Ncert Maths Class 7 Chapter 9 Solutions
NCERT Solutions for Class 7 Maths Chapter 9 Rational Numbers download the pdf provided below. Our specialist tutors formulate these exercises to help you with your exam preparation to attain good marks in Maths. Students who wish to score good marks in Maths are advised to practise NCERT Solutions for Class 7 Maths.
Chapter 9 – Rational Numbers consists of 2 exercises, and here we provide the NCERT Solutions to all the questions present in these exercises. Given below are some of the concepts present in this chapter.
- Need For Rational Numbers
- What Are Rational Numbers
- Positive And Negative Rational Numbers
- Rationals Numbers On a Number Line
- Rational Numbers in Standard Form
- Comparison of Rational Numbers
- Rational Numbers Between Two Rational Numbers
- Operations On Rational Numbers
- Addition of Rational Numbers
- Subtraction of Rational Numbers
- Multiplication of Rational Numbers
- Division of Rational Numbers
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Exercise 9.1 Solutions
Exercise 9.2 Solutions
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Exercise 9.1 Page: 182
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 the multiples of 3 and 5.
= (-3 × 1)/ (5 × 1), (-3 × 2)/ (5 × 2), (-3 × 3)/ (5 × 3), (-3 × 4)/ (5 × 4)
Then, 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 the multiples of 1 and 4.
= (-1 × 1)/ (4 × 1), (-1 × 2)/ (4 × 2), (-1 × 3)/ (1 × 3)
Then, 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 the multiples of 1 and 6.
= (-1 × 1)/ (6 × 1), (1 × 2)/ (-6 × 2), (1 × 3)/ (-6 × 3), (1 × 4)/ (-6 × 4)
Then, 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 the multiples of 2 and 3.
= (-2 × 1)/ (3 × 1), (2 × 1)/ (-3 × 1), (2 × 2)/ (-3 × 2), (2 × 3)/ (-3 × 3)
Then, 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 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 number line as,
(iii) -7/4
Solution:-
Now 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 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 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 represent the same rational number?
(i) (-7/21) and (3/9)
Solution:-
We have to check if 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 if 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 if 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 if 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 if 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 if 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 if 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 the 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 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 … [∵ 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 the 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:-
Since 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 the 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 the 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 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:-
The given fraction is like friction,
So, -¼ < ¼
Hence ¼ is greater,
(v)
,
Solution:-
First we have to convert mixed fraction into 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 form of like fraction,
Hence,
(-3/5)< (-2/5) < (-1/5)
(ii) -1/3, -2/9, -4/3
Solution:-
To convert the given rational numbers into like 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 like 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)
Exercise 9.2 Page: 190
1. Find the sum:
(i) (5/4) + (-11/4)
Solution:-
We have:
= (5/4) – (11/4)
= [(5 – 11)/4] … [∵ denominator is same in both the rational numbers]
= (-6/4)
= -3/2 … [∵ Divide both numerator and denominator by 3]
(ii) (5/3) + (3/5)
Solution:-
Take the LCM of the denominators of the given rational numbers.
LCM of 3 and 5 is 15
Express each of the given rational numbers with the above LCM as the common denominator.
Now,
(5/3)= [(5×5)/ (3×5)] = (25/15)
(3/5)= [(3×3)/ (5×3)] = (9/15)
Then,
= (25/15) + (9/15) … [∵ denominator is same in both the rational numbers]
= (25 + 9)/15
= 34/15
(iii) (-9/10) + (22/15)
Solution:-
Take the LCM of the denominators of the given rational numbers.
LCM of 10 and 15 is 30
Express each of the given rational numbers with the above LCM as the common denominator.
Now,
(-9/10)= [(-9×3)/ (10×3)] = (-27/30)
(22/15)= [(22×2)/ (15×2)] = (44/30)
Then,
= (-27/30) + (44/30) … [∵ denominator is same in both the rational numbers]
= (-27 + 44)/30
= (17/30)
(iv) (-3/-11) + (5/9)
Solution:-
We have,
= 3/11 + 5/9
Take the LCM of the denominators of the given rational numbers.
LCM of 11 and 9 is 99
Express each of the given rational numbers with the above LCM as the common denominator.
Now,
(3/11)= [(3×9)/ (11×9)] = (27/99)
(5/9)= [(5×11)/ (9×11)] = (55/99)
Then,
= (27/99) + (55/99) … [∵ denominator is same in both the rational numbers]
= (27 + 55)/99
= (82/99)
(v) (-8/19) + (-2/57)
Solution:-
We have
= -8/19 – 2/57
Take the LCM of the denominators of the given rational numbers.
