Question

NCERT Solutions Class 8 Maths Chapter 1 Rational Numbers here are made by experts in the field. Our experts have prepared these solutions in very simple language to make it easy for students to learn and comprehend. These solutions are precise and it is according to the latest syllabus provided by the CBSE. Our NCERT solutions are the best way to assist students in their CBSE exam preparation as well as for competitive exams like JEE Mains, JEE Advance, or any other similar exams. These solutions are explained in the step-wise method.

You can find Class 8 Maths Chapter 1 Rational Numbers here to help you understand the fundamental concepts. It will keep you interested and useful in solving a variety of problems. Revision Notes for Class 8 Maths are created by Sarthaks experts who have extensive knowledge of the subject. It will undoubtedly improve your performance and enable you to achieve high grades. If you want to clear your doubts and boost your confidence, NCERT Solutions for Class 8 Maths is essential.

Our NCERT Solutions Class 8 is all you need for your overall preparation for all the topics needed. Important topics mentioned here are: 

• Rational numbers on a number line
• Rational numbers between any two given rational numbers
• Explain natural numbers, whole numbers, integers, and rational number
• Closure property
• Commutativity
• Associativity
• Additive identity and multiplicative identity
• Distributive property
• Additive inverse of a number
• Reciprocal or multiplicative inverse

These topics are very important according to the exam perspective. Experts at Sarthaks advise learners to go through all the topics of NCERT Solutions Class 8 Maths to gain complete clarity of this chapter. Every year number of questions are asked from this chapter which carries a significant weightage in the CBSE board examination. One must refer to our solutions for better understanding, solving questions, revision, completing assignments, and doing homework with ease. Students can also find solutions for NCERT intext questions, exercises, and back-of-chapter questions. These solutions give not only provide the required solutions but also give you a deep understanding of all the related concepts. Now all the solutions and practice questions are at your fingertip get started now.

Answer 1

NCERT Solutions Class 8 Maths Chapter 1 Rational Numbers

1. Using appropriate properties find:

(i) \(-\frac23 \times \frac35 + \frac 52 - \frac35\times \frac16\)

(ii) \(\frac25\times(\frac{-3}7) - \frac16 \times \frac32 + \frac1{14} \times \frac25\)

Solution:

(i) We have \(-\frac23 \times \frac35 + \frac 52 - \frac35\times \frac16\) 

Thus, the required value = 2.

(ii) We have \(\frac25\times(\frac{-3}7) - \frac16 \times \frac32 + \frac1{14} \times \frac25\) 

Thus, the required value = \(\frac{-11}{28}\).

2. Write the additive inverse of each of the following:

(i) \(\frac28\)

(ii) \(\frac{-5}9\)

(iii) \(\frac{-6}{-5}\)

(iv) \(\frac2{-9}\)

(v) \(\frac{19}{-6}\) 

Solution:

3. Verify that: -(-x) = x for.

(i) x = \(\frac{11}{15}\)

(ii) x = \(\frac{-13}{17}\)

Solution:

4. Find the multiplicative inverse of the following:

(i) -13

(ii) \(\frac{-13}{19}\)

(iii) \(\frac15\)

(iv) \(\frac{-5}8\times \frac{-3}7\)

(v) \(-1 \times \frac{-2}5\)

(vi) -1

Solution:

(i) Multiplicative inverse of -13 = \(\frac{-1}{13}\)

(ii) Multiplicative inverse of \(\frac{-13}{19} = \frac{-19}{13}\)

(iii) Multiplicative inverse of \(\frac15 = 5\)

(iv) Multiplicative inverse of \(\frac{-5}8\times \frac{-3}7 = \frac{15}{56} = \frac{56}{15}\)

(v) Multiplicative inverse of \(-1 \times \frac{-2}5 = \frac25 = \frac52\)

(vi) Multiplicative inverse of \(-1 = -1\)

5. Name the property under multiplication used in each of the following:

(i) \(\frac{-4}{5} \times 1 = 1 \times \frac{-4}5 = \frac{-4}5\) 

(ii) \(\frac{-13}{17} \times \frac{-2}7 = \frac{-2}7 \times \frac{-13}{17}\)

(iii) \(\frac{-19}{29} \times \frac{29}{-19} = 1\) 

Solution:

(i) Commutative property of multiplication

(ii) Commutative property of multiplication

(iii) Multiplicative inverse property

6. Multiply \(\frac6{13}\) by the reciprocal of \(\frac{-7}{16}\)

Solution:

Reciprocal of \(\frac{-7}{16}\) = \(\frac{16}{-7}\) = \(\frac{-16}{7}\)

According to the question,

\(\frac6{13}\) × (Reciprocal of \(\frac{-7}{16}\))

\(\frac6{13}\) × (\(\frac{-16}{7}\)) = \(\frac{-96}{91}\)

7. Tell what property allows you to compute 1/3 × (6 × 4/3) as (1/3 × 6) × 4/3

Solution:

1/3 × (6 × 4/3) = (1/3 × 6) × 4/3

Here, the way in which factors are grouped in a multiplication problem, supposedly, does not change the product. Hence, the Associativity Property is used here.

