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Exercise 1.6 in Chapter 1 of RD Sharma's Class 8 mathematics textbook marks a significant progression in the study of rational numbers, building upon the concepts and skills developed in previous exercises. This section delves into more advanced applications and properties of rational numbers, challenging students to apply their knowledge in increasingly complex scenarios. The focus of Exercise 1.6 is on deepening students' understanding of the relationships between rational numbers, exploring their properties in greater depth, and introducing more sophisticated problem-solving techniques. Students are expected to work with rational numbers in various forms, including fractions, decimals, and percentages, seamlessly converting between these representations as needed. In this section, we delve into Chapter 1 of the Class 8 RD Sharma textbook, which focuses on Rational Numbers. Exercise 1.6 is designed to reinforce students' understanding of operations involving rational numbers, helping them build a strong foundation in basic arithmetic and number theory.
This section provides detailed solutions for Exercise 1.6 from Chapter 1 of the Class 8 RD Sharma textbook. These solutions aim to guide students through various problems involving rational numbers, ensuring they grasp the essential concepts and techniques needed to solve them effectively.
(i) x = -1/3, y = 2/7
Solution:
x × y = (-1/3) × 2/7
= (-1 × 2)/(3 × 7)
= -2/21
y × x = 2/7 × (-1/3)
= (2 × -1)/(7 × 3)
= -2/21
Hence, x × y = y × x, property is verified
(ii) x = -3/5, y = -11/13
Solution:
x × y = (-3/5) × (-11/3)
= (-3 × -11)/(5 × 13)
= 33/65
y × x = (-11/13) × (-3/5)
= (-11 × -3)/(13 × 5)
= 33/65
Hence, x × y = y × x, property is verified
(iii) x = 2, y = 7/-8
Solution:
x × y = 2/1 × (-7/8)
= (2 × -7)/(1 × 8)
2 is the common factor
= -7/4
y × x = (-7/8) × (2/1)
= (-7 × 2)/(8 × 1)
2 is the common factor
= -7/4
Hence, x × y = y × x, property is verified
(iv) x = 0, y = -15/8
Solution:
x × y = 0 × (-15/8)
= 0
y × x = (-15/8) × 0
= 0
Hence, x × y = y × x, property is verified
(i) x = -7/3, y = 12/5, z = 4/9
Solution:
x × (y × z)
= -7/3 × (12/5 × 4/9)
= -7/3 × ((12 × 4)/(5 × 9))
3 is the common factor of 12 and 9
= -7/3 × ((4 × 4)/(5 × 3))
= -7/3 × (16/15)
= (-7 × 16)/(3 × 15)
= -112/45
(x × y) × z
= (-7/3 × 12/5) × 4/9
= ((-7 × 12)/(3 × 5)) × 4/9
= ((-7 × 4)/(5)) × 4/9
= -28/5 × 4/9
= -112/45
Hence, x × (y × z) = (x × y) × z Property is verified
(ii) x = 0, y = -3/5, z = -9/4
Solution:
x × (y × z)
= 0 × (-3/5 × -9/4)
= 0 × (27/20)
= 0(Any number multiplied with zero is zero)
(x × y) × z
= (0 × -3/5) × -9/4
= 0 × -9/4
= 0(Any number multiplied with zero os zero)
Hence, x × (y × z) = (x × y) × z Property is verified
(iii) x = 1/2, y = 5/-4, z = -7/5
Solution:
x × (y × z)
= 1/2 × (-5/4 × -7/5)
= 1/2 × ((-5 × -7)/(4 × 5))
Common factor 5
= 1/2 × (7/4)
= 7/8
(x × y) × z
= (1/2 × -5/4) × -7/5
= -5/8 × -7/5
Common factor 5
= 7/8
Hence, x × (y × z) = (x × y) × z Property is verified
(iv) x = 5/7, y = -12/13, z = -7/18
Solution:
x × (y × z)
= 5/7 × (-12/13 × -7/18)
= 5/7 × ((-12 × -7)/(13 × 18))
Common factor 6 of 12 and 18
= 5/7 × ((-2 × -7)/(13 × 3))
= 5/7 × (14/39)
= (5 × 14)/(7 × 39)
7 is the common factor of 7 and 14
= 5 × 2/39
= 10/39
(x × y) × z
= (5/7 × -12/13) × -7/18
= ((5 × -12)/(7 × 13)) × -7/18
= (5 × -12 × -7)/(7 × 13 × 18)
Common factor 7 and 6
= (5 × -2 × -1)/(1 × 13 × 3)
= 10/39
Hence, x × (y × z) = (x × y) × z Property is verified
(i) x = -3/7, y = 12/13, z = -5/6
Solution:
