NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers

NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers provide step-by-step answers to the questions and exercises in the chapter. Students can use these solutions to understand rational numbers, their properties, additive and multiplicative inverses, number line representation, and operations on rational numbers.

This page covers Class 8 Maths Chapter 1 questions and answers, including Exercise 1.1 and Exercise 1.2, with simple explanations for revision and practice.

Class 8 Maths Chapter 1 Rational Numbers Questions and Answers

The chapter covers rational numbers and their properties, including closure, commutativity, associativity, distributivity, additive identity, multiplicative identity, additive inverse, and multiplicative inverse.

Class 8 Maths Chapter 1 Exercise 1.1 Solutions

1. Using appropriate properties, find:

(i) −23×35+52−35×16-\frac{2}{3}\times\frac{3}{5}+\frac{5}{2}-\frac{3}{5}\times\frac{1}{6}

Group the terms containing 35\frac{3}{5}:

−23×35−35×16+52-\frac{2}{3}\times\frac{3}{5}-\frac{3}{5}\times\frac{1}{6}+\frac{5}{2} =35(−23−16)+52=\frac{3}{5}\left(-\frac{2}{3}-\frac{1}{6}\right)+\frac{5}{2} =35(−56)+52=\frac{3}{5}\left(-\frac{5}{6}\right)+\frac{5}{2} =−12+52=2=-\frac{1}{2}+\frac{5}{2}=2

Answer: 2

(ii)

25×(−37)−16×32+114×25\frac{2}{5}\times\left(-\frac{3}{7}\right)-\frac{1}{6}\times\frac{3}{2}+\frac{1}{14}\times\frac{2}{5} =25(−37+114)−14=\frac{2}{5}\left(-\frac{3}{7}+\frac{1}{14}\right)-\frac{1}{4} =25(−514)−14=\frac{2}{5}\left(-\frac{5}{14}\right)-\frac{1}{4} =−17−14=−1128=-\frac{1}{7}-\frac{1}{4} =-\frac{11}{28}

Answer: −1128-\frac{11}{28}

2. Write the additive inverse of each of the following

(i) 28\frac{2}{8}

Additive inverse = −28-\frac{2}{8}

(ii) −59-\frac{5}{9}

Additive inverse = 59\frac{5}{9}

(iii) −6−5=65\frac{-6}{-5}=\frac{6}{5}

Additive inverse = −65-\frac{6}{5}

(iv) 2−9=−29\frac{2}{-9}=-\frac{2}{9}

Additive inverse = 29\frac{2}{9}

(v) 19−16=−1916\frac{19}{-16}=-\frac{19}{16}

Additive inverse = 1916\frac{19}{16}

3. Verify that −(−x)=x-(-x)=x for:

(i) x=1115x=\frac{11}{15}

−x=−1115-x=-\frac{11}{15}

Therefore,

−(−x)=−(−1115)=1115=x-(-x)=-\left(-\frac{11}{15}\right)=\frac{11}{15}=x

(ii) x=−1317x=-\frac{13}{17}

−x=1317-x=\frac{13}{17}

Therefore,

−(−x)=−1317=x-(-x)=-\frac{13}{17}=x

Hence, −(−x)=x-(-x)=x in both cases.

4. Find the multiplicative inverse of the following

(i) −13-13

Multiplicative inverse = −113-\frac{1}{13}

(ii) −1319-\frac{13}{19}

Multiplicative inverse = −1913-\frac{19}{13}

(iii) 15\frac{1}{5}

Multiplicative inverse = 55

(iv)

−58×−37=1556-\frac{5}{8}\times-\frac{3}{7}=\frac{15}{56}

Multiplicative inverse = 5615\frac{56}{15}

(v)

−1×−25=25-1\times-\frac{2}{5}=\frac{2}{5}

Multiplicative inverse = 52\frac{5}{2}

(vi) −1-1

Multiplicative inverse = −1-1

5. Name the property of multiplication used

(i)

−45×1=1×−45=−45-\frac45\times1=1\times-\frac45=-\frac45

Answer: Multiplicative identity.

