Explanation:
Let \(a\) be the common acceleration of the system. The equations of motion of masses \(M_{1}\) and \(M_{2}\) are
\({T_1} = {M_1}a\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( 1 \right)\)
\({T_2} - {T_1} = {M_2}{\rm{ }}a\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( 2 \right)\)
Adding (1) and (2), we get
\(T_{2}=\left(M_{1}+M_{2}\right) a\)
\(\therefore \dfrac{T_{1}}{T_{2}}=\dfrac{M_{1}}{M_{1}+M_{2}}\)