Important Questions for CBSE Class 8 Maths Chapter 1 – Rational Numbers

Class 8 Maths  Chapter 1 Important Questions – Rational Numbers

We need Maths in everything we do. It is one of the fundamentals for everything in our daily lives, including mobile devices, computers, software, architecture, art, money, engineering and even sports. The concepts, theories and formulas that we learn in Maths textbooks have tremendous applications in real-life. Solutions to problems can be found using formulas and concepts. Therefore, it is necessary to learn this subject to understand its various applications and significance.

Chapter 1 of Class 8 Maths is about ‘Rational Numbers’.  We are already familiar with some types of numbers. We have studied natural numbers, whole numbers as well as integers. Natural numbers are those numbers that begin with 1 and go on endlessly up to infinity. Whole numbers are those that start from 0 to infinity. And an integer is a whole number that can be positive, negative, or zero. A rational number is a kind of number that is represented as the ratio of any two integers, in which the denominator cannot be equal to zero. In contrast, an irrational number cannot be expressed in the form of fractions. 

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Rational Numbers Class 8 Test Paper with Answers

Our in-house Maths faculty experts have collated an entire list of Important Questions Class 8 Maths Chapter 1 by referring to various sources. For each question, the team has prepared a step-by-step explanation that will help students understand the concepts used in each question. Also, the questions are chosen in a way that would cover full chapter topics. So by practicing from our question bank, students will be able to revise the chapter and understand their strong and weak points. And improve their preparation by further focusing on weaker sections of the chapter.

Given below are a few of the questions and answers from our question bank of Important Questions Class 8 Maths Chapter 1:

Question 1: Mention the commutativity, associative and distributive properties of rational numbers. Also, check a × b = b × a and a + b = b + a for a = ½ and b = ¾

Answer 1: Commutative property:

Let a and b be the two rational numbers, 

a + b = b + a.

Let a and b be the two rational numbers, 

a × b = b × a.

Associative Property:

For any three rational numbers a, b and c,

(a + b) + c = a + (b + c)

Distributive property states that for any three numbers x, y and z,

x × ( y + z ) = (x × y) + ( x × z)

a*b = b*a

a*b = ½ * ¾ = 3/8

b*a = ¾ * ½ = 3/8

a + b = ¾ + ½ = 5/4

b + a = ½ + ¾ = 5/4

Question 2: Write the additive inverse of the following-

        a)-27 b) -58 c)9-4

Answer 2: a) The additive inverse of 27 is -27.

58. b) The additive inverse of -58 is 58.
59. c) The additive inverse of 9-4 is -94.

Question 3: Mention a rational number that has no reciprocal.

Answer 3: The rational number “0” has no reciprocal or multiplicative inverse.

Question 4: Verify commutativity of addition of rational numbers 49 and -712.

Answer 4: We need to show that ab+cd=cd+ab

In the question a=4, b=9, c=-7 and d=12

49+(-712)=4494+(-73)123=1636+(-21)36=16-2136=(-536)

-712+49=(-73)123+4494=-2136+1636=-21+1636=(-536)

49+(-712) = -712+49

Hence verified.

Question 5: Find a rational number between 3 and 4.

Answer 5: The rational number between any two numbers can be calculated as (p+q)2.

Let’s assume p=3 and q=4.

So, the required rational number is (3+4)2 = 72;

Therefore, 72 is the required answer.  

Question 6: The equivalent rational number of 7/9, whose denominator is 45, is.

Answer 6: From the given question,

 The equivalent rational number of 7/9 = Numerator/45

To get 45 in the denominator,

It is essential to multiply both the numerator and denominator by 5,

= (7 × 5)/ (9 × 5)

= 35/45

So, the equivalent rational number of 7/9, whose denominator is 45, is (35/45)

Question 7: What are the multiplicative and additive identities of rational numbers?

Answer 7: 0 and 1 are the additive and multiplicative identities of rational numbers, respectively.

Question 8.  If the product of any rational numbers is 2 and one of them is  15, find the other.

Answer 8: Consider the two rational numbers as “a” and “b”.

Given, a= 15 and  a × b = 2

So, 15 × b = 2

⇒ b = 5 × 2 = 10

Therefore, the other rational number is 10.                  

Question 9: Write five rational numbers smaller than 2.