8. Is 8/9 the multiplication inverse of \(-1\frac18\) ? Why or why not?

Solution:

\(-1\frac18 = \frac{-9}8\)

[Multiplicative inverse ⇒ product should be 1]

According to the question,

8/9 × (-9/8) = -1 ≠ 1

Therefore, 8/9 is not the multiplicative inverse of \(-1\frac18\).

9. If 0.3 the multiplicative inverse of \(3\frac13\)? Why or why not?

Solution:

\(3\frac13\) = 10/3

0.3 = 3/10

[Multiplicative inverse ⟹ product should be 1]

According to the question,

3/10 × 10/3 = 1

Therefore, 0.3 is the multiplicative inverse of \(3\frac13\).

10. Write

(i) The rational number that does not have a reciprocal.

(ii) The rational numbers that are equal to their reciprocals.

(iii) The rational number that is equal to its negative.

Solution:

(i) The rational number that does not have a reciprocal is 0.

Reason:

0 = 0/1

Reciprocal of 0 = 1/0, which is not defined.

(ii) The rational numbers that are equal to their reciprocals are 1 and -1.

Reason:

1 = 1/1

Reciprocal of 1 = 1/1 = 1 Similarly, Reciprocal of -1 = – 1

(iii) The rational number that is equal to its negative is 0.

Reason:

Negative of 0 = -0 = 0

11. Fill in the blanks.

(i) Zero has _______reciprocal.

(ii) The numbers ______and _______are their own reciprocals

(iii) The reciprocal of – 5 is ________.

(iv) Reciprocal of 1/x, where x ≠ 0 is _________.

(v) The product of two rational numbers is always a ________.

(vi) The reciprocal of a positive rational number is _________.

Solution:

(i) Zero has no reciprocal.

(ii) The numbers -1 and 1 are their own reciprocals

(iii) The reciprocal of – 5 is -1/5.

(iv) Reciprocal of 1/x, where x ≠ 0 is x.

(v) The product of two rational numbers is always a rational number.

(vi) The reciprocal of a positive rational number is positive.

Answer 2

12. Represent these numbers on the number line.

(i) 7/4

(ii) -5/6

Solution:

(i) 7/4

Divide the line between the whole numbers into 4 parts. i.e., divide the line between 0 and 1 to 4 parts, 1 and 2 to 4 parts and so on.

Thus, the rational number 7/4 lies at a distance of 7 points away from 0 towards positive number line.

(ii) -5/6

Divide the line between the integers into 4 parts. i.e., divide the line between 0 and -1 to 6 parts, -1 and -2 to 6 parts and so on. Here since the numerator is less than denominator, dividing 0 to – 1 into 6 part is sufficient.

Thus, the rational number -5/6 lies at a distance of 5 points, away from 0, towards negative number line

13. Represent -2/11, -5/11, -9/11 on a number line.

Solution:

Divide the line between the integers into 11 parts.

Thus, the rational numbers -2/11, -5/11, -9/11 lies at a distance of 2, 5, 9 points away from 0, towards negative number line respectively.

14. Write five rational numbers which are smaller than 2.

Solution:

The number 2 can be written as 20/10

Hence, we can say that, the five rational numbers which are smaller than 2 are:

2/10, 5/10, 10/10, 15/10, 19/10

15. Find the rational numbers between -2/5 and 1/2.

Solution:

Let us make the denominators 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 1/2 = 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

16. Find five rational numbers between.

(i) 2/3 and 4/5

(ii) -3/2 and 5/3

(iii) 1/4 and 1/2

Solution:

(i) 2/3 and 4/5

Let us make the denominators same, say 60

i.e., 2/3 and 4/5 can be written as:

2/3 = (2 × 20)/(3 × 20) = 40/60

4/5 = (4 × 12)/(5 × 12) = 48/60

Five rational numbers between 2/3 and 4/5 = five rational numbers between 40/60 and 48/60

Therefore, Five rational numbers between 40/60 and 48/60 = 41/60, 42/60, 43/60, 44/60, 45/60.

(ii) -3/2 and 5/3

Let us make the denominators same, say 6

i.e., -3/2 and 5/3 can be written as:

-3/2 = (-3 × 3)/(2× 3) = -9/6

5/3 = (5 × 2)/(3 × 2) = 10/6

Five rational numbers between -3/2 and 5/3 = five rational numbers between -9/6 and 10/6

Therefore, Five rational numbers between -9/6 and 10/6 = -1/6, 2/6, 3/6, 4/6, 5/6.

(iii) 1/4 and 1/2

Let us make the denominators same, say 24.

i.e., 1/4 and 1/2 can be written as:

1/4 = (1 × 6)/(4 × 6) = 6/24

1/2 = (1 × 12)/(2 × 12) = 12/24

Five rational numbers between 1/4 and 1/2 = five rational numbers between 6/24 and 12/24

Therefore, Five rational numbers between 6/24 and 12/24 = 7/24, 8/24, 9/24, 10/24, 11/24.

17. Write five rational numbers greater than -2.

Solution:

-2 can be written as – 20/10

Hence, we can say that, the five rational numbers greater than -2 are

-10/10, -5/10, -1/10, 5/10, 7/10.

18. Find ten rational numbers between 3/5 and 3/4,

Solution:

Let us make the denominators 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 3/4 = 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.