x × (y + z)
= -3/7 × (12/13 + -5/6)
LCM of 13 and 6 is 78
= -3/7 × ((12 × 6 - 5 × 13)/78)
= -3/7 × ((72 - 65)/78)
= -3/7 × 7/78
= -3 × 7/7 × 78
Common factor 7 and 3
= -1/26
x × y + x × z
= -3/7 × 12/13 + -3/7 × -5/6
= (-3 × 12)/(7 × 13) + (-3 × -5)/(6 × 7)
= -36/91 + 15/42
= (-36 × 6 + 15 × 13)/546
= 196 - 216/546
= -21/546
= -1/26
Hence, the property x × (y + z) = x × y + x × z is verified
(ii) x = -12/5, y = -15/4, z = 8/3
Solution:
x × (y + z)
= -12/5 × (-15/4 + 8/3)
LCM is 12
= -12/5 × ((-15 × 3 + 8 × 4)/12)
= -12/5 × ((-45 + 32)/12)
= -12/5 × (-13)/12
= (-12 × -13)/(5 × 12)
12 is the common factor
= 13/5
x × y + x × z
= -12/5 × -15/4 + -12/5 × 8/3
= (-12 × -15)/(5 × 4) + (-12 × 8)/(5 × 3)
Common factor 4 and 5, 3
= 9 + -32/5
LCM is 5
= (9 × 5 - 32)/5
= 45 - 32/5
= 13/5
Hence, the property x × (y + z) = x × y + x × z is verified
(iii) x = -8/3, y = 5/6, z = -13/12
Solution:
x × (y + z)
= -8/3 × (5/6 + -13/12)
LCM is 12
= -8/3 × (5 × 2 - 13)/(12)
= -8/3 × (10 - 13)/12
= -8/3 × (-3/12)
= (-8 × -3)/(3 × 12)
Common factor 4 and 3
= 2/3
x × y + x × z
= -8/3 × 5/6 + -8/3 × -13/12
= (-8 × 5)/(3 × 6) + (-8 × -13)/(3 × 12)
Common factor 2 and 4
= (-4 × 5)/(3 × 3) + (-2 × -13)/(3 × 3)
= -20/9 + 26/9
= (-20 + 26)/9
= 6/9
Common factor is 3
= 2/3
Hence, the property x × (y + z) = x × y + x × z is verified
iv) x = -3/4, y = -5/2, z = 7/6
Solution:
x × (y + z)
= -3/4 × (-5/2 + 7/6)
LCM is 6
= -3/4(-5 × 3 + 7)/6
= -3/4 × (-15 + 7)/6
= -3/4 × -8/6
= (-3 × -8)/(4 × 6)
Common factor 3 and 4
= 1
x × y × + x × z
= -3/4 × -5/2 + -3/4 × 7/6
= (-3 × -5)/(4 × 2) + (-3 × 7)/(4 × 6)
= 15/8 + -7/8
= (15 - 7)/8
= 8/8
= 1
Hence, the property x × (y + z) = x × y + x × z is verified
(i) 3/5 × ((35/24) + (10/1))
Solution:
= 3/5 × 35/24 + 3/5 × 10/1
= (3 × 35)/(5 × 24) + (3 × 10)/(5 × 1)
Common factor is 5 and 3
= (1 × 7)/(1 × 8) + (3 × 2)/(1)
= 7/8 + 6/1
LCM is 8
= (7 + 6 × 8)/8
= 7 + 48/8
= 55/8
(ii) -5/4 × ((8/5) + (16/5))
Solution:
= -5/4×8/5+-5/4×16/5
= (-5×8)/(4×5)+(-5×16)/(4×5)
Common factor is 4, 5
= -2/1+-4/1
= -6
(iii) 2/7 × ((7/16) — (21/4))
Solution:
= 2/7×7/16-2/7×21/4
= (2×7)/(7×16)-(2×21)/(7×4)
Common factor 2 and 7
= 1/8-3/2
LCM is 8
= (1-3×4)/8
= (1-12)/8
= -11/8
(iv) 3/4 × ((8/9) — 40)
Solution:
= 3/4×8/9-3/4×40
= (3×8)/(4×9)-(3×40)/(4×1)
Common factor 3 and 4
= 2/3-30/1
LCM is 3
= (2-90)/3
= -88/3
(i) 9
Solution:
Multiplicative inverse of 9/1 is 1/9
(ii) -7
Solution:
Multiplicative inverse of -7/1 is 1/-7 or -1/7
(iii) 12/5
Solution:
Multiplicative inverse of 12/5 is 5/12
(iv) -7/9
Solution:
Multiplicative inverse of -7/9 is 9/-7 or -9/7
(v) -3/-5
Solution:
Multiplicative inverse of -3/-5 is -5/-3 or 5/3
(vi) 2/3 × 9/4
Solution:
(2×9)/(3×4)
2 is common factor of 2 and 4, 3 is common factor of 3 and 9
=3/2
Multiplicative inverse is 2/3
(vii) -5/8 × 16/15
Solution:
(-5×16)/(8×15)
5 is the common factor of 5 and 15, 8 is the common factor of 8 and 16
=-2/3
Multiplicative inverse is 3/-2 or -3/2
(viii) -2 × -3/5
Solution:
=(-2×-3)/(1×5)
=6/5
Multiplicative inverse is 5/6
(ix) -1
Solution:
Multiplicative inverse is -1
(x) 0/3
Solution:
Multiplicative inverse is 3/0 which does not exist
(xi) 1
Solution:
Multiplicative inverse is 1
(i) -5/16 × 8/15 = 8/15 × -5/16
Solution:
a×b=b×a
This is commutative property
(ii) -17/5 ×9 = 9 × -17/5
Solution:
a×b=b×a
This is commutative property
(iii) 7/4 × (-8/3 + -13/12) = 7/4 × -8/3 + 7/4 × -13/12
Solution:
a×(b+c)=a×b+a×c
This is distributive property of multiplication over addition
(iv) -5/9 × (4/15 × -9/8) = (-5/9 × 4/15) × -9/8
Solution:
a×(b×c)=(a×b)×c
