divide the same rectangle into

three equal parts atrd shade one of them,

1We get one.tJrird, i.e., f nart of the bottle ofJUlCe.

So, by comparing the two rectangles, his sistergets more.

r;rlorI'-' 10 10

t7 L2(iii) ,* or -

11 13(ii)

^or ^

Why are these comparieons easy to make?

sor. t;1fr m ftHere, denominators ofthe two fractions aresamesndT<8

.'. ft i"ru"sotlruo *a

11

11

rIITI

III

B:

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.... 11 13\tL) or

Here, denominators of the two fractions aresame and 11 < 13

13. _ 11.'.

24 ls larger than

,Z..... 17 t2llu')

lO2 or

-Ifere, denominators oftJre two fractions aresame and 17 > 12

t7 L2

102 t02These comparisons ar€ easy to make as thedenominptors ofeach pair offractions is same.

Q2. Write these in 6scending and als6 il flssceldingorder.

Sol. (a) Qiygn thal; 1, 538'8

Here, the denominators of each fractions iasame and 1, 3 and 5 are in ascending order.

lL7 43b bbb

! are in ascending order

are in descending order.

(b) Given that: !,+,!,2,!b O bba)lIere, the denominator.s of each fractions issame and l, 3,4, 7 and 11 are in asceld;''g

, f are in ascending order.

1g are in descendi"g order.

.. 131311 7

7'7' 7 ', 7'7Here, the denoninators of each fraction issame and 1,3,7,17 sntl lg arg in asceadingorder.

(b) l,+bb

13, 7 are m aaoen(U-Dg order.

73 7

7, 7 , 7 are in descending order.

151)

l1

g8

13

T7

I

(a

(c

168' 8'13i'i'

138'8

6318'8'8

437=,='=DOD

order.

1347bbbb

13;,;'

11'v'

77L;,;

13

T'nv Tuese (Prer

Q1. Arrange the following in ascending anddesconiling order:

I60

1

I1

120

_.1 1la) 12' 2B'

11b'l 'r7

Descending order is f,,

MATHEMATICS-VI

3 3 33 3 3 3,a\ - _'"' t' tt' 6' z' ts' 4' t7(c) Write 3 more similar exanples and arra-nge

tJrem in ascending and descending order.Sol. We know that if the numerators of all the

fractions are aame in unlike fractions, thensmaller value of the denomi[ator, the greaterthe value ofthe fractions.(o) Here, nunerator of each fractions is 1.

6,7,9, L2, 17, 23 and 60 are in ascendingorder.

111" 60'23'17't2'ascending order.1111 1 1

are rn

6'7'9'12'L7'23 and are ln

descending order.(b) Here, numerator of each fraction is 3.

2,4,5,7 ,1\ LA a:a.d 17 are in ascentling order.

333t 1z'1g'rr'ing order.

3333i,Z,E,i'order.

(c) Three additional examples ofrrnliLe fractionswith same numerators are;

333i'E,Z

2t

anil I are in ascend-

ard f are in rlescendins

I 1

6

1ts'i and

I50

o'i,Ascending order is #, *, iand descending order is

,a) 2

' 11' 13 15

.... 6 5\tL) -, -

Ascendins order is #, *, *, *Pssceaa;ng order is f,,

..... 7 7ltLll

-. -.' 11'g',

Ascen,ri.g order is fr,

26

9

3311' 13

22 2 2

6' 13' 11' 16

bbt)6' 13' 17

77 7

r7'%'^

22i'E an

.)d-

3

222oo I

and

6

6

5

t7

and

andbbb8'12'13

and

and

777?o't7'n777n'r7'lr

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Ql, Write shaded porbion as fraction. Arange themin ascending and descending order using corr€ctgiF'd, '=', '>'between the fractioDs.

(a)

(6)

(c) Show Z,t,Z * f on the number line.

Put appropriate sieFs between the fractionggiven.

6

6

236

6

6 6

1 8

6

5

60,

6Sol, (a) Total number of divisions = I

(i) Number of shaded Parts = 33

.'. Fraction = :8(ii) Tota] n"mbor of divisions = 8

Number of shaded Parts = 66

.. Fraction = =8(iii) Tota] number of divisions = 8

Nunber of shaded Pares = a4

... Fraction = 6(iu) Tota] mrmber of divisions = 8

Number of shaded Pari = 1

1... Itaction = A

Now the fractions are:364 -1:. :. : and :withsamed.enoyninetor.8',8',8 8

AscendinB order: | <

psscetr.Ii"g order 3

>

(b)(i) Total number of divisions = 9Number of shaded parts = 8

8... Fraction = g

(ii) Total number of divisiong = 9

Number of shaded Parts = 44

... Fraction = g=

(lii) Total number of divisions = 9

Number of shaded Parts = a3

... Fraction = d

346888481888

IIIrtt!TI

FRACNONS

(a

6 I

121

isEE*qP-.e.7,.4,.(iu) Total number of divisions = 9

Number of shaded parts = 6

.'. Fractions = $843

.'. FYaCUOnS Are -. -. -denominator. 9' 9' I

Ascendins order: $

<

Descendin8 order f,

>

6

III4

9

4

96

9

143

i

1

i3b

with same

..2tc)

U

48E,6

and6

6

8

93

I

686' 6

-l 0 2o

46

6866

No* |236'6

0.1'66

6

6

6

5

6

Q2. Compare the fractions and put an appmpriatesign.

Sol

Here, denominators ofthe two fractions aresameand3<6.

t6

6=t)5

6

g64=D3

6

(c

(a

3

6

(b) I 1

ZHere, numerators of tJre fractions are sameandT>4.

7

1

i,

1

Z4r-t6(c):ll=b-bHere, denominators ofthe two fractions aresameand4<6.

