Chapter 1 : Number Systems - Rd Sharma Solutions for Class 9 Maths CBSE

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Chapter 1 - Number Systems Excercise Ex. 1.1

Question 1
Is zero a rational number? Can you write it in the form , where p and q are integers and q  0?
Solution 1
Yes zero is a rational number as it can be represented in the   form,  where p and q are integers and q  0 as  etc.

Concept Insight: Key idea to answer this question is "every integer is a rational number and zero is a non negative integer".  Also 0 can be expressed in form in various ways as 0 divided by any number is 0. simplest is   .

Question 2
Find five rational numbers between 1 and 2.
Solution 2
Question 3
Find six rational numbers between 3 and 4.
Solution 3
There are infinite rational numbers in between 3 and 4.  
3 and 4 can be represented as respectively.
 
Now rational numbers between 3 and 4 are    
    

Concept Insight:  Since there are infinite number of rational numbers between any two numbers so the answer is not unique here.  The trick is to convert the number to equivalent  form by multiplying and dividing by the number atleast 1 more than the rational numbers to be inserted.

Question 4
Find five rational numbers between .
Solution 4
There are infinite rational numbers between
        
        
Now rational numbers between are  
         
 
Concept Insight: Since there are infinite number of rational numbers between any two numbers so the answer is not unique here.  The trick is to convert the number to equivalent   form by multiplying and dividing by the number at least 1 more than the rational numbers required.

Alternatively for any two rational numbers a and b,    is also a rational number which lies between a and b.


Question 5
Are the following statements true or false? Give reasons for you answer.

(i) Every whole number is a natural number.

(ii) Every integer is a rational number.

(iii) Every rational number is an integer.

(iv) Every natural number is a whole number.

(v) Every integer is whole number.

(vi) Every rational number is whole number.
Solution 5
(i) False

(ii) True

(iii) False

(iv)True

(v) False

(vi) False

Chapter 1 - Number Systems Excercise Ex. 1.2

Question 1
Solution 1


Question 2
Express the follwoing rational numbers as decimals:

Solution 2

(i)



(ii)



(iii)



(iv)



(v)



(vi)

Question 3
Solution 3

Chapter 1 - Number Systems Excercise Ex. 1.3

Question 1
Solution 1


Question 2

begin mathsize 12px style Express space each space of space the space followimg space decimals space in space the space form space straight p over straight q
open parentheses straight i close parentheses space 0 top enclose.4 end enclose
open parentheses ii close parentheses space 0. top enclose 37
open parentheses iii space close parentheses 0. top enclose 54
open parentheses iv close parentheses space 0. top enclose 621
open parentheses straight v close parentheses space 125. top enclose 3
open parentheses vi space close parentheses 4. top enclose 7
open parentheses vii space close parentheses 0.47 end style

Solution 2

(i)



(ii)



(iii)



(iv)



(v)



(vi)



(vii)

Chapter 1 - Number Systems Excercise Ex. 1.4

Question 1
Solution 1
Question 2

Solution 2

Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5

Question 6
Solution 6

Question 7
Solution 7

Question 8

Solution 8

Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12

Solution 12

As decimal expansion of this number is non-terminating non recurring. So it is an irrational number.

Question 13
Solution 13


Rational number as it can be represented in   form.
Question 14
Examine whether 0.3796 is rational or irrational.
Solution 14
0.3796

As decimal expansion of this number is terminating, so it is a rational number.
Question 15
Examine whether 7.478478... is rational or irrational.
Solution 15

As decimal expansion of this number is non terminating recurring so it is a rational number.

Question 16
Examine whether 1.101001000100001... is rational or irrational.
Solution 16
Question 17
Solution 17
Question 18
Solution 18
Question 19
Solution 19
Question 20
Solution 20
Question 21
Solution 21
Question 22
Solution 22
Question 23
Solution 23


Question 24

Solution 24

Question 25

Solution 25

Question 26

 

Solution 26

Question 27

Solution 27

Question 28
Find three different irrational numbers between the rational numbers
Solution 28
 
3 irrational numbers are -
0.73073007300073000073 ... ... ...
0.75075007500075000075 ... ... ...
0.79079007900079000079 ... ... ...

Concept Insight: There is infinite number of rational and irrational numbers between any two rational numbers. Convert the number into its decimal form to find irrationals between them.

