Waterways is the cheapest mode of transport among the given options. Here's why:

1. Cost of Infrastructure:
- Waterways mainly utilize natural water bodies such as rivers, canals, and seas, which do not require significant investment in infrastructure development.
- In contrast, roadways and railways require extensive construction and maintenance of roads, tracks, bridges, and tunnels, which results in higher costs.

2. Fuel Costs:
- Water transport generally uses fuel-efficient engines, such as diesel or LNG-powered vessels, which consume less fuel per ton-kilometer compared to road and rail transport.
- Road and rail transport, on the other hand, rely on vehicles that consume more fuel per unit of distance covered.

3. Capacity and Efficiency:
- Waterways have the advantage of accommodating large cargo vessels, barges, and ships that can carry a substantial amount of goods in a single trip.
- Railways also have a relatively high capacity, but roadways have limitations in terms of the size and weight of vehicles, resulting in the need for multiple trips to transport the same amount of goods.
- Water transport is also more energy-efficient compared to road and rail transport, as ships have a lower energy consumption per ton-kilometer.

4. Maintenance and Operational Costs:
- Water transport requires less maintenance and operational costs compared to road and rail transport.
- Roadways and railways need regular maintenance of tracks, signals, vehicles, and other infrastructure, leading to higher operational expenses.

5. External Costs:
- Waterways have fewer negative externalities compared to road and rail transport.
- Road transport contributes significantly to traffic congestion, air pollution, and accidents, which impose costs on society.
- Rail transport also has some external costs associated with land acquisition and noise pollution.
- Water transport, on the other hand, has minimal impact on congestion and pollution, making it a more environmentally friendly option.

In conclusion, waterways are the cheapest mode of transport due to lower infrastructure costs, fuel efficiency, larger capacity, lower maintenance and operational costs, and reduced negative externalities compared to road and rail transport.