4;b

33

5Here, numerators of the two fractions aresameand6<7.

3

(b)

@)

(a)

)

)

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Q3. Make five more such pairs and put appropriate8rgtra

Sol. (a)

(e)

sol. ta) |

2

3

56

m

2

749

4

10

c

6

81

I

3

86

9

2

6

3

5

g4

6

6

Q4. Look at the figures and write '<', or 5', '='between the given pairs offractions.

01

02

03

_s.

4

06

0

1

1

46

56

34

2

46

36

1

2

24

-c.

26

1

3

26

22

33

44

66

;2

a3

3

1

4

1

6

1

6

b1

3

2

ZD

B

1

52

4b

E

L;o2

56

6

(a

(c

(e

G)i

,,8

Make five more such problems aad solve themwith your friends

Make five more such problems yourselfand solvethem witJr your friends.

Q5. How quickly can you do this? FiU appropriatesip. ('<', '=' , '>').

)

@)(c)

(e

Zo

b

6

2t3

=o(c

122

@,;Z| rar?13 Here, 30 < 4210

MATHEMATIGS-VI

@[

,r1

2

8

2

8

15

2L

16

@)1

[email protected])i

Sol. (u) ;

3 ,---- 6(e); i l=

(l,)*n;3

9

8

Wehavelx6=6and.lx2=21 I

2

4

6Here, 2 < 5

2

2 ,--- 3ror; LJ IWe have 2 x6=L2a:o;d?x4=12

Here,

G): I 2;o

t2 t2 3

6

WehaveBxS=9and2x6=10

Here, 9 < 10 .'. *ralE I

8

Here,2a<S .'. I

We have 3 x 8 = 24 and,2 x 4 = 8

,5

2

8

(e

(n

6

6 6

WehaveSxS=15and6x

3

9

6=30

95

Here, 16 < 30 .'. I

o-or7

3

I

4=404=b

WehaveTx9=63and3xI

2

8

9Here, 63 < 27

erlnWehavelx8=8and2x4=8

Here,8<8 .'. i(ft)*E t

6

2

8

Wehave6xS=30and10xo

)

(b)

@)

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inlnWehave 3 x 8 =24 and(x 7 =28

7

8

3

4 8

@;n 15

2L

We have 6 x 2l = 105 and 7 x 16 = 105

Here, 106 < 10615

2t

Q6. The following fractions represent just threedifferent nu.mbers. Separate them into tbreegroups ofequivalent fractions, by changing eachone to its simplest form.

Ilerc,24<28

,\a) i (b) *

8(c) 50

..12g) 60

(ft) *

16

100(a

)(e

Sol. (o)

(b

(c)

10

60

t276

2

L2

3

16

8

6o

..10(e) 60'

@#

t2@) ao

&)t#

... L2lt) =='I t)

v)15

76t2an

2+2 I

(o)#lo*

72+2 63*3 1

['.' HCF of 2 and 12 is 2]

['.' HCF of 3 and 15 is 3]

['.' HCF of 16 and100 is 4l

['.' HCF of 10 and 60is 101

['.' HCF of 15 and 75is 15I

['.' HCF of 12 and 60is 121

['.' HCF of 16 and 96is 16I

['.' HCF of 12 and 76 is 3]

['.' HCF of 12 and 72is 121

15+3 6

= :- = i t... HCF of 8 and 6o is 2.t6O +2 25'

_ 16 L6+4 4(.11

- ='-' 100 100 + 4 26

10+10 1

60+10 6

l[+15 1

fg+16 6

L2+L2 I60+L2 5

16+16 1

96+16 6

72+3 4

75+3 26

12 L2-r L2 1ril _ =

-=_

'' 72 72+12 6

(ft) r8g = ** = * ["' HCF or3 and 18 is 3]

6 a = !.7r. =1 t... ncr ora urra zr i" tt*'26 25+l 26'

5

i

FRACNONS 123

Now grouping the above frastions into equivalentfractions, we have

2 to 16 t2 3r .11(,') -=-=-=-=-

leach- |"'L2 60 96 72 18L 6l

.... 3 15 t2'oo' L6- 76- 60

..... 8 16 L2 4'""' 60 LOO 76 26

Q7. Find answera to the following. Write andinilicate how you solved them.

tal Is I equar to f ?

ar rs 9 eoual to !?16' 9

r"l Is 4 eo,ol to 19?520

td Isfreruatofir

6 _4Sol. (a)

9 and

EBy cross-multiplying, we get

5x6=25and4x9=36Since 25 ;a 36

54.'. , is not eCual to

U .

rlr 9 and !*'16 9

By cross-multiplying, we get

9x9=81 and16x5 =80Since 81 + 80

.'. ft i"r"t"cr"rt" i.4 -L6(r!) - and

-*6 20

By cross-multiplying, we get4x20=80ard6x16=80Since 80 = 80

4. .. 16.'. 5 l8 equa.t to

20 .

each

each

1

=b

4%

tar fi "na SBy cross-multiplying, sre get

1x30=30and4x16=60Since 30 * 60

t4.'. ,6 is not eQual to , .

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Q8.

So1.

Qe.

Sol.

IIa read 25 pages ofa book containing 100 pages.,

T.alita rpad f of the s"-e book. Who read less?

tra reatls 26 pages out of 100 pages.

26 26+26 1.'. ltacuolrs = 100 = 100* % =Z

,Lalita reads i of the same book.

comparing ] ""a f ,r"s"t1x6=5and2x4=8

Sinces<8 i.?Hence Ila reatls less pages.

Rafiq exercisetl for I ofan hour, while Rohit

3exercised for 7 ofan hour. Who exercised for a

longer lime?3

Rafiq exercised for U

ofan hour.

RRohit exercised for f, ofan hour.

