KSEEB Solutions for Class 8 Maths Chapter 1 Rational Numbers InText Questions

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KSEEB Solutions for Class 8 Maths Chapter 1 Rational Numbers InText Questions

Try These (Page 6)

Question 1.
Complete the following table:

NumbersCommutative for
AdditionSubtractionMultiplicationDivision
Rational NumbersYes__________________
Integers______No____________
Whole Numbers____________Yes______
Natural Numbers__________________No

Solution:

a
NumbersCommutative for
AdditionSubtractionMultiplicationDivision
Rational NumbersYesNoYesNo
IntegersYesNoYesNo
Whole NumbersYesNoYesNo
Natural NumbersYesNoYesNo

Try These (Page 7)

(i) Whole Numbers

Question 1.
Recall the associativity of the four operations for whole numbers through this table:

OperationNumbersRemarks
Addition_________Addition is associative
Subtraction_________Subtraction is not associative
MultiplicationIs 7 × (2 × 5) = (7 × 2) × 5?
Is 4 × (6 × 0) = (4 × 6) × 0?
For any three whole numbers a, b and c,
a × (b × c) = (a × b) × c
Multiplication is associative
Division_________The division is not associative

Fill in this table and verify the remarks given in the last column.
Check for yourself the associativity of different operations for natural numbers.
Solution:

OperationNumbersRemarks
AdditionLet us take an example to show that addition is associative associative
Is (7 + 2) + 6 = 7 + (2 + 6)
9 + 6 = 15
7 + (2 + 6) = 7 + 8 = 15
15 = 15,
∴ (7 + 2) + 6 = 7 + (2 + 6)
Is (4 + 0) + 13 = 4 + (0 + 13)
Let us take
(4 + 0) + 13 = 4 + 13 = 17
4 + (0 + 13) = 4 + 13 = 17
∴ (4 + 0) + 13 = 4 + (0 + 13)
So, for any three whole numbers say a, b and c
(a + b) + c = a + (b + c)
Addition is associative
SubtractionIs 19 – (7 – 2) = (19 – 7) – 2
19 – (7 – 2) = 19 – 5 = 14
(19 – 7) – 2 = 12 – 2 = 10
14 ≠ 10
∴ 19 – (7 – 2) = (19 – 7) – 2
Is 23 – (0 – 5) = (23 – 0) – 5
23 – (0 – 5) = 23 – (- 5) = 23 + 5 = 28
(23 – 0) – 5 = 23 – 5 = 18
28 ≠ 18
∴ 23 – (0 – 5) ≠ (23 – 0) – 5
So, for any three whole numbers a, b and c
a – (b – c) ≠ (a – b) – c
Subtraction is not associative
MultiplicationIs 7 × (2 × 5) = (7 × 2) × 5
7 × (2 × 5) = 7 × 10 = 70
(7 × 2) × 5 = 14 × 5 = 70
70 = 70
∴ 7 × (2 × 5) = (7 × 2) × 5
Is 4 × (6 × 0) = (4 × 6) × 0
4 × (6 × 0) = 4 × 0 = 0
(4 × 6) × 0 = 24 × 0 = 0
0 = 0
∴ 4 × (6 × 0) = (4 × 6) × 0
Hence, for any three whole numbers a, b and c
a × (b × c) = (a × b) × c
Multiplication is associative
DivisionIs 18 ÷ (6 ÷ 3) = 18 ÷ (6 ÷ 3)
18 ÷ (6 ÷ 3) = \(\left(\frac{18}{6}\right) \times \frac{1}{3}=1\)
18 ÷ (6 ÷ 3) = 18 ÷ \(\frac{6}{3}\) = \(\frac{3}{6}\) = 3 × 3 = 9 = 3 × 3 = 9
1 ≠ 9
Hence, (18 ÷ 6) ÷ 3 ≠ 18 ÷ (6 ÷ 3)
So, for any three whole numbers a, b and c
(a ÷ b ) ÷ c ≠ a ÷ (b ÷ c)
Division is not associative

(ii) Integers

Question 1.
The associativity of the four operations for integers can be seen from this table:

OperationNumbersRemarks
AdditionIs (-2) + [3 + (-4)] = [(-2) + 3] + (-4)?
Is (-6) + [(-4) + (-5)] = [(-6) + (-4)] + (-5)?
For any three integers a, b and c,
a + (b + c) = (a + b) + c
Addition is associative
SubtractionIs 5 – (7 – 3) = (5 – 7) – 3?Subtraction is not associative
MultiplicationIs 9 × [(-5) × (-3)] = [9 × (-5)] × (-3)?
Is (-13) × [(-17) × (-5)] = [(-13) × (-17)] × (-5)?
For any three integers a, b and c,
a × (b × c) = (a × b) × c
Multiplication is associative
DivisionIs [(-15) ÷ 5] ÷ (-3) = (-15) ÷ [5 ÷ (-3)]?Division is not associative

