A body is floating in a liquid partly immersed. The body and the liquid are taken to the moon. The body

According to the principle of floatation, a floating body displaces a weight of liquid equal to its own weight. Let $V_{\text{in}}$ be the volume of the body immersed, ${\rho }_{\text{body}}$ be the density of the body, ${\rho }_{\text{liquid}}$ be the density of the liquid, $V$ be the total volume of the body, and $g$ be the acceleration due to gravity:
$$V_{\text{in}}\cdot {\rho }_{\text{liquid}}\cdot g=V\cdot {\rho }_{\text{body}}\cdot g$$

Canceling $g$ from both sides gives:

$$\frac{V_{\text{in}}}{V}=\frac{{\rho }_{\text{body}}}{{\rho }_{\text{liquid}}}$$

Because the fraction of volume immersed depends purely on the ratio of the densities of the body and the liquid, changing the ambient gravity (taking it to the moon where gravity is $\frac{1}{6}$ of Earth's) does not affect the fraction immersed. Thus, it will continue to float as before.