Important Questions: Rational Numbers | Mathematics (Maths) Class 7 PDF Download

Q1: List any five rational numbers between:
(a) -1 and 0
Ans: The five rational numbers present between the numbers -1 and 0 are as follows,
-1< (-2/3) < (-3/4) < (-4/5) < (-5/6) < (-6/7) < 0

(b) -2 and -1
Ans: The five rational numbers present between the numbers -2 and -1 are,
-2 < (-8/7) < (less than) (-9/8) < (-10/9) < (-11/10) < (-12/11) < -1

(c) -4/5 and -2/3
Ans: The five rational numbers present between the numbers -4/5 and -2/3 are,
-4/5 < (less than) (-13/12) < (-14/13) < (-15/14) < (-16/15) < (-17/16) < -2/3

(d) -1/2 and 2/3
Ans: The five rational numbers present between -1/2 and 2/3 are,
-1/2 < (less than) (-1/6) < (0) < (1/3) < (1/2) < (20/36) < 2/3

Q2: Write any four more rational numbers in each of these following patterns:
(a) -3/5, -6/10, -9/15, -12/20, …..
Ans: In the above given question, we can easily observe that the numerator and the denominator are the multiples of numbers three and five.
= (-3 × 1)/ (5 × 1) and (-3 × 2)/ (5 × 2), (-3 × 3)/ (5 × 3), (-3 × 4)/ (5 × 4)
Thus, the next four rational numbers present in this same pattern are as follows,
= (-3 × 5)/ (5 × 5) and (-3 × 6)/ (5 × 6), (-3 × 7)/ (5 × 7), (-3 × 8)/ (5 × 8)
= -15/25, -18/30, -21/35, -24/40 ….

(b) -1/4, -2/8, -3/12, …..
Ans: In the above given question, we can easily observe that the numerator and the denominator are the multiples of the numbers one and four.
= (-1 × 1)/ (4 × 1) and (-1 × 2)/ (4 × 2), (-1 × 3)/ (1 × 3)
Then we get, the next four rational numbers present in this pattern will be,
= (-1 × 4)/ (4 × 4) and (-1 × 5)/ (4 × 5), (-1 × 6)/ (4 × 6), (-1 × 7)/ (4 × 7)
= -4/16, -5/20, -6/24, -7/28 and so on.

(c) -1/6, 2/-12, 3/-18, 4/-24 and so on.
Ans: In the above given question, we can easily observe that the numerator and the denominator are the multiples of numbers one and six.
= (-1 × 1)/ (6 × 1) and (1 × 2)/ (-6 × 2), (1 × 3)/ (-6 × 3) and (1 × 4)/ (-6 × 4)
Then, the next four rational numbers present in this pattern are as follows,
= (1 × 5)/ (-6 × 5) and (1 × 6)/ (-6 × 6), (1 × 7)/ (-6 × 7) and (1 × 8)/ (-6 × 8)
= 5/-30, 6/-36, 7/-42, 8/-48 ….

(d) -2/3, 2/-3, 4/-6, 6/-9 …..
Ans: In the above given question, we can easily observe that the numerator and the denominator are the multiples of numbers two and three.
= (-2 × 1)/ (3 × 1) and (2 × 1)/ (-3 × 1), (2 × 2)/ (-3 × 2) and (2 × 3)/ (-3 × 3)
Then, the next four rational numbers present in this pattern are as follows,
= (2 × 4)/ (-3 × 4) and (2 × 5)/ (-3 × 5), (2 × 6)/ (-3 × 6) and (2 × 7)/ (-3 × 7)
= 8/-12, 10/-15, 12/-18 and 14/-21 ….

