NCERT Exemplar Problems Class 8 Mathematics Rational Numbers

NCERT  Exemplar Problems Class 8 Mathematics Chapter 1 Rational Numbers

Multiple Choice Questions
Question . 1 A number which can be expressed as \(\frac { p }{ q }\) , where p and q are integers and \( q\neq 0\) is
(a) natural number (b) whole number
(c) integer (d) rational number
Solution. (d) A number which can be expressed as \(\frac { p }{ q }\), where p and q are integers and \( q\neq 0\) is a rational number.

Question . 2 A number of the form \(\frac { p }{ q }\) is said to be a rational number, if
(a) p, q are integers (b) p, q are integers and \( q\neq 0\)
(c) p, q are integers and \( p\neq 0\) (d) p, q are integers and \( p\neq 0\), also \( q\neq 0\)
Solution. (b) A number of the form \(\frac { p }{ q }\) is said to be a rational number, if p and q are integers and

Question . 3 The numerical expression \( \frac { 3 }{ 8 } +\frac { (-5) }{ 7 } =\frac { -19 }{ 56 }\) 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
Solution. (b) We have \( \frac { 3 }{ 8 } +\frac { (-5) }{ 7 } =\frac { -19 }{ 56 }\)
Show that rational numbers are closed under addition.
[\(\frac { 3 }{ 8 }\) and\( \frac { -5 }{ 7 }\) are rational numbers and their addition is \(\frac { -19 }{ 56 }\) which is also a rational number]
Note The sum of any two rational numbers is always a rational number.

Question . 4 Which of the following is not true?
(a) rational numbers are closed under addition
(b) rational numbers are closed under subtraction
(c) rational numbers are closed under multiplication
(d) rational numbers are closed under division
Solution. (d) Rational numbers are not closed under division.
As, 1 and 0 are the rational numbers but \( \frac { 1 }{ 0 }\) is not defined.

Question . 5 \(-\frac { 3 }{ 8 } +\frac { 1 }{ 7 } =\frac { 1 }{ 7 } +[\frac { -3 }{ 8 } ]\) is an example to show that
(a) addition of rational numbers is commutative
(b) rational numbers are closed under addition
(c) addition of rational numbers is associative
(d) rational numbers are distributive under addition
Solution.

Clearly, a + b = b + a
So, addition is communication for rational numbers

Question . 6 Which of the following expressions shows that rational numbers are associative under multiplication.

Solution.

So, a x (b  x c) = (a x b) x c
Hence, the given expression shows that rational numbers are associative under multiplication.

Question . 7 Zero (0) is
(a) the identity for addition of rational numbers
(b) the identity for subtraction of rational numbers
(c) the identity for multiplication of rational numbers
(d) the identity for  division of rational numbers
Solution . (a) Zero (0) is the identity for addition of rational numbers.
That means,
If a is a rational number.
Then, a+0=0+a = a
Note Zero (0) is also the additive identity for integers and whole number as well.

Question . 8 One (1) is
(a) the identity for addition of rational numbers
(b) the identity for subtraction of rational numbers
(c) the identity for multiplication of rational numbers
(d) the identity for division of rational numbers
Solution . (c) One (1) is the identity for multiplication of rational numbers.
That means,
If a is  a rational number.
Then, a-1 = 1-a = a
Note One (1) is the multiplication identity for integers and whole number also.

Question . 9 The additive inverse  of \(\frac { -7 }{ 19 }\) is

Solution . (b) We know that, if a and b are the additive inverse of each other, then a + b = 0
Suppose, x is the additive inverse of \(\frac { -7 }{ 19 }\)

Question . 10 Multiplicative inverse of a negative rational number is
(a) a positive rational number (b) a negative rational number
(c) 0 (d) 1
Solution. (b) We know that, the product of two rational numbers is 1, taken they are multiplication inverse of each other, e.g.
Suppose, p is negative rational number, i.e.
\(\frac { 1 }{ p }\) is the  multiplicative inverse of-p, then, -p x \(\frac { 1 }{ -p }\) = 1
Hence, multiplicative inverse of a negative rational number is a negative rational number.

