The correct answer is option 3: 1 and 3 only.

Let`s analyze each statement to understand why the answer is option 3:

1. The density of the liquid: The fraction of a ball floating inside a liquid depends on the density of the liquid. This is because the buoyant force exerted on an object in a fluid depends on the density of the fluid. If the density of the liquid is greater than the density of the ball, the ball will float partially. If the density of the liquid is less than the density of the ball, the ball will sink.

2. The mass of the ball: The mass of the ball does not directly affect the fraction of the ball floating inside the liquid. The buoyant force exerted on the ball depends on its volume and the density of the liquid, not its mass.

3. The density of the ball: The fraction of a ball floating inside a liquid also depends on the density of the ball. As mentioned earlier, if the density of the ball is less than the density of the liquid, it will float partially. If the density of the ball is greater than the density of the liquid, it will sink.

Therefore, the correct statements are 1 (density of the liquid) and 3