Why are gear ratios critical in a helicopter's transmission system?

Gear ratios are critical in a helicopter's transmission system because they allow the engine, which typically operates at high RPM (revolutions per minute), to drive the main and tail rotors at their optimal, much lower RPM. Turboshaft engines are designed to produce maximum power at high rotational speeds. However, the rotor blades need to rotate at a significantly slower speed to maintain aerodynamic efficiency and prevent exceeding their structural limits. The transmission system uses a series of gears with different sizes to reduce the engine's high RPM to the appropriate RPM for the main and tail rotors. For example, an engine might operate at 20,000 RPM, while the main rotor needs to rotate at around 300-500 RPM. The gear ratios within the transmission system accomplish this significant speed reduction. Without these gear ratios, the rotor blades would either not generate enough lift (if the engine RPM was too low) or would be damaged or destroyed due to excessive speed (if the engine RPM was directly applied to the rotors). The transmission also splits the engine's power to drive both the main and tail rotors simultaneously.