Q3: Give any four rational numbers equivalent to:
(a) -2/7
Ans: The four rational numbers present whic are equivalent to the fraction -2/7 are,
= (-2 × 2)/ (7 × 2) and (-2 × 3)/ (7 × 3), (-2 × 4)/ (7 × 4) and (-2 × 5)/ (7× 5)
= -4/14, -6/21, -8/28 and -10/35

(b) 5/-3
Ans: The four rational numbers present which are equivalent to the fraction 5/-3 are,
= (5 × 2)/ (-3 × 2), (5 × 3)/ (-3 × 3), (5 × 4)/ (-3 × 4) and (5 × 5)/ (-3× 5)
= 10/-6, 15/-9, 20/-12 and 25/-15

(c) 4/9
Ans: The four rational numbers present which are equivalent to the fraction 5/-3 are,
= (4 × 2)/ (9 × 2), (4 × 3)/ (9 × 3), (4 × 4)/ (9 × 4) and (4 × 5)/ (9× 5)
= 8/18, 12/27, 16/36 and 20/45

Q4: Which of these following pairs represents the same rational number?
(i) (-7/21) and (3/9)
Ans:
We have to check whether the given pair represents the same rational number.
Then,
-7/21 = 3/9
-1/3 = 1/3
∵ -1/3 ≠ 1/3
Hence -7/21 ≠ 3/9
So, the given pair do not represent the same rational number.

(ii) (-16/20) and (20/-25)
Ans: We have to check whether the given pair represents the same rational number.
Then,
-16/20 = 20/-25
-4/5 = 4/-5
∵ -4/5 = -4/5
Hence -16/20 = 20/-25
So, the given pair represents same rational number.

(iii) (-2/-3) and (2/3)
Ans:
We have to check whether the given pair represents the same rational number.
Then,
-2/-3 = 2/3
2/3= 2/3
∵ 2/3 = 2/3
Hence, -2/-3 = 2/3
So, the given pair represents same rational number.

(iv) (-3/5) and (-12/20)
Ans:
We have to check whether the given pair represents the same rational number.
Then,
-3/5 = – 12/20
-3/5 = -3/5
∵ -3/5 = -3/5
Hence -3/5= -12/20
So, the given pair represents same rational number.

(v) (8/-5) and (-24/15)
Ans: We have to check whether the given pair represents the same rational number.
Then,
8/-5 = -24/15
8/-5 = -8/5
∵ -8/5 = -8/5
Hence 8/-5 = -24/15
So, the given pair represents same rational number.

(vi) (1/3) and (-1/9)
Ans: We have to check whether the given pair represents the same rational number.
Then,
1/3 = -1/9
∵ 1/3 ≠ -1/9
Hence, 1/3 ≠ -1/9
So, the given pair does not represent same rational number.

(vii) (-5/-9) and (5/-9)
Ans:
We have to check if these given pairs represent the same rational number.
Then,
-5/-9 equals to 5/-9
Therefore, 5/9 ≠ -5/9
Hence -5/-9 ≠ 5/-9
So, these given pairs do not represent the same rational number.

Q5: Rewrite the following rational numbers given below in the simplest form:
(i) -8/6
Ans: The given above rational numbers can be simplified further,
Then,
= -4/3 … [∵ Divide both the numerator and denominator by 2]

(ii) 25/45
Ans: The given above rational numbers can be simplified further,
Then,
= 5/9 … [∵ Divide both the numerator and denominator by 5]

(iii) -44/72
Ans: The given above rational numbers can be simplified further,
Then,
= -11/18 … [∵ Divide both the numerator and denominator by 4]

(iv) -8/10
Ans: The given above rational numbers can be simplified further,
Then,
= -4/5 … [∵ Divide both the numerator and denominator by 2]

Q6: Fill in the below boxes with the correct symbol of >, <, and =.
(a) -5/7 [ ] 2/3
Ans: The LCM of the denominators of numbers 7 and 3 is the number 21
Therefore, (-5/7) = [(-5 × 3)/ (7 × 3)] is = (-15/21)
And (2/3) = [(2 × 7)/ (3 × 7)] equals to (14/21)
Now,
-15 < 14
So, (-15/21) < (14/21)
 -5/7 [<] 2/3

(b) -4/5 [ ] -5/7
Ans: The LCM of the denominators of 5 and 7 is the number 35
Therefore (-4/5) = [(-4 × 7)/ (5 × 7)] is = (-28/35)
And (-5/7) = [(-5 × 5)/ (7 × 5)] equals to (-25/35)
Now,
-28 < -25
So, (-28/35) < (- 25/35)
 -4/5 [<] -5/7

