Verify the associative property for addition and multiplication for the rational number.
2/3, -1/15, 4/15
Answers
Step-by-step explanation:
1/2, 2/3 and -1/6
Show that:
1/2 + (2/3 + (-1/6) = (1/2 + 2/3) + (-1/6)
∵ 1/2 + (2/3 + (-1/6)
∴ LCM of 3 and 6 = 6
= 1/2 + (2 × 2/3 × 2) + (-1 × 1/6 ×1)
= 1/2 + (4/6 – 1/6)
= 1/2 + [(4-1)/6]
=1/2 + (3/6)
= (1 × 3/2 × 3) + (3 × 1/6 × 1) (∵ LCM of 2 and 6 = 3)
= (3 + 3)/6 = 6/6 = 1
And, (1/2 + 2/3) + (-1/6)
∴ LCM of 2 and 3 = 6
= (1 × 3/2 × 3) + (2 × 2/3 × 2) + (-1/6)
= (3 + 4)/6 + -1/6
= (7 – 1)/6 = 6/6 = 1
∴ 1/2 + (2/3 + (-1/6) = (1/2 + 2/3) + (-1/6)
This verifies associative property of the addition of rational numbers.
(ii) -2/5, 4/15 and -7/10
Show that:
-2/5 + (4/15 + (-7/10) = (-2/5 + 4/15) + (-7/10)
∵ -2/5 + (4/15 + (-7/10)
∴ LCM of 15, 10 = 2 × 3 × 5 = 30
= -2/5 + (4 × 2/15 × 2) + (-7 × 3/10 × 3)
(∵ LCM of 15 and 10 = 30)
= -2/5 + [(8 – 21)/30]
= -2/5 – 13/30 = (-2 × 6/5 × 6) – (13 × 1/30 × 1)
= (-12 -13)/30 = -25/30 = -5/6
And, (-2/5 + 4/15) + -7/10
∴ LCM of 5 and 15 = 3 × 5 = 15
= (-2 × 3/5 × 3) + (4 × 1/15 ×1) + -7/10
∴ LCM of 5 and 15 = 15
= (-6 + 4)/15 + (-7/10)
= -2/15 + (-7/10)
= (-2 × 2/15 × 2) – (7 × 3/10 × 3)
= -4/30 – 21/30 = -25/30 = -5/6
∴ -2/5 + (4/15 + (-7/10) = (-2 /5 + 4/15) + (-7/10)
This verifies associative property of the addition of rational numbers.
(iii) -7/9, 2/ -3 and -5/18
Show that:
-7/9 + (2/-3 + (-5/18) = (-7/9 + 2/3) + (-5/18)
∵ -7/9 + (2/-3 + (-5/18)
∴ LCM of 3 and 18 = 2 × 3 × 3 = 18
= -7/9 + (-2 × 6/3 × 6) + (-5 ×1/18 ×1)
(∵ LCM of 3 and 18 = 18)
= -7/9 + [(-12 -5)/18]
= -7/9 + -17/18
= (-7 × 2/9 ×2) – (17 ×1/18 × 1)
(∵ LCM of 9 and 18 = 18)
= (-14 -17)/18 = -31/18
And, (-7/9 + 2/-3) + (-5/18)
∴ LCM of 3 and 9 = 3
= (-7 × 1/9 × 1) + (-2 × 3/3 × 3) + (-5/18)
(∵ LCM of 9 and 3= 9)
= (-7 -6)/9 + (-5/18)
= -13/9 + (-5/18)
= (-13 × 2/9 × 2) + (-5 × 1/18 × 1) = (-26 -5)/18 = -31/18
∴ -7 /9 + (2/-3 + -5/18) = (-7/9 + 2/-3) + (-5/18)
This verifies associative property of the addition of rational numbers.
(iv) -1, 5/6 and -2/3
Show that:
This verifies associative property of the addition of rational numbers.
-1/1 + (5/6 + -2/3) = (-1/1 + 5/6) + (-2/3)
∵ -1/1 + (5/6 + -2/3)
∴ LCM of 6 and 3 = 6
= -1/1 + (5 × 1/6 × 1) + (-2 × 2/3 ×2)
(∵ LCM of 6 and 3= 6)
= -1/1 + [(5 – 4)/6]
= -1/1 + 1/6
= (-1 × 6/1 × 6) + (1 × 1/6 × 1) (∵ LCM of 1 and 6= 1)
= (-6 + 1)/6 = -5/6
And, (-1/1 + 5/6) + (-2/3)
= (-1 × 6/1 × 6 + 5 × 1/6 ×1) + (-2/3)
(∵ LCM of 1 and 6= 6)
= (-6 + 5)/6 + (-2/3)
= -1 /6 + (-2/3)
= (-1 × 1)/6 × 1 + (-2 × 2/3 ×2) (∵ LCM of 6 and 3= 6)
= (-1 -4)/6 = -5/6
∴ -1/1 + (5/6 + -2/3) = (-1/1 + 5/6) + -2/3