Question

A bus travels \(87.5 \mathrm{km}\) to another town at a speed of \(72.0 \mathrm{km} / \mathrm{h} .\) What must be its return rate if the total time for the round trip is to be 2.50 hours?

Short answer

The return rate must be 87.5 km divided by the remainder of the total trip time (2.5 hours) after subtracting the time spent on the outbound trip.

Step-by-step solution

Calculate the time taken to reach the town

To find the time for the outbound trip, use the formula for time, which is time = distance / speed. The distance to the town is 87.5 km and the speed is 72.0 km/h. Thus time = 87.5 km / 72.0 km/h.

Subtract the outbound trip time from the total time

Once the time for the outbound trip is calculated, subtract it from the total round trip time of 2.50 hours to find the time left for the return trip.

Calculate the return speed

Use the time remaining for the return trip and the formula speed = distance / time. The distance is the same (87.5 km) and the time is the remainder from step 2.

Key concepts

Understanding the Time-Distance-Speed Relationship

One of the fundamental concepts in motion-related problems is understanding the relationship between time, distance, and speed. This trio of variables is interconnected, where the speed is how fast something is moving, distance is how much ground is covered, and time is the duration of the travel. The relationship can be simplified in the formula: \[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \] Or, rearranged depending on the variable you need to find: \[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \] and \[ \text{Distance} = \text{Speed} \times \text{Time} \]
With this relationship, solving various travel problems becomes more straightforward. In our bus problem, by knowing the distance and speed, we easily find the time for the outbound trip by dividing the distance by speed. It's essential to ensure that units are consistent, for instance, using hours for time and kilometers per hour for the speed.

Round Trip Calculations

A round trip involves going from an origin point to a destination and then returning back to the original starting point. The total time spent on a round trip is the sum of the time taken to travel to the destination and the time taken to return.
When tackling round trip problems, a key step is determining the time spent on each leg of the journey separately before dealing with the return rate or speed. Usually, you are provided with the overall round trip time, like in our bus example where it's 2.50 hours. After finding the time taken for the outbound trip, you subtract it from the total to find the remainder for the return trip.

Algebraic Problem Solving

Algebraic problem solving is all about formulating and solving equations based on the problems' conditions. It allows us to find unknown values by using known quantities. In our exercise, once we know the time taken for the outward part of the trip, which is an algebraic manipulation itself (\[ \text{Time Outbound} = \frac{\text{Distance}}{\text{Speed Outbound}} \]), we need to employ further algebraic steps. First, ascertain the total time remaining for the round trip (\[ \text{Time Remaining} = \text{Total Round Trip Time} - \text{Time Outbound} \]). Finally, solve for the unknown return rate using the formula for speed, where \[ \text{Speed Return} = \frac{\text{Distance}}{\text{Time Remaining}} \].
The use of algebra simplifies the process of problem solving, allowing us to fill in the values we know and find those we don't. Algebra teaches us to isolate variables and rearrange equations, which is a crucial skill in not just bus travel rate problems but in a wide array of scientific and mathematical problems.