Question

100 marks
HURRY UP

✂For a = 2/3 , b= 4/5 ,c =-5/6, verify that ,
(a+b)+c=a+(b+c)
(ab)c=a(bc)
a(b-c)=ab-ac
​

Answer 1

$\textbf{\underline{\underline{Answer :-}}}$

$\textsf{\underline{\underline{Given :}}}$

$\rightarrow$ $\tt{a = \dfrac{2}{3}}$

$\rightarrow$ $\tt{b = \dfrac{4}{5}}$

$\rightarrow$ $\tt{c = \dfrac{-5}{6}}$

$\textsf{\large{Statement I Verification :-}}$

${\tt{\implies(a+b)+c=a+(b+c)}}$

${\tt\rightarrow\left(\frac{2}{3} + \frac{4}{5}\right) + \frac{ - 5}{6} = \frac{2}{3} + \left( \frac{4}{5} + \frac{ - 5}{6}\right) }$


To add the brackets we need to find the LCM of the denominators, 3 - 5(15) and 5 - 6(30)


${\tt\implies \frac{10 + 12}{15} + \frac{ - 5}{6} = \frac{2}{3} + \frac{24 + ( - 25)}{30}}$


$\tt{\implies \dfrac{22}{15} + \dfrac{ - 5}{6} = \dfrac{2}{3} + \dfrac{ - 1}{30} }$


Again find the LCM of 15 - 6 and 3 - 30


$\tt{\implies \dfrac{44 + ( - 25)}{30}} = \dfrac{20 + ( - 1)}{30}$


$\tt{\implies \dfrac{19}{30} = \dfrac{19}{30} }$


${\boxed{\boxed{\bf{\dfrac{19}{30} = \dfrac{19}{30}}}}}$


${\boxed{\sf\:{LHS = RHS}}}$


$\textbf{\large{Statement I Verified ! }}$



$\textsf{\large{Statement II Verification :-}}$


${\tt{\implies (ab)c = a(bc)}}$


$\tt{\implies\left( \frac{2}{3} \times \frac{4}{5}\right) \times \frac{ - 5}{6} = \frac{2}{3}\left( \frac{4}{5} \times \frac{ - 5}{6}\right)}$


$\tt{\implies \dfrac{8}{15} \times \dfrac{ - 5}{6} = \dfrac{2}{3} \times \dfrac{ - 20}{30} }$


$\tt{\implies \dfrac{ - 40}{90} = \dfrac{ - 40}{90} }$


${\boxed{\boxed{\bf{\dfrac{ - 40}{90} = \dfrac{ - 40}{90}}}}}$


${\boxed{\sf\:{LHS = RHS}}}$


$\textbf{\large{Statement II Verified ! }}$



$\textsf{\large{Statement III Verification :- }}$


${\tt{\implies a(b-c) = ab - ac}}$


$\tt{ \frac{2}{3} \times \left( \frac{4}{5} - \frac{ - 5}{6}\right) = \frac{2}{3} \times \frac{4}{5} - \frac{2}{3} \times \frac{ - 5}{6} }$


LCM of 5 and 6 to add them = 30


$\tt{\implies \frac{2}{3} \times \left( \frac{24 + 25}{30}\right) = \frac{8}{15} + \frac{10}{18} }$


LCM of 15 and 18 = 90


$\tt{\implies \dfrac{2}{3} \times \dfrac{49}{30} = \dfrac{48}{90} + \dfrac{50}{90} }$


$\tt{\implies \dfrac{98}{90} = \dfrac{98}{90}}$


${\boxed{\boxed{\bf{\dfrac{98}{90} = \dfrac{98}{90}}}}}$


${\boxed{\sf\:{LHS = RHS}}}$


$\textbf{\large{Statement III Verified ! }}$


$\therefore$$\text{\large{All statements are Verified}}$

Answer 2

Given,

For a = 2/3 , b= 4/5 and c =-5/6.

There are 3 condition and we will prove them one by one=>

1)(a+b) +c = a+(b+c)

RHS

(a+b) + c

=>(2/3 + 4/5) - 5/6

=>{(10+12)/15} - 5/6

=>22/15 - 5/6

=>(44-25)/30

=>19/30.

LHS

a+(b+c)

=>2/3 + (4/5 - 5/6)

=>2/3 + {(24-25)/30}

=>2/3 - 1/30

=>19/30.

Hence LHS = RHS and proved.

2)(ab)c=a(bc)

RHS

ab(c)

=>(2/3 × 4/5)×(-5/6)

=>(8/15) × (-5/6)

=>-40/90

=>-4/9.

LHS

a(bc)

=>2/3 ×{4/5 ×(-5/6)}

=>2/3 × (-20/30)

=>-4/9.

Hence LHS=RHS and proved.

3)a(b-c)=ab-ac

LHS

a(b-c)

=>2/3(4/5 + 5/6)

=>2/3 {(24+25)/30}

=>2/3 × (49/30)

=>98/90

RHS

ab-ac

=>{2/3 × 4/5} + (2/3 × 5/6)

=>8/15 + 10/18

=>8/15 + 5/9

=>(50+48)/90

=>98/90.

Hence, LHS = RHS and hence proved.