Comparing | "oa f ,*"s"t3x4=12andBx6=18

Si[ce 12 < 18

Hence Rohit exercised for longer time.

Q10. In a dass A of 25 students, 20 passed in firstclass, il another class B of 30 studente, 24passed in first class. In whi& class was a great€rfraction of studenta getting first class?

So1. In class A, 20 students passed in first class outof26 students..'. Fraction ofstudents getting first class

20 20+6 426 26+6 5

In class B, 24 studonts passed in first dass out,of30 students..'. Fraction ofstudents getting first class

24 2A+-B 430 30+6 6

Comparing the two fractions, we get | = !DttHence, both the class A and B have the samefractions.

3346

'124 MATHEMATICS-VI

T'nv Tsese (Faee 155)

QI. My mother divided aa apple into 4 equal parts.She gave me two parts and my brother onepart. How much apple did she give to both ofus together?

Sol. Apple was tlivided in 4 equal parts.I got 2 parts.

,,... Fracrion = f,My brother got I part.

'I

.'. Fraction = :4

.'. Fractions got by both together

2L2+l 3

44 4 4

3Hence, both ofus got 7 ofthe apple.

Mother asked Neelu and her brother to pickstones from the wheat. Neelu picked one fourthof the total stones in it and her brother alsopicked up one fourth ofthe stones. What fractionof the stones alid both pick up together?

1

Neelu picked up I th of the stones.

IHer brother picked uD : th of the stonss.' '4.'. Fractions of stones picked up by both

= 1+1=3=144 4 2

Hence, tJre stones picked up by bo6 = 1 ofttestones. 2

Sohan was putting covers on his notebooks.He put one fourth of the covers on Monday.He put anotJrer one fourth on T\resday and thelg6eining on Wednesday. What fraction of thecovers did he put on Wedlesday?

1

Sohan put ; tJr ofthe covers on Monday.4

1

He put 7

th of the covers on T\resday.

He put the remaining covers on Wednesday.

q2.

SoL

Q3.

Sol.

f,p111qining covers = 1- 1l-+-44

=Fz=42L42

Hence on Wednesday, he put I of the covers.

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ITTTTTII

Tnv Tnese (Paae 156)

Q1. Add lrith the help of a iliagram.

,, *.* ot) ?.3 uiit f,+11-+-66

Sol. (i) +

+ 1

b

31=n =T111666

1

T

2T

oi) +

+26 E 9=r

b

Gii) +

Q2. Add fi * fi . ff"* *U we show this pictorially?

Using paper folding.

1 1 1+1 2 tsiol- -+-=-t2 72 12 t2 6

1lTo show fr+fr by pictograph, we get

1

+

+1 2 1

72!2 72

Using paper folding is an activity.Students will do it themself.

Q3. Make 6 more examples of problems given in 1

and 2 above. Solve them witJr your friends.Sol. Example 1: Add with the help of diagram

ot !*?*2 <ut l*?ODDIIEranple 2: Add with the help of diagram

",1.1 *,:.*Example 3:Atld | * | . H"* *iU rou show this

pictorialy? Usitrg PaPer folding.

prarnFle 4rAdd |*f . H"* *iU rou show tldg

pictorially? Using paper folding.

Examples:Atld $*fr *itltlr" heln ofdiagram

solve tJxe above examples witJe your friends.

Irlrttt

ETTIIIIIIIIIA III

FRACNONS

16

125

Tnv TrrEsE (Paee 157)

Ql. Finil the difference betweetr I *U 3.s"r.n""",Ir]

7 3 7-3 4 I"8 8 8 12thus, the diference between * *, 3 = i

Q2. Mother made a gud patti in a mund shape. Sheilivided it into 6 parts. Seema at€ one piece fromit. If I eat another piece then how much wouldbe left?

Sol. Total number of equal parts ofgud patti = 5

Number ofparts eaten by Seema = 1

Fraction of eaten part =

Number of parts eaten by me =

15

1

.'. Fraction of eaten part = 16

.'. Fraction ofgud patti eaten by

1 1 1+1 2Sesma and me = 6*6= 6 =6

.'. Fraction of gud patti left2=b

L 2 1x5-2x1_116 5

Q3.

Sol.

6-2 3

553

Hence. the left fraction = ; .'tMy elder sister divided the watermelon into 16parts. I at 7 out of them. My friend ate 4. Howmuc.h did we eat betweon us? How much more ofthe watermelon did I eat than my friend? Whatportion of the watermelon remained?

Total number of parts of wat€rmelon = 16

Number of parts eaten bY me = 7

.'. Fraction of watermelon eaten bY me = ,alNumber of parts eaten by mY friend = 4

.'. Fraction of watermelon eaten by my friend416

Fraction of watermelon eat€n by me and myfriend

7 4 7+4 11= -+-=-=-16 16 16 16

. . Fraction of watermelon eaten by botJr of us11

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Fraction of watermelon eaten by me- Fraction ofwatermelon eaten by my friend

7 4 7-4 3

QI. Write these Sactions appmpriately as additionsor subtractions.(a)

(b)

(c)

665given diagrams can be represented

Sol. (o) Tlee given figure represents tJre addition ofI -2: and =.bb.12L+231.e., ; +

D

Thus theaa

16 16 16 163

So, I ate ,a part more tb.an my friend-

Portion of watermelon left now

- 11 1 11 1x16-1x11-^ 16-1 16- 16

6(b) The given figure represents the difrerence

between l and l.b

66

+

+

3=b

2b

3b

1

I

1 3 1x6-3x1 5-3 2i.e., 1- 15 6 55thus, the given diagrams can be representedas

(c) The

and

given figure r€presents ad<lition of fr3

ITTIT TITTTTITT

oooooo

oooooo

oooooo

IIIII tllttITIIT

1266

(r) 3DIb 6

16-11 6

16 16

Hence, the left part ofthe watermelon = * .