Alternatively following result can be used to answer

Irrational number between two numbers x and y
 
Question 29

Solution 29


Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Chapter 1 - Number Systems Excercise Ex. 1.5

Question 1
Complete the following sentences:

(i) Every point on the number line corresponds to a ___ number which may be either ____ or_____.

(ii) The decimal form of an irrational number is neither ______ nor ______.

(iii) The decimal representation of a rational number is either ____ or _____.

(iv) Every real number is either ______ number or ______ number.
Solution 1
(i) Real, rational, irrartional.

(ii) terminating, repeating.

(iii) terminating, non-terminating and reccuring.

(iv) rational, an irrational.
Question 2

Find whether the following sentences are true or false:

(i) Every real number is either rational or irrational.

(ii) is an irrational number.

(iii) Irrational numbers cannot be represented by points on the number line.

Solution 2

(i) True

(ii) True

(iii) False

Question 3

Solution 3




Question 4

Solution 4



Chapter 1 - Number Systems Excercise Ex. 1.6

Question 1

Solution 1








Question 2

Solution 2

Chapter 1 - Number Systems Excercise 1.40

Question 1

Which one of the following is a correct statement?

(a) Decimal Expansion of a Rational number is terminating.

(b) Decimal Expansion of a Rational number is Non-terminating.

(c) Decimal Expansion of a irrational number is terminating.

(d) Decimal Expansion of a irrational number is Non-terminating and Non-Repeating.

Solution 1

Decimal space Expansion space of space straight a space Rational space number space is space not space only space terminating comma
it space can space be space either space terminating space like space 1 half equals 0.5 space or space non minus terminating space Repeating space like
1 third equals 0.3333333.... space So space option space left parenthesis straight a right parenthesis space is space not space true space alone.
Now space we space know space that space Non minus Terminating space numbers space are space of space two space types space colon
one space is space Non minus Terminating space Repeating space and space other space is space Non minus terminating space Non minus Repeating.
The space Decimal space expansion space of space straight a space Rational space number space matches space one space of space it apostrophe straight s space Kind
straight i. straight e space Non minus terminating space Repeating space of space Non minus terminating space numbers.
So space Rational space number space does space not space consist space both space the space kinds space of space Non minus terminating space numbers
Hence comma space they space are space not space Non minus terminating space numbers.
An space irrational space number space is space always space Non minus terminating space in space nature comma but space again space not space of space both space of space it apostrophe straight s space kinds. space
The space Decimal space Expansion space of space an space irrational space number space is space Non minus terminating space Non minus Repeating space in space Nature.
So space From space all space above space points space and space theory space we space can space conclude space an space Irrational space number space is
Non minus terminating space but space Non minus Repeating space in space nature
straight i. straight e. space square root of 2 equals 1.4142135623730 space......
So comma space option space left parenthesis straight d right parenthesis space is space correct.

Question 2

Which one of the following statement is true?

(a) The sum of two irrational numbers is always an irrational number.

(b) The sum of two irrational numbers is always a rational number.

(c) The sum of two irrational numbers may be a rational number or an irrational number

(d) The sum of two irrational numbers is always an Integer.

Solution 2

Error converting from MathML to accessible text.

Question 3

Which of the following is a correct statement?

(a) Sum of two irrational numbers is always irrational.

(b) Sum of a rational and irrational number is always irrational.

(c) Square of an irrational number is always a Rational number.

(d) Sum of two Rational numbers can never be an integer.

Solution 3

Error converting from MathML to accessible text.

Question 4

Which of the following statements is true?

(a) Product of two irrational numbers is always irrational

(b) Product of a rational and an irrational number is always irrational

(c) Sum of two irrational numbers can never be irrational

(d) Sum of an integer and a rational number can never be an integer

Solution 4

Error converting from MathML to accessible text.

Question 5

Which of the following is irrational?

left parenthesis straight a right parenthesis space square root of 4 over 9 end root
left parenthesis straight b right parenthesis space 4 over 5
left parenthesis straight c right parenthesis space square root of 7 space
left parenthesis straight d right parenthesis space square root of 81

Solution 5

Option space left parenthesis straight a right parenthesis space is space incorrect space because space square root of 4 over 9 end root equals fraction numerator square root of 4 over denominator square root of 9 end fraction equals plus-or-minus 2 over 3 space open parentheses Rational close parentheses
Option space left parenthesis straight b right parenthesis space is space also space incorrect space space as space 4 over 5 space is space in space the space form space of space straight P over straight Q space open parentheses straight Q space not equal to 0 close parentheses comma space left parenthesis Rational right parenthesis
Option space left parenthesis straight c right parenthesis space is space correct space beacuse space square root of 7 space is space straight a space non minus terminating space and space Non minus Repeating space number.
Option space left parenthesis straight d right parenthesis space is space also space incorrect space because space square root of 81 equals plus-or-minus 9 space left parenthesis Rational right parenthesis
Hence comma space correct space option space is space left parenthesis straight c right parenthesis.