Question. 11 If x + 0 = 0 + x = x, which is rational number, then 0 is called
(a) identity for addition of rational numbers
(b) additive inverse of  x
(c) multiplicative inverse of x
(d) reciprocal of x
Solution . (a) We know that, the sum of any rational number and zero (0) is the rational number itself.
Now, x + 0 = 0+ x= x, which is a rational number, then 0 is called identity for addition of rational numbers.

Question . 12 To get the product 1,  we should multiply \( \frac { 8 }{ 21 }\) by

Solution .

Question . 13 – (-x) is same as
(a)-x (b)x (c)\(\frac { 1 }{ x }\) (d)\(\frac { -1 }{ x }\)
Solution . (b) -(-x) = x
Negative of negative rational number is equal to positive rational number.

Question . 14 The multiplicative inverse of \( -1\frac { 1 }{ 7 }\) is

Solution . (d) We know that, if the product of two rational numbers is 1, then they are multiplicative inverse of each other.

Question . 15 If x is any rational number, then x + 0 is equal to
(a)x (b) 0 (c)-x (d) Not defined
Solution . (a) If x is any rational number, then x + 0 = x [0 is the additive identity]

Question . 16  The reciprocal of 1 is ;
(a) 1 (b) -1 (c) 0 (d) Not defined
Solution . (a) The reciprocal of 1 is the number itself.

Question . 17  The reciprocal of -1 is
(a) 1 (b) -1 (c) 0 (d) Not defined
Solution . (b) The reciprocal of -1 is the number itself.

Question . 18 The reciprocal of 0 is
(a) 1 (b) -1 (c) 0 (d) Not defined
Solution . (d) The reciprocal of 0 is not defined.

Question . 19 The reciprocal of any rational number \(\frac { p }{ q }\) , where p and q are integers and \( q\neq 0\) is
(a)\(\frac { p }{ q }\) (b)1 (c)0 (d)\(\frac { q }{ p }\)
Solution . (d) The reciprocal of any rational number \(\frac { p }{ q }\), where p and q are integers and \( q\neq 0\) is \(\frac { q }{ p }\)

Question . 20 If y is the reciprocal of rational number x, then the reciprocal of y will be
(a)x (b) y (c) \(\frac { x }{ y }\) (d) \(\frac { y }{ x }\)
Solution . (a) If y be the reciprocal of rational number x, i.e. y = \(\frac { 1 }{ x }\) or x = \(\frac { 1 }{ y }\).
Hence, the reciprocal of y will be x.

Question .21

Solution .

Question . 22 Which of the following is an example of distributive property of multiplication over addition for rational numbers.

Solution . We know that, the distributive property of multiplication over addition for rational numbers can be expressed as a x (b + c) = ab + ac, where a, b and c are rational numbers.

is the example of distributive property of multiplication over addition for rational numbers.

Question . 23 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
Solution . (d) We can find infinite many rational numbers between two given rational numbers.

Question .24

(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
Solution .

Question . 25 Which of the following statements is always true?

Solution .

Fill in the Blanks
In questions 26 to 47, fill in the blanks to make the statements true.
Question . 26 The equivalent of \( \frac { 5 }{ 7 } \) whose numerator is 45, is —.
Solution .

Question . 27 The equivalent rational number of \( \frac { 7 }{ 9} \) , whose denominator is 45 is——————.
Solution .

Question . 28 Between the numbers \(\frac {15 }{ 20} \) and \(\frac { 35 }{ 40} \), the greater number is———————-.
Solution .

Question . 29 The reciprocal of a positive rational number is—————.
Solution .

Question . 30 The reciprocal of a negative rational number is——————–.
Solution .

Question. 31 Zero has————reciprocal.
Solution .

Question. 32 The numbers ————–and————–are their own reciprocal.
Solution .

Question . 33 If y is the reciprocal of x, then the reciprocal of \({ y }^{ 2 }\) in terms of x will be—————-.
Solution .

Question . 34

Solution .

Question . 35

Solution .

Question . 36 The negative of 1 is—————-.
Solution . -1 The negative of 1 is -1.

Question . 37

Solution .

Question . 38 \(\frac { -5 }{ 7 }\) is———————than -3.
Solution .

