*According to the CBSE Syllabus 2023-24, this chapter has been renumbered as Chapter 8.
NCERT Solutions for Class 7 Maths Exercise 9.1 Chapter 9 Rational Numbers in simple PDF are given here. Students are going to learn about key topics in this exercise of NCERT Maths Solutions for Class 7 Chapter 9. The topics are rational numbers, positive and negative rational numbers, rational numbers on the number line, rational numbers in standard form, comparison of rational numbers and rational numbers between two rational numbers. These NCERT Solutions for Class 7 Maths Chapter 9 Rational Numbers are prepared by our expert tutors to help the students to score good marks in Maths.
NCERT Solutions for Class 7 Maths Chapter 9 Rational Numbers – Exercise 9.1
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1. List five rational numbers between
(i) -1 and 0
Solution:
The five rational numbers between -1 and 0 are
-1< (-2/3) < (-3/4) < (-4/5) < (-5/6) < (-6/7) < 0
(ii) -2 and -1
Solution:
The five rational numbers between -2 and -1 are
-2 < (-8/7) < (-9/8) < (-10/9) < (-11/10) < (-12/11) < -1
(iii) -4/5 and -2/3
Solution:
The five rational numbers between -4/5 and -2/3 are
-4/5 < (-13/12) < (-14/13) < (-15/14) < (-16/15) < (-17/16) < -2/3
(iv) -1/2 and 2/3
Solution:
The five rational numbers between -1/2 and 2/3 are
-1/2 < (-1/6) < (0) < (1/3) < (1/2) < (20/36) < 2/3
2. Write four more rational numbers in each of the following patterns.
(i) -3/5, -6/10, -9/15, -12/20, …..
Solution:
In the above question, we can observe that the numerator and denominator are multiples of 3 and 5.
= (-3 × 1)/ (5 × 1), (-3 × 2)/ (5 × 2), (-3 × 3)/ (5 × 3), (-3 × 4)/ (5 × 4)
Then, the next four rational numbers in this pattern are
= (-3 × 5)/ (5 × 5), (-3 × 6)/ (5 × 6), (-3 × 7)/ (5 × 7), (-3 × 8)/ (5 × 8)
= -15/25, -18/30, -21/35, -24/40 ….
(ii) -1/4, -2/8, -3/12, …..
Solution:
In the above question, we can observe that the numerator and denominator are multiples of 1 and 4.
= (-1 × 1)/ (4 × 1), (-1 × 2)/ (4 × 2), (-1 × 3)/ (1 × 3)
Then, the next four rational numbers in this pattern are
= (-1 × 4)/ (4 × 4), (-1 × 5)/ (4 × 5), (-1 × 6)/ (4 × 6), (-1 × 7)/ (4 × 7)
= -4/16, -5/20, -6/24, -7/28 ….
(iii) -1/6, 2/-12, 3/-18, 4/-24 …..
Solution:
In the above question, we can observe that the numerator and denominator are multiples of 1 and 6.
= (-1 × 1)/ (6 × 1), (1 × 2)/ (-6 × 2), (1 × 3)/ (-6 × 3), (1 × 4)/ (-6 × 4)
Then, the next four rational numbers in this pattern are
= (1 × 5)/ (-6 × 5), (1 × 6)/ (-6 × 6), (1 × 7)/ (-6 × 7), (1 × 8)/ (-6 × 8)
= 5/-30, 6/-36, 7/-42, 8/-48 ….
(iv) -2/3, 2/-3, 4/-6, 6/-9 …..
Solution:
In the above question, we can observe that the numerator and denominator are multiples of 2 and 3.
= (-2 × 1)/ (3 × 1), (2 × 1)/ (-3 × 1), (2 × 2)/ (-3 × 2), (2 × 3)/ (-3 × 3)
Then, the next four rational numbers in this pattern are
= (2 × 4)/ (-3 × 4), (2 × 5)/ (-3 × 5), (2 × 6)/ (-3 × 6), (2 × 7)/ (-3 × 7)
= 8/-12, 10/-15, 12/-18, 14/-21 ….
3. Give four rational numbers equivalent to
(i) -2/7
Solution:
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
Solution:
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
Solution:
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
4. Draw the number line and represent the following rational numbers on it.
(i) ¾
Solution:
We know that 3/4 is greater than 0 and less than 1.
∴ it lies between 0 and 1. It can be represented on the number line as
(ii) -5/8
Solution:
We know that -5/8 is less than 0 and greater than -1.
∴ it lies between 0 and -1. It can be represented on the number line as
(iii) -7/4
Solution:
Now, the above question can be written as
= (-7/4) =
We know that (-7/4) is less than -1 and greater than -2.
∴ it lies between -1 and -2. It can be represented on the number line as
(iv) 7/8
Solution:
We know that 7/8 is greater than 0 and less than 1.
∴ it lies between 0 and 1. It can be represented on the number line as
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:
By observing the figure, we can say that
The distance between A and B = 1 unit
And it is divided into 3 equal parts = AP = PQ = QB = 1/3
P = 2 + (1/3)
= (6 + 1)/ 3
= 7/3
Q = 2 + (2/3)
= (6 + 2)/ 3
= 8/3
Similarly,
The distance between U and T = 1 unit
And it is divided into 3 equal parts = TR = RS = SU = 1/3
R = – 1 – (1/3)
= (- 3 – 1)/ 3
= – 4/3
S = – 1 – (2/3)
= – 3 – 2)/ 3
= – 5/3
6. Which of the following pairs represents the same rational number?
(i) (-7/21) and (3/9)
Solution:
We have to check the given pair represents the same rational number.
