Verify commutativity of addition of Rational numbers for each of the following pair of Rational numbers 4/9 and 7/-12
Answers
Answer:
i) -8/7 and 5/14
To show that: -8/7 + 5/4 = 5/14 + (-8/7)
∵ -8/7 + 5/14
∴ LCM of 2 and 7 = 14
= (-8 × 2/ 7 × 2) + (5 × 1/14 × 1)
= (-16 + 5)/14 = -11/14
And, 5/14 + -8/7
= (5 × 1/14 × 1) + (-8 × 2/7 ×2)
= (5 – 16)/14 = -11/14
∴ -8/7 + 5/14 = 5/14 + -8/7
This verifies the commutative property for the addition of rational numbers.
(ii) 5/9 and 5/12
To show that: 5/9 + 5/-12 = 5/-12 + 5/9
∵ 5/9 + 5/-12
∴ LCM of 9 and 12 = 2 × 2 × 3 × 3 = 36
= (5 × 4/9 ×4) – (5 × 3/12 × 3)
= (20 – 15)/36 = 5/36
And, 5/-12 + 5/9
= (5 × 3/-12 × 3) + (5 × 4/9 × 4)
= (-15 + 20)/36 = 5/36
∴ 5/9 + 5/-12 = 5/-12 + 5/9
This verifies the commutative property for the addition of rational numbers.
(iii) -4/5 and -13/-15
To show that:
-4/5 and -13/-15 = -13/-15 + (-4/5)
∵ -4/5 + 13/15
∴ LCM of 5 and 15 = 5 × 3 = 15
= (-4 × 3/5 × 3) + (13 × 1/15 × 1)
= (-12 + 13)/15 = 1/15
And, 13/15 + (-4/5)
= (13 × 1/15 × 1) + (-4 × 3/5 × 3)
= (13 – 12)/15 = 1/15
∴ -4/5 + -13/-15 = -13/-15 + -4/5
This verifies the commutative property for the addition of rational numbers.
(iv) 2/-5 and 11/-15
Show that: 2/-5 + 11/-15 = 11/-15 + 2/ -5
= 2/-5 + 11/-15
∴ LCM of 5 and 15 = 15
= (-2 × 3/5 × 3) – (11 × 1/15 × 1)
= (-6 -11)/15 = -17/15
And, 11/-15 + 2/-5
= (-11 × 1/15 × 1) – (2 × 3/5 × 3) = (-11 -6)/15 = -17/15
∴ 2/-5 + 11/-15 = 11/-15 + 2/ -5
This verifies the commutative property for the addition of rational numbers.
(v) 3 and -2/7
Show that: 3/1 + -2/7 = -2/7 + 3/1
= 3/1 + (-2/7) (∵ LCM of 1 and 7 = 7)
= (3 × 7 /1 × 7 – 2 × 1/7 × 1)
= (21 – 2)/7 = 19/7
And, -2/7 + 3/1
= (-2 × 1/7 × 1) + (3 × 7/1 × 7)
= (-2 + 21)/7 = 19/7
∴ 3/1 + -2/7 = -2/7 + 3/1
This verifies the commutative property for the addition of rational numbers.
(vi) -2 and 3/-5
Show that: -2/1 + (-3/5) = -3/5 + (-2/1)
= -2/1 + (-3/5) (∵ LCM of 1 and 5 = 5)
= (-2 × 5/1 × 5) + (-3 × 1/5 × 1)
= (-10 -3)/5 = -13/5
And, -3/5 + -2/1
= (-3 × 1/ 5 × 1) + (-2 × 5/1 × 5)
= (-3 -10)/5 = -13/5
∴ -2/1 + (-3/5) = (-3/5) + (-2/1)
This verifies the commutative property for the addition of rational numbers.