Question . 39 There are rational numbers between any two rational numbers.
Solution . Infinite
There are infinite rational numbers between any two rational numbers.

Question . 40 The rational numbers \(\frac { 1 }{ 3 }\) and \(\frac { -1 }{ 3 }\) are on the sides of zero on the number line.
Solution .

Question . 41 The negative of a negative rational number is always a—————-rational
number.
Solution. positive
Let x be a positive rational number.
Then, – x be a negative rational number.
Now, negative of a negative rational number = – (- x)= x =positive rational number.

Question . 42 Rational numbers can be added or multiplied in any————-.
Solution . order
Rational numbers can be added or multiplied in any order and this concept is known as commutative property.

Question . 43 The reciprocal of \(\frac { -5 }{ 7 }\) is——————.
Solution .

Question . 44 The multiplicative inverse of \(\frac { 4 }{ 3 }\) is———–.
Solution .

Question . 45 The rational number 10.11 in the form \(\frac { p }{ q }\) is ——–.
Solution .

Question .46

Solution .

Question . 47 The two rational numbers lying between -2 and -5 with denominator as 1 are———–and————.
Solution .

True/False
In questions 48 to 99, state whether the given statements are True or False.
Question . 48 If \(\frac { x }{ y }\)is a rational number, then y is always a whole number.
Solution .

Question . 49 If \(\frac { p }{ q }\) is a rational number, then p Cannot be equal to zero.
Solution .

Question . 50 If \(\frac { r }{ s }\) is a rational number, then s cannot be equal to zero.
Solution .

Question . 51 \(\frac { 5 }{ 6 }\) lies between \(\frac { 2 }{ 3 }\) and 1.
Solution .

Question . 52 \(\frac { 5 }{ 10 }\) lies between \(\frac { 1 }{ 2 }\) and 1.
Solution .

Question . 53 \(\frac { 5 }{ 10 }\) lies between -3 and 4.
Solution .

Question . 54 \(\frac { 9 }{ 6 }\) lies between 1 and 2.
Solution .

Question . 55 If \(a\neq 0\) the multiplicative inverse of \(\frac { a }{ b }\) is \(\frac { b }{ a }\) .
Solution .

Question . 56 The multiplicative inverse of \(\frac { -3 }{ 5 }\) is \(\frac { 5 }{ 3 }\) .
Solution .

Question . 57 The additive inverse of \(\frac { 1 }{ 2 }\) is -2.
Solution .

Question . 58

Solution .

Question . 59 For every rational number x, x + 1 = x.
Solution . False
For every rational number , x + 0 = x

Question . 60

Solution .

Question . 61 The reciprocal of a non-zero rational number \(\frac { q }{ p }\) is the rational number \(\frac { q }{ p }\).
Solution .False
The reciprocal of a non-zero rational number \(\frac { q }{ p }\) . is the rational number\(\frac { p }{ q }\)

Question . 62 If x + y = 0, then -y is known as the negative of x, where x and y are rational numbers.
Solution . False
If x and y are rational numbers and x+ y = 0.
Then, y is known as the negative of x.

Question . 63 The negative of the negative of any rational number is the number itself.
Solution . True
Let x be a positive rational number. Then, -x be a negative rational number.
Now, negative of negative rational number = -(-x)= x = Positive rational number

Question . 64 The negative of 0 does not exist.
Solution . True
Since, zero is neither a positive integer nor a negative integer.

Question . 65 The negative of 1 is 1 itself.
Solution . False
The negative of 1 is -1.

Question . 66 For all rational numbers x and y,x-y = y- x
Solution . False
For all rational numbers x and y,
x-y = -(y-x)

Question . 67 For all rational numbers x and y, x x y = y x x.
Solution . True
For all rational numbers x and y,
x x y= y x x

Question . 68 For every rational number x, x x 0 = x.
Solution . False
For every rational number x,
x x 0 = 0

Question . 69 For every rational numbers x, y and z, x + (y x z) = (x + y) x (x + z)
Solution . False
For all rational numbers a, b and c.
a(b + c) = ab+ ac

Question . 70 For all rational numbers a, b and c,a (b + c) = ab + bc.
Solution . False
As, addition is not distributive over multiplication.