Solution:

OperationNumbersRemarks
AdditionIs (-2) + [3 + (-4)] = [(-2) + 3] + (-4)
(-2) + [3 + (-4)] = (-2) + [3 – 4]
= -2 + (-1)
= -2 – 1
= -3
[(-2) + 3] + (-4) = [-2 + 3] + (-4)
= (1) + (-4)
= 1 – 4
= -3
Is (-6) + [(-4) + (-5)] = [(-6) + (-4)] + (-5)
(-6) + [(-4) + (-5)] = (-6) + [-4 – 5]
= -6 + (-9)
= -6 – 9
= -15
[(-6) + (-4)] + (-5) = [-6 – 4] + (-5)
= [-10] + (-5)
= -10 – 5
= -15
-15 = -15
∴ (-6) + [(-4) + (-5)] = [(-6) + (-4)] + (-5)
So, for any three integers a, b and c,
a + (b + c) = (a + b) + c
Addition is associative
SubtractionIs 5 – (7 – 3) = (5 – 7) – 3
5 – (7 – 3) = 5 – 4 = 1
(5 – 7) – 3 = – 2 – 3 = -5
1 ≠ -5
5 – (7 – 3) ≠ (5 – 7) – 3
So, for any three integers a, b and c,
a – (b – c) ≠ (a – b) – c
Subtraction is not associative
MultiplicationIs 9 × [(-5) × (-3)] = [9 × (-5)] × (-3)
9 × [(-5) × (-3)] = 9 × [15] = 135
[9 × (-5)] × (-3) = -45 × -3 = 135
135 = 135
9 × [(-5) × (-3)] = [(9) × (-5)] × (-3)
Is (-13) × [(17) × (-5)] = [(-13) × 17] × (-5)
(-13) × [(17) × (-5)] = (-13) × (-85) = 1105
[(-13) × 17] × (-5) = [-221] × (-5)
= -221 × -5
= 1105
1105 = 1105
∴ (-13) × [17 × (-5)] = [(-13) × 17] × (-5)
So, for any three integers a, b and c,
a × (b × c) = (a × b) × c
Multiplication is associative
DivisionIs [(-15) ÷ 5] ÷ (-3) = (-15) ÷ [5 ÷ (-3)]
[(-15) ÷ 5] ÷ (-3) = \(\left[\frac{-15}{5}\right]\) ÷ (-3)
= -3 ÷ (-3)
= 1
[(-15) ÷ 5] ÷ (-3) = (-15) ÷ \(\left[\frac{5}{-3}\right]\)
= -15 × \(\frac{-3}{5}\)
= 9
\(\frac{5}{3}\) ≠ 9
[(-15) ÷ 5] ÷ (-3) ≠ (-15) ÷ [5 ÷ (-3)]
So, for any three integers a, b and c,
(a ÷ b) ÷ c ≠ a ÷ (b ÷ c)
Division is not associative

Try These (Page 10)

Question 1.

NumbersAssociative for
AdditionSubtractionMultiplicationDivision
Rational numbers__________________No
Integers____________Yes______
Whole numbersYes__________________
Natural numbers______No____________

Solution:

NumbersAssociative for
AdditionSubtractionMultiplicationDivision
Rational numbersYesNoYesNo
IntegersYesNoYesNo
Whole numbersYesNoYesNo
Natural numbersYesNoYesNo

Try These (Page 13)

Question 1.
Find using distributivity.
(i) \(\left\{\frac{7}{5} \times\left(\frac{-3}{12}\right)\right\}+\left\{\frac{7}{5} \times \frac{5}{12}\right\}\)
(ii) \(\left\{\frac{9}{16} \times \frac{4}{12}\right\}+\left\{\frac{9}{16} \times \frac{-3}{9}\right\}\)
Solution:
(i) \(\left\{\frac{7}{5} \times\left(\frac{-3}{12}\right)\right\}+\left\{\frac{7}{5} \times \frac{5}{12}\right\}\)
KSEEB Solutions for Class 8 Maths Chapter 1 Rational Numbers InText Questions Page 13 Q1

Try These (Page 17)

Question 1.
Write the rational number for each point labeled with a letter.
Solution:
(i) A = \(\frac{1}{5}\)
B = \(\frac{4}{5}\)
C = \(\frac{5}{5}\)
D = \(\frac{8}{5}\)
E = \(\frac{9}{5}\)
KSEEB Solutions for Class 8 Maths Chapter 1 Rational Numbers InText Questions Page 17 Q1

(ii) -J = \(\frac{-11}{6}\)
I = \(\frac{-8}{6}\)
H = \(\frac{-7}{6}\)
G = \(\frac{-5}{6}\)
F = \(\frac{-2}{5}\)
KSEEB Solutions for Class 8 Maths Chapter 1 Rational Numbers InText Questions Page 17 Q1.1

a