The magnification created when an object is placed 30 cm away from a convex mirror is 1/2. What position should the object be in to get a 1/4 magnification?

Object distance, u₁ = -30 cm (The object distance is always taken as negative.)
Magnification, m₁ = 1/2
Magnification, m₂ = 1/4
Object distance, u₂  =?

Magnification is given by the negative ratio of image distance to object distance.

Magnification = $\frac{-\mathrm{Image} \mathrm{distance} }{\mathrm{Object} \mathrm{distance}}=\frac{-\mathrm{v}}{\mathrm{u}}$

For magnification, m₁ = 1/2

$\frac{1}{2}=\frac{-{\mathrm{v}}_1}{-30}$

$\Rightarrow {\mathrm{v}}_1=\frac{30}{2}=15 \mathrm{cm}$

$\mathrm{Mirror} \mathrm{formula}: \frac{1}{\mathrm{f}}=\frac{1}{\mathrm{v}}+\frac{1}{\mathrm{u}}$

$\therefore \frac{1}{\mathrm{f}}=\frac{1}{15}+\frac{1}{(-30)}= \frac{1}{30}$

$\therefore \mathrm{f}= 30 \mathrm{cm}$

For magnification, m₂ = 1/4

$\frac{1}{4}=\frac{-{\mathrm{v}}_2}{{\mathrm{u}}_2}$

$\Rightarrow {\mathrm{v}}_2=\frac{-{\mathrm{u}}_2}{4}$

Substituting in the mirror formula, we get,
$\frac{1}{30}=\frac{1}{-{\displaystyle \frac{{\mathrm{u}}_2}{4}}}+\frac{1}{{\mathrm{u}}_2}$

$\Rightarrow \frac{1}{30}=\frac{-4 +1}{{\displaystyle {\mathrm{u}}_2}}=\frac{-3}{{\displaystyle {\mathrm{u}}_2}}$

$\therefore {\mathrm{u}}_2= -90 \mathrm{cm}$
 Therefore, when the magnification is 1/4 the object distance should be -90 cm.