**NCERT Exemplar Problems Class 8 Mathematics Chapter 1 Rational Numbers**

**Question . 1 A number which can be expressed as , where p and q are integers and is**

** (a) natural number (b) whole number**

** (c) integer (d) rational number**

** Solution.** (d) A number which can be expressed as , where p and q are integers and is a rational number.

**Question . 2 A number of the form is said to be a rational number, if**

** (a) p, q are integers (b) p, q are integers and **

** (c) p, q are integers and (d) p, q are integers and , also **

** Solution.** (b) A number of the form is said to be a rational number, if p and q are integers and

**Question . 3 The numerical expression 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

Show that rational numbers are closed under addition.

[ and are rational numbers and their addition is 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 is not defined.

**Question . 5 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 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

**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.

is the multiplicative inverse of-p, then, -p x = 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 by**

** **

** Solution .**

**Question . 13 – (-x) is same as**

** (a)-x (b)x (c) (d)**

** Solution .** (b) -(-x) = x

Negative of negative rational number is equal to positive rational number.

**Question . 14 The multiplicative inverse of 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 , where p and q are integers and is**

** (a) (b)1 (c)0 (d)**

** Solution .** (d) The reciprocal of any rational number , where p and q are integers and is

**Question . 20 If y is the reciprocal of rational number x, then the reciprocal of y will be**

** (a)x (b) y (c) (d) **

** Solution .** (a) If y be the reciprocal of rational number x, i.e. y = or x = .

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 whose numerator is 45, is —.**

** Solution .**

**Question . 27 The equivalent rational number of , whose denominator is 45 is——————.**

** Solution .**

**Question . 28 Between the numbers and , 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 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 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 and 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 is——————.**

** Solution .**

**Question . 44 The multiplicative inverse of is———–.**

** Solution .**

**Question . 45 The rational number 10.11 in the form 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 is a rational number, then y is always a whole number.**

** Solution .**

**Question . 49 If is a rational number, then p Cannot be equal to zero.**

** Solution .**

**Question . 50 If is a rational number, then s cannot be equal to zero.**

** Solution .**

**Question . 51 lies between and 1.**

** Solution .**

**Question . 52 lies between and 1.**

** Solution .**

**Question . 53 lies between -3 and 4.**

** Solution .**

**Question . 54 lies between 1 and 2.**

** Solution .**

**Question . 55 If the multiplicative inverse of is .**

** Solution .**

**Question . 56 The multiplicative inverse of is .**

** Solution .**

**Question . 57 The additive inverse of 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 is the rational number .**

** Solution .**False

The reciprocal of a non-zero rational number . is the rational number

**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.

where, a and b are rational numbers.

**Question . 79 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 and .**

** Solution.**

**Question. 84**

**Solution.**

**Question. 85 The rational number lies to the left of zero on the number line.**

** Solution.** False

Since, is a positive rational number.

So, it lies on the right of zero on the number line.

**Question .86 The rational number lies to the right of zero on the number line.**

** Solution .** False

Since, is a negative rational number.

So, it lies on the left of zero on the number line.

**Question .87 The rational number lies neither to the right nor to the left of zero on the number line.**

** Solution .** False

= is a positive rational number.

Hence, it lies on the right of zero on the number line.

**Question . 88 The rational numbers 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, 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 , where p, q are integers and .

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 and 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

**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, as it is smaller than -1 whereas rest of the numbers are greater than -1.

**Question. 114 The cost of m of wire is Rs Find the cost of one metre of the wire.**

** Solution .**

**Question. 115 A train travels km in 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 = m = m = 1.5 m

Hence, 1.5 m cloth is needed for making one shirt.

**Question. 117 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 m long rope is cut into equal pieces measuring 7 m each. How many such small pieces are these?**

** Solution .**

**Question. 119 of the class students are above average, 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 = of the class students

Number of average students = of the class students

**Question. 120 of total number of students of a school come by car while of ****students come by bus to school. All the other students walk to school of which 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 .**

**Question. 121 Huma, Hubna and Seema received a total of Rs 2016 as monthly allowance from their mother such that Seema gets of what Hubna gets and Huma gets 1 times Seema’s share. How much money do the three sisters get individually?**

** Solution .**

**Question. 122 A mother and her two daughters got a room constructed for Rs 62000. The elder daughter contributes of her mother’s contribution while the younger daughter contributes of her mother’s share. How much do the three contribute individually?**