Q4. MsLe five problems of this type and solve themwith your friends.

Sol. Try yourself with the help of Q. No. 2.

.232+351.e., 6+- = 6 =6Thus, the given diegrarne can be representedas

26

+

+ 66;

SolQ2. VE:

(a)

Sol. (a)

(b)

(c)

@)

(e)

11-+-18 1876

*,*.1?

77c ,, *.#rn 9+9"'8 8

@)i.I

..12 7(e) 16

- 16

*,,-3('=;)(,) 3-1?

1 1 1+1 2 2+2 I----18 18 18 18 78+2 I8 3 8+3 11

15 16 15 16

7 5 7-6 2

77 7 7l2l L+zL 22,22' 22- 22 - 22- -

72 7 12-7 5 6+6 1

16 15 15 15 15+6 3

16 q*9=5*3=9=r

-'8 8 8 8

- 2(- 3'\ S 2 3-2 1G)'-5['=5J=5-E= s =E

.-. I 0 1+0 1th) _+_=_=_'44 4 4

L2 3 LZ 3x5-12x1 L5-12 3Db

IUATI{EMATIGS-VI

oooooo

oooooo

oooooo

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Q3.

Sol.

Q4.

,Shubham nfited i of the wall space in his

room. His sister Madhavi helped and pqintefl1

! of the wall space. How much did they paint

together?

Fraction ofwall painted bf Snubhan = JFraction of wall painted bf Madbavi = |Fraction of wall painted by Shubham and

Madhavi= 3.+=+=3=,Thus the fraction of wall painted by both = If'jll in f[s ynissing fractions.

tal r-L_J 10

35(b)

33@)(c) 66

Sol. (a)Ihe difference between 110

.'. Missing fraction

7 3 7-3 4

10 10 10 10

(c) The ilifference between

3

Ihus,2

6

(b) The ilifrerence between f-l anil3

n

21 2l612

' 27- 27

ald 1S

4+2 2

10+ 2 5

la

and

3 3+3 6

3

10

2L'5

I

6 3 5+3 8.'. Missiag fractiot = 2t+ ZL=

- = -

'Lhus,8

2Loo-]a-66

.'. Missing fraction 66 66Thus, =1

(d) Sum of and6L2

- Ig

--27 27

Ql. Solve

2t(a) ,+ ,(a 51-+-73

*,*.*2L(e) =+=bb

*,:-*

(c)

q)

2

7

2

531-+-42

4-+9

4;*b

..31@)z- g

FRACTIONS

2(i) 5+ 2L 17

Hence, ,+7 =;12t 2L

127

t2 6 t2-6 7

m*,E27

Q5. Javed was given f ofa basket oforanges. What

fraction of oranges wa6 left in the basket?

Sol. Fraction ofbasket of orar 6

'Eies = tFraction ofbasket as a whole can be taken as 1.

.'. Fraction ofbasket of oranges left

-6151x7-lx67 L7 7

7 -6 2-77

Thus, the required frtrctro = ? .

Tnv Tnese (Paee 159)

.. 1y1lssing fraction 27 27 27 27

36 36

2x7+lx3 l4+3 L7

Ql. Add and

Therefore

6Hence

6x6 2x7 26 74 11

'7x6 6x7 35 36 36

2LL7 5 36'

3

i,=b

2SSol. fls hays - .1. -

LCM ofE and 7 is 36

2x7 3x6 L4 16Tlrerefore, =-'-=

i =--=-----'-'-'6x7 7x6 36 35

14+16 29

23 29

57 35Hence

e2. subrract ?*"^X.sor. wenave |-f

LCMofTa-nd6=36

,;.*.* ta r|+aJL67 4l

(m) =-; (z) ;-;DONZ

at +f;+*

sa. tarf+|= 2L

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to) *+*LCM of 10 and 15 = 30

. 3+7_3xg _7x2 _9,t4_Zg" 10 16 10x3'16x2 - 30 ' g0 g0

_3723Hence, 15+16=-

..4 2 4x7 2x9tal _+_=_-l-_'"'9 7 9x7 7xg28 18 28+LA 4663 63 63 63

42 46Hence, g+7=-

,.,6.1 6xg,lx7 L6.7 16+7 22"7 3 7x3 3x7 2t 2L Zt 2t5122fience,7+E=-

..2 L 2x6 1x6(el -+_=-+-" 6 6 6x6 6x672 6 72+ 6 t730 30 30 30

2l L7Hence, S+-=-

,^ 4 .2 _4x3 . 2xBv'E-5-ErB-ar672 .10 12+ 10 2215 16 15 15

__4222H"r*- -+-=-'5 3 15

,,31 3xB 1x4 I 4 9-4 E"' 4 3 4xB 3x4 L2 t2 L2 L2

315Henc€,;-3=12

51(h) ---"6 3LCM of6 and 3 = 6

6 1 5x1 1x26 3 6x1 3x2

5-2 3 I25

6 6 6 62Hence,

51163 2

...2 3 L\L)

z+ Z+ -LCM of 3, 4 sad,2 = 72

128 MATHEMANGS-VI

23-+-341 2x4 3xB 1x6+-=-+-+-2 3x4 4xB 2x6

8 9 6 8+9+6 23

72Hence,

_- 1 1u) ,+

"+

12 L2 t2 t223

72

23L-+-+-342I6

LCMof2,3and6=6lx3 1x2 lxl 3 2 L=-+-+-2xB 3x2 LxG 6 6 6

_ 3+2+1 6 .66I+- = 1Hence,

11-+-236ttl+s? = r+1+e+? = ++!*?3 3 3 3 33

.L+23= 4 * ---=- = 4+- =4+l=533

tt.r""- t1+ a3 = r'3 3 -

al aZ+sL = ++?+s+L = l,+s+?+!" 3 4 3 4 34- 2x4 lx3-83

3x4 4x3 12 L2

-8+3 -11 771= l+- L2t2tL2_ 7 xlz + lxLl 84+11 95

t2

H"o"..42*g1=95'3 4L2L6 7 L6-7 9

12 t2

(rn)

