Introduction:
In simultaneous heat and mass transfer, both heat and mass are being transferred between two mediums. The heat transfer coefficient and mass transfer coefficient are used to quantify the rates of heat transfer and mass transfer, respectively. The ratio of these coefficients is a function of the ratio of certain dimensionless numbers that characterize the flow and transport processes.

Explanation:
The heat transfer coefficient is related to the Nusselt number (Nu) and the mass transfer coefficient is related to the Sherwood number (Sh). These dimensionless numbers are defined as follows:

1. Nusselt number (Nu): It represents the ratio of convective heat transfer to conductive heat transfer across a fluid boundary layer. It is given by the formula:
Nu = hL/k
where h is the heat transfer coefficient, L is a characteristic length, and k is the thermal conductivity of the fluid.

2. Sherwood number (Sh): It represents the ratio of convective mass transfer to diffusive mass transfer across a fluid boundary layer. It is given by the formula:
Sh = kL/D
where k is the mass transfer coefficient, L is a characteristic length, and D is the diffusivity of the species being transferred.

The ratio of heat transfer coefficient to mass transfer coefficient (h/k) can be expressed as the ratio of Nusselt number to Sherwood number (Nu/Sh).

To determine the relationship between the ratio of heat transfer coefficient to mass transfer coefficient and the ratio of dimensionless numbers, we need to consider the relevant dimensionless numbers:

1. Reynolds number (Re): It represents the ratio of inertial forces to viscous forces in a fluid flow. It is given by the formula:
Re = ρVL/μ
where ρ is the density of the fluid, V is the velocity of the fluid, L is a characteristic length, and μ is the dynamic viscosity of the fluid.

2. Schmidt number (Sc): It represents the ratio of momentum diffusivity to mass diffusivity in a fluid. It is given by the formula:
Sc = μ/ρD
where μ is the dynamic viscosity of the fluid, ρ is the density of the fluid, and D is the diffusivity of the species being transferred.

Conclusion:
The ratio of heat transfer coefficient to mass transfer coefficient (h/k) is a function of the ratio of Schmidt number to Reynolds number (Sc/Re). Therefore, option 'A' is the correct answer. The relationship between these dimensionless numbers determines the relative rates of heat transfer and mass transfer in simultaneous heat and mass transfer processes.