(c) -7/8 [ ] 14/-16
Ans:
14/-16 can simplified further,
Then,
7/-8 … [∵ Divide both the numerator and denominator by 2]
So, (-7/8) = (-7/8)
Hence, -7/8 [=] 14/-16

(d) -8/5 [ ] -7/4
Ans:
The LCM of the denominators of 5 and 4 is the no 20
Therefore (-8/5) = [(-8 × 4)/ (5 × 4)] = (-32/20)
And (-7/4) = [(-7 × 5)/ (4 × 5)] is equal to (-35/20)
Now,
-32 > – 35
So, (-32/20) > (- 35/20)
 -8/5 [>] -7/4

(e) 1/-3 [ ] -1/4
Ans:
The LCM of the denominators of 3 and 4 is the no 12
Hence, (-1/3) = [(-1 × 4)/ (3 × 4)] is = (-4/12)
And (-1/4) = [(-1 × 3)/ (4 × 3)] is equal to (-3/12)
Now,
-4 < – 3
So, (-4/12) is less than (- 3/12)
Hence, 1/-3 [<] -1/4

(f) 5/-11 [ ] -5/11
Ans:
Since, (-5/11) = (-5/11)
Hence, 5/-11 [=] -5/11

(g) 0 [ ] -7/6
Ans: Since every negative rational number is said to be less than zero.
We have:
= 0 [>] -7/6

Q7: Which is greater in each of these following:
(a) 2/3, 5/2
Ans:
The LCM of the denominators of 3 and 2 is 6
(2/3) = [(2 × 2)/ (3 × 2)] equals to (4/6)
And (5/2) equals to [(5 × 3)/ (2 × 3)] = (15/6)
Now,
4 < 15
So, (4/6) < (15/6)
∴ 2/3 < 5/2
Hence, 5/2 is greater.

(b) -5/6, -4/3
Ans:
The LCM of the denominators of 6 and 3 is 6
∴ (-5/6) = [(-5 × 1)/ (6 × 1)] is = (-5/6)
And (-4/3) = [(-4 × 2)/ (3 × 2)] equals to (-12/6)
Now,
-5 > -12
So, (-5/6) > (- 12/6)
∴ -5/6 > -12/6
Hence, – 5/6 is greater.

(c) -3/4, 2/-3
Ans:
The LCM of the denominators of 4 and 3 is the number 12
∴ (-3/4) = [(-3 × 3) divided by (4 × 3)] is = (-9/12)
And (-2/3) = [(-2 × 4)/ (3 × 4)] equals to (-8/12)
Now,
-9 < -8
So, (-9/12) is less than (- 8/12)
Therefore -3/4 < 2/-3
Hence, 2/-3 is greater.

(d) -¼, ¼
Ans: The given fraction is like friction,
So, -¼ < ¼
Hence ¼ is greater,

Q8: Write the following rational numbers in an ascending order:
(i) -3/5, -2/5, -1/5
Ans:
The above given rational numbers are in the form of like fraction,
Hence,
(-3/5)< (-2/5) < (-1/5)

(ii) -1/3, -2/9, -4/3
Ans: To convert the above given rational numbers into the like fraction we have to first find LCM, LCM of the numbers 3, 9, and 3 is 9
Now,
(-1/3)is equal to [(-1 × 3)/ (3 × 9)] = (-3/9)
(-2/9) is equal to [(-2 × 1)/ (9 × 1)] = (-2/9)
(-4/3) is equal to [(-4 × 3)/ (3 × 3)] = (-12/9)
Clearly,
(-12/9) < (-3/9) < (-2/9)
So,
(-4/3) < (-1/3) < (-2/9)

(iii) -3/7, -3/2, -3/4
Ans: To convert the given rational numbers into the like fraction we have to first find LCM, LCM of 7, 2, and 4 is the number 28
Now,
(-3/7)= [(-3 × 4)/ (7 × 4)] is equal to (-12/28)
(-3/2)= [(-3 × 14)/ (2 × 14)] is equal to (-42/28)
(-3/4)= [(-3 × 7)/ (4 × 7)] is equal to (-21/28)
Clearly,
(-42/28) < (-21/28) < (-12/28)
Hence,
(-3/2) < (-3/4) < (-3/7)