(n)

56Hence,

4t

65167Ibbb4x2 lxg 8 3 8-3 5

Hence,

2 3x2475326

3 .x.a 6 6 6 6

Q2. Sarita bought f mehe ofribbon and Lafita Imetre of ribbon. What is the total tength of the

ribbon they botghf

SoI. l,ength ofrib[ea fu6rrghf !y garita = f netre

Length of ribbon bought by Lalita = | -"C"

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.'. l,ength ofribbon bought by Sarita and Lalita

= fr-"**|-""" = (f .|)-"*

=(

=t

2x4 3x6

-+-6x4 4x6

metre=

b

61 (6)84

816-+-20 20

metre

1

4

8+15\ 23

zo J t"'"" = - mef,re

Hence, the required length = ffi mere'I

Q3. Nqina was Siven 1i piece of cake and Najma

1

was given 1f niece of cake. Find the total

amount of ca}e was given to both of them.

Sol. Piece of cake Siven to Naina = 1|

Piece of cake given to Najma = r|

Piece of cake given to Naina and Najma

= 11+11 = r+1+r+1= r+r+1+1-2 -3 ' 2 3 2 3

= 2*l'3 *!*2 = 2*2*? = 2*!-2x}3x2666

26

Hence the total amount of pieee given to both

= 29.6

Q4. FilI in the boxes:

(a)1162

,'1tc) r-

1

6

61Sol. (z)

-or" tnuo f,.

84Here, missing number is

6

8

D

8

8

1+-4

xL +Lxz

5+28

Hence

FRACNONS

8

129

(b)1l62

Here, micsi',g nunU"" i" | .o"" tUo | .

1162lxz LxB

= -+-6x2 2xE2510 10

2+5 7

10 10

7Hence

10

1@) r-

1

61l

sing number is f lesstl'"":.Here, mis

1126lxB 1x12xB 6xl31663-1 2

661

31

3Hence

Q5. Complete the addition-subtraction box

(a)

-€=---+(b) --------@------>

2 4 2+4Sol. (a) Addition: I Row ;+- =

-3B 3

6 6+3 2 ^3 3+3 I

l2l+23-IIRow5+5= B =E='

Subtraction: I C"l"-.t :-+ = ? = +4 2 4-2 2

If Column: 3rr33

2 43

1 2

1

21

31 1

4

www.ncert.info

Ihus the box may be completed as follows:___€l------->

(b) Addition" I Row:1 I lx3+1x2 3+2 623 6 6 6

1 I 1x4+1x3 4+3 7ll ltow: -+-=34t2t2t2Subtraction: I Col 'rnn:

1 1 1x3-lx2 3-2 I2311

664-3 1

6

1x4-1xBII Col 34 L2 L2 t2Thus, the box may be completed as follows:

€1

21

\,I 1

411\\6/

11\\12./

1

4

Q6. A piece of wire metre long broke into two7

8pieces. One piece was I metre long. How long

is the other piece?

Sol. Total lsngth of the wire = | -"*Length ofone piece ofwire = 1 metre

4

.'. Length of the oth"" ei"* = ; - 1LCM of8 and 4 = 8

777xLlx27Z7-26" 8-a = 8"L- 4"2 = 8-s = 8 =8

Hence, the length ofthe other piece = f, net *

Nanrlini's house is ft t- m- n", .chool. She

walked some distance and then took a bus for

I U "

ru."l m" school. How far did she walk?

Totsl distance from house to schoo, = * *.

Distance travelled by Nanrlini by b* = i *

Q7.

SoI.

2 4

1 2

r\3/

12\\s/ 1

130

65

MATHEMATICS-VI

.'. DistaD.ce travelled by her on foot9 - 1 19 1\

= ro k__; h= [m_AJ

LCM of 10 a.nd 2 = 10

(#, km( gxt 1x6\(ro, r zx6 )

km

le:!)t.\10/( 9 6'\-

=t---tkm=\10 10/4 - 2..= _km=E}m

Hence, the distance travelled by her on foot =

km

2

6km

Q8.

Sol.

Qe.

Sol.

Asha and Samuel have bookshelves of the samesize partly fiIIed with books. Asha's shelf is

f rl ru "oa

su-uel's shelf is f ,t n l. wto."bookshelf is more frrll? By what fraction?

Ashatshelfis f,tnflila-nd Samuel's shelf i" I A n U

Comparing 5 *U ?LCM of6 and 6 = 30

6x5 26" 6x6 30

. 2x6 12and

-=-5x6 30

__62Hence, 26 > 12, So

U is more than i .

Hence, Asha's shelf is fuIl more than Samuel'sshelf.

_. 6 2 26 t2 26-12 13iow- ---'6 5 30 30 30 30

ff"o.., ffi tn m"tion is more frrll of Asba,s shelf.

Jaidev takes 2l -;''u1gg to walk across theb-school ground. Rahul takes 1 minutes to do the-4same. Who tales less b''ne and by what fraction?

1

Jaidev tqkes 2: minut€s

Rahul +rLes

Conparing 2

minutes

minutes aad f -roo*"-l -I2L2x6 lxL L0 1Z-- = Z+ -:- =- +- =-+- - -+-5 6 1 5 lx6 6x1 5 5

10+l 11

Z1

=b

@

o

,!,J2,

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Now, the given fractio* ,t" f *a ILI 4M6420

Here, 35 < 44 .',

and635620

,

l. Venv Suonr Aruswen (VSA) Quesrols

Q1. Represent the following fractions on numberline.

