NCERT Class 7 Maths Chapter 8 ‘Rational Numbers’ Exercise 8.1

Introduction:

Welcome to Online Tutorial Classes! We are excited to present a valuable resource for NCERT students studying Class 7 Mathematics. In this post, we will delve into the NCERT class 7 Maths chapter 8 ‘Rational Numbers’ Exercise 8.1 from the NCERT curriculum and provide you with a comprehensive notes and worksheets to help you master this important topic.

NCERT Class 7 Maths Chapter 8, ‘Rational Numbers’, Exercise 8.1 introduces students to the concept of rational numbers, their properties, and operations. It covers topics like representation on a number line, comparison, finding rational numbers between two numbers.

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NCERT class 7 Maths Chapter 8 ‘Rational Numbers’ Exercise 8.1 Overview

CBSE Class 7 Maths Chapter 8, titled ‘Rational Numbers’, Exercise 8.1 is a comprehensive introduction to the concept of rational numbers. The chapter begins with an introduction to rational numbers, defined as numbers that can be expressed in the form of p/q, where p and q are integers and q ≠ 0. It then delves into the properties of rational numbers, including their representation on a number line, and the rules for adding, subtracting, multiplying, and dividing them.

Exercise 8.1 of the chapter CBSE Class 7 Maths Chapter 8, titled ‘Rational Numbers’, also explains the concept of equivalent rational numbers, which are rational numbers that are equal to each other. It introduces the idea of rational numbers in standard form, where the denominator is a positive integer and the numerator and denominator have no common factor other than 1. The chapter also covers the topic of finding rational numbers between two given rational numbers. The chapter concludes with exercises that allow students to apply their understanding of these concepts.

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NCERT class 7 Maths Chapter 8 ‘Rational Numbers’ Solutions

Question 1

List five rational numbers between:
(i) –1 and 0 (ii) –2 and –1 (iii) -4/5 and -2/3 (iv) -1/2 and 2/3.

Solution:

(i) -1 and 0

Let us write –1 rational number with denominator 6

-1 = -6/6

The five rational numbers will be:

-5/6< -4/6< -3/6<-2/6< -1/6

(ii) –2 and –1

Let us write –2 and –1 as rational numbers with denominator 10.

-2 = -20/10 and -1 =-10/10

The five rational numbers between -2 and -1 will be

-19/10 < -18/10 < -17/10 < -16/10 < -15/10

or -19/10 < -9/5 < -17/10 < -8/5 < -3/2

(iii) -4/5 and -2/3

LCM of denominators 5 and 3 is 15

Let us write -4/5 and -2/3 with denominator 60 (multiple of 15)

-4/5 = -48/60 and -2/3 = -40/60

The five rational numbers between them will be

-47/60 < -46/60 < -45/60 < -44/60 < -43/60

Or -47/60 < -23/30 < -3/5 < -11/15 < -43/60

(iv) -1/2 and 2/3.

LCM of 2 and 3 is 6

Lets write -1/2 and 2/3 with deniminator 6

-1/2 = -3/6 and 2/3 = 4/6

Therefore the fractions between them are:

-2/6 < -1/6 < 1/6 < 2/6 3/6

Or -1/3 <-1/6 < 1/6 < 1/3 1/2

Question 2

Write four more rational numbers in each of the following patterns:

Solution:

(i) in this pattern the numerator and denominator are multiples of -3 and 5

Hence four more ratioanl numbers will be

(ii) in this pattern the numerator and denominator are multiples of -1 and 4

Hence four more ratioanl numbers will be

(iii) in this pattern the numerator and denominator are multiples of -1 and 6

Hence four more ratioanl numbers will be

(iv) In this pattern the numerator and denominator are multiples of 2 and -3

Hence four more ratioanl numbers will be

Question 3

Give four rational numbers equivalent to:

Solution:

(i) -2/7

The four rational numbers equivalent to -2/7 are,

= (-2 × 2)/ (7 × 2), (-2 × 3)/ (7 × 3), (-2 × 4)/ (7 × 4), (-2 × 5)/ (7× 5)