This is associative property of multiplication
(v) 13/-17 × 1 = 13/-17 = 1 × 13/-17
Solution:
a×1=a=1×a
This is multiplicative identity
(vi) -11/16 × 16/-11 = 1
Solution:
a×1/a=1
This is multiplicative inverse
(vii) 2/13 × 0 = 0 = 0 × 2/13
Solution:
a×0=0=0×a
Any number multiplied with 0 is 0
(viii) -3/2 × 5/4 + -3/2 × -7/6 = -3/2 × (5/4 + -7/6)
Solution:
a×b+a×c=a×(b+c)
This is distributive law of multiplication over addition
(i) The product of two positive rational numbers is always…
Solution:
The product of two positive rational numbers is always positive.
(ii) The product of a positive rational number and a negative rational number is always….
Solution:
The product of a positive rational number and a negative rational number is always negative
(iii) The product of two negative rational numbers is always…
Solution:
The product of two negative rational numbers is always positive
(iv) The reciprocal of a positive rational number is…
Solution:
The reciprocal of a positive rational numbers is positive
(v) The reciprocal of a negative rational number is…
Solution:
The reciprocal of a negative rational numbers is negative
(vi) Zero has …. Reciprocal.
Solution:
Zero has no reciprocal.
(vii) The product of a rational number and its reciprocal is…
Solution:
The product of a rational number and its reciprocal is 1
(viii) The numbers … and … are their own reciprocals.
Solution:
The numbers 1 and -1 are their own reciprocals.
(ix) If a is reciprocal of b, then the reciprocal of b is.
Solution:
If a is reciprocal of b, then the reciprocal of b is a
(x) The number 0 is … the reciprocal of any number.
Solution:
The number 0 is not the reciprocal of any number.
(xi) reciprocal of 1/a, a ≠ 0 is …
Solution:
Reciprocal of 1/a, a ≠ 0 is a
(xii) (17×12)-1 = 17-1 × …
Solution:
(17×12)-1 = 17-1 × 12-1
(i) -4 × 79 = 79 × …
Solution:
-4 × 79= 79 × -4
By using commutative property.
(ii) 5/11 × -3/8 = -3/8 × …
Solution:
5/11 × -3/8 = -3/8 × 5/11
By using commutative property.
(iii) 1/2 × (3/4 + -5/12) = 1/2 × … + … × -5/12
Solution:
1/2 × (3/4 + -5/12) = 1/2 × 3/4 + 1/2 × -5/12
By using distributive property.
(iv) -4/5 × (5/7 + -8/9) = (-4/5 × …) + -4/5 × -8/9
Solution:
-4/5 × (5/7 + -8/9) = (-4/5 × 5/7) + -4/5 × -8/9
By using distributive property.
Exercise 1.6 in Chapter 1 of RD Sharma's Class 8 mathematics textbook presents an advanced exploration of rational numbers, building upon the foundation laid in previous exercises. This section challenges students with more complex problems involving rational number operations, conversions between different representations of rational numbers, and applications in real-world scenarios. The exercise covers a wide range of topics, including solving equations with rational coefficients, working with ratios and proportions, applying percentage calculations, and exploring properties of rational expressions. Students are introduced to more sophisticated problem-solving techniques and are encouraged to develop logical reasoning and analytical skills. The problems in this exercise often require multi-step solutions, integrating various concepts related to rational numbers. By working through these questions, students not only reinforce their understanding of rational number operations but also gain insights into how these concepts connect to more advanced areas of mathematics, such as algebra and mathematical proofs. This comprehensive approach prepares students for higher-level mathematical thinking and problem-solving, setting a strong foundation for future mathematical studies.
