A sum of money lent at simple interest becomes double of itself in $8$ years. Then the sum will triple itself in:

Let the initial principal amount be $P$.
If the sum doubles itself, the amount becomes $2P$.

$$\text{Simple Interest earned }({\text{SI}}_1)=\text{Amount}-\text{Principal}=2P-P=P$$

This interest is accumulated in $T_1=8$ years.

For the sum to triple itself, the target amount becomes $3P$.

$$\text{Required Simple Interest }({\text{SI}}_2)=3P-P=2P$$

Since Simple Interest ($\text{SI}$) is directly proportional to time ($T$) when principal and rate are constant:

$$\frac{{\text{SI}}_1}{{\text{SI}}_2}=\frac{T_1}{T_2}$$

$$\frac{P}{2P}=\frac{8}{T_2}$$

$$\frac{1}{2}=\frac{8}{T_2}⟹T_2=16\text{ years}$$