= -4/14, -6/21, -8/28, -10/35

(ii) 5/-3

The four rational numbers equivalent to 5/-3 are,

= (5 × 2)/ (-3 × 2), (5 × 3)/ (-3 × 3), (5 × 4)/ (-3 × 4), (5 × 5)/ (-3× 5)

= 10/-6, 15/-9, 20/-12, 25/-15

(iii) 4/9

The four rational numbers equivalent to 5/-3 are,

= (4 × 2)/ (9 × 2), (4 × 3)/ (9 × 3), (4 × 4)/ (9 × 4), (4 × 5)/ (9× 5)

= 8/18, 12/27, 16/36, 20/45

Question 4

Draw the number line and represent the following rational numbers on it:

 (i)3/4 (ii) -5/8 (iii) -7/4 (iv) 7/8 

Solution:

(i)3/4 lies between 0 and 1

It can be represented on the number line between 0 and 1 as under:

(ii)

 -5/8 lies between 0 and -1

It can be represented on the number line between 0 and 1 as under:

(iii)

-7/4 lies between -1 and -2

It can be represented on the number line between 0 and 1 as under:

(iv) 

7/8 lies between 0 and 1

It can be represented on the number line between 0 and 1 as under:

Question 5

The points P, Q, R, S, T, U, A and B on the number line are such that, TR = RS = SU and AP = PQ = QB. Name the rational numbers represented by P, Q, R and S.

Solution:

P = 7/3

Q = 8/3

R = -4/3

S = -5/3

Question 6

Which of the following pairs represent the same rational number?

Solution:

To find if the given pairs represent the same rational number we will simplify them to see if they are equivalent rational numbers:

(i) In the given pair -7/21 is a negative rational number and 3/9 is a positive rational number and hence cannot be same.

(ii) -16/20

= -4/5 (dividing numerator and denominator by 4)

and

20/-25

= -20/25 (dividing numerator and denominator by 5)

= -4/5

Hence -16/20 and 20/-25 represent the same rational number.

(iii) -2/-3 and 2/3

-2/-3

= 2/3

Hence -2/-3 and 2/3 represent the same rational number

(iv) (-3/5) and (-12/20)

-3/5

= (-3 x 4)/(5 x 4) (multiplying numerator and denominator by 4)

= -12/20

∴ -3/5= -12/20

Hence -3/5 and -12/20 represent the same rational number.

(v) (8/-5) and (-24/15)

8/-5

= (8 x 3)/-5 x 3)

= -24/15

∴ 8/-5 = -24/15

Hence 8/-5 and -24/15 represent the same rational number.

(vi) (1/3) and (-1/9)

In the given pair 1/3 is a positive rational number and -1/9 is a negative rational number and hence cannot be same.

(vii) (-5/-9) and (5/-9)

-5/-9

= 5/9

5/-9

= -5/9

∵ 5/9 ≠ -5/9

Hence -5/-9 and 5/-9 do not represent the same rational number.

Question 7

Rewrite the following rational numbers in the simplest form:

Solution:

(i) -8/6

-8/6

= -4/3 (By dividing both numerator and denominator by HCF of 8 and 6 which is 2)

∴ -4/3 is the simplest form of -8/6

(ii) 25/45

By dividing both numerator and denominator by HCF of 25 and 45 which is 5

= 5/9

∴ 5/9 is the simplest form of 25/45

(iii) -44/72

By dividing both numerator and denominator by HCF of 44 and 72 which is 4

= -11/18

∴ -11/18 is the simplest form of -44/72.

(iv) -8/10

By dividing both numerator and denominator by HCF of 8 and 10 which is 2 we get

= -4/5

∴ -4/5 is the simplest form of -8/10.

Question 8

Fill in the boxes with the correct symbol out of >, <, and =.