Q9: Find the sum:
(a) (5/4) + (-11/4)
Ans: We have:
= (5/4) – (11/4)
= [(5 – 11)/4] … [∵ the denominator is same in both the rational numbers]
= (-6/4)
= -3/2 … [∵ Divide both the numerator and denominator by 3]

(b) (5/3) + (3/5)
Ans:Take the LCM of these denominators of the above given rational numbers.
LCM of 3 and 5 is 15
Express each of the above given rational numbers with the above found LCM as common denominator.
So,
(5/3)= [(5×5)/ (3×5)] equals to (25/15)
(3/5) equals to [(3×3)/ (5×3)] = (9/15)
Then,
= (25/15) + (9/15) … [∵ the denominator is same in both the rational numbers]
= (25 + 9)/15
= 34/15

(c) (-9/10) + (22/15)
Ans: Take the LCM of these denominators of the above given rational numbers.
LCM of 10 and 15 is 30
Express each of the above given rational numbers with the above LCM, by taking 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) … [∵ the denominator is same in both the rational numbers]
= (-27 + 44)/30
= (17/30)

(d) (-3/-11) + (5/9)
Ans:
We have,
= 3/11 + 5/9
Take the LCM of these denominators of the above given rational numbers.
Hence, the LCM of 11 and 9 is the number 99
Express each of these given rational numbers while taking the above LCM by taking it as the common denominator.
Now,
(3/11) equals to [(3×9)/ (11×9)] = (27/99)
(5/9) equals to [(5×11)/ (9×11)] = (55/99)
Then,
= (27/99) + (55/99) … [∵ the denominator is same in both the rational numbers]
= (27 + 55)/99
= (82/99)

(e) (-8/19) + (-2/57)
Ans:
We have
= -8/19 – 2/57
Take the LCM of the following denominators of the given rational numbers.
LCM of the numbers 19 and 57 is 57
Express each of the given rational numbers by taking 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) … [∵ the denominator is same in both the rational numbers]
= (-24 – 2)/57
= (-26/57)

(f) -2/3 + 0
Ans:
We know that when any number or fraction is added to the number zero the answer will be the same number or fraction.
Hence,
= -2/3 + 0
= -⅔

Q10: Find
(a) 7/24 – 17/36
Ans: Take the LCM of these denominators of the above given rational numbers.
LCM of the numbers 24 and 36 is 72
Express each of these given rational numbers with the above LCM, by taking it as the common denominator.
Now,
(7/24) is equal to [(7×3)/ (24×3)] = (21/72)
(17/36) equals to [(17×2)/ (36×2)] = (34/72)
Then,
= (21/72) – (34/72) … [∵ the denominator is same in both the rational numbers]
= (21 – 34)/72
= (-13/72)

(b) 5/63 – (-6/21)
Ans:
We can also write that -6/21 = -2/7
= 5/63 – (-2/7)
We have,
= 5/63 + 2/7
Now Take the LCM of the denominators of the above given rational numbers.
LCM of the numbers 63 and 7 is 63
Express each of these given rational numbers with the above LCM, by taking it as the common denominator.
Now,
(5/63)= [(5×1)/ (63×1)] equals to (5/63)
(2/7)= [(2×9)/ (7×9)] equals to (18/63)
Then,
= (5/63) + (18/63) … [∵ the denominator is same in both the rational numbers]
= (5 + 18)/63
= 23/63

(c) -6/13 – (-7/15)
Ans:
We have,
= -6/13 + 7/15
LCM of the numbers 13 and 15 is 195
Express each of these given rational numbers with the above LCM, by taking it as the common denominator.
Now,
(-6/13)= [(-6×15)/ (13×15)] = (-90/195)
(7/15) equals to [(7×13)/ (15×13)] = (91/195)
Then,
= (-90/195) + (91/195) … [∵ the denominator is same in both the rational numbers]
= (-90 + 91)/195
= (1/195)