Question 6

Which of the following is irrational?

left parenthesis straight a right parenthesis space 0.14
left parenthesis straight b right parenthesis space 0.14 top enclose 16
left parenthesis straight c right parenthesis space 0. top enclose 1416
left parenthesis straight d right parenthesis space 0.1014001400014..

Solution 6

Correct space option space left parenthesis straight d right parenthesis
Option space left parenthesis straight a right parenthesis colon space 0.14 equals 14 over 100 comma space which space is space straight a space Rational space number
Option space left parenthesis straight b right parenthesis colon space 0.14 top enclose 16 space is space non minus terminating space but space repeating comma space hence space straight a space rational space number
Option space left parenthesis straight c right parenthesis colon space 0. top enclose 1416 space is space Non minus terminating space but space repeating comma space hence space straight a space rational space number
Now comma space option space left parenthesis straight d right parenthesis colon space 0.1014001400014.... space is space non minus terminating space as space well space as space non minus repating space number comma space
which space is space irrational space in space nature.
Hence comma space correct space option space is space left parenthesis straight d right parenthesis. space

Question 7

Which of the following is rational?

left parenthesis straight a right parenthesis space square root of 3
left parenthesis straight b right parenthesis space straight pi
left parenthesis straight c right parenthesis space 4 over 0
left parenthesis straight d right parenthesis space 0 over 4

Solution 7

Option space left parenthesis straight a right parenthesis colon space square root of 3 equals 1.732..... space equals Non minus terminating space and space non minus repeating space number comma space hence space irrational
Option space left parenthesis straight b right parenthesis colon space straight pi space equals space 3.14..... space also space can space not space be space terminated space to space straight p over straight q space form comma space
and space is space non minus terminating space and space non minus repeating space in space nature.
Hence comma space irrational.
Option space left parenthesis straight c right parenthesis colon space 4 over 0 space is space not space straight a space rational space number space because space this space is space in space the space form space straight p over straight q comma space
where space straight p space and space straight q space are space integers space but space straight q equals 0
Option space left parenthesis straight d right parenthesis colon space 0 over 4 space follows space the space difin i tion space of space rational space number.
Hence comma space correct space option space is space left parenthesis straight d right parenthesis.

Question 8

The number 0.318564318564318564......... is:

(a) a natural number

(b) an integer

(c) a rational number

(d) an irrational number

Solution 8

0.318564318564318564..... space equals space 0. top enclose 318564 space is space straight a space Non minus terminating space Repeating space Number.
Hence comma space it space is space straight a space rational space number.
So comma space correct space option space is space left parenthesis straight c right parenthesis.

Question 9

If space straight n space is space straight a space natural space number comma space then space square root of straight n space is

(a) always a natural number

(b) always an rational number

(c) always an irrational number

(d) sometimes a natural number and sometimes an irrational number.

