Density of unknown solution? A glass vessel weighs $\pu{20.2367 g}$ when empty and $\pu{20.3102 g}$ when filled to the etched mark with water at $\pu{4 °C}$. The same vessel was then dried and filled to the same mark with a solution at $\pu{4 °C}$, the vessel was now found to weigh $\pu{20.3300 g}$. What is the density of the solution? (Assume the density of water is $\pu{1.00 g/ml}$)


Here’s what I did:

The mass of the water is just $\pu{0.0735 g}$ when you subtract the weight of (water + vessel) and (empty vessel). Then I found how many $\pu{ml}$ of water there was in the vessel by realizing that the density ($m/V$) must be equal to $1$, so $\pu{0.0735 g}$ of water must be $\pu{0.0735 ml}$ of water too. So with that, I figured out how many grams of the unknown solution were in the vessel ($\pu{0.0993 g}$), and knowing that it was filled to the same “etched mark,” we can say that the density of the unknown solution is $\pu{0.0933 g}/\pu{0.0735 ml} = \pu{1.27 g/ml}$. I went through all of that because I am not sure if my thoughts are on track with logic and I am pretty confused.
What you did in your explanation is correct. First find the difference in the weight:
$$20.3102-20.2367=0.0735 \ \mathrm{g}$$

We know that from your density of water:
$\mathrm{\frac{1\ g}{1\ mL}\ \text{that }\ 0.0735 \ g = 0.0735\ mL}$

Then the difference between the unknown and the vessel is calculated as:
$$20.300-20.2367 = 0.0933\ \mathrm{g}$$

and then density is expressed as mass over volume, so:
$$\mathrm{\frac{0.0933\ g}{0.0735\ mL}=1.2693_8\ g\ mL^{-1}}$$