The here is that you need to use the density of water to find the volume of the tube.
Since you know what the tube weighs when it is empty and when it is filled with water, which means that you can figure out the mass of water that's needed to fill it
#m_"filled" = m_"empty" + m_"water"#
#m_"water" = m_"filled" - m_"empty"#
#m_"water" = "105.6 g" - "46.3 g" = "59.3 g"#
Water's density at room temperature is
#59.3color(red)(cancel(color(black)("g"))) * "1 mL"/(0.9982071color(red)(cancel(color(black)("g")))) = "59.41 mL"#
The trick now is to realize tha the volume of the tube is the same for both liquids.
The mass of the second liquid is
#m_"liquid" = "96.72 g" - "46.3 g" = "50.42 g"#
Density is defined as mass per unit of volume, which means that the second liquid will have a density of
#rho = m/V#
#rho = "50.42 g"/"59.42 mL" = color(green)("0.849 g/mL")#
Now, does this result make sense? Think about it like this - if you have the same tube and fill it with exactly the same volume of water and mistery liquid, then you would expect the liquid that weight more to be denser.
Simply put, a denser liquid can pack more mass in the same volume. Since the same volume of liquid is heavier for water than for the mistery liquid, it follows that water's denser.