 |
VOOZH | about |
|
VOOZH | about |
Rational numbers are a fundamental concept in mathematics, representing numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. In Chapter 1 of RD Sharma's Class 8 mathematics textbook, Exercise 1.4 delves deeper into the properties and operations of rational numbers. This exercise builds upon the foundational knowledge established in previous sections, challenging students to apply their understanding to more complex problems.The focus of Exercise 1.4 is on the various operations that can be performed with rational numbers, including addition, subtraction, multiplication, and division. Students will learn how to manipulate these numbers in different contexts, solve equations involving rational numbers, and explore their relationships on the number line. This exercise also emphasizes the importance of understanding the properties of rational numbers, such as closure, commutativity, associativity, and the existence of identity and inverse elements for various operations.
(i) 3/4 + 5/6 + -7/8
Solution:
2 = 2 × 2
6 = 2 × 3
8 = 2 × 2 × 2
LCM is 2 × 2 × 2 × 3 = 24
= (3 × 6 + 5 × 4 + (-7 × 3)) / 24
= (18 + 20 - 21) / 24
= (38 - 21) / 24
= 17 / 24
(ii) 2/3 + -5/6 + -7/9
Solution:
LCM of 3, 6 and 9 is 18
= (2 × 6 + (-5 × 3) + (-7 × 2)) / 18
= (12 - 15 - 14) / 18
= (12 - 29) / 18
= -17 / 18
(iii) -11/2 + 7/6 + -5/8
Solution:
2 = 2 × 1
6 = 2 × 3
8 = 2 × 2 × 2
LCM is 2 × 2 × 2 × 3 = 24
= (-11 × 12 + 7 × 4 + (-5 × 3)) / 24
= (-132 + 28 - 15) / 24
= (-147 + 28) / 24
= -119 / 24
(iv) -4/5 + -7/10 + -8/15
Solution:
10 = 5 × 2
15 = 3 × 5
LCM is 5 × 2 × 3 = 30
= (-4 × 6 + (-7 × 3) + (-8 × 2)) / 30
= (-24 - 21 - 16) / 30
= -61 / 30
(v) -9/10 + 22/15 + 13/-20
Solution:
This can be written as
-9/10 + 22/15 + -13/20
10 = 2 × 5
15 = 3 × 5
20 = 2 × 2 × 5
LCM is 2 × 2 × 3 × 5 = 60
= (-9 × 6 + 22 × 4 + (-13 × 3)) / 60
= (-54 + 88 - 39) / 60
= (-93 + 88) / 60
= -5 / 60
= -1 / 12
(vi) 5/3 + 3/-2 + -7/3 + 3
Solution:
This can be written as
5 / 3 + -3 / 2 + -7 / 3 + 3 / 1
LCM is 6
= (5 × 2 + (-3 × 3) + (-7 × 2) + 3 × 6) / 6
= (10 - 9 -14 + 18) / 6
= (28 - 23) / 6
= 5 / 6
(i) -8/3 + -1/4 + -11/6 + 3/8 + -3
Solution:
4 = 2 × 2
6 = 2 × 3
8 = 2 × 2 × 2
LCM is 2 × 2 × 2 × 3 = 24
= (-8 × 8 + (-1 × 6) + (-11 × 4) + 3 × 3 + (-3 × 24)) / 24
= (-64 - 6 - 44 + 9 - 72) / 24
= (-186 + 9) / 24
= -177 / 24
= -59 / 8
(ii) 6/7 + 1 + -7/9 + 19/21 + -12/7
Solution:
(6 / 7 + -12 / 7) + (-7 / 9) + 19 / 21 + 1(Taking numbers with same denominators together)
= (6 - 12) / 7 + (-7 / 9) + 19 / 21+1
= -6 / 7 + -7 / 9 + 19 / 21 + 1 / 1
9 = 3 × 3
21 = 3 × 7
LCM of 7, 1, 9 and 21 is 63
= (-6 × 9 + (-7 × 7) + 19 × 3 + 1 × 63) / 63
= (-54 - 49 + 57 + 63) / 63
= (-103 + 120) / 63
= 17 / 63
(iii) 15/2 + 9/8 + -11/3 + 6 + -7/6
Solution:
15 / 2 + 9/8 + (-11 / 3) + 6 / 1 + (-7 / 6)
LCM of 2, 8, 3, 1 and 6 is 24
= (15 × 12 + 9 × 3 + (-11 × 8) + 6 × 24 + (-7 × 4)) / 24
= (180 + 27 - 88 + 144 - 28) / 24
= (351 - 116) / 24
