NCERT Solutions for Class 8 Maths Chapter 1 – Rational Numbers

Class 8 Maths Chapter 1 – Rational Numbers introduces students to the properties and operations of rational numbers. A rational number is any number that can be expressed in the form p/q where p and q are integers and q ≠ 0. This chapter covers the properties of rational numbers including closure, commutativity, associativity, and distributivity, along with the role of 0 and 1 as additive and multiplicative identities. Students also learn to represent rational numbers on the number line and find rational numbers between any two given rational numbers.

This chapter is fundamental to the entire Class 8 Maths syllabus and is regularly tested in school examinations. A thorough understanding of rational numbers helps students handle algebraic operations, comparisons, and problem-solving with confidence throughout their mathematical journey in Class 9 and beyond.

NCERT Solutions for Class 8 Maths Chapter 1 – Rational Numbers

Rational Numbers

Difficult

Q.

Using appropriate properties find:
$\begin{array}{l}\left(\mathrm{i}\right) \frac{-2}{3}\times \frac{3}{5}+\frac{5}{2}-\frac{3}{5}\times \frac{1}{6}\\ \left(\mathrm{ii}\right) \frac{2}{5}\times \left(-\frac{3}{7}\right)-\frac{1}{6}\times \frac{3}{2}+\frac{1}{14}\times \frac{2}{5}\end{array}$

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Rational Numbers

Easy

Q.

Fill in the blanks.

(i) Zero has ________ reciprocal.
(ii) The numbers ________ and ________ are their own reciprocals.
(iii) The reciprocal of – 5 is ________.
$\left(\mathrm{iv}\right) \mathrm{Reciprocal} \mathrm{of} \frac{1}{\mathrm{x}}, \mathrm{where} \mathrm{x}\ne 0 \mathrm{is}\_\_\_\_\_\_\_\_\_\_.$
(v) The product of two rational numbers is always a _______.
(vi) The reciprocal of a positive rational number is ________.

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Rational Numbers

Easy

Q.

Write five rational numbers greater than –2.

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Rational Numbers

Easy

Q.

$\begin{array}{l}\text{Find five rational numbers between}\\ \left(\mathrm{i}\right)\frac{2}{3} \mathrm{and} \frac{4}{5} \left(\mathrm{ii}\right)\frac{-3}{2} \mathrm{and} \frac{5}{3} \left(\mathrm{iii}\right)\frac{1}{4}\mathrm{and}\frac{1}{2}\end{array}$

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Rational Numbers

Easy

Q.

$\mathrm{Find} \mathrm{ten} \mathrm{rational} \mathrm{numbers} \mathrm{between}\frac{-2}{5}\mathrm{and}\frac{1}{2}.$

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Rational Numbers

Medium

Q.

$\mathrm{Find} \mathrm{ten} \mathrm{rational} \mathrm{numbers} \mathrm{between} \frac{-2}{5} \mathrm{and} \frac{1}{2}.$

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Rational Numbers

Easy

Q.

Write five rational numbers which are smaller than 2.

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Rational Numbers

Medium

Q.

$\mathrm{Represent} \frac{-2}{11},\frac{-5}{11},\frac{-9}{11} \mathrm{on} \mathrm{the} \mathrm{number} \mathrm{line}.$

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Rational Numbers

Medium

Q.

Represent these numbers on the number line.

$\left(\mathrm{i}\right)\frac{7}{4} \left(\mathrm{ii}\right)\frac{-5}{6}$

 

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Rational Numbers

Easy

Q.

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.

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Rational Numbers

Easy

Q.

Write the additive inverse of each of the following.
$\left(\mathrm{i}\right)\frac{2}{8} \left(\mathrm{ii}\right)\frac{-5}{9} \left(\mathrm{iii}\right)\frac{-6}{-5} \left(\mathrm{iv}\right)\frac{2}{-9} \left(\mathrm{v}\right)\frac{19}{-6}$

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Rational Numbers

Medium

Q.

$\mathrm{Is} 0.3 \mathrm{the} \mathrm{multiplicative} \mathrm{inverse} \mathrm{of} 3\frac{1}{3}?\mathrm{Why} \mathrm{or} \mathrm{why} \mathrm{not}?$

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Rational Numbers

Medium

Q.

$\mathrm{Is} \frac{8}{9} \mathrm{the} \mathrm{multiplicative} \mathrm{inverse} \mathrm{of}-1\frac{1}{8}?\mathrm{Why} \mathrm{or} \mathrm{why} \mathrm{not}?$

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Rational Numbers

Easy

Q.

$\begin{array}{l}\mathrm{Tell} \mathrm{what} \mathrm{property} \mathrm{allows} \mathrm{you} \mathrm{to} \mathrm{compute}\\ \frac{1}{3}\times (6\times \frac{4}{3})\mathrm{as}(\frac{1}{3}\times 6)\times \frac{4}{3}.\end{array}$

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Rational Numbers

Easy

Q.