,.1\a) =b

1(d:b

Sol. <-----t-re----r--t--->0123461366

Point A represents

7-x4

11-=.t)

ABCDE

t:b

g5

Point C represents

Q2. Write the fractions showing the shaded portions:

(a) (b)

Sol. (c) Shaded portion repr€sents

Shaded portion represents

Q3. Colour the part according to the fraction given:

14,6

_.3\a) i (b)*

sol. (o) f, (b)+

Q4. Identify the pmper and improper fractions

51uod9Sol. Proper firactions are: 6,, 4

7 tL -6lmproper fractions are: ,,; and

6

Q5. What fraction of tJrese circles have'x'in them?

@ @@o@@oc

5 7 1 3 11

6'r'r'z'E6=tt

IIII II

FRACNONS

1

1

131

So, the time take to cover the sa.Ee distance byRahul is less than that ofJaidev.

tt7 M36 M-36 I=-=-[unutes6420202020Hence, Rahul ** * minutes less to across

the school ground.

LearningMoreQ&A

Q7.

Sol. Fraction ofthe circles with'x'in the given figure5

Q6.

Sol.

8

Write all the natura] numbers from 1 to 16.

What fraction of them are prime numbers?

Natural numbers from 1 to 15 are

r,2,3,4,6,6,7,8,9, rO,lt,12,13, 14 and 15

Prirne numbers from 1to 16 are 2, 3, 6, 7, 11,13, i.e., 6 prime numbers.

. . Fraction of prim" o*b"* = 916

Identify the like fractions from tJre following:21636e'B'6'Z'E

,1: a-nd I have the same denominator.33

. f r"a I arethelikefractions.

Q8. Identi-ff the unlike fractione from the following:

Sol.

35';,;

22oa

1

61

6Sol. and have different denominators.

22itlol

2

b,? *, * are udike fractions.

Q9. Convert the following improper fractions intomixed fraction.

37(a) a

13(b) T6Is7( 6

-36

37?Sol. (o)

37 ^L-e=o6(b) * 2E5(6z -72

(mired fraction)

www.ncert.info

13 1

26 (mixed fraction)

2

Ql0. Convert the following mixed fractions intoimproper fracbions:

,.-2(o) D5 (b) 69o

^".--2 - 2 6 2 6x3 2x1Dol' (O) o:- = b+-=-+-=-+-U 3 1 3 1x3 3x176 2 16+2 17

-3'3- 3 -3_2 L7

... 6- = - (inproper fraction)33...^5 ^ 6 6 6 6x6 5x1(O) b= = b+-=-+-=-+-6 6 1 6 1x6 6x1

36 6 36+5 4l6666

^6 4L- Ue = ? (improper fraction)

ll. Snonr Ar.rswen (SA) Quesrror,rs

Q11. Write the following fractions in ascending order:223'i and

,11

2=b

2

9

Sol, Here, the nurneratore of all the fraction-s aresame.

2 2222... 6gcelding order * n,;,2,;,i

Q12. Write any(o) three proper and three improper fractiotrs

with denominator 7.

(b) two proper and two improper fractiom \rithnumerator 9.

Sol. (a) Proper fractions with denominator 7 are:23 _6-.- and -7'7 7

Improper fractions with denominator 7 are:

9 11i'n 7

13and

(b) Proper fractions with numerator 9 are99_ alld _11 L7

lmproper fractions with numerator 9 are9 -9-and-25

Q13. Compare the following fractions:

to)fanaZ ,u,i*u?

132 MATI{EMANCS-VI

sor. <ol f ana ILCM of5 ard 6 = 30

4=bq6

4x6 246x6bxb

30

26and

Here,

and

6x6 30

or|""af

%<26+?0.2i

6 6x4 20

rr>s= E>g20 20

46DO

LCM of4 aad 5 = 20

3 3x5 15

4 4xE 20

2x4 82

Here,

Q14. Aad #,* *uI%

Sol. LCM of 12, 16 and 24 is a18

324,8

2 L2 16 24

4 62 3

LCM=2 x2x2x2xB=48,1,7

7 7x4 2812 l2x4 ,i18

5 6x3 15

16 16xB II 9x2 18

24 24x2 I

e 4a+P=4)

('.' 48 + 16 = 3)

* 4a+?tL=2)

7 6 9 28 16 18

-+-+_ - _+_+_12 t6 24 484f34a

I4s 16-I(_48

28+16+1848

61 _13484a

,, ^2Q15. Find the s"'n of 1! and 3i .

sor. 13+ s? =r+?+s+Z =t+e+?+?3 6 3 5 36

13

|tl

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=,.(*=*#3)=,.(i3.*u)_ 4+(10+6) = ++rj=++r+ L

15 15 16

- 1 -1= h+-=h-16 16

nence, rf +sf = s1

GQ16. subtract zln" +f,.

3 (4x8)+1 (2x4)+3 32+1 8+348484Sol. 4 1

82

33 11

84LCMof8and4isS

33x1 11x2 33 22

8x1 4x2 8 8

33-22 11 ,3888

Hence, 4 21 3

84

7 7

13

8

Q17. Insert > or < to make each of the following true.

(a

(a

6

i6;

D

;D

i

(b),1qEl; ,;tr3Sol.