(d) -3/8 – 7/11
Ans:
Take the LCM as the denominators of the above given rational numbers.
LCM of the numbers 8 and 11 is 88
Express each of the above given rational numbers with the above LCM, taking them as the common denominator.
Now,
(-3/8)= [(-3×11)/ (8×11)] equals to (-33/88)
(7/11) equal to [(7×8)/ (11×8)] = (56/88)
Then,
= (-33/88) – (56/88) … [∵ the denominator is same in both the rational numbers]
= (-33 – 56)/88
= (-89/88)

Q11: Find the given product:
(a) (9/2) × (-7/4)
Ans:
The product of the two rational numbers is equal to = (product of their numerator) divided (product of their denominator)
The above question can also be written as (9/2) × (-7/4)
We have,
= (9×-7)/ (2×4)
= -63/8

(b) (3/10) × (-9)
Ans:
The product of two rational numbers is equal to (product of their numerator) divided by (product of their denominator)
The above question can also be written as (3/10) × (-9/1)
We have,
= (3×-9)/ (10×1)
= -27/10

(c) (-6/5) × (9/11)
Ans:
The product of two rational numbers is equal to = (product of their numerator)/ (product of their denominator)
We have,
= (-6×9)/ (5×11)
= -54/55

(d) (3/7) × (-2/5)
Ans:
The product of two rational numbers is equal to = (product of their numerator)/ (product of their denominator)
We have,
= (3×-2)/ (7×5)
= -6/35

(e) (3/11) × (2/5)
Ans: The product of two rational numbers is equal to = (product of their numerator)/ (product of their denominator)
We have,
= (3×2)/ (11×5)
= 6/55

(f) (3/-5) × (-5/3)
Ans: The product of two rational numbers is equal to = (product of their numerator)/ (product of their denominator)
We have,
= (3×-5)/ (-5×3)
On simplifying,
= (1×-1)/ (-1×1)
= -1/-1
= 1

Q12: Find the value of:
(a) (-4) ÷ (2/3)
Ans:
We have,
= (-4/1) × (3/2) … [∵ the reciprocal of the fraction (2/3) is (3/2)]
The product of two rational numbers is equal to = (product of their numerator) divided by (product of their denominator)
= (-4×3) / (1×2)
= (-2×3) / (1×1)
= -6

(b) (-3/5) ÷ 2
Ans:
We have,
= (-3/5) × (1/2) … [the reciprocal of (2/1) is (1/2)]
The product of the two rational numbers = (product of their numerator) divided by (product of their denominator)
= (-3×1) / (5×2)
= -3/10

(c) (-4/5) ÷ (-3)
Ans: We have,
= (-4/5) × (1/-3) … [∵ the reciprocal of (-3) is (1/-3)]
The product of two rational numbers is equal to = (product of their numerator)/ (product of their denominator)
= (-4× (1)) / (5× (-3))
= -4/-15
= 4/15

(d) (-1/8) ÷ 3/4
Ans:
We have,
= (-1/8) × (4/3) … [the reciprocal of the fraction (3/4) is (4/3)]
The product of these two rational numbers is equal to = (product of their numerator)/ (product of their denominator)
= (-1×4) / (8×3)
= (-1×1) / (2×3)
= -1/6

(e) (-2/13) ÷ 1/7
Answer :-
We have,
= (-2/13) × (7/1) … [∵ the reciprocal of (1/7) is (7/1)]
The product of the following two rational numbers is equal to = (product of their numerator)/ (product of their denominator)
= (-2×7) / (13×1)
= -14/13

(f) (-7/12) ÷ (-2/13)
Ans:
We have,
= (-7/12) × (13/-2) … [the reciprocal of (-2/13) is (13/-2)]
The product of two rational numbers is equal to = (product of their numerator)/ (product of their denominator)
= (-7× 13) / (12× (-2))
= -91/-24
= 91/24

(g) (3/13) ÷ (-4/65)
Ans: We have,
= (3/13) × (65/-4) … [∵ the reciprocal of (-4/65) is (65/-4)]
The product of two rational numbers is equal to = (product of their numerator)/ (product of their denominator)
= (3×65) / (13× (-4))
= 195/-52
= -15/4