A truck covers 550 metres in one minute, while a bus covers 33 kilometres in three quarters of an hour. What is the ratio of the speed of the truck to the speed of the bus?

Introduction / Context:
This is a speed comparison problem involving a truck and a bus travelling different distances in different times. We are asked to compute the ratio of their speeds. To compare speeds correctly, they must be expressed in the same units, for example kilometres per hour. Once both speeds are converted to a common unit, forming the ratio is straightforward.

Given Data / Assumptions:
- The truck covers 550 metres in one minute.
- The bus covers 33 kilometres in three quarters of an hour which is 45 minutes.
- We need the ratio speed of truck : speed of bus.

Concept / Approach:
Speed is distance divided by time. The truck speed is given in metres per minute and the bus speed in kilometres per hour, so we convert both to kilometres per hour for an easy comparison. After conversion, we will have speeds S_truck and S_bus in the same units. The required ratio is then S_truck : S_bus, which we simplify to smallest integer terms.

Step-by-Step Solution:
Step 1: Truck distance = 550 metres in 1 minute. Step 2: Convert 550 metres to kilometres: 550 metres = 0.55 kilometres. Step 3: Time for the truck is 1 minute, which is 1/60 hour. Step 4: Truck speed = distance / time = 0.55 / (1/60) = 0.55 * 60 = 33 km per hour. Step 5: Bus distance = 33 kilometres in three quarters of an hour, which is 0.75 hour. Step 6: Bus speed = 33 / 0.75 = 44 km per hour. Step 7: Speed ratio truck : bus = 33 : 44. Step 8: Simplify 33 : 44 by dividing both terms by 11 to get 3 : 4.

Verification / Alternative check:
We can also work in metres per minute for both vehicles and confirm the same ratio. The truck speed is 550 metres per minute. The bus travels 33,000 metres in 45 minutes, so its speed is 33,000 / 45 = 733.33 metres per minute approximately. Then the ratio 550 : 733.33 approximately simplifies to 3 : 4, consistent with the km per hour method. This double check confirms the correctness of our ratio.

Why Other Options Are Wrong:
1 : 3, 2 : 3 and 1 : 4 represent much smaller truck speed relative to the bus than what the calculations show. Since the truck speed is 33 km per hour and the bus speed is 44 km per hour, the fraction is 33/44 which equals 3/4, not any of the other options. Therefore the only correct ratio is 3 : 4.

Common Pitfalls:
A common error is mixing units, for example comparing metres per minute directly with kilometres per hour without conversion. Another mistake is to mis convert three quarters of an hour as 3/4 minutes instead of 45 minutes or 0.75 hour. Careful unit conversion and consistent use of either kilometres per hour or metres per minute avoid these issues.

Final Answer:
The ratio of the speed of the truck to the speed of the bus is 3 : 4.