LCM of 19 and 57 is 57
Express each of the given rational numbers with the above LCM as the common denominator.
Now,
(-8/19)= [(-8×3)/ (19×3)] = (-24/57)
(-2/57)= [(-2×1)/ (57×1)] = (-2/57)
Then,
= (-24/57) – (2/57) … [∵ denominator is same in both the rational numbers]
= (-24 – 2)/57
= (-26/57)
(vi) -2/3 + 0
Solution:-
We know that any number or fraction is added to zero the answer will be the same number or fraction.
Hence,
= -2/3 + 0
= -2/3
(vii) 
+ 
Solution:-
First we have to convert mixed fraction into improper fraction.
=
= -7/3
=
= 23/5
We have, -7/3 + 23/5
Take the LCM of the denominators of the given rational numbers.
LCM of 3 and 5 is 15
Express each of the given rational numbers with the above LCM as the common denominator.
Now,
(-7/3)= [(-7×5)/ (3×5)] = (-35/15)
(23/5)= [(23×3)/ (15×3)] = (69/15)
Then,
= (-35/15) + (69/15) … [∵ denominator is same in both the rational numbers]
= (-35 + 69)/15
= (34/15)
2. Find
(i) 7/24 – 17/36
Solution:-
Take the LCM of the denominators of the given rational numbers.
LCM of 24 and 36 is 72
Express each of the given rational numbers with the above LCM as the common denominator.
Now,
(7/24)= [(7×3)/ (24×3)] = (21/72)
(17/36)= [(17×2)/ (36×2)] = (34/72)
Then,
= (21/72) – (34/72) … [∵ denominator is same in both the rational numbers]
= (21 – 34)/72
= (-13/72)
(ii) 5/63 – (-6/21)
Solution:-
We can also write -6/21 = -2/7
= 5/63 – (-2/7)
We have,
= 5/63 + 2/7
Take the LCM of the denominators of the given rational numbers.
LCM of 63 and 7 is 63
Express each of the given rational numbers with the above LCM as the common denominator.
Now,
(5/63)= [(5×1)/ (63×1)] = (5/63)
(2/7)= [(2×9)/ (7×9)] = (18/63)
Then,
= (5/63) + (18/63) … [∵ denominator is same in both the rational numbers]
= (5 + 18)/63
= 23/63
(iii) -6/13 – (-7/15)
Solution:-
We have,
= -6/13 + 7/15
LCM of 13 and 15 is 195
Express each of the given rational numbers with the above LCM as the common denominator.
Now,
(-6/13)= [(-6×15)/ (13×15)] = (-90/195)
(7/15)= [(7×13)/ (15×13)] = (91/195)
Then,
= (-90/195) + (91/195) … [∵ denominator is same in both the rational numbers]
= (-90 + 91)/195
= (1/195)
(iv) -3/8 – 7/11
Solution:-
Take the LCM of the denominators of the given rational numbers.
LCM of 8 and 11 is 88
Express each of the given rational numbers with the above LCM as the common denominator.
Now,
(-3/8)= [(-3×11)/ (8×11)] = (-33/88)
(7/11)= [(7×8)/ (11×8)] = (56/88)
Then,
= (-33/88) – (56/88) … [∵ denominator is same in both the rational numbers]
= (-33 – 56)/88
= (-89/88)
(v) –
– 6
Solution:-
First we have to convert the mixed fraction into improper fraction,
–
= -19/9
We have, -19/9 – 6
Take the LCM of the denominators of the given rational numbers.
LCM of 9 and 1 is 9
Express each of the given rational numbers with the above LCM as the common denominator.