(ii)

−1317×−27=−27×−1317-\frac{13}{17}\times-\frac27 = -\frac27\times-\frac{13}{17}

Answer: Commutative property of multiplication.

(iii)

−1929×29−19=1-\frac{19}{29}\times\frac{29}{-19}=1

Answer: Multiplicative inverse.

6. Multiply 613\frac{6}{13} by the reciprocal of −716-\frac{7}{16}

The reciprocal of −716-\frac{7}{16} is −167-\frac{16}{7}.

613×−167=−9691\frac{6}{13}\times-\frac{16}{7} = -\frac{96}{91}

Answer: −9691-\frac{96}{91}

7. Name the property used

13×(6×43)=(13×6)×43\frac13\times\left(6\times\frac43\right) = \left(\frac13\times6\right)\times\frac43

The grouping of the factors changes, but their order remains the same.

Answer: Associative property of multiplication.

8. Is 89\frac89 the multiplicative inverse of −98-\frac98? Why or why not?

For two numbers to be multiplicative inverses, their product must be 1.

89×−98=−1\frac89\times-\frac98=-1

Since −1≠1-1\neq1, 89\frac89 is not the multiplicative inverse of −98-\frac98.

9. If 0.3 is the multiplicative inverse of 103\frac{10}{3}, why or why not?

0.3=3100.3=\frac{3}{10}

Now,

310×103=1\frac{3}{10}\times\frac{10}{3}=1

Therefore, 0.3 is the multiplicative inverse of 103\frac{10}{3}.

10. Write:

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

Zero has no reciprocal because its reciprocal would require division by zero.

(ii) Rational numbers equal to their reciprocals:
1 and -1

(iii) The rational number equal to its negative:
0

11. Fill in the blanks

(i) Zero has no reciprocal.

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

(iii) The reciprocal of −5-5 is −15-\frac15.

(iv) The reciprocal of 1x\frac1x, where x≠0x\neq0, is xx.

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

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

Class 8 Maths Chapter 1 Exercise 1.2 Solutions

1. Represent these numbers on the number line

(i) 74\frac74

Since the denominator is 4, divide each unit interval into four equal parts. Starting from 0, move seven equal parts towards the positive side.

Therefore, 74\frac74 is located between 1 and 2.

(ii) −56-\frac56

Divide the interval between 0 and -1 into six equal parts. Move five parts towards the negative side.

Therefore, −56-\frac56 lies between 0 and -1.

2. Represent −211,−511,−911-\frac2{11},-\frac5{11},-\frac9{11} on a number line

Divide the interval between 0 and -1 into 11 equal parts.

The numbers are located:

• −211-\frac2{11}: second point from 0

• −511-\frac5{11}: fifth point from 0

• −911-\frac9{11}: ninth point from 0

All three lie on the negative side of the number line.

3. Write five rational numbers smaller than 2

We can write:

2=20102=\frac{20}{10}

Five rational numbers smaller than 2 are:

210,510,1010,1510,1910\frac2{10},\frac5{10},\frac{10}{10},\frac{15}{10},\frac{19}{10}

4. Find ten rational numbers between −25-\frac25 and 12\frac12

Convert both numbers to a common denominator:

−25=−2050-\frac25=-\frac{20}{50} 12=2550\frac12=\frac{25}{50}

Ten rational numbers between them include:

−1850,−1550,−550,−250,450,550,850,1250,1550,2050-\frac{18}{50},-\frac{15}{50},-\frac5{50},-\frac2{50}, \frac4{50},\frac5{50},\frac8{50},\frac{12}{50},\frac{15}{50},\frac{20}{50}

5. Find five rational numbers between:

(i) 23\frac23 and 45\frac45

Using denominator 60:

23=4060,45=4860\frac23=\frac{40}{60},\qquad \frac45=\frac{48}{60}

Five numbers are:

4160,4260,4360,4460,4560\frac{41}{60},\frac{42}{60},\frac{43}{60},\frac{44}{60},\frac{45}{60}