Here, denominators are same, i.e., 7 and 6 > 5

6 b

c)#tri;Here, numorators are same, i.e., 10 and2L>12

10:' 2l3 r---r 3

tc) 7 L_J a

10

L2

Here, numerators are same, i.e., 3 and 7 < 8

3

I

3

8

lll. Hronen Onoen Tnnrrr.rc Srtlls (HOTS) Quesrtoxs

Q18. Find the difference between the greatest andthe smallest fractiong.

s2.zL.6' 7'19 18

a'TFRACNONS 133

sor. welave a!,2f,, 19 18

6,73

b

4

i

(3x5)+3 16+3 183

2

6

(2x7) + 4

66l4+4 18

77

18

TD

56

I

and

22

35

18 18 19

Dlo

Improper form of all the fractions are

LCM =2x2x2x 3x5x7=840LCM of5, 7, 6 and 8 = 840

Making the denominators same, we bave

18 18x168 3024:: = ::_ji::= :-r--- [... 940+ 5 = 168]5 5x168 840

18 18 x 120 2160['.'840+7=120]

7 xl2019 x 14019

['.'840+6=140]

1

I

71I

842

11

633

840

2660

618

8 5x105 840['.'840+8=105]

6x14018 x 105

8401890

u"r" ffi o, f i" the greatest fraction aod

# * f is the smallest fraction.

Difference

18 18 18x8 18x5 L44 90

5

6440

8 5x8 8x5 40 40

27

Hence the required difference = T * t*20 20,Simran painted ! of the wall sPace in her room.

Her brother Rahul helrcd anrl painted I of the

wall space. How much did they paint togpther?

Wbat part ofthe whols space i8 left unpainted?

Space of the wall painted by Simran = f,

Spaco ofthe wall painted bv Rahul = ]

20

Qle.

Sol.

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I-+3

3

I 1 13 1x15 13x11 16 1x15 15x116 13 16-13 2

16 16 15 15,

Hence f th of the wall space is unpainted.Ib

Total space painted by both =

2x5 1x3 10

5

10+3 13

3x5 6xB 16 16 l5 15

Unpainted space ofthe wall = 1-p16

[. Veny Sronr Ar'rswen (VSA) Quesnorus

Q1. What fractions of an hour is 2Q minulpg9

Q2. What fractions of these circles have Ys in tJxem?

@ o@o

5 6 bb

,Q3. Represent a fraction I on number line.

Q4. Fill up the blcnt gsing one of these'<' or 5'.

(")+I1 @zEJl5 @XZZQ5. Express X[g rnixgfl fractions 6] as mproper

fraction.

Q6. Express the improper fractions f * Or" -tafractions.

Q7. Write three equivalent fractions of f.Q8. Find the equivalent fraction of I with

nulerator 6.

Q9. Fintl the simplest form of the fra"ti"" H .

Q10. Which is the larger fraction?22-or-35

ll. Snonr ArusweR (SA) QuEsloNS

Q11. Write the given fractions in ascending order anddescending order.1713 11 8

Q12. Find tJle difference between 3 *U #.Q13. Two thick wires are respectively O1m and

-46 ; m long. Find the total lengths of the twob-wires.

Q14. Roshni travelled zOltm'f cer, t+fr m t, U*

ana Zf f fr h by train. Find the total distance

travelled by her.

Q15. A man spencb I ofhis salary on rations, I ofJlit on clothes and ft ofit on rent. What part of

his salary does he spend?

Q16.Simp[-fy the following:

cr sf -af <iit tf,+aftQl7. Reduce to simplest form:

.. 260 324(o) -

to) r*Q18. Represent the following on number line.

...7u)e Qil1

Qle. What should be added to ! " s" f,tQ20, Subtract the smallest from the greatest

@

Test Yourself

ANSWERS

6

26363'6'8'9

1

3

4. (a)

,. + u.n|

'. * ,.*

rr. Ascendine order: |, fDescending order: 13

6

3I5

_4 6 8" 10' 16' 20

210.:30

8 11 13

bb b11 871-=r=,='=b 5bt)

134

(b) (c)

MA-I1IEMATICS-VI

www.ncert.info

12.

lh.

b

%293o

ra. rrfim

16.02#

13.14. 249

20 Em

Gi)8k

18. (i)

ANSWERS

0t2s466

012

8

Qi)

7;

(b) +

4

u. @fi -81(b) 1oo 3,". m

lnternal Assessment

ar. shade fr

711,o. u

Ql. Complete the addition-subtraction box ?ItI

a,"ri#

--------H

(b)

(d) proper

(fl equivalent

6. (i) +> (c)(iii) <+ (b)

(u)ef)

(d) None ofthese

4. (a)

(ii) <+ @)(iu) e (d)(ui) o (e)

Q2. FilI in the blnnkc.

tol ,1

i. ... uu" ].(b) Sum of l *u i is equal to

(c)A........... is a nunber representing a partofwhole.

(d In a ........... fraction, the numerator i8always less than the denominator.

(e) The mrmerato. of fr i"(D Fractions havins the same value are called

........... fractions.

Q3. Which of the following is an improper fraction?2 _ 7 ..6 -1(a) , ta) g (c) U \a) ,

Q4. Which of the following is the simplest form of24,108 '

of the following.

Q6. Match the equivalent fractions.3610835

10512060060n3680

...46(u A0

(,)

(ii)

(iii)

(tu)

(u)

1(o) g

,.. 1lb) =b,.3\c) d

@z3

(e) 7o(n;

l. (a)

2. (a) greatf,r

(c) fraction

(e) 7

(b ------H 3. (6)

5.

b5

6

26 61

6 6

1

21

1

32

6 61

1

626

36

1

61

626

12

I 32

1 21

1 1 1

2

FRACTIONS 135

)

-_@-------> (c

www.ncert.info

<p Half Yearly AssessmentSET.I

Timz: 3 hpur

Getrcral Instructians

" All questinns are cornpulsory. Howeuer th.ere i* an internal choite.. Sectinn A consbts of 4 questinrc carrying I mnrk each.

" Section B cornists of 5 questiotts carrying 2 mnrks anh.

" Sectian C consists of 10 quzstians carrying 3 marks eath,

" Sectinn D corxists of I questintx carrying 4 marks earh.