$\mathrm{Multiply} \frac{6}{13} \mathrm{by} \mathrm{the} \mathrm{reciprocal} \mathrm{of}\frac{-7}{16}.$

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Rational Numbers

Easy

Q.

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

$\begin{array}{l}\left(\mathrm{i}\right)\frac{-4}{5}\times 1=1\times \frac{-4}{5}=\frac{-4}{5}\left(\mathrm{ii}\right)-\frac{13}{17}\times \frac{-2}{7}=\frac{-2}{7}\times \frac{-13}{17}\\ \left(\mathrm{iii}\right)\frac{-19}{29}\times \frac{29}{-19}=1\end{array}$

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Rational Numbers

Easy

Q.

Find the multiplicative inverse of the following.
$\begin{array}{l}\left(\mathrm{i}\right)-13 \left(\mathrm{ii}\right)\frac{-13}{19} \left(\mathrm{iii}\right)\frac{1}{5} \left(\mathrm{iv}\right)\frac{-5}{8}\times \frac{-3}{7}\\ \left(\mathrm{v}\right)-1\times \frac{-2}{5}\left(\mathrm{vi}\right)-1\end{array}$

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Rational Numbers

Easy

Q.

Verify that – (– x) = x for.
$\left(\mathrm{i}\right) \mathrm{x}=\frac{11}{15}\left(\mathrm{ii}\right)\mathrm{x}=-\frac{13}{17}$

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Rational Numbers

Easy

Q.

$\mathrm{Find} \mathrm{ten} \mathrm{rational} \mathrm{numbers} \mathrm{between} \frac{3}{5} \mathrm{and} \frac{3}{4}.$

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Important Questions and Answers – Chapter 1: Rational Numbers

Question 1. Using appropriate properties find: -2/3 × 3/5 + 5/2 – 3/5 × 1/6

Answer:

= -2/3 × 3/5 + 5/2 – 3/5 × 1/6

= 3/5 × (-2/3 – 1/6) + 5/2 (using distributive property)

= 3/5 × (-4/6 – 1/6) + 5/2

= 3/5 × (-5/6) + 5/2

= -1/2 + 5/2

= 4/2 = 2

Question 2. Write the additive inverse of each of the following: (i) 2/8 (ii) -5/9 (iii) -6/-5 (iv) 2/-9 (v) 19/-6

Answer:

The additive inverse of a/b is -a/b such that a/b + (-a/b) = 0.

(i) Additive inverse of 2/8 = -2/8 (ii) Additive inverse of -5/9 = 5/9 (iii) -6/-5 = 6/5 → Additive inverse = -6/5 (iv) 2/-9 = -2/9 → Additive inverse = 2/9 (v) 19/-6 = -19/6 → Additive inverse = 19/6

Question 3. Verify that -(-x) = x for: (i) x = 11/15 (ii) x = -13/17

Answer:

(i) x = 11/15 -x = -11/15 -(-x) = -(-11/15) = 11/15 = x ✓

(ii) x = -13/17 -x = -(-13/17) = 13/17 -(-x) = -13/17 = x ✓

Hence verified that -(-x) = x.

Question 4. Find the multiplicative inverse (reciprocal) of the following: (i) -13 (ii) -13/19 (iii) 1/5 (iv) -5/8 × (-3/7) (v) -1

Answer:

The multiplicative inverse of a/b is b/a such that a/b × b/a = 1.

(i) Multiplicative inverse of -13 = -1/13 (ii) Multiplicative inverse of -13/19 = -19/13 (iii) Multiplicative inverse of 1/5 = 5 (iv) -5/8 × (-3/7) = 15/56 → Multiplicative inverse = 56/15 (v) Multiplicative inverse of -1 = -1

Question 5. Name the property used in each of the following: (i) -2/5 × 3/7 = 3/7 × (-2/5) (ii) -2/3 + 0 = 0 + (-2/3) = -2/3 (iii) 1/3 × (6/7 × 8/9) = (1/3 × 6/7) × 8/9

Answer:

(i) The order of multiplication is changed but the result is the same. Commutativity of multiplication

(ii) Adding 0 to any rational number gives the same number. 0 is the additive identity. Existence of additive identity

(iii) The grouping of numbers in multiplication is changed but the result is the same. Associativity of multiplication

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

Answer:

The grouping of numbers is changed while the result remains the same. This is the Associativity of multiplication.

1/3 × (6 × 4/3) = (1/3 × 6) × 4/3

Both sides give the same result, confirming the associative property.

Question 7. Is 8/9 the multiplicative inverse of -1(1/8)? Why or why not?

