Home experiment to measure specific gravity of liquids 
The problem states: invent to a way to measure specific gravity of various liquids in assorted bottles given
1 test tube
1 small vial of mercury
1 pen
1 ruler
1 beaker of water (500 ml)

I know this problem will involve Archimedes' Principle and that specific gravity is the ratio of density of liquid/ density of water. My assumption is that the density of water is known (997 kg/m^3).
My proposed solution is:
step 1: label starting point of water and hold ruler next to beaker
step 2: pour mercury into the tube and put the tube into the beaker of water and note the displacement of water with the pen.
The volume of water displaced times its density is equal to the mass of water displaced.
I am not sure where to go from here..... I could dump out the mercury and pour the liquids into the test tube one at a time to measure the change in height of water. But I wouldn't be able to know the volume of the liquids...
Thanks.
 A: Your suggestion would work but it would not be accurate. 
You would put the vial of mercury into the test tube to act as ballast. (Mercury is toxic so the vial should not be opened except with laboratory precautions.) Put the various liquids into the beaker one at a time, put the test tube containing the vial of mercury in the liquid, and measure the rise in the level of liquid in the beaker. According to Archimedes' Principle, if the test tube floats then the weight of the liquid displaced equals the weight of the test tube, which is constant. The only quantities which change are the density of liquid and the rise in its level in the beaker. Comparing the change in height for the liquids of unknown density with that for water will give you the relative density of the liquid.
The drawback of this method is that the beaker is quite wide so the rise in the liquid level in the beaker could be quite small and difficult to measure accurately. 
A better method, as used in the hyrdometer, is to compare the volume of the test tube which is below the liquid surface. This is also equal to the volume of liquid displaced. However, because the test tube is much narrower than the beaker, the depth of the test tube varies much more than the level in the beaker. You could glue a strip of card with a marked scale on it inside the test tube against which to measure the depth at which the tube floats.  
See also Measure liquids density: hydrometer.