Solution 9

Option space left parenthesis straight a right parenthesis space is space incorrect space because space square root of straight n space can space not space be space always space straight a space natural space number
space space space space space space space space space space space space space space space space space space straight i. straight e. space space space if space straight n space equals space 2 comma space space space square root of straight n space equals space square root of 2 space left parenthesis not space straight a space natural space no. right parenthesis
Option space left parenthesis straight b right parenthesis space is space incorrect space similiarly comma space if space straight n space equals space 2 comma space 5 comma space...... space or space any space odd space no comma space or space not space perfect space square comma
space space space space space space space space space space space space space space space space space space square root of straight n space equals space square root of 2 comma space square root of 5 comma space square root of 7 space are space Non minus terminating space and space non minus repeating comma space so space irrational space in space nature comma space
space space space space space space space space space space space space space space space space space So comma space not space always space straight a space rational space number.
Option space left parenthesis straight c right parenthesis space is space also space incorrect. space square root of straight n space can space also space be space rational space or space say space straight a space natural space number. space
space space space space space space space space space space space space space space space space space space space If space straight n space equals space 4 comma space 9 comma space 16...... space or space any space perfect space square space number space then space square root of straight n space equals space 2 comma space 3 comma space 4 space..... space natural space numbers.
Option space left parenthesis straight d right parenthesis space is space fully space correct space because space if space straight n space is space any space odd space number space or space non minus perfect space square space number
space space space space space space space space space space space space space space space space space then space square root of straight n space space would space be space irrational comma space but space if space straight n space is space straight a space perfect space square space number space then space square root of straight n space
space space space space space space space space space space space space space space space space space will space be space straight a space natural space number.
space space space space space space space space space space space space space space space space space if space straight n space equals space 2 comma space 3 comma space 5 comma space 8 comma space........ space space square root of straight n space equals space square root of 2 comma space square root of 3 comma space square root of 8 space space......... left parenthesis irrational right parenthesis
space space space space space space space space space space space space space space space space space if space straight n space equals space 4 comma space 9 comma space 16............ space space equals space 2 comma space 3 comma space 4 space space space space space......... left parenthesis Natural space number right parenthesis
Hence comma space correct space option space is space left parenthesis straight d right parenthesis.

Chapter 1 - Number Systems Excercise 1.41

Question 1

Which of the following numbers can be represented as non-terminating repeating decimals?

left parenthesis straight a right parenthesis space 39 over 24
left parenthesis straight b right parenthesis space 3 over 16
left parenthesis straight c right parenthesis space 3 over 11
left parenthesis straight d right parenthesis space 137 over 25

Solution 1

O p t i o n space left parenthesis a right parenthesis colon space 39 over 24 equals 1.625 space equals space T e r m i n a t i n g space D e c i m a l
O p t i o n space left parenthesis b right parenthesis colon space 3 over 16 equals 0.1875 equals T e r m i n a t i n g space D e c i m a l
O p tion space left parenthesis straight c right parenthesis colon space 3 over 11 equals 0.27272727....... space equals space N o n minus t e r m i n a t i n g space d e c i m a l
O p t i o n space left parenthesis d right parenthesis colon space 137 over 25 equals 5.48 equals T e r m i n a t i n g space D e c i m a l
H e n c e comma space o p t i o n space left parenthesis c right parenthesis space i s space c o r r e c t.

Question 2

Every point on a number line represents:

(a) a unique real number

(b) a natural number

(c) a rational number

(d) an irrational number

Solution 2

On space number space line comma space we space have space minus infinity space to space infinity space numbers comma space
consisting space minus infinity..... negative 4 comma space minus 3 comma space minus 2 comma space minus 1 comma space 0 comma space 1 comma space 2 space comma space 3 space comma space 4...... infinity comma space space
1.12 comma space 1.14 space and space 1.41406532 comma space 3.146201286295......... space etc.
That space means space on space number space line comma space there space are space natural space numbers space left parenthesis 1 comma space 2 comma space 3 comma space 4.... right parenthesis comma space integers comma space
rational space numbers space 1 half comma space 1 third comma space 1.33333 comma space space irrational space numbers space 1.4148625385...
But space if space we space see space every space number space as space straight a space complete space family comma space it space becomes space
Real space numbers space left parenthesis any space number space which space can space be space represent space on space Real space axes right parenthesis
So comma space every space point space on space the space number space line space represents space straight a space unique space real space number space
which space contains space every space type.
Hence comma space correct space option space is space left parenthesis straight a right parenthesis.

Question 3

Which of the following is irrational?

(a) 0.15

(b) 0.01516

(c) 0. top enclose 1516

(d) 0.5015001500015

Solution 3

RD Sharma Solution for Class 9 Mathematics Chapter 1 - Number Systems Excercise 1.41

Question 4

The space number space 1. top enclose 27 space in space the space form space straight p over straight q comma space where space straight p space and space straight q space are space integers space and space straight q space not equal to 0 comma space is
left parenthesis straight a right parenthesis space 14 over 9
left parenthesis straight b right parenthesis space 14 over 11
left parenthesis straight c right parenthesis space 14 over 13
left parenthesis straight d right parenthesis space 14 over 15

Solution 4

Let space straight x space equals space 1. top enclose 27 space equals space 1.272727.... space space space space space space space space space space space space space left parenthesis 1 right parenthesis
Now comma space 100 straight x space equals space 127.272727... space equals space 127. top enclose 27 space space space space space space space space left parenthesis 2 right parenthesis
Subtracting space equation space left parenthesis 1 right parenthesis space from space left parenthesis 2 right parenthesis comma space we space get
99 straight x space equals space 126
rightwards double arrow straight x equals 126 over 99 equals 14 over 11
Hence comma space option space left parenthesis straight b right parenthesis space is space correct.