= (235) / 24
(iv) -7/4 + 0 + -9/5 + 19/10 + 11/14
Solution:
4 = 2 × 2
5 = 5 × 1
10 = 2 × 5
14 = 2 × 7
LCM is 2 × 2 × 5 × 7 is 140
= (-7 × 35 + (-9 × 28) + 19 × 14 + 11 × 10) / 140
= (-245 - 252 + 266 + 110) / 140
= (-497 + 376) / 140
= (-121) / 140
(v) -7/4 + 5/3 + -1/2 + -5/6 + 2
Solution:
LCM of 4, 3, 2 and 6 is 12
= (-7 × 3 + 5 × 4 + (-1 × 6) + (-5 × 2) + 2 × 12) / 12
= (-21 + 20 - 6 - 10 + 24) / 12
= (-37 + 44) / 12
= 7 / 12
(i) -3/2 + 5/4 + -7/4
Solution:
Taking numbers with the same denominators together
= -3 / 2 + (5 - 7) / 4
= -3 / 2 - 2 / 4
LCM of 2 and 4 is 4
= (-3 × 2 - 2 × 1) / 4
= (-6 - 2) / 4
= (-8) / 4
= -2
(ii) 5/3 + -7/6 + -2/3
Solution:
Taking numbers with same denominators together
(5 / 3 + -2 / 3) + -7 / 6
= (5 - 2) / 3 + -7 / 6
= 3 / 3 + (-7 / 6)
LCM of 3 and 6 is 6
= (3 × 2 + (-7 × 1)) / 6
= (6 - 7) / 6
= -1 / 6
(iii) 5/4 - 7/6 - (-2/3)
Solution:
This can be written as
5 / 4 - 7 / 6 + 2 / 3
LCM of 4,6 and 3 is 12
= (5 × 3 - 7 × 2 + 2 × 4) / 12
= (15 - 14 + 8) / 12
= (23 - 14) / 12
= 9 / 12
= 3 / 4
(iv) -2/5 - (-3/10) - (-4/7)
Solution:
This can be written as:
-2 / 5 + 3 / 10 + 4 / 7
LCM of 5,10 and 7 is 70
= (-2 × 14 + 3 × 7 + 4 × 10) / 70
= (-28 + 21 + 40) / 70
= (-28 + 61) / 70
= 33 / 70
(v) 5/6 + -2/5 - (-2/15)
Solution:
This can be written as
5 / 6 + -2 / 5 + 2 / 15
6 = 2 × 3
5 = 5 × 1
15 = 3 × 5
LCM is 2 × 3 × 5 = 30
= (5 × 5 + (-2 × 6) + 2 × 2) / 30
= (25 - 12 + 4) / 30
= (29 - 12) / 30
= 17 / 30
(vi) 3/8 - (-2/9) + (-5/36)
Solution:
This can be written as
3 / 8 + 2 / 9 - 5 / 36
8 = 2 × 2 × 2
9 = 3 × 3
36 = 2 × 2 × 3 × 3
LCM is 2 × 2 × 2 × 3 × 3 = 72
= (3 × 9 + 2 × 8 - 5 × 2) / 72
= (27 + 16 - 10) / 72
= (43 - 10) / 72
= 33 / 72
= 11 / 24
Exercise 1.4 in Chapter 1 of RD Sharma's Class 8 mathematics textbook provides a comprehensive exploration of rational numbers and their properties. This exercise builds upon foundational knowledge, challenging students to apply their understanding to more complex problems involving rational numbers. It covers key concepts such as performing basic arithmetic operations (addition, subtraction, multiplication, division) with rational numbers, simplifying rational expressions, and representing these numbers on a number line. Students learn to solve equations involving rational numbers, find rational numbers between given values, and apply various properties of rational numbers to solve complex problems. The exercise also touches on real-world applications, helping students connect mathematical concepts to practical situations. By mastering the concepts presented in this exercise, students develop a solid foundation for more advanced mathematical topics and improve their ability to solve problems involving fractions and rational numbers. This comprehensive approach ensures that students not only understand the theoretical aspects of rational numbers but also gain the skills to apply this knowledge in diverse mathematical contexts.
