Light propagates $2cm$ distance in glass of refractive index $1.5$ in time ${t_0}$. At the same time ${t_0}$, light propagates a distance of $2.25cm$ in a medium. The refractive index of the medium is:
A. $\dfrac{4}{3}$
B. $\dfrac{3}{2}$
C. $\dfrac{8}{3}$
D. none of these

Hint
First using the refractive index formula, calculate the value of ${t_0}$ for glass. Then using the very same formula also calculate the value of ${t_0}$ for the unknown medium. Finally equate the two values of time.
$\Rightarrow n = \dfrac{c}{v}$ where $c$is the velocity of light in vacuum and $v$ is the velocity of light in the given media.
$\Rightarrow v = \dfrac{d}{t}$ where $d$ is the distance travelled in time $t$ and $v$ is the velocity.

Complete step by step answer
When light travels from one medium to another its direction changes. This optical phenomenon is known as the refraction of light. Now the degree up to which light bends as it changes medium is measured by a factor known as the refractive index. The refractive index $n$ for a given medium is given as
$\Rightarrow n = \dfrac{c}{v}$ where $c$is the velocity of light in vacuum and $v$ is the velocity of light in the given media.
So the refractive index of a medium is essentially the ratio of the velocity of light in free space to its velocity in that given medium.
We are given that the refractive index of glass is $1.5$ and light travels a distance of $2cm$ in time ${t_0}$while travelling through it.
So, from here we can say that
$\Rightarrow n = \dfrac{c}{v} $
$\Rightarrow n = \dfrac{c}{{\dfrac{d}{t}}} = \dfrac{{c \times t}}{d} $
Substituting the values in this equation we get,
$\Rightarrow 1.5 = \dfrac{{c \times {t_0}}}{2} $
$\Rightarrow {t_0} = \dfrac{{1.5 \times 2}}{c} $
Now, when light travels through a different medium of refractive index say ${n_0}$, it travels a distance of $2.25cm$ in time ${t_0}$.
So, we have,
$\Rightarrow {n_0} = \dfrac{{c \times {t_0}}}{{2.25}} $
$\Rightarrow {t_0} = \dfrac{{{n_0} \times 2.25}}{c}$
Equating the two values of ${t_0}$ we get,
$\Rightarrow \dfrac{{{n_0} \times 2.25}}{c} = \dfrac{{1.5 \times 2}}{c}$
$\Rightarrow {n_0} \times 2.25 = 1.5 \times 2 $
$\Rightarrow {n_0} = \dfrac{{1.5 \times 2}}{{2.25}} $
$\Rightarrow {n_0} = \dfrac{4}{3} $
Therefore, the refractive index of the medium is $\dfrac{4}{3}$
So, the correct option is (A).

Note
When light travels from rarer to denser medium, its direction changes towards the normal and when it travels from denser to rarer medium, its direction changes away from the normal. There is no change in direction when it changes medium along the normal.