**Mole Fraction of Solute**
The mole fraction of a solute in a solution is defined as the ratio of the number of moles of the solute to the total number of moles in the solution. Mathematically, it is represented as:

Mole fraction of solute (Xsolute) = moles of solute / total moles in solution

Given that the mole fraction of the solute is 0.1, we can write:

0.1 = moles of solute / total moles in solution

**Molarity and Molality**
Molarity (M) is defined as the number of moles of solute present in one liter of solution. On the other hand, molality (m) is defined as the number of moles of solute present per kilogram of solvent.

In this case, it is given that at 298K, the molarity is the same as molality. Therefore:

Molarity (M) = molality (m)

**Density of the Solution**
Density (ρ) is defined as the mass per unit volume of a substance. It is given that the density of the solution is 2.0 g/cm³.

**Calculating Molecular Weight Ratio**
To find the ratio of the molecular weight of the solute to the solvent, we can use the following steps:

Step 1: Convert the density of the solution from g/cm³ to kg/m³.
Density (ρ) = 2.0 g/cm³ = 2000 kg/m³ (since 1 g/cm³ = 1000 kg/m³)

Step 2: Calculate the mass of the solution.
Mass of the solution = density × volume
Let's assume the volume of the solution is V. Therefore:
Mass of the solution = 2000 kg/m³ × V

Step 3: Calculate the number of moles of the solute using the molarity.
Moles of solute = Molarity × volume
Since the molarity is the same as molality in this case, we can write:
Moles of solute = molality × volume

Step 4: Calculate the number of moles of the solvent.
Moles of solvent = Moles of solute / Mole fraction of solute
Using the given mole fraction of solute (0.1) and the moles of solute calculated in Step 3, we can write:
Moles of solvent = moles of solute / 0.1

Step 5: Calculate the mass of the solvent.
Mass of solvent = Moles of solvent × molecular weight of solvent

Step 6: Calculate the ratio of the molecular weight of the solute to the solvent.
Ratio = Molecular weight of solute / Molecular weight of solvent

By following these steps, you should be able to calculate the desired ratio of the molecular weight of the solute to the solvent.