If the absolute refractive indices of two media X and Y are $\frac{6}{5}$ and $\frac{4}{3}$ respectively, then the refractive index of Y with respect to X will be:

The chapter defines the refractive index of medium 2 with respect to medium 1 as $n_{21}=\dfrac{v_1}{v_2}$, and the absolute refractive index of a medium as $n_m=\dfrac{c}{v}$ (speed of light in air $\div$ speed of light in the medium), so $v=\dfrac{c}{n_m}$.

Given $n_X=\dfrac{6}{5}$ and $n_Y=\dfrac{4}{3}$.

Refractive index of Y with respect to X:

$$n_{YX}=\frac{v_X}{v_Y}=\frac{c/n_X}{c/n_Y}=\frac{n_Y}{n_X}=\frac{4/3}{6/5}=\frac{4}{3}\times\frac{5}{6}=\frac{20}{18}=\frac{10}{9}$$

(A) is correct: direct application of $n_{YX}=n_Y/n_X$.

(B) $\frac{9}{10}$ is the reciprocal — this would be $n_{XY}$ (refractive index of X w.r.t. Y), a common inversion error.

(C) $\frac{9}{8}$ results from wrongly multiplying $n_X\times n_Y$ instead of taking the ratio.

(D) $\frac{8}{9}$ comes from inverting both the ratio and swapping numerator/denominator incorrectly.