Question 5

The space number space 0. top enclose 3 space in space the space form space straight p over straight q comma space where space straight p space a n d space q space are space integers space and space straight q not equal to 0 comma space is
left parenthesis straight a right parenthesis space 33 over 100
left parenthesis straight b right parenthesis space 3 over 10
left parenthesis straight c right parenthesis space 1 third
left parenthesis straight d right parenthesis space 3 over 100

Solution 5

Let space straight x space equals space 0. top enclose 3 space equals space 0.33333....... space space space space space space space space space space space space space space space space space left parenthesis 1 right parenthesis
Now comma space 10 straight x equals 3.33333...... equals 3. top enclose 3..... space space space space space space space space left parenthesis 2 right parenthesis
Subtracting space equation space left parenthesis 1 right parenthesis space from space left parenthesis 2 right parenthesis comma space we space get
9 straight x space equals space 3
rightwards double arrow straight x equals 3 over 9 equals 1 third
rightwards double arrow 0. top enclose 3 equals 1 third
Hence comma space option space left parenthesis straight c right parenthesis space is space correct.

Question 6

0.3 top enclose 2 space when space expressed space in space form space of space straight p over straight q space left parenthesis p comma space q space are space integers comma space q not equal to space 0 right parenthesis comma space is
left parenthesis straight a right parenthesis space 8 over 25
left parenthesis straight b right parenthesis space 29 over 90
left parenthesis straight c right parenthesis space 32 over 99
left parenthesis straight d right parenthesis space 32 over 199

Solution 6

Let space straight x space equals space 0.3 top enclose 2 space equals space 0.322222....... space space space space space space space space space left parenthesis 1 right parenthesis
Now comma space 10 straight x space equals space 3.22222...... equals space 3. top enclose 2 space space space space space space space space left parenthesis 2 right parenthesis comma
100 straight x equals 32.2222....... equals 32. top enclose 2 space space space space space space space space space space space space space space space left parenthesis 3 right parenthesis
subtracting space equation space left parenthesis 2 right parenthesis space from space left parenthesis 3 right parenthesis comma space we space get
90 straight x space equals space 29
rightwards double arrow straight x equals 29 over 90
Hence comma space option space left parenthesis straight b right parenthesis space is space correct.

Question 7

23. top enclose 43 space when space expressed space in space the space form space straight p over straight q comma space left parenthesis straight p comma space straight q space are space integers comma space q not equal to space 0 right parenthesis space is
left parenthesis straight a right parenthesis space 2320 over 99
left parenthesis straight b right parenthesis space 2343 over 100
left parenthesis straight c right parenthesis space 2343 over 999
left parenthesis straight d right parenthesis space 2320 over 199

Solution 7

Let space straight x space equals space 23. top enclose 43 space equals space 23.434343........ space space space space space space space space left parenthesis 1 right parenthesis
Now comma space 100 straight x space equals space 2343.43333........ space space space space space space space space space space left parenthesis 2 right parenthesis
Subtracting space equation space left parenthesis 1 right parenthesis space from space left parenthesis 2 right parenthesis comma space we space get
99 straight x space equals space 2320
rightwards double arrow straight x equals 2320 over 99
Hence comma space option space left parenthesis straight a right parenthesis space is space correct.

Question 8

0. top enclose 001 space when space expressed space in space the space form space straight p over straight q space left parenthesis p comma space q space are space integers comma space q not equal to 0 right parenthesis space is
left parenthesis straight a right parenthesis space 1 over 1000
left parenthesis straight b right parenthesis space 1 over 100
left parenthesis straight c right parenthesis space 1 over 1999
left parenthesis straight d right parenthesis space 1 over 999

Solution 8

Let space straight x space equals space 0. top enclose 001 space equals space 0.001001001..... space space space space space space left parenthesis 1 right parenthesis
Now comma space 1000 straight x equals 001.001001001..... space space space space space space space left parenthesis 2 right parenthesis
Subtracting space equation space left parenthesis 1 right parenthesis space from space left parenthesis 2 right parenthesis comma space we space get
999 straight x space equals space 1
rightwards double arrow straight x equals 1 over 999
Hence comma space correct space option space is space left parenthesis straight d right parenthesis.