Answer 9: Five rational numbers smaller than 2 are 1, 12, 0, -1, – 12.

Question 10: The reciprocal of any rational number pq, where p and q are integers and q ≠ 0, is

(a) pq (b) 1  (c) 0  (d) qp

Answer 10 : (d) qp

The reciprocal of pq = qp

Question 11: Multiply 613 by the reciprocal of -710.

Question 11: Reciprocal of -710 is -107

613 -107= -6091

Therefore, the required answer is -6091. 

Question 12:  2/5 × (- 3 / 7 ) – 1 / 6 × 3 / 2 + 1 / 14 × 2 / 5

Answer 12: 2 / 5 × ( – 3/ 7) – 1/6 × 3/ 2 + 1 / 14 × 2 / 5

                 = 2/5 × (- 3/7) + 1/14 × 2/5 – (1/6 × 3/2) (by commutativity)

                 = 2/5 × (- 3 / 7 + 1 / 14) – 3/12

                 = 2/5 × {(- 6 + 1)/14} – 3/12

                 = 2 / 5 × ((- 5)/14)) – 1/4

                 = (-10/70) – 1/4

                 = – 1/7 – 1/4

                 = (– 4– 7)/28

                 = – 11/28

Question 13: Find the rational numbers between -2/5 and 1/2.

Answer 13: Let us make the denominators the same, say 50.

                      -2/5 = (-2 × 10)/(5 × 10) = -20/50

                       1/2 = (1 × 25)/(2 × 25) = 25/50

                      Ten rational numbers between -2/5 and ½ = ten rational numbers between -20/50 and 25/50

  Therefore, ten rational numbers between -20/50 and 25/50 = -18/50, -15/50, -5/50, -2/50, 4/50, 5/50, 8/50, 12/50, 15/50, 20/50

Question 14: Verify associativity of addition of rational numbers when, x=12, y=13, z=-15

Answer 14: We need to show that, (x + y) + z = x + (y + z).

                       x=12, y=13, z=-15

                       (x + y) + z = (12+13) + -15                                                   

                    = (36 + 26) + -15    (taking LCM)

                    = 56 + (-15)

                    = 2530 + (-630)     (taking LCM)

                    = 25-630

                    = 1930

 x + (y + z) = 12 + (13 + -15)

                   = 12 + (515 + -315)     (taking LCM)

                   = 12 + 215

                   = 1530 + 430                 (taking LCM)

                    = 15+430

                     = 1930

Therefore, (12+13) + -15 = 12 + (13 + -15) 

Hence verified.

Question 15:  Find: (-149) 35 (-47) 1516

Answer 15: (-149 35) (-47 1516)

                     = (-143 15) (-17 154)

                     = -1415 ( -1528)

                     = 1428

                    = 12

Question 16: Three numbers are in the ratio 2 : 3: 4. The sum of their cubes is 0.334125. Find the numbers.

Answer 16: Given, the ratio of the three numbers is 2 : 3: 4.

Let 2x, 3x and 4x be the three numbers.

According to the given details, 

( 2x )³ + ( 3x)³  + ( 4x )³ =  0.334125 

8x³  +   27x³   + 64x³ = 0.334125 

99x³  = 0.334125 

x³ = 334125/(1000000 × 99) 

= 3375/1000000

x = ∛(3375/1000000) 

= ∛[(15 × 15 × 15)/(100 × 100 × 100)] 

= 15/100 

= 0.15 

2x = 2(0.15) = 0.3

3x = 3(0.15) = 0.45

4x = 4(0.15) = 0.6

Therefore, the three numbers are 0.3, 0.45 and 0.6.

Question 17: Subtract the sum of 3a + 2b – c and 5b – 6a + 2ab from the sum of 9b – 11ab and -4a + 21ab.

Answer 17: 3a + 2b – c + 5b – 6a + 2ab

                     = (3a – 6a) + (2b + 5b) + 2ab

                     = -3a + 7b + 2ab….(i) 

                      9b -11ab + (-4a + 21ab)

                     = 9b + (-11ab + 21ab) – 4a

                     = 9b + 10ab – 4a….(ii)

                     Subtracting (i) from (ii),

                     9b + 10ab – 4a – (-3a + 7b + 2ab)

                     = 9b + 10ab – 4a + 3a – 7b – 2ab

                     = 8ab – a + 2b

Question 18: Verify – (-x) = x for  x = 35

Answer 18:  – x = -35

                       – (-x) = – (-35)

                             x = 35

                        Hence verified

Question 19: Tell what property allows you to compute 13(643) as (136)43 ?

