Step-by-step Solution:

1. Analyze the given distribution curve: The problem provides a plot of the fraction of molecules versus molecular speed. We need to determine the ratio of the most probable speed ($u_{mp}$), the average speed ($u_{av}$), and the root mean square speed ($u_{rms}$) from the shape of this curve.

   The key features of the given distribution curve are:

   • Symmetry: The curve is symmetric about its peak. It resembles a normal (Gaussian) distribution rather than the skewed Maxwell-Boltzmann distribution.
   • Finite Width: The curve is not infinitely sharp (i.e., not a Dirac delta function). It has a definite spread, which means there is a variance in the molecular speeds.

2. Relate the curve's properties to the speeds:

   • Most Probable Speed ($u_{mp}$): This is the speed possessed by the maximum fraction of molecules. Graphically, it corresponds to the peak of the distribution curve. Let's denote this speed as $u_0$.

   • Average Speed ($u_{av}$): This is the arithmetic mean of the speeds of all molecules. For any symmetric probability distribution, the mean (average value) is equal to the mode (the most probable value). Since the given distribution is symmetric, we have: $u_{av} = u_{mp} = u_0$

   • Root Mean Square Speed ($u_{rms}$): This is the square root of the mean of the squares of the molecular speeds. The relationship between the root mean square value, the mean value, and the variance ($"sigma^2"$) of a distribution is given by: $u_{rms}^2 = u_{av}^2 + \sigma^2$ Since the curve has a finite width, the distribution of speeds has a non-zero variance ($\sigma^2 > 0$). Therefore: $u_{rms}^2 > u_{av}^2$ $u_{rms} > u_{av}$

3. Determine the ratio: From our analysis, we have the relationship: $u_{mp} = u_{av} < u_{rms}$ The ratio of the speeds is: $u_{mp} : u_{av} : u_{rms} = u_0 : u_0 : u_{rms}$ Dividing by $u_{mp} (= u_0)$, we get: $1 : 1 : \frac{u_{rms}}{u_{mp}}$ Since $u_{rms} > u_{mp}$, the third term in the ratio must be greater than 1. So, the ratio must be of the form $1 : 1 : x$, where $x > 1$.

4. Evaluate the given options:

   • A: 1 : 1 : 1: This implies $u_{mp} = u_{av} = u_{rms}$. This would only be true if the distribution had zero width ($\sigma = 0$), which contradicts the graph. So, A is incorrect.
   • B: 1 : 1 : 1.224: This ratio is of the form $1 : 1 : x$ where $x = 1.224 > 1$. This is consistent with our deductions from the graph.
   • C: 1 : 1.128 : 1.224: This implies $u_{mp} \neq u_{av}$. This is the ratio for a standard Maxwell-Boltzmann distribution, which is skewed, not symmetric. The given graph is symmetric, so C is incorrect.
   • D: 1 : 1.128 : 1: This implies $u_{mp} \neq u_{av}$ and $u_{rms} = u_{mp}$, which contradicts both properties of the graph. So, D is incorrect.

5. Conclusion: The only option that correctly reflects the properties of the given symmetric distribution with a finite width is B. The distribution is symmetric, so $u_{mp} = u_{av}$. The distribution has a finite width, so $u_{rms} > u_{av}$.

Therefore, the correct ratio is $1 : 1 : 1.224$.