Question . 71 1 is the only number which is its own reciprocal.
Solution . False
Reciprocal of 1 is 1 and reciprocal of -1 is -1.

Question . 72 -1 is not the reciprocal of any rational number.
Solution . False
-1 is the reciprocal of -1.

Question . 73 For any rational number x, x + (-1) = – x.
Solution . False
For every rational number x,
x x (-1) = – x

Question . 74 For rational numbers x and y, if x < y, then x – y is a positive rational number.
Solution .

Question . 75 If x and y are negative rational numbers, then so is x + y.
Solution .

Question . 76 Between any two rational numbers there are exactly ten rational . numbers.
Solution . False
There are infinite rational numbers between any two rational numbers.

Question . 77 Rational numbers are closed under addition and multiplication but not under subtraction.
Solution . False
Rational numbers are closed under addition, subtraction and multiplication.

Question . 78 Subtraction of rational number is commutative.
Solution . False
Subtraction of rational numbers is not commutative, i.e. \(a-b\neq b-a\)
where, a and b are rational numbers.

Question . 79 \(-\frac { 3 }{ 4 }\) is smaller than -2 .
Solution .

Question . 80 0 is a rational number.
Solution .

Question . 81 All positive rational numbers lie between 0 and 1000.
Solution . False
Infinite positive rational numbers lie on the right side of 0 on the number line.

Question. 82 The population of India in 2004-05 is a rational number.
Solution. True
The population of India in 2004-05 is a rational number.

Question. 83 There are countless rational numbers between \(-\frac { 5 }{ 6 }\) and \(-\frac { 8 }{ 9 }\).
Solution.

Question. 84

Solution.

Question. 85 The rational number \(-\frac { 57 }{ 23 }\) lies to the left of zero on the number line.
Solution. False
Since,\(-\frac { 57 }{ 23 }\) is a positive rational number.
So, it lies on the right of zero on the number line.

Question .86 The rational number \(-\frac { 7 }{ -4 }\) lies to the right of zero on the number line.
Solution . False
Since, \(-\frac { 7 }{ -4 }\) is a negative rational number.
So, it lies on the left of zero on the number line.

Question .87 The rational number \(-\frac { -8 }{ -3 }\) lies neither to the right nor to the left of zero on the number line.
Solution . False
\(-\frac { -8 }{ -3 }\) = \(-\frac { 8 }{ 3 }\) is a positive rational number.
Hence, it lies on the right of zero on the number line.

Question . 88 The rational numbers \(-\frac { 1 }{ 2 }\) and -1 are on the opposite sides of zero on the number tine.
Solution . True
Since, positive rational number and negative rational number are on the opposite sides of zero on the number line.’
Hence, \(-\frac { 1 }{ 2 }\) and -1 are on the opposite sides of zero on the number line.

Question . 89 Every fraction is a rational number.
Solution .

Question . 90 Every integer is a rational number.
Solution . True
Every integer is a rational number whose denominator remain 1.

Question . 91 The rational numbers can be represented on the number line.
Solution . True
The rational numbers can be represented on the number line.

Question . 92 The negative of a negative rational number is a positive rational number.
Solution . True
Let be a positive rational number.
Then, – x be the negative rational number.
Hence, negative of negative rational number = – (- x)= x = Positive rational number

Question . 93 If x and y are two rational numbers such that x > y, then x – y is always a positive rational number.
Solution . True
If x and y are two rational numbers such that x > y.
Then, there are three possible cases, i.e.
Case I x and y both are positive. ‘
Case II x is positive and y is negative.
Case III x and y both are negative.
In all three cases, x – y is always a positive rational number.

Question . 94 0 is the smallest rational number.
Solution . False
As the smallest rational number does not exist.

Question .95 Every whole number is an integer.
Solution .True
W (whole numbers) = {0,1,2, 3 }
Z (integers) = {…- 3, – 2, -1, 0,1,2, 3,…}
Every whole number is an integer, but every integer is not a whole number.

Question .96 Every whole number is a rational number.
Solution .True
Every whole number can be written in the form of \(-\frac { p }{ q }\), where p, q are integers and \(q\neq 0\).
Hence, every whole number is a rational number.

