\(\frac{{dT}}{{dt}} = K\left( {T - {T_s}} \right)\)

For two cases,

\(\frac{{d{T_1}}}{{dt}} = K\left( {{T_1} - {T_s}} \right)\,and\,\frac{{d{T_2}}}{{dt}} = K\left( {{T_2} - {T_s}} \right)\)

\(Here,\,{T_s} = T,\,{T_1} = \frac{{3T + 2T}}{2} = 2.5\,T\)

\(and\frac{{d{T_1}}}{{dt}} = \frac{{3T -2T}}{{10}} = \frac{T}{{10}}\)

\({T_2} = \frac{{2T + T'}}{{2}}and\frac{{d{T_2}}}{{dt}} = \frac{{2T - T'}}{{10}}\)

\(So,\,\frac{T}{{10}} = K\left( {2.5\,T - T} \right)\)                     \(...(i)\)

\(\frac{{2T - T'}}{{10}} = K\left( {\frac{{2T + T'}}{2} - T} \right)\)                \(...(ii)\)

Dividing eqn. \((i)\) by eqn. \((ii)\), we get 

\(\frac{T}{{2T - T'}} = \frac{{\left( {2.5T - T} \right)}}{{\left( {\frac{{2T + T'}}{2} - T} \right)}}\)

\(\frac{{2T + T'}}{2} - T = \left( {2T - T'} \right) \times \frac{3}{2}\)

\(T' = 3\left( {2T - T'} \right)\,\,or,\,\,4T' = 6T\,\,\therefore \,\,T' = \frac{3}{2}T\)