Weigh the beaker.
Add 50 milliliters of the solution you want to test to the beaker and record its mass. The mass of the solution is equal to the total mass minus the mass of the beaker.
Divide the mass of the solution by its volume -- in this case, 50 milliliters -- to find its density.
Empty, rinse and dry the beaker. Add 4.3 grams of sugar to the empty beaker, then add water until the contents reach 50 milliliters in volume. Stir until the sugar dissolves, then measure the mass of the sugar solution and its volume. Divide mass by volume to find density and record this figure.
Repeat this process, but now using 8.6 grams of sugar instead of 4.3; again, record the density. Repeat the process again with 17.1 grams and record the density. Repeat the process with 34.2 grams of sugar and record the density.
Write down your data in the form of a table, where density is your y-value and concentration is your x-value. The 4.3-gram solution is 0.025 molar, while the 8.6, 17.1 and 34.2-gram solutions are 0.05, 0.1 and 0.2 molar, respectively. This gives you four data points you can use to do linear regression.
Fit a line to your data using linear regression. If you have a spreadsheet program like Excel, you can do this easily in the program; otherwise, do it with a calculator and the following formulas:
Equation for the line: y = Bx + A
B = ( 4 'xy - 'x 'y ) / (4 'x squared - ( 'x)squared )
A = ( 'x squared 'y - 'x'xy ) / (4 'x squared - ( 'x)squared )
When calculating these formulas, remember that x is the concentration at each data point and y is the density. Sum up your x's and y's using the formula as shown to find your A and B and your equation.
Plug the density of your unknown solution in as y then solve for x. Your answer is the sugar concentration in moles per liter of the unknown solution.