(ii) −32-\frac32 and 53\frac53

Using denominator 6:

−32=−96,53=106-\frac32=-\frac96,\qquad \frac53=\frac{10}{6}

Five numbers are:

−16,26,36,46,56-\frac16,\frac26,\frac36,\frac46,\frac56

(iii) 14\frac14 and 12\frac12

Using denominator 24:

14=624,12=1224\frac14=\frac6{24},\qquad \frac12=\frac{12}{24}

Five numbers are:

724,824,924,1024,1124\frac7{24},\frac8{24},\frac9{24},\frac{10}{24},\frac{11}{24}

6. Write five rational numbers greater than -2

We can write:

−2=−2010-2=-\frac{20}{10}

Five rational numbers greater than -2 are:

−1010,−510,−110,510,710-\frac{10}{10},-\frac5{10},-\frac1{10},\frac5{10},\frac7{10}

7. Find ten rational numbers between 35\frac35 and 34\frac34

Using denominator 80:

35=4880\frac35=\frac{48}{80} 34=6080\frac34=\frac{60}{80}

Ten rational numbers between them are:

4980,5080,5180,5280,5380,5480,5580,5680,5780,5880\frac{49}{80},\frac{50}{80},\frac{51}{80},\frac{52}{80}, \frac{53}{80},\frac{54}{80},\frac{55}{80}, \frac{56}{80},\frac{57}{80},\frac{58}{80}

Rational Numbers Class 8: Important Concepts

What Is a Rational Number?

A rational number is a number that can be written in the form:

pq\frac pq

where pp and qq are integers and q≠0q\neq0.

Examples include:

23,−57,4,0\frac23,\quad-\frac57,\quad4,\quad0

Properties of Rational Numbers

The important properties covered in Class 8 Maths Chapter 1 include:

• Closure property: The sum, difference, or product of two rational numbers is rational. The quotient is rational when the divisor is non-zero.

• Commutative property: a+b=b+aa+b=b+a and a×b=b×aa\times b=b\times a.

• Associative property: (a+b)+c=a+(b+c)(a+b)+c=a+(b+c) and (a×b)×c=a×(b×c)(a\times b)\times c=a\times(b\times c).

• Distributive property: a(b+c)=ab+aca(b+c)=ab+ac.

• Additive identity: 0.

• Multiplicative identity: 1.

Additive and Multiplicative Inverses

The additive inverse of a number is the number that gives 0 when added to it.

For example:

23+(−23)=0\frac23+\left(-\frac23\right)=0

The multiplicative inverse, or reciprocal, is the number that gives 1 when multiplied by the original number.

For example:

23×32=1\frac23\times\frac32=1

Zero does not have a multiplicative inverse.

Rational Numbers on a Number Line

Positive rational numbers are represented to the right of 0, while negative rational numbers are represented to the left.

To represent 34\frac34, divide the distance between 0 and 1 into three equal parts and mark the third point.

Finding Rational Numbers Between Two Numbers

To find rational numbers between two given rational numbers, first convert them to equivalent fractions with a suitable common denominator. The fractions lying between them give the required rational numbers.

Class 8 Rational Numbers Questions for Practice

Students can revise the chapter using questions such as:

1. What is a rational number?

2. Find the additive inverse of −78-\frac78.

3. Find the multiplicative inverse of 59\frac59.

4. Represent −34-\frac34 on a number line.

5. Write five rational numbers between 12\frac12 and 34\frac34.

6. Which rational number does not have a reciprocal?

7. State the commutative and associative properties of rational numbers.

8. Find the rational numbers between two given fractions.

Quick Revision of Class 8 Maths Chapter 1

• A rational number is written as pq\frac pq, where q≠0q\neq0.

• The additive inverse of aa is −a-a.

• The reciprocal of a non-zero rational number is its multiplicative inverse.

• Zero has no reciprocal.

• 0 is the additive identity.

• 1 is the multiplicative identity.

• Rational numbers can be represented on a number line.

• Rational numbers can be compared by converting them into suitable equivalent fractions.