Question 9

The space value space of space 0. top enclose 23 space plus space 0. top enclose 22 space is
left parenthesis straight a right parenthesis space 0. top enclose 45
left parenthesis straight b right parenthesis space 0. top enclose 43
left parenthesis straight c right parenthesis space 0.4 top enclose 5
left parenthesis straight d right parenthesis space 0.45

Solution 9

Let space straight x space equals space 0. top enclose 23 equals space 0.232323...... space space space space space space space left parenthesis 1 right parenthesis
space space space space space space space straight y equals 0. top enclose 22 equals 0.222222..... space space space space space space space left parenthesis 2 right parenthesis
Adding space equations space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis comma space we space get
straight x plus straight y equals 0.454545 equals 0. top enclose 45
rightwards double arrow 0. top enclose 23 plus 0. top enclose 22 equals 0. top enclose 45
Hence comma space option space left parenthesis straight a right parenthesis space is space correct.

Question 10

An irrational number between 2 and 2.5 is

left parenthesis straight a right parenthesis space square root of 11
left parenthesis straight b right parenthesis space square root of 5
left parenthesis straight c right parenthesis space square root of 22.5 end root
left parenthesis straight d right parenthesis space square root of 12.5 end root

Solution 10

square root of 4 equals 2 space and space square root of 6.25 end root space equals space 2.5
Option space left parenthesis straight a right parenthesis comma space left parenthesis straight c right parenthesis space and space left parenthesis straight d right parenthesis colon space square root of 11 comma space square root of 22.5 end root space and space square root of 12.5 end root comma space all space are space greater space than space square root of 6.25 end root
rightwards double arrow Out space of space interval space left parenthesis 2 comma space 2.5 right parenthesis
Option space left parenthesis straight b right parenthesis colon space square root of 4 less than square root of 5 space less than square root of 6.25 end root space rightwards double arrow space lies space in space the space interval space left parenthesis 2 comma space 2.5 right parenthesis
Hence comma space option space left parenthesis straight b right parenthesis space is space correct.

Question 11

The space number space of space consecutive space zeros space in space 2 cubed cross times 3 to the power of 4 cross times 5 to the power of 4 cross times 7 comma space is

(a) 3

(b) 2

(c) 4

(d) 5

Solution 11

5 cross times 2 equals 10 rightwards double arrow one space 5 space and space one space 2 space make space one space zero comma space so space 5 cross times 2 cross times 5 cross times 2 equals 100
Number space of space pairs space of space 5 space and space 2 space will space be space equal space to space the space number space of space consecutive space
zeros space in space the space given space number.
In space the space given space number comma space there space are space three space 2 apostrophe straight s space and space four space 5 apostrophe straight s.
So space number space of space pairs space of space 5 space and space 2 space are space only space three.
So space there space will space be space three space consecutive space zeros space in space the space given space number.
So comma space option space left parenthesis straight a right parenthesis space is space correct.

Chapter 1 - Number Systems Excercise 1.42

Question 1

The space smallest space rational space number space by space which space 1 third space should space be space multiplied space so space that space its space decimal
expansion space terminates space after space one space place space of space decimal space is
left parenthesis straight a right parenthesis space 1 over 10
left parenthesis straight b right parenthesis space 3 over 10
left parenthesis straight c right parenthesis space 3
left parenthesis straight d right parenthesis space 30

Solution 1

1 third equals 0.333333......... left parenthesis straight a space Non minus terminating space number right parenthesis
Now comma space if space we space remove space 3 space from space denominator space it space will space terminate.
So comma space if space we space multiply space by space 3 over 10
straight i. straight e. space fraction numerator 1 over denominator up diagonal strike 3 end fraction cross times fraction numerator up diagonal strike 3 over denominator 10 end fraction equals 1 over 10 equals 0.1 space space left parenthesis terminates space after space one space place space of space decimal right parenthesis
By space multiplying space by space 1 over 10 comma space 3 space does space not space replaces.
By space multiplying space by space 3 comma space we space get space 1 comma space which space is space not space terminating space after space one space place space of space decimal
And comma space by space multiplying space by space 30 comma space we space get space 10 comma space again space not space terminating space after space one space place
of space decimal.
So comma space correct space answer space is space left parenthesis straight b right parenthesis.

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