Question

Q9.Verify associative property of addition of rational numbers (x + y) + z = x + (y + z), when x =1/5 , y = 1/3 , z = −2/5
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Answer 1

Answer:

As the property states (x+y)+z=x+(y+z)(x+y)+z=x+(y+z)

We use value as (12+23)+−15=12+(23+−15)(12+23)+−15=12+(23+−15)

Let us consider L.H.S (12+23)+−15(12+23)+−15 L.C.M

for 2 and 3 is 6 (1×3)(2×3)+(2×2)(3×2)=36+46=76=76+−15(1×3)(2×3)+(2×2)(3×2)=36+46=76=76+−15

(Since the denominators are same we add it) L.C.M for 6 and 5 is 30 (7×5)(6×5)+(−1×6)(5×6)=3530+−630=(35+(−6))30=(35−6)30=2930(7×5)(6×5)+(−1×6)(5×6)=3530+−630=(35+(−6))30=(35−6)30=2930

(Since denominators are same we add it) Let us consider R.H.S =12+(23+−15)=12+(23+−15)

L.C.M for 3 and 5 is 15 23+−15=(2×5)(3×5)+(−1×3)(5×3)=1015+−31523+−15=(2×5)(3×5)+(−1×3)(5×3)=1015+−315 1015+−315=(10−3)15

=7151015+−315=(10−3)15=715

(Since the denominators are same we add it) L.C.M for 2 and 15 is 30 12+715=(1×15)(2×15)+(7×2)(15×2)=1530+1430=293012+715=(1×15)(2×15)+(7×2)(15×2)=1530+1430=2930

(Since denominator is same we add it) ∴∴ L.H.S = R.H.S (associativity of addition of rational number is verified.

ii) As the property states (x+y)+z=x+(y+z)(x+y)+z=x+(y+z) We use value as (−25+43)+−710=−25+(43+−710)(−25+43)+−710=−25+(43+−710)

Let us consider L.H.S (−25+43)+−710(−25+43)+−710 LCM for 5 and 3 is 15 (−2×3)(5×3)+(4×5)(3×5)=−615+2015=−615+2015=(−6+20)15=1415(−2×3)(5×3)+(4×5)(3×5)=−615+2015=−615+2015=(−6+20)15=1415

( Since denominators are same we add it)

Now, 1415+−7101415+−710 L.C.M for 15 and 30 is 30 (14×2)(15×2)+(−7×3)(10×3)=2830+−2130=(28+(−21))30=(28−21)30=730(14×2)(15×2)+(−7×3)(10×3)=2830+−2130=(28+(−21))30=(28−21)30=730

(Since the denominators are same we add it) Let us take R.H.S −25+(43+−710)−25+(43+−710)

L.C.M for 3 and 10 is 30 43+−710=(4×10)(3×10)+(−7×3)(10×3)=4030+−2130=4030+−2130=(40−21)30=193043+−710=(4×10)(3×10)+(−7×3)(10×3)=4030+−2130=4030+−2130=(40−21)30=1930

( Since the denominators are same we add it)

Now, −25+1930−25+1930 LCM for 5 and 30 is 30 −25+1930=(−2×6)(5×6)+(19×1)(30×1)=−1230+1930=(−12+19)30=730−25+1930=(−2×6)(5×6)+(19×1)(30×1)=−1230+1930=(−12+19)30=730

(Since denominators are same we add it)

∴∴ L.H.S = R.H.S (associativity of addition of rational number is verified.

iii) As the property states (x+y)+z=x+(y+z)(x+y)+z=x+(y+z)

We use value as (−21+35)+−43=−21+(35+−43)(−21+35)+−43=−21+(35+−43)

Let us consider L.H.S (−21+35)+−43(−21+35)+−43

LCM for 1 and 5 is 5 (−2×5)(1×5)+(3×1)(5×1)=−105+35=(−10+3)5=−75(−2×5)(1×5)+(3×1)(5×1)=−105+35=(−10+3)5=−75

Now, −75+−43−75+−43 LCM for 5 and 3 is 15 (−7×3)(5×3)+(−4×5)(3×5)=−2115+−2015=(−21+(−20))15=−21−2015=−4115(−7×3)(5×3)+(−4×5)(3×5)=−2115+−2015=(−21+(−20))15=−21−2015=−4115

(Since the denominators are same) Let us consider R.H.S −21+(35+−43)−21+(35+−43)

LCM for 5 and 3 is 15 −35+−43=(3×3)(5×3)+(−4×5)(3×5)=915+−2015=(9−20)15=−1115−35+−43=(3×3)(5×3)+(−4×5)(3×5)=915+−2015=(9−20)15=−1115

(Since denominators are same)

Now, −21+−1115=(−2×15)(1×15)+(−11×1)(15×1)=−3015+−1115=−4115−21+−1115=(−2×15)(1×15)+(−11×1)(15×1)=−3015+−1115=−4115

∴∴ L.H.S = R.H.S (associativity of addition of rational number is verified)

Answer 2

Step-by-step explanation:

associatively of addition of rational is verified