Exercise 1.5 in Chapter 1 of RD Sharma's Class 8 mathematics textbook delves deeper into the world of rational numbers, building upon the foundational concepts established in previous exercises. This section challenges students to apply their understanding of rational numbers to more complex scenarios, emphasizing the properties of rational numbers and their applications in various mathematical contexts. The exercise focuses on advanced operations with rational numbers, including working with negative rationals, exploring the relationships between fractions and decimals, and solving multi-step problems involving rational numbers.
Class 8 RD Sharma - Chapter 1 Rational Numbers
"Rational Numbers" This chapter mainly deals with problems based on rational numbers, whole numbers, natural numbers, and the representation of rational numbers on the number line. Exercise 1.5 is about multiplying the numerator and denominator or different rational numbers.
Question 1. Multiply:
(i) 7/11 by 5/4
Solution:
7/11 × 5/4
Multiplying numerator with numerator of other rational number and denominator with denominator
= (7 × 5)/(11 × 4)
= 35/44
(ii) 5/7 by -3/4
Solution:
(5/7) × (-3/4)
Multiplying numerator with numerator of other rational number and denominator with denominator
= (5 × -3)/(7 × 4)
= -15/28
(iii) -2/9 by 5/11
Solution:
-2/9 × 5/11
Multiplying numerator with numerator of other rational number and denominator with denominator
= (-2 × 5)/(9 × 11)
= -10/99
(iv) -3//17 by -5/-4
Solution:
-3/17 × 5/4
Multiplying numerator with numerator of other rational number and denominator with denominator
= (-3 × 5)/(17 × 4)
= -15/68
(v) 9/-7 by 36/-11
Solution:
(9/-7) × (36/-11)
Multiplying numerator with numerator of other rational number and denominator with denominator
= (-9 × -36)/(7 × 11)
= 324/77
(vi) -11/13 by -21/7
Solution:
Multiplying numerator with numerator of other rational number and denominator with denominator
= (-11 × -21)/(13 × 7)
= 231/91
(vii) -3/5 by -4/7
Solution:
Multiplying numerator with numerator of other rational number and denominator with denominator
= (-3 × -4)/(5 × 7)
= 12/35
(viii) -15/11 by 7
Solution:
-15/11×7/1
Multiplying numerator with numerator of other rational number and denominator with denominator
=(-15×7)/(11×1)
=-105/11
Question 2. Multiply
(i) -5/17 by 51/-60
Solution:
(-5/17) × (51/-60)
Multiplying numerator with numerator of other rational number and denominator with denominator
= (-5 × -51)/(17 × 60)
Common factor of 5 and 60
= 51/17 × 12
51 and 12 have 3 as common factor
= 17/17 × 4
= 1/4
(ii) -6/11 by -55/36
Solution:
(-6/11) × (-55/36)
Multiplying numerator with numerator of other rational number and denominator with denominator
= (-6 × -55)/(11 × 36)
Common factor of 6 and 36, 55 and 11
= 5/6
(iii) -8/25 by -5/16
Solution:
(-8/25) × (-5/16)
Multiplying numerator with numerator of other rational number and denominator with denominator
= (-8 × -5)/(25 × 16)
Common factor of 8 and 16, 5 and 25
= 1/5 × 2
= 1/10
(iv) 6/7 by -49/36
Solution:
(6/7) × (-49/36)
Multiplying numerator with numerator of other rational number and denominator with denominator