Solution:

(i) -5/7 [ ] 2/3

-5/7 is a -ve rational number and 2/3 is a positive rational number

Hence -5/7 [<] 2/3

(ii) -4/5 [ ] -5/7

The LCM of the denominators 5 and 7 is 35

∴ (-4/5) = [(-4 × 7)/ (5 × 7)] = (-28/35)

And (-5/7) = [(-5 × 5)/ (7 × 5)] = (-25/35)

Now,

-28 < -25

So, (-28/35) < (- 25/35)

Hence, -4/5 [<] -5/7

(iii) -7/8 [ ] 14/-16

Solution:-

14/-16 can be simplified further,

Then,

7/-8 … [Dividing both numerator and denominator by 2]

So, (-7/8) = (-7/8)

Hence, -7/8 [=] 14/-16

(iv) -8/5 [ ] -7/4

The LCM of the denominators 5 and 4 is 20

∴ (-8/5) = [(-8 × 4)/ (5 × 4)] = (-32/20)

And (-7/4) = [(-7 × 5)/ (4 × 5)] = (-35/20)

Now,

-32 > – 35

So, (-32/20) > (- 35/20)

Hence, -8/5 [>] -7/4

(v) 1/-3 [ ] -1/4

Here the numerator is same and the denominators are different

-3 > -4

∴ 1/-3 [<] -1/4

(vi) 5/-11 [ ] -5/11

Solution:-

Since, (-5/11) = (-5/11)

Hence, 5/-11 [=] -5/11

(vii) 0 [ ] -7/6

Solution:-

Since every negative rational number is less than 0,

We get:

= 0 [>] -7/6

Question 9

Which is greater in each of the following:

Solution:

(i) 2/3, 5/2

In rational number 2/3 denominator is greater than the numerator.

∴ its value is < 1

In 5/2 numerator is greater than the denominator

∴ its value is greater than 1

∴ 2/3 < 5/2

Hence, 5/2 is greater.

(ii) -5/6, -4/3

Solution:-

The LCM of the denominators 6 and 3 is 6

∴ (-4/3) = [(-4 × 2)/ (3 × 2)] = (-12/6)

Now,

-5 > -12

So, (-5/6) > (- 12/6)

Hence, – 5/6 is greater.

(iii) -3/4, 2/-3

Solution:-

The LCM of the denominators 4 and 3 is 12

∴ (-3/4) = [(-3 × 3)/ (4 × 3)] = (-9/12) and

(-2/3) = [(-2 × 4)/ (3 × 4)] = (-8/12)

Now,

-9 < -8

So, (-9/12) < (- 8/12)

∴ -3/4 < 2/-3

Hence, 2/-3 is greater.

(iv) -¼, ¼

Solution:-

-¼ < ¼

Hence ¼ is greater,

(v) , and

Both have -3 as the whole number

∴ we will compare -2/7 and -4/5

The LCM of the denominators 7 and 5 is 35

∴ (-2/7) = [(-2 × 5)/ (7 × 5)] = (-10/35)

And (-4/5) = [(-4 × 7)/ (5 × 7)] = (-28/35)

-28 < -10

∴ -28/35 < -10/35

Or -10/35 > -28/35

Or -2/7 > -4/5

∴ >

Hence,  is greater

Question 10

Write the following rational numbers in ascending order:

Solution:

(i) -3/5, -2/5, -1/5

The given rational numbers are in the form of like fractions,

Hence,

(-3/5)< (-2/5) < (-1/5)

(ii) -1/3, -2/9, -4/3

Solution:-

To convert the given rational numbers into like fractions, we have to find the LCM,

The LCM of 3, 9, and 3 is 9

Now,

(-1/3)= [(-1 × 3)/ (3 × 9)] = (-3/9)

(-2/9)= [(-2 × 1)/ (9 × 1)] = (-2/9)

(-4/3)= [(-4 × 3)/ (3 × 3)] = (-12/9)

We have:

(-12/9) < (-3/9) < (-2/9)

Hence,

(-4/3) < (-1/3) < (-2/9)

(iii) -3/7, -3/2, -3/4

Solution:-

To convert the given rational numbers into like fractions, we have to find LCM,

The LCM of 7, 2, and 4 is 28

Now,

(-3/7)= [(-3 × 4)/ (7 × 4)] = (-12/28)

(-3/2)= [(-3 × 14)/ (2 × 14)] = (-42/28)

(-3/4)= [(-3 × 7)/ (4 × 7)] = (-21/28)

We have:

(-42/28) < (-21/28) < (-12/28)

Hence,

(-3/2) < (-3/4) < (-3/7)

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