Class 7 Maths Chapter 1 HOTS Questions - Rational Numbers

Q1: Which is greater in each of these following:
(i) 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.

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

L.C.M of 6 and 3 is 6

- 5/6 = (−5 × 1)/(6 × 1) = −5/6

- 4/3 = (−4 × 2)/(3 × 2) = −8/6

Since,−5/6 > −8/6

So,−5/6 > −4/3

(iii) -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.

(iv) -¼, ¼
Ans: The negative number is always smaller than the positive number
So, -¼ < ¼
Hence ¼ is greater.

Q2: Find the sum:
(i) (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]

(ii) (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

(iii) (-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)

(iv) (-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)

(v) (-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)

(vi) -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

Q3: Find the given product:
(i) (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

(ii) (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

(iii) (-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

(iv) (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

(v) (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

(vi) (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

Q4: 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)

Q5: Find
(i) 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)

(ii) 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

(iii) -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)

(iv) -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)

Q6: Find the value of:
(i) (-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

(ii) (-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

(iii) (-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

(iv) (-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

(iv) (-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

(vi) (-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

(vii) (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