= (6 × -49)/(7 × 36)
Common factor of 6 and 36, 49 and 7
= -7/6
(v) 8/-9 by -7/-16
Solution:
-8/9 × 7/16
Multiplying numerator with numerator of other rational number and denominator with denominator
= (-8 × 7)/(9 × 16)
Common factor of 8 and 16
= -7/9 × 2
= -7/18
(vi) -8/9 by 3/64
Solution:
-8/9 × 3/64
Multiplying numerator with numerator of other rational number and denominator with denominator
= (-8 × 3)/(9 × 64)
Common factor of 8 and 64, 3 and 9
= -1/3 × 8
= -1/24
Question 3. Simplify each of the following and express the result as a rational number in standard form:
(i) (-16/21) × (14/5)
Solution:
Multiplying numerator with numerator of other rational number and denominator with denominator
= (-16 × 14)/(21 × 5)
Common factor of 21 and 14
= (-16 × 2)/(3 × 5)
= -32/15
(ii) (7/6) × (-3/28)
Solution:
Multiplying numerator with numerator of other rational number and denominator with denominator
= (7 × -3)/(6 × 28)
Common factor of 7, 28, 3 and 6
= -1/2 × 4
= -1/8
(iii) (-19/36) × 16
Solution:
Multiplying numerator with numerator of other rational number and denominator with denominator
= (-19 × 16)/(36 × 1)
Common factor of 16 and 36
= (-19×4)/(9 × 1)
=-76/9
(iv) (-13/9) × (27/-26)
Solution:
Multiplying numerator with numerator of other rational number and denominator with denominator
=(-13× -27)/(9×26)
Common factor of 27 and 9 , 13 and 26
=(-1 × -3)/(2)
=3/2
(v) (-9/16) × (-64/-27)
Solution:
Multiplying numerator with numerator of other rational number and denominator with denominator
= (-9 × 64) / (16 × 27)
Common factor of 9 and 27, 64 and 16
= (-1 × 4) / (1 × 3)
= -4/3
(vi) (-50/7) × (14/3)
Solution:
Multiplying numerator with numerator of other rational number and denominator with denominator
= (-50 × 14)/(7 × 3)
Common factor of 14 and 7
= (-50 × 2)/(3)
= -100/3
(vii) (-11/9) × (-81/-88)
Solution:
Multiplying numerator with numerator of other rational number and denominator with denominator
= (-11 × 81)/(9 × 88)
Common factor of 1, 88, 9 and 81
= (-1 × 9)/(1 × 8)
= -9/8
(viii) (-5/9) × (72/-25)
Solution:
Multiplying numerator with numerator of other rational number and denominator with denominator
= (-5 * -72)/(9 × 25)
Common factor of 5 and 25, 9 and 72
= (-1 × -8)/(1 × 5)
= 8/5
4. Simplify:
(i) ((25/8) × (2/5)) - ((3/5) × (-10/9))
Solution:
= ((25 × 2)/(8 × 5)) - ((3 × -10)/(5 × 9))
= 50/40 - (-30)/45
= 5/4 + 2/3
LCM of 4 and 3 is 12
= (5 × 3 + 2 × 4)/12
= (15 + 8)/12
= 23/12
(ii) ((1/2) × (1/4)) + ((1/2) × 6)
Solution:
= ((1 × 1)/(2 × 4)) + ((1 × 6)/(1 × 2))
= 1/8 + 3/1
LCM of 8 and 1 is 8
= (1 × 1 + 3 × 8)/8
= (1 + 24)/8
= 25/8
(iii) (-5 × (2/15)) - (-6 × (2/9))
Solution:
= ((-5 × 2)/(15 × 1)) - ((-6 × 2)/(1 × 9))
= (-10/15) - (-12/9)
Common factor of 10 and 15, 12 and 9
= -2/5 + 4/3
LCM of 5 and 3 is 15
= (-2 × 3 + 4 × 4)/15
= (-6 + 16)/15
= 10/15
Common factor of 10 and 15
= 2/3
(iv) ((-9/4) × (5/3)) + ((13/2) × (5/6))
Solution:
= (-9 × 5)/(4 × 3) + (13 × 5)/(2 × 6)
Common factor of 9 and 3
= (-45/12) + (65/12)
As denominators are same
= (-45 + 65)/12
= (20)/12
Common factor of 20 and 12
= 5/3
(v) ((-4/3) × (12/-5)) + ((3/7) × (21/15))
