Explanation:
\({T_2} - {T_1} = {M_2}a\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( 1 \right)\)
\({T_1} = {M_1}a\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( 2 \right)\)
\(T_{2}-T_{1}=M_{2} \times \dfrac{T_{1}}{M_{1}}(\) from eq (2))
\(T_{2}=\dfrac{T_{1} M_{2}}{M_{1}}+T_{1} \Rightarrow T_{2}=T_{1}\left(1+\dfrac{M_{2}}{M_{1}}\right)\)
\(\dfrac{T_{1}}{T_{2}}=\dfrac{M_{1}}{M_{1}+M_{2}}\)