A straight vertical conductor carries a current. At a point 5cm due north of it, the magnetic induction is found to be $20\mu T$ due east. The magnetic induction at a point 10cm east of it will be

As the wire is vertical and the direction of magnetic induction a  point north of the wire is towards east, we can conclude that the direction of current is along negative z-axis.

$\mathrm{B}=\frac{{\mathrm{\mu }}_0\mathrm{i}}{2\mathrm{\pi r}}\Rightarrow \mathrm{B}\propto \frac{1}{\mathrm{r}}$

$\frac{{\mathrm{B}}_1}{{\mathrm{B}}_2}=\frac{{\mathrm{r}}_2}{{\mathrm{r}}_1}\Rightarrow \frac{20\mathrm{\mu T}}{{\mathrm{B}}_2}=\frac{10}{5}\Rightarrow {\mathrm{B}}_2=10\mathrm{\mu T}$

As we know the direction of the current is along negative z-axis, the direction of magnetic induction at a point 10cm east of the wire will be towards south.

So the final magnetic induction at a point 10cm east of the wire will be $10\mathrm{\mu T} \mathrm{due} \mathrm{south}$