According to Newton's Law of Cooling, the rate of cooling is proportional to the temperature difference between the body and the surroundings. For small temperature intervals, we use the average form: $\frac{T_1 - T_2}{t} = K \left[ \frac{T_1 + T_2}{2} - T_s \right]$

Case 1: Cooling from $90^{\circ}\text{C}$ to $80^{\circ}\text{C}$ in time $t$. $\frac{90 - 80}{t} = K \left[ \frac{90 + 80}{2} - 20 \right]$ $\frac{10}{t} = K$ $\frac{10}{t} = 65K \implies K = \frac{10}{65t} = \frac{2}{13t}$

Case 2: Cooling from $80^{\circ}\text{C}$ to $60^{\circ}\text{C}$ in time $t'$. $\frac{80 - 60}{t'} = K \left[ \frac{80 + 60}{2} - 20 \right]$ $\frac{20}{t'} = K$ $\frac{20}{t'} = 50K$

Substitute the value of $K$ from Case 1: $\frac{20}{t'} = 50 \left( \frac{2}{13t} \right)$ $\frac{20}{t'} = \frac{100}{13t}$ $t' = \frac{20 \times 13t}{100} = \frac{13}{5}t$