In an amplitude modulation, a modulating signal having amplitude of...

Modulating index is stated as $\mu = \dfrac{{A}_{m}}{{A}_{c}}$, where, ${A}_{m} =$amplitude of modulating signal and ${A}_{c} =$amplitude of carrier signal.

Now, for first case, ${\mu}_{1} = \dfrac{X}{Y}$

For second case, ${\mu}_{2} = \dfrac{X}{2Y}$

Thus, the ratio of $\dfrac{{\mu}_{1}}{{\mu}_{2}} = \dfrac{\dfrac{X}{Y}}{\dfrac{X}{2Y}} = \dfrac{2}{1} ⇒ {\mu}_{1}: {\mu}_{2} = 2: 1$