Let $m$ be the mass of an object flying with velocity $v$ in air. When it split into two pieces of masses in ratio $1: 5$, the mass of smaller piece is $\mathrm{m} / 6$ and of bigger piece is $\frac{5 \mathrm{~m}}{6}$.
This situation can be interpreted diagrammatically as below




As the object breaks in two pieces, so the momentum of the system will remains conserved i.e. the total momentum (before breaking) = total momentum (after breaking)
$$
\begin{aligned}
& \mathrm{mv}=\frac{\mathrm{m}}{6} \mathrm{v}_1+\frac{5 \mathrm{~m}}{6} \mathrm{v}_2 \Rightarrow \mathrm{v}=\frac{\mathrm{v}_1}{6}+\frac{5 \mathrm{v}_2}{6} \\
& \text { Here, } \quad \begin{aligned}
\mathrm{v} & =20 \hat{\mathrm{i}}+25 \hat{\mathrm{j}}-12 \hat{\mathrm{k}} \\
\mathrm{v}_1 & =100 \hat{\mathrm{i}}+35 \hat{\mathrm{j}}+8 \hat{\mathrm{k}}
\end{aligned}
\end{aligned}
$$

$$
\begin{array}{rlrl}
\Rightarrow & 20 \hat{\mathrm{i}}+25 \hat{\mathrm{j}}-12 \hat{\mathrm{k}} & =\frac{(100 \hat{\mathrm{i}}+35 \hat{\mathrm{j}}+8 \hat{\mathrm{k}})}{6}+\frac{5 \mathrm{v}_2}{6} \\
\Rightarrow & (120 \hat{\mathrm{i}}+150 \hat{\mathrm{j}}-72 \hat{\mathrm{k}}) & =(100 \hat{\mathrm{i}}+35 \hat{\mathrm{j}}+8 \hat{\mathrm{k}})+5 \mathrm{v}_2 \\
\Rightarrow & \mathrm{v}_2 & =\frac{1}{5}(20 \hat{\mathrm{i}}+115 \hat{\mathrm{j}}-80 \hat{\mathrm{k}}) \\
& =4 \hat{\mathrm{i}}+23 \hat{\mathrm{j}}-16 \hat{\mathrm{k}}
\end{array}
$$