Now,
(-19/9)= [(-19×1)/ (9×1)] = (-19/9)
(6/1)= [(6×9)/ (1×9)] = (54/9)
Then,
= (-19/9) – (54/9) … [∵ denominator is same in both the rational numbers]
= (-19 – 54)/9
= (-73/9)
3. Find the product:
(i) (9/2) × (-7/4)
Solution:-
The product of two rational numbers = (product of their numerator)/ (product of their denominator)
The above question can be written as (9/2) × (-7/4)
We have,
= (9×-7)/ (2×4)
= -63/8
(ii) (3/10) × (-9)
Solution:-
The product of two rational numbers = (product of their numerator)/ (product of their denominator)
The above question can be written as (3/10) × (-9/1)
We have,
= (3×-9)/ (10×1)
= -27/10
(iii) (-6/5) × (9/11)
Solution:-
The product of two rational numbers = (product of their numerator)/ (product of their denominator)
We have,
= (-6×9)/ (5×11)
= -54/55
(iv) (3/7) × (-2/5)
Solution:-
The product of two rational numbers = (product of their numerator)/ (product of their denominator)
We have,
= (3×-2)/ (7×5)
= -6/35
(v) (3/11) × (2/5)
Solution:-
The product of two rational numbers = (product of their numerator)/ (product of their denominator)
We have,
= (3×2)/ (11×5)
= 6/55
(vi) (3/-5) × (-5/3)
Solution:-
The product of two rational numbers = (product of their numerator)/ (product of their denominator)
We have,
= (3×-5)/ (-5×3)
On simplifying,
= (1×-1)/ (-1×1)
= -1/-1
= 1
4. Find the value of:
(i) (-4) ÷ (2/3)
Solution:-
We have,
= (-4/1) × (3/2) … [∵ reciprocal of (2/3) is (3/2)]
The product of two rational numbers = (product of their numerator)/ (product of their denominator)
= (-4×3) / (1×2)
= (-2×3) / (1×1)
= -6
(ii) (-3/5) ÷ 2
Solution:-
We have,
= (-3/5) × (1/2) … [∵ reciprocal of (2/1) is (1/2)]
The product of two rational numbers = (product of their numerator)/ (product of their denominator)
= (-3×1) / (5×2)
= -3/10
(iii) (-4/5) ÷ (-3)
Solution:-
We have,
= (-4/5) × (1/-3) … [∵ reciprocal of (-3) is (1/-3)]
The product of two rational numbers = (product of their numerator)/ (product of their denominator)
= (-4× (1)) / (5× (-3))
= -4/-15
= 4/15
(iv) (-1/8) ÷ 3/4
Solution:-
We have,
= (-1/8) × (4/3) … [∵ reciprocal of (3/4) is (4/3)]
The product of two rational numbers = (product of their numerator)/ (product of their denominator)
= (-1×4) / (8×3)
= (-1×1) / (2×3)
= -1/6
(v) (-2/13) ÷ 1/7
Solution:-
We have,
= (-2/13) × (7/1) … [∵ reciprocal of (1/7) is (7/1)]
The product of two rational numbers = (product of their numerator)/ (product of their denominator)
= (-2×7) / (13×1)
= -14/13
(vi) (-7/12) ÷ (-2/13)
Solution:-
We have,
= (-7/12) × (13/-2) … [∵ reciprocal of (-2/13) is (13/-2)]
The product of two rational numbers = (product of their numerator)/ (product of their denominator)
= (-7× 13) / (12× (-2))
= -91/-24
= 91/24
(vii) (3/13) ÷ (-4/65)
Solution:-
We have,
= (3/13) × (65/-4) … [∵ reciprocal of (-4/65) is (65/-4)]
The product of two rational numbers = (product of their numerator)/ (product of their denominator)
= (3×65) / (13× (-4))
= 195/-52
= -15/4
Frequently Asked Questions on NCERT Solutions for Class 7 Maths Chapter 9
How can we score full marks in Class test of NCERT Solutions for Class 7 Maths Chapter 9?
By using the NCERT Solutions for Class 7 Maths Chapter 9 provided in BYJU'S website makes you reach full marks in class tests as well as in board exams. These solutions are highly imperative for easy and quick revision during the class tests and exams. Meanwhile, this serves as the best study material for learners.
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What are the main concepts covered in NCERT Solutions for Class 7 Maths Chapter 9?
The main topics of NCERT Solutions for Class 7 Maths Chapter 9 are
1. Need For Rational Numbers
2. What Are Rational Numbers
3. Positive And Negative Rational Numbers
4. Rationals Numbers On a Number Line
5. Rational Numbers in Standard Form
6. Comparison of Rational Numbers
7. Rational Numbers Between Two Rational Numbers
8. Operations On Rational Numbers
9. Addition of Rational Numbers
10. Subtraction of Rational Numbers
11. Multiplication of Rational Numbers
12. Division of Rational Numbers.
Ncert Maths Class 7 Chapter 9 Solutions
Source: https://byjus.com/ncert-solutions-class-7-maths/chapter-9-rational-numbers/