To design a gear reducer, start by calculating the gear ratio using the speeds provided, which can be expressed as the inverse of the speed ratio. For example, if the input speed is 500 rpm and the output speed is 250 rpm, the velocity ratio is 1/2, allowing for various gear combinations to achieve this ratio. Consider factors like power requirements, torque loads, and physical constraints when determining suitable pitch and diameter for the gears. Reference materials like "Shigley's Mechanical Engineering" can provide guidance on power input and chain sizing, particularly in sections related to roller chains and spur gearing. Understanding these principles is essential for effective gear design.