** Solution .**

**Question. 123 Tell which property allows you to compare**

** **

** Solution .**

**Question. 124 Name the property used in each of the following:**

** **

** Solution .** (i) Commutative property over multiplication

(ii) Distributive property over addition

(iii) Associative property over addition

(iv) Existence of additive identity

(v) Existence of multiplicative identity

**Question. 125 Find the multiplicative inverse of(i)-1 (ii)3 **

** Solution .**

**Question. 126 Arrange the numbers is , , in the descending order.**

** Solution .**

**Question. 127 The product of two rational numbers is If one of the numbers be ****find the other.**

** Solution .**

**Question. 128 By what numbers should we multiply so that the product may be ?**

** Solution .**

**Question. 129 By what number should we multiply so that the product may be 24?**

** Solution .**

**Question. 130 The product of two rational numbers is -7. If one of the number is -5, find the other?**

** Solution .**

**Question. 131 Can you find a rational number whose multiplicative inverse is -1?**

** Solution .** No, we cannot find a rational number whose multiplicative inverse is -1.

**Question. 132 Find five rational numbers between 0 and 1.**

** Solution .**

**Question. 133 Find the two rational numbers whose absolute value is .**

** Solution .**

**Question. 134 From a rope 40 m long, pieces of equal size are cut. If the length of one piece is m, find the number of such pieces.**

** Solution .**

**Question. 135 5 m long rope is cut into 12 equal pieces. What is the length of each piece?**

** Solution .**

**Question. 136 Write the following rational numbers in the descending order.**

** **

** Solution .**

**Question. 137 Find**

** **

** Solution .**

**Question. 138 On a winter day the temperature at a place in Himachal Pradesh was -16°C. Convert it in degree Fahrenheit (°F) by using the formula**

** **

** Solution .**

**Question. 139 Find the sum of additive inverse and multiplicative inverse of 7.**

** Solution .**

**Question. 140 Find the product of additive inverse and multiplicative inverse of .**

** Solution .**

**Question. 141 The diagram shows the wingspans of different species of birds. Use the diagram to answer the question given below**

** (a) How much longer is the wingspan of an Albatross than the wingspan of a Sea gull?**

** (b) How much longer is the wingspan of a Golden eagle than the wingspan of a Blue jay?**

** Solution .**

**Question. 142 Shalini has to cut out circles of diameter 1 cm from an aluminium strip of dimensions 8 cm by 1 cm. How many full circles can Shalini cut? Also, calculate the wastage of the aluminium strip.**

**Solution .**

**Question. 143 One fruit salad recipe requires cup of sugar. Another recipe for the same fruit salad requires 2 tablespoons of sugar. If 1 tablespoon is 1 equivalent to cup, how much more sugar does the first recipe require?**

** Solution .**

**Question. 144 Four friends had a competition to see how far could they hop on one foot. The table given shows the distance covered by each.**

** **

** (a) How farther did Soni hop than Nancy?**

** (b) What is the total distance covered by Seema and Megha?**

** (c) Who walked farther, Nancy or Megha?**

** Solution .**

**Question. 145 The table given below shows the distances, in kilo metres, between four villages of a state. To find the distance between two villages, locate the square, where the row for one village and the column for the other village intersect.**

** **

** (a) Compare the distance between Himgaon and Rawalpur to Sonapur and Ramgarh?**

** (b) If you drove from Himgaon to Sonapur and then from Sonapur to Rawalpur, how far would you drive?**

** Solution .**

**Question. 146 The table shows the portion of some common materials that are recycled.**

** **

** (a) Is the rational number expressing the amount of paper recycled more than or less than ?**

** (b) Which items have a Recycled amount less than ?**

** (c) Is the quantity of aluminium fans recycled more (or less) than half of the quantity of aluminium cans?**

** (d) Arrange the rate of recycling the materials from the greatest to the smallest.**

** Solution .**

**Question. 147 The overall width in cm of several wide-screen televisions are 97.28 cm,98 cm, 98 cm and 97.94 cm. Express these numbers as rational numbers in the form and arrange the widths in ascending order.**

** Solution .**

**Question. 148 Roller coaster at an amusement park is m high. If a new roller coaster is built that is times the height of the existing coaster, what will be the height of the new roller coaster?**

** Solution .**

**Question. 149 Here is a table which gives the information about the total rainfall for several months compared to the average monthly rains of a town. Write each decimal in the form of rational number .**

** **

** Solution .**

**Question. 150 The average life expectancies of males for several states are shown in the table. Express each decimal in the form and arrange the states from the least to the greatest male life expectancy.**

** State-wise data are included below; more indicators can be found in the “FACTFILE” section on the homepage for each state.**

**Source Registrar General of India (2003) SRS Based Abridged Lefe Tables. SRS Analytical Studies, Report No. 3 of 2003, New Delhi: Registrar General of India.**

** The data are for the 1995-99 period; states subsequently divided are therefore included in their pre-partition states (Chhatisgarh in MP, Uttaranchal in UP and Jharkhand in Bihar)**

** Solution .**

Arrangement of the states from the least to the greatest male life expectancy, Haryana, Tamil Nadu, West Bengal, Karnataka, Gujarat, AndhraPradesh, Bihar, Rajasthan, Uttar Pradesh, Orissa, Assam, Madhya Pradesh.

**Question. 151 A skirt that is 35 cm long has a hem of 3 cm. How tong will the skirt . be if the hem is let down?**

** Solution .**

**Question. 152 Manavi and Kuber each receives an equal allowance. The table shows the fraction of their allowance each deposits into his/her saving account and the fraction each spends at the mall. If allowance of each is Rs 1260, find the amount left with each.**

** **

** Solution .**