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:

Numbers Commutative for
Addition Subtraction Multiplication Division
Rational Numbers Yes ______ ______ ______
Integers ______ No ______ ______
Whole Numbers ______ ______ Yes ______
Natural Numbers ______ ______ ______ No

Solution:

a
Numbers Commutative for
Addition Subtraction Multiplication Division
Rational Numbers Yes No Yes No
Integers Yes No Yes No
Whole Numbers Yes No Yes No
Natural Numbers Yes No Yes No

Try These (Page 7)

(i) Whole Numbers

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

Operation Numbers Remarks
Addition _________ Addition is associative
Subtraction _________ Subtraction is not associative
Multiplication Is 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:

Operation Numbers Remarks
Addition Let 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
Subtraction Is 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
Multiplication Is 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
Division Is 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:

Operation Numbers Remarks
Addition Is (-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
Subtraction Is 5 – (7 – 3) = (5 – 7) – 3? Subtraction is not associative
Multiplication Is 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
Division Is [(-15) ÷ 5] ÷ (-3) = (-15) ÷ [5 ÷ (-3)]? Division is not associative

Solution:

Operation Numbers Remarks
Addition Is (-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
Subtraction Is 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
Multiplication Is 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
Division Is [(-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.

Numbers Associative for
Addition Subtraction Multiplication Division
Rational numbers ______ ______ ______ No
Integers ______ ______ Yes ______
Whole numbers Yes ______ ______ ______
Natural numbers ______ No ______ ______

Solution:

Numbers Associative for
Addition Subtraction Multiplication Division
Rational numbers Yes No Yes No
Integers Yes No Yes No
Whole numbers Yes No Yes No
Natural numbers Yes No Yes No

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