Answer 19: 13(643) = 83

(136)43 = 83

We can see that the product is unaffected by the arrangement of the factors in the multiplication problem. Thus, the associativity property is applied in this case. 

Question 20:  Simplify each of the following by using suitable properties. Also, name the property.

1. a) (1214) + (126)

Answer 20: 12(14+6)           (taking out 12 as common

                       = 12(1+244)

                       = 12(254)

                       = 12254

                        = 258

                       The distributive property of multiplication over the distributive property of addition is used in this part.

Question 21:  Simplify each of the following by using suitable properties. Also, name the property.

(15215)-(1525)

Answer 21: 15(215–25)              (taking out 15 common)

                       215=21151=215                       (LCM of the denominators 15 and 5 is 15)

                    and 25=2353=615

                     = 15(215–615)

                       = 15(2-615)

                     = 15(-415)

                     = 15-415

                      = -475

The distributive property of multiplication over the distributive property of subtraction is used in this part.

Question 22: Find ten rational numbers between -13 and 13.

Answer 22: Multiply the numerator and denominator of both the fractions by any whole number, say 10.

                      We get, -131010=-1030

                       131010=1030                                                                                                                                                                                           

    Therefore, the equivalent fractions are -1030 and 1030.

Ten rational numbers between them are -930, -830, -730, -630, -530, 530, 630, 730, 830 and930.

Question 23: Divide the sum of 7/8 and 15/24 by their difference. Express your answer in standard form.

Answer 23: Sum of 7/8 and 15/24: (7/8) + (15/24)

                                                              = (21 + 15)/24

                                                              = 36/24

                                                              = 3/2

                    Difference between 7/8 and 15/24: (7/8) – (15/24)

                                                                               = (21 – 15)/24

                                                                                = 6/24

                                                                                = 1/4

                     Now, dividing the sum by the difference,

                      (3/2) / (1/4)

                    = (3/2) × 4

                     = 6                                           

Question 24: Find ten rational numbers between 3/5 and ¾.

Answer 24: Let us make the denominators the same, say 80.

3/5 = (3 × 16)/(5× 16) = 48/80

3/4 = (3 × 20)/(4 × 20) = 60/80

Ten rational numbers between 3/5 and ¾ = ten rational numbers between 48/80 and 60/80

Therefore, ten rational numbers between 48/80 and 60/80 = 49/80, 50/80, 51/80, 52/80, 54/80, 55/80, 56/80, 57/80, 58/80, 59/80

Question 25: 7/11 of the money in Hamid’s bank account altogether is ₹ 77,000. How much money does Hamid have in his bank account?

Answer 25: In the question, 

               it is given that

               7/11 part of all the money in the bank account of Hami = ₹ 77,000.    

               Let the money in the bank account of Hamid be ₹ x.

               so,

               ( 7 / 11 ) × ( x ) = ₹ 77,000

               x = 77,000 / (7/11)

               x = 77000 × (11/7)

               x = 11000 × 11

               x = ₹ 1,21,000

∴The total money in the bank account of Hamid is ₹ 121000.

Question 26: 2/5 of the number of students of a school come by car, while 1/4 of students travel by bus to school. The remaining students walk to school, out of which 1/3 walk on their own accord and the rest are escorted by their own parents. If 224 out of all students come to school walking on their own accord, how many students study in that school?

Answer 26: Let the total number of students in school be x.

                      From the question, it is given that,

                      The number of students coming by car = (2/5) × x

                      The number of students coming by bus = (1/4) × x

                      The remaining students walk to school = x – ((2x/5) + (1x/4))

                                                                                            = x – ((8x – 5x)/20)

                                                                                            = x – (13x/20)

                                                                                            = (20x – 13x)/20

                                                                                            = 7x/20

Then, the number of students who come to school walking on their own= (1/3) of (7x/20)

                                                                                                                                  = 7x/60

Since 224 students come to school walking on their own.