Answer:

-1(1/8) = -9/8

Product = 8/9 × (-9/8) = -72/72 = -1

Since the product is -1 and not 1, 8/9 is NOT the multiplicative inverse of -1(1/8).

The multiplicative inverse of -9/8 would be -8/9.

Question 8. Is 0.3 the multiplicative inverse of 3(1/3)? Why or why not?

Answer:

0.3 = 3/10 3(1/3) = 10/3

Product = 3/10 × 10/3 = 30/30 = 1

Since the product is 1, 0.3 IS the multiplicative inverse of 3(1/3). ✓

Question 9. 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.

Answer:

(i) 0 — Zero does not have a reciprocal because 1/0 is undefined.

(ii) 1 and -1 — Reciprocal of 1 is 1 itself, and reciprocal of -1 is -1 itself.

(iii) 0 — The negative of 0 is -0 = 0, so 0 is equal to its own negative.

Question 10. Fill in the blanks: (i) Zero has ______ reciprocal. (ii) The numbers 1 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 ______.

Answer:

(i) No (ii) -1 (iii) -1/5 (iv) x (v) Rational number (vi) Positive

Question 11. Represent -2/11, -5/11, and -9/11 on the number line.

Answer:

All three numbers are negative rational numbers with denominator 11. On the number line: -9/11 is the leftmost, followed by -5/11, then -2/11, all lying between -1 and 0.

-9/11 < -5/11 < -2/11 < 0

Divide the segment between -1 and 0 into 11 equal parts. Mark the 2nd, 5th, and 9th points to the left of 0 to represent -2/11, -5/11, and -9/11 respectively.

Question 12. Find five rational numbers between: (i) 2/3 and 4/5 (ii) -3/2 and 5/3 (iii) 1/4 and 1/2

Answer:

(i) Between 2/3 and 4/5: Convert to same denominator: 2/3 = 20/30, 4/5 = 24/30 Five rational numbers: 21/30, 22/30, 23/30, 41/60, 43/60

(ii) Between -3/2 and 5/3: Convert: -3/2 = -9/6, 5/3 = 10/6 Five rational numbers: -8/6, -5/6, 0, 2/6, 7/6

(iii) Between 1/4 and 1/2: Convert: 1/4 = 2/8, 1/2 = 4/8 Five rational numbers: 9/40, 10/40, 11/40, 12/40, 13/40

Question 13. Write four more rational numbers in the pattern: -1/5, -2/10, -3/15, -4/20, ......

Answer:

The pattern shows: -1/5 = -2/10 = -3/15 = -4/20 (all equivalent to -1/5)

The numerators and denominators are increasing by 1 and 5 respectively.

Next four: -5/25, -6/30, -7/35, -8/40

Question 14. Give four rational numbers equivalent to: (i) -2/7 (ii) 5/-3 (iii) 4/9

Answer:

(i) -2/7: -4/14, -6/21, -8/28, -10/35

(ii) 5/-3 = -5/3: -10/6, -15/9, -20/12, -25/15

(iii) 4/9: 8/18, 12/27, 16/36, 20/45

Question 15. Using appropriate properties find: 2/5 × (-3/7) – 1/6 × 3/2 + 1/14 × 2/5

Answer:

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

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

= 2/5 × (-6/14 + 1/14) – 1/4

= 2/5 × (-5/14) – 1/4

= -10/70 – 1/4

= -1/7 – 1/4

= -4/28 – 7/28 = -11/28

FAQs – Chapter 1 Rational Numbers

Q1. What is a rational number? A rational number is any number that can be written in the form p/q where p and q are integers and q is not equal to zero. Examples include 1/2, -3/4, 0, 5, and -7. All integers and fractions are rational numbers.

Q2. What are the properties of rational numbers? Rational numbers follow closure, commutativity, associativity, and distributivity under addition and multiplication. Zero is the additive identity and one is the multiplicative identity. Every rational number has an additive inverse and every non-zero rational number has a multiplicative inverse.

Q3. What is the difference between additive inverse and multiplicative inverse? The additive inverse of a rational number a/b is -a/b such that their sum is 0. The multiplicative inverse (reciprocal) of a/b is b/a such that their product is 1. Zero has no multiplicative inverse but its additive inverse is 0 itself.

Q4. How do you find rational numbers between two given rational numbers? To find rational numbers between two rational numbers, convert them to the same denominator and list the numbers between them. Alternatively, use the formula: the average of two rational numbers always lies between them. You can always find infinitely many rational numbers between any two rational numbers.

Q5. Why is Chapter 1 Rational Numbers important for Class 8? Chapter 1 establishes the complete number system that students use throughout Class 8 and beyond. The properties of rational numbers — commutativity, associativity, distributivity — are used in every algebraic topic. Understanding rational numbers thoroughly makes topics like linear equations, algebraic expressions, and comparing quantities much easier to handle.