Question

Solve:
$\frac{1}{2} (x - \frac{2}{3}) + \frac{1}{6} (x - \frac{5}{2}) = \frac{1}{5} (2x - \frac{1}{6} )$
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Answer 1

GIVEN :-

$\\ \sf \: \dfrac{1}{2} \left(x - \dfrac{2}{3} \right) + \dfrac{1}{6} \left(x - \dfrac{5}{2} \right) = \dfrac{1}{5} \left(2x - \dfrac{1}{6} \right)$

TO FIND :-

• Value of x.

SOLUTION :-

$\\ \sf \: \dfrac{1}{2} \left(x - \dfrac{2}{3} \right) + \dfrac{1}{6} \left(x - \dfrac{5}{2} \right) = \dfrac{1}{5} \left(2x - \dfrac{1}{6} \right) \\ \\ \\ \sf \implies \: \dfrac{1}{2} \left( \dfrac{3x - 2}{3} \right) + \dfrac{1}{6} \left( \dfrac{2x - 5}{2} \right) = \dfrac{1}{5} \left( \dfrac{2x(6) - 1}{6} \right) \\ \\ \\ \implies \sf \: \dfrac{3x - 2}{6} + \dfrac{2x - 5}{12} = \dfrac{12x - 1}{30}$

Multiplying whole equation by 6,

$\\ \\ \implies \sf \: \cancel6\left( \dfrac{3x - 2}{ \cancel6} \right) + \cancel6\left( \dfrac{2x - 5}{ \cancel{12}} \right) = \cancel6\left( \dfrac{12x - 1}{ \cancel{30}} \right) \\ \\ \\ \implies \sf \: 3x - 2 + \dfrac{2x - 5}{2} = \dfrac{12x - 1}{5}$

$\implies \sf \: 3x + \dfrac{2x - 5}{2} - \dfrac{12x - 1}{5} = 2 \\ \\ \\ \implies \sf \: \dfrac{10(3x) + 5(2x - 5) - 2(12x - 1)}{10} = 2 \\ \\ \\ \implies \sf \: 30x + 10x - 25 - 24x + 2 = 20 \\ \\ \\ \implies \sf \: 16x = 20 - 2 + 25 \\ \\ \\ \implies \sf \: 16x = 43 \\ \\ \\ \implies \sf \: x = \dfrac{43}{16} \\ \\ \\ \implies \underline{\boxed{ \mathfrak{ x = \frac{43}{16} }}}$

Answer 2

Solution :

$\frac{1}{2} (x - \frac{2}{3}) + \frac{1}{6} (x - \frac{5}{2}) = \frac{1}{5} (2x - \frac{1}{6} ) \\ \\ \Longrightarrow \: \: \frac{1}{2} (x - \frac{2}{3}) + \frac{1}{6}(x - \frac{5}{2}) = \frac{1}{5} (2x - \frac{1}{6} ) \\ \\ \Longrightarrow \: \: \frac{1}{2} ( \frac{3x - 2}{3} ) + \frac{1}{6} ( \frac{2x - 5}{2} ) = \frac{1}{5} (\frac{12x - 1}{30} ) \\ \\ \ \Longrightarrow \: \: \frac{3x - 2}{6} + \frac{2x - 5}{12} = \frac{12x - 1}{30}$

LCM of 6,12,30 = 60

Multiplying both side of the equation by 60 we get

$60( \frac{3x - 2}{6} ) + 60( \frac{2x - 5}{12} ) = 60( \frac{12x - 1}{30} ) \\ \\ \longrightarrow \: \: 30x - 20 + 10x - 25 = 24x - 2 \\ \\ \longrightarrow \: \: 40x - 45 = 24x - 2 \\ \\ \longrightarrow \: \: 40x - 24x = - 2 + 45 \\ \\ \longrightarrow \: \: 16x = 43 \\ \\ \longrightarrow \: \: x = \frac{43}{16}$