As per the given conditions,

= (7x/60) = 224

x = (224 × 60)/7

x = 32 × 60

x = 1920

∴ The total number of students in that particular school is 1920. 

Question 27: Find the product of the additive and multiplicative inverse of 1/3.

Answer 27: Additive inverse of -1/3 = 1/3

                      Multiplicative inverse of -1/3 = -3/1

                      Then,

                       The product of additive and multiplicative inverse of 1/3 = 1/3 × (-3)

                                                                                                                            = -1

Question 28: How much longer is the wingspan of a golden eagle than the wingspan of a blue jay?

Answer 28: We have to find out the difference between the wingspan of a golden eagle and the wingspan of a blue jay.

Length of the wingspan of a golden eagle = 2½ = 5/2 m

Length of the wingspan of a blue jay = 41/100 m

Difference of both = (5/2) – (41/100)

= (250 – 41)/ 100

= 209/100 m

∴The wingspan of a golden eagle is 209/100 m longer than the wingspan of a blue jay.

Question 29: Select the correct answer based on the following:

What should be subtracted from -5/3 to obtain  5/6?

1. a) 5/2
2. b) 3/2
3. c) 5/4
4. d) -5/2

Answer 29: A number is represented in the form of p/q, where q≠0 is referred to as a rational number in maths. Subtracting rational numbers works in the same way that adding them does.

Let us consider one of the numbers as x

-5/3 –x = 5/6

-5/3 – 5/6 = x

x = (-5/3 ) – 5/6

By taking the least common multiple  for 3 and 6 as 6, we get

x = (-5×2 – 5×1)/6

= (-10-5)/6

= -15/6

By further dividing by 3, we get

-15/6 = -5/2

Question 30: Find the rational number between 2 and 3

Answer 30: Let us consider the rational number as x

                    So to find the rational number between 2 and 3

                    By using the formula x= ½ (a/b + c/d)

                    x = ½(2 + 3)

                    x = ½(5)

                        = 5/2

∴ The rational number between 2 and 3 is 5/2.

Question 31: Praful reads 15 pages of a storybook in a day. Raju reads 10 pages of the same given book in a day. How many days are required for Raju to finish the given book if Praful finishes in 13 days?

Answer 31:The number of pages read by Praful in a day = 15

               The number of pages read by Raju in a day = 10

               The number of days required for Praful to finish the book = 13

               That means the total number of pages = 15 × 13 

                                                                                  = 195

              Hence, the number of days Raju will take to finish the book 

                                                                                                   = 195/10 

                                                                                                    = 19.5 days

Question 32: A rectangular piece of paper 12 cm x 5 cm is folded without overlapping to make a cylinder of height 4 cm. Find the volume of the cylinder.

Answer 32: The length of the rectangular paper = 12 cm

                      The breadth of the rectangular paper = 5 cm

                      The circumference of the circular part of the given cylinder = 2πr

                      Also, given that the paper is rolled along its length.

                     The circumference of the circular part of the given cylinder = Length of the rectangular  

                      paper

                      2πr = 12 cm

                      2 × (22/7) × r = 12

                      r = (12 × 7)/44

                      r = 84/44

                      r = 21/11 cm

                      Height of cylinder = Breadth of the rectangular paper = h = 4 cm

                      Volume of cylinder = πr2h

                                                         =  (22/7) × (21/11) × (21/11) × 4

                                                         = 45.82 cm³

Question 33:State whether the following statements are true or false.

(A) Every whole number is a rational number.

(B) Every integer is a rational number.

(C) 0 is a whole number, but it is not a rational number.

Answer 33:

(A) Every whole number is a rational number.

Answer: This statement is true. 

Explanation: By the definition of rational number, p/q where q≠ 0,

We also know that each whole number can also be represented as a/1​

∴ From the two statements given above, we can conclude that every whole number is a rational number.

• Every integer is a rational number.

Answer: True

Explanation: By the definition of rational number, p/q​ where q≠ 0,

We know that every integer can also be represented as a/1​.

∴ From the two statements above, we can conclude that every whole number is a rational number.

(c) 0 is a whole number, but it is not a rational number.

Answer: This statement is false. Since, by the definition of a rational number, p/q​ where q≠ 0.

We know that 0 can also be represented as a/1.

∴ From the two statements above, we can conclude that 0 is a whole number and a rational number.

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