Solution:
= (-4 * -12)/(3 × 5) + ((3 × 21)/(7 × 15))
= 48/15 + 3/5 (Common factor 3 and 15, 21 and 7)
LCM of 15 and 5 is 15
= (48 + 3 × 3)/15
= (48 + 9)/15
= 57/15
Common factor of 57 and 15
= 19/5
(vi) ((13/5) × (8/3)) - ((-5/2) × (11/3))
Solution:
= (13 × 8)/(5 × 3) - ((-5 × 11)/(2 × 3))
= 104/15 - 55/6
LCM of 15 and 6 is 3 × 5 × 2 = 30
= (104 × 2 + 55 × 5)/30
= (208 + 275)/30
= 483/30
(vii) ((13/7) × (11/26)) — ((-4/3) × (5/6))
Solution:
= ((13 × 11)/(7 × 26)) - ((-4 × 5)/(3 × 6))
Common factor of 13 and 26, 4 and 6
= 11/7 × 2 - (-2 × 5/3 × 3)
= 11/14 + 10/9
LCM of 14 and 9 is 126
= (11 × 9 + 10 × 14)/126
= (99 + 140)/126
= 239/126
Question 5. Simplify:
(i) ((3/2) × (1/6)) + ((5/3) × (7/2) - (13/8) × (4/3))
Solution:
= (3 × 1)/(2 × 6) + (5 × 7)/(3 × 2) - (13 × 4)/(8 × 3)
Common factor of 3 and 6, 4 and 8
= 1/4 + 35/6 - 13/6
LCM of 4 and 6 is 12
= (1 × 3 + 35 × 2 - 13 × 2)/12
= (3 + 70 - 26)/12
= (73 - 26)/12
= 47/12
(ii) ((1/4) × (2/7)) — (5/14) × (-2/3) + (3/7) × (9/2)
Solution:
= (1 × 2)/(4 × 7) - (5 × -2)/(14 × 3) + (3 × 9)/(7 × 2)
Common factor of 2 and 4, 2 and 14
= 1/14 - (-5/21) + 27/14
LCM of 21 and 14 is 7 × 2 × 3 = 42
= 1/14 + 5/21 + 27/14
LCM of 14 and 21 is 2 × 7 × 3 = 42
= (1 × 3 + 5 × 2 + 27 × 3)/42
= (3 + 10 + 81)/42
= (94)/42
(iii) ((13/9) × (-15/2)) + ((7/3) × (8/5) + (3/5) × (1/2))
Solution:
= (13 × -15)/(9 × 2) + ((7 × 8)/(3 × 5) + (3 × 1)/(5 × 2))
Common factor of 9 and 15
= (13 × -5)/(3 × 2) + ((56/15) + 3/10)
= -65/6 + 56/15 + 3/10
6 = 2 × 3
15 = 3 × 5
10 = 2 × 5
LCM is 2 × 3 × 5 = 30
= (-65 × 5 + 56 × 2 + 3 × 3)/30
= (-325 + 112 + 9)/30
= (-325 + 121)/30
= -204/30
(iv) ((3/11) × (5/6)) - (9/12) × (4/3) + (5/13) × (6/15)
Solution:
= (3 × 5)/(11 × 6) - ((9 × 4)/(12 × 3) + (5 × 6)/(13 × 15))
Common factor of 3 and 6, 9 and 12, 5 and 15
= 5/22 - 1/1 + 2/13
= 5/22 - 1/1 + 2/13
LCM of 22,1 and 13 is 286
= (5 × 13 - 286 + 2 × 22)/286
= (65 - 286 + 44)/286
= (65 - 330)/286
= -177/286
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Summary
Exercise 1.5 in Chapter 1 of RD Sharma's Class 8 mathematics textbook expands on the concept of rational numbers, challenging students with more advanced problems and applications. This exercise covers a wide range of topics, including converting between fractions and recurring decimals, exploring the density property of rational numbers, and solving complex problems involving rational number operations. Students are tasked with simplifying expressions, evaluating formulas with rational numbers, and proving mathematical statements related to rational numbers. The problems also incorporate real-world applications, helping students connect abstract mathematical concepts to practical scenarios. By working through these questions, students develop critical thinking skills, enhance their problem-solving abilities, and gain a deeper understanding of the properties and behavior of rational numbers. This comprehensive approach prepares students for more advanced mathematical concepts they will encounter in future studies, particularly in algebra and calculus.
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