Question . 97 0 is whole number but it is not a rational number.
Solution . False
0 is a whole number and also a rational number.

Question . 98 The rational numbers \(-\frac { 1 }{ 2 }\)and \(-\frac -{ 5 }{ 2}\) are on the opposite sides of zero on the number line.
Solution . True
Positive rational number and negative rational number remain on opposite sides of zero on the number line.

Question .99 Rational numbers can be added (or multiplied) in any order

Solution .

Question . 100 Solve the following, select the rational numbers from the list which are also the integers.

Solution . From the given rational numbers, the numbers whose denominator is 1 and the numbers whose numerator is the multiple of denominator are the integers.

Question . 101 Select those which can be written as a rational number with denominator 4 in their lowest form

Solution . From the given rational numbers, the number with denominator 4 in their lowest form is \(-\frac { 5 }{ -4 }\)

Question . 102 Using suitable rearrangement and find the sum

Solution .

Question . 103 Verify – (-x) = x for

Solution .

Question . 104 Give one example each to show that the rational numbers are closed under addition, subtraction and multiplication. Are rational numbers closed under division? Give two examples in support of your answer.
Solution . We know that, rational numbers are closed under addition, subtraction and multiplication. We can understand this from the following examples.
Rational numbers are closed under addition


But rational are not closed under division. If zero is excluded from the collection of rational numbers, then we can say that rational numbers are closed under division.

Question . 105 Verify the property x + y = y + x of rational numbers by taking

Solution .

Question . 106 Simplify each of the following by using suitable property. Also, name the property.

Solution .

Question . 107

Solution .

Question . 108 Verify the property x x y = y x x of rational numbers by using

Solution .

Question . 109 Verify the property x x (y x z)=i.(x x y) x z of rational numbers by using

Solution .

Question . 110 Verify the property x x (y + z) = x x y + x x z of rational numbers by taking

Solution .

Question . 111 Use the distributivity of multiplication of rational numbers over addition to simplify

Solution .

Question. 112 Simplify

Solution .

Question. 113 Identify the rational number that does not belong with the other three. Explain your reasoning

Solution . does not belong with the other three. Since,\( \frac { -7 }{ 3 }\) as it is smaller than -1 whereas rest of the numbers are greater than -1.

Question. 114 The cost of \( \frac { 19 }{ 4 }\) m of wire is Rs \( \frac { 171 }{ 2 }\) Find the cost of one metre of the wire.
Solution .

Question. 115 A train travels \( \frac { 1445 }{ 2 }\) km in \( \frac { 17 }{ 2 }\) h. Find the speed of the train in km/h.
Solution .

Question. 116 If 16 shirts of equal size can be made out of 24m of cloth, how much cloth is needed for making one shirt?
Solution . If 16 shirts are to be made by cloth of 24 m
Then, 1 shirt is to be made by cloth of = \( \frac { 24 }{ 16 }\) m = \( \frac { 3 }{ 2 }\) m = 1.5 m
Hence, 1.5 m cloth is needed for making one shirt.

Question. 117 \( \frac { 7 }{ 11 }\) of all the money in Hamid’s bank account is Rs 77000. How much money does Hamid have in his bank account?
Solution .

Question. 118 A 117\( \frac { 1 }{ 3 }\) m long rope is cut into equal pieces measuring 7\( \frac { 1 }{ 3 }\) m each. How many such small pieces are these?
Solution .

Question. 119 \( \frac { 1 }{ 6 }\) of the class students are above average, \( \frac { 1 }{ 4 }\) are average and rest are below average. If there are 48 students in all, how many students are below average in the class?
Solution . Number of above average students = \( \frac { 1 }{ 6 }\) of the class students
Number of average students = \( \frac { 1 }{ 4 }\)of the class students

Question. 120 \( \frac { 2 }{ 5 }\)of total number of students of a school come by car while \( \frac { 1 }{ 4 }\) of students come by bus to school. All the other students walk to school of which \( \frac { 1 }{ 3 }\)walk on their own and the rest are escorted by their parents. If 224 students come to school walking on their own, how many students study in that school?
Solution .

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