Solve the following

$\frac{x-5}{2}-\frac{x-3}{5}=\frac{1}{2}$

$\Rightarrow 10\left(\frac{x-5}{2}\right)-10\left(\frac{x-3}{5}\right)=10\left(\frac{1}{2}\right)$  (multiplying throughout by 10, which is the L.C.M. of 2,2 and 5 )

$\Rightarrow 5(x-5)$

$-2(x-3)=5$

$\Rightarrow 5 x-25-2 x+6=5$

$\Rightarrow 5 x-2 x-25+6=5$

$\Rightarrow 3 x-19=5$

$\Rightarrow 3 x=5+19$

$\Rightarrow 3 x=24$

$\Rightarrow x=$$\frac{24}{3}=8$

$\therefore x=8$