Then,
-7/21 = 3/9
-1/3 = 1/3
∵ -1/3 ≠ 1/3
∴ -7/21 ≠ 3/9
So, the given pair does not represent the same rational number.
(ii) (-16/20) and (20/-25)
Solution:
We have to check the given pair represents the same rational number.
Then,
-16/20 = 20/-25
-4/5 = 4/-5
∵ -4/5 = -4/5
∴ -16/20 = 20/-25
So, the given pair represents the same rational number.
(iii) (-2/-3) and (2/3)
Solution:
We have to check the given pair represents the same rational number.
Then,
-2/-3 = 2/3
2/3= 2/3
∵ 2/3 = 2/3
∴ -2/-3 = 2/3
So, the given pair represents the same rational number.
(iv) (-3/5) and (-12/20)
Solution:
We have to check the given pair represents the same rational number.
Then,
-3/5 = – 12/20
-3/5 = -3/5
∵ -3/5 = -3/5
∴ -3/5= -12/20
So, the given pair represents the same rational number.
(v) (8/-5) and (-24/15)
Solution:
We have to check the given pair represents the same rational number.
Then,
8/-5 = -24/15
8/-5 = -8/5
∵ -8/5 = -8/5
∴ 8/-5 = -24/15
So, the given pair represents the same rational number.
(vi) (1/3) and (-1/9)
Solution:
We have to check the given pair represents the same rational number.
Then,
1/3 = -1/9
∵ 1/3 ≠ -1/9
∴ 1/3 ≠ -1/9
So, the given pair does not represent the same rational number.
(vii) (-5/-9) and (5/-9)
Solution:
We have to check the given pair represents the same rational number.
Then,
-5/-9 = 5/-9
∵ 5/9 ≠ -5/9
∴ -5/-9 ≠ 5/-9
So, the given pair does not represent the same rational number.
7. Rewrite the following rational numbers in the simplest form.
(i) -8/6
Solution:
The given rational numbers can be simplified further.
Then,
= -4/3 … [∵ Divide both numerator and denominator by 2]
(ii) 25/45
Solution:
The given rational numbers can be simplified further.
Then,
= 5/9 … [∵ Divide both numerator and denominator by 5]
(iii) -44/72
Solution:
The given rational numbers can be simplified further.
Then,
= -11/18 … [∵ Divide both numerator and denominator by 4]
(iv) -8/10
Solution:
The given rational numbers can be simplified further.
Then,
= -4/5 … [∵ Divide both numerator and denominator by 2]
8. Fill in the boxes with the correct symbol out of >, <, and =.
(i) -5/7 [ ] 2/3
Solution:
The LCM of denominators 7 and 3 is 21.
∴ (-5/7) = [(-5 × 3)/ (7 × 3)] = (-15/21)
And (2/3) = [(2 × 7)/ (3 × 7)] = (14/21)
Now,
-15 < 14
So, (-15/21) < (14/21)
Hence, -5/7 [<] 2/3
(ii) -4/5 [ ] -5/7
Solution:
The LCM of 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 … [∵ Divide both numerator and denominator by 2]
So, (-7/8) = (-7/8)
Hence, -7/8 [=] 14/-16
(iv) -8/5 [ ] -7/4
Solution:
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
Solution:
The LCM of denominators 3 and 4 is 12.
∴ (-1/3) = [(-1 × 4)/ (3 × 4)] = (-4/12)
And (-1/4) = [(-1 × 3)/ (4 × 3)] = (-3/12)
Now,
-4 < – 3
So, (-4/12) < (- 3/12)
Hence, 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:
Every negative rational number is less than 0.
We have
= 0 [>] -7/6
9. Which is greater in each of the following?
(i) 2/3, 5/2
Solution:
The LCM of denominators 3 and 2 is 6.
(2/3) = [(2 × 2)/ (3 × 2)] = (4/6)
And (5/2) = [(5 × 3)/ (2 × 3)] = (15/6)
Now,
4 < 15
So, (4/6) < (15/6)
∴ 2/3 < 5/2
Hence, 5/2 is greater.
(ii) -5/6, -4/3
Solution:
The LCM of denominators 6 and 3 is 6.
∴ (-5/6) = [(-5 × 1)/ (6 × 1)] = (-5/6)
And (-4/3) = [(-4 × 2)/ (3 × 2)] = (-12/6)
Now,
-5 > -12
So, (-5/6) > (- 12/6)
∴ -5/6 > -12/6
Hence, – 5/6 is greater.
(iii) -3/4, 2/-3
Solution:
The LCM of 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:
The given fraction is like friction.
So, -¼ < ¼
Hence, ¼ is greater.
(v)
,
Solution:
First, we have to convert the mixed fraction into an improper fraction,
= -23/7
= -19/5
Then,
The LCM of the denominators 7 and 5 is 35.
∴ (-23/7) = [(-23 × 5)/ (7 × 5)] = (-115/35)
And (-19/5) = [(-19 × 7)/ (5 × 7)] = (-133/35)
Now,
-115 > -133
So, (-115/35) > (- 133/35)
∴
>
Hence,
is greater.
10. Write the following rational numbers in ascending order.
(i) -3/5, -2/5, -1/5
Solution:
The given rational numbers are in the form of a fraction,
Hence,
(-3/5)< (-2/5) < (-1/5)
(ii) -1/3, -2/9, -4/3
Solution:
To convert the given rational numbers into a fraction, we have to find LCM.
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)
Clearly,
(-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 a fraction, we have to find LCM.
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)
Clearly,
(-42/28) < (-21/28) < (-12/28)
Hence,
(-3/2) < (-3/4) < (-3/7)
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