(a) rB16

M.M.:80

SECTION-A

I. Write the number correspronding to each of the following:(a) 7000 + 500 + I (b) 6000 + 30 + 6

2" A cricket player so far scored 6986 runs in test matches. He wishes to complete 10,000 ruas. How manymore runs does he need?

3. What is the successor of the following:(a) 39,999 (b) 3000

4. Give three examples of three even prime numbers.

SECTION-B

5. Find the HCF of 70, 105 aad 175.6. Find the LCM of 24, rlt} a-nd 80.7. Represent (--4) + (+7) on the n,'mber line.

8. Show the fraction I on number line.5

9. Convert the foilowing in improper fraction:

ft)4 (c) 1 (il sq8

SECTION-C

10. What should be added to $? b eet Zg! ?-3 " 2

11. solve:81 +gq-1.4 4 612. Rewrite the following in ascending order 8.2, 8.02, 8.7, 8.17, B.O0B13. What should be added to the difference of 5.24 and 2.163 to get 8.b?14' Think of a nu.mber. Multiply it by 6 and add 7 to the result. Subtract r from tho result. What is the final

outcome?15. lf x = 2, I = - 1 and z = 3, find the value of 2rya - 5* + z2 + ry,16. The length of rectangle is 3 cm more tJran its breadth and its perimeter is 34 cm. Find the length and

breadth of the rectangle.17. Rohan worLs in a factory and earns ( 3375 per month. He saves ( 250 every month. Find the ratio ofhis

savinge to his income.

3

Z5;

136

www.ncert.info

18. Find the mea.n proportious between 16 and 441.

19. Pradeep pays ( 9600 as rent for 3 months. How much does he have to pay for a whole year?

SECTION-D

20, the boys is to girls ratio in a school is 11:10. How many girls students are there, if 605 boys are enrolledin school?

21. Ifthe first three terms of proportions are 9, 8 and 54 respectively, ffnd the fourth term.

22. Divitte ( 4230 o-ong Rqieev, Rohan and fuJ'at so that their shares are in the ratio ]23. Solve: * -' -- 6 and c.heck the a.nswer.s424. Ravi purchased 6 kg 400 g rice, 3 kg 50 g sugar and 12 kg 760 g flour. Find the total weight of his

purchases.OR

Roshan buys exercise books worth t 56.75, pencils for ( 26.30 and geom etry box for 7 42.25. How much heh"o to pay for purchases?

25. Arrange the following in descending order: Z, h,#,#

26. The HCF a-ntl LCM oftwo numbers are 8 a.nd 576 respectively. Ifone number is 64, fnd the other number.

2?. Find the smallest number which when diminished by 6 is divisible by 12' 75' 20 and27,28. the length, breadth and height of a room are 826 cm, 676 cm and 450 cm respectively. Find the longest

tape which can measure the three tlimensions of the room exactly.

114'6

137

www.ncert.info

o Half Yearly AssessmentSET.2

Timz: 3 hnur

Generol Instructinna: Samt as paper-l

SECTION-A,

1. Find the value ofthe following using distributive pmperties.(6524x69)+(6624x37)

2, Which of the follorning is the greatest of all:

M.M.:80

3321E'7'i''

a. sinnurr: sf +a|

4. Write the foltowing as decimals: 30 + 8 + -!

* -J-

10 100

SECTION-B

5. Solve for v: 6Y-4 =z"3

ro

,7

I

Compare the given ratio 5:12 and 9:16.Find the mean proportions of 100, 256.The cost of9 m of cloth is ? 288. Find the cost ofS m of cloth.Find z:

tu+443r'

SECTION-C

10. Write the following in Roman Numerals:(i) 98 (ii) 140

1 1 . Find the value of the following using distributive property:(429 x LO x 661) - (461 x 4290)

12. Ravi had ( 52000. He gave ? 9250 to .dieet, ? 12428 to R^qian and ? 24962 to Rajeev. How much money wasleft with him?

13. What is the greatest number that can divide ?81 and 468 leaving remainders of 1 and 3 respectively?14. Find the LCM of30, 45 and 90.

15. Find the HCF of 24, 36 and,72.

16. The HCF of two numbers is 146 and their LCM is 2175. If one of the numbers is 4135, fnd the otJrernumber.

17. Subtract 411 from the sun of-325 and -176.

138

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18. \Mrite four fractions equivalent to each ofthe following:

,r;19. Simplify the following:

3(ii) s

(i)

SECTION-D

20. Ankit covers 48 lrm 340 m by car,4 km 70 m by rickshaw, and 40 m on foot. Find the total distance

covered bY hirn.21. lf a=2,y =- l and z = 3, fnd the valiue of ?.r!a - 5* + z2 + ry.22. Harish cycles a distance of 18 tm in 3 hours and Akhtar covers a diatance of 64 Lrrr in 2 hours by car. Find

the ratio of their speeds.

23. Arrange the following fractions in ascencling order.

1 6 13 ,3I'L8'24 n

oR

simplify,: az1-zl.zl24. Find the least number from which if 35 is subtracted, the result is exactly tlivisible by 12, L8'20'2L'24

and 30.25. Find the least number which when divided by 6, 15 and 18 leave remainder 6 in each case.

!g. tlrc artnbel of students in each class of a school is 60. The fee paid by each student is ( 406 per month.

If there are 20 classes in the school, what is the total fee collections in a month?

27. Match the following:(a) r25(b) 331(c) 248(d) 479

28. Medicine is packed in boxes, each weighing 4 kg 500 g. How many such boxes can be loaded in a van

which can not carry beyond 800 kg?

6116'4 2

,...7 7 4(r1t __-+-"510 5

(il CDDO(D((ii) ccxl-ur(iii) C)ow(iv) CCC)OOfl

139

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