Explanation:
Equation of motion for block \({C}\) is
\({{m_C}g - T = {m_C}{a_C}}\)
Or \({180 - T = 18{a_C}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( 1 \right)}\)
\({T - 45 = 9{a_A}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( 2 \right)}\)
and \({2T - 45 = 9{a_B} = 9\left( {\frac{{{a_C} - {a_A}}}{2}} \right)\,\,\left( 3 \right)}\)
or \({4T - 90 = 9{a_C} - 9{a_A}}\)
or \({4T - 90 = 9\left( {\frac{{180 - T}}{{18}}} \right) - 9\left( {\frac{{T - 45}}{9}} \right)}\)
\({4T + \frac{T}{2} + T = 90 + 45 = 135}\)
\({\therefore \,\,\,\,\,\,\,\,\,\,T = \frac{{135}}{{5.5}} = \frac{{1350}}{{55}} = \frac{{1350}}{{11n}}}\)
Comparing with the given value, we have \({n=5}\)