Two buses travel the same route to a destination at speeds of 45 km/h and 60 km/h. If the faster bus takes 5.5 hours less than the slower bus, what is the total length of the journey?

Introduction / Context:
This problem is similar to earlier time and distance questions with two vehicles moving at constant but different speeds over the same distance. The important information is the time difference between the two journeys. Converting this information into an algebraic equation is the key step toward finding the length of the journey.

Given Data / Assumptions:
Speed of the slower bus = 45 km/h.
Speed of the faster bus = 60 km/h.
Both buses cover the same distance from the starting city to the destination.

Given Data / Assumptions:
The faster bus takes 5.5 hours less time than the slower bus to cover the journey.
We are required to find the length of the journey in kilometres.

Concept / Approach:
Let the distance be D km. Then time taken by the slower bus is D / 45 hours and by the faster bus is D / 60 hours. The condition given is that the difference between these two times equals 5.5 hours. This gives the equation D / 45 − D / 60 = 5.5. Solving this equation for D gives the required distance.

Step-by-Step Solution:
Step 1: Let the journey length be D km.Step 2: Time taken by the slower bus = D / 45 hours.Step 3: Time taken by the faster bus = D / 60 hours.Step 4: According to the problem, D / 45 − D / 60 = 5.5.Step 5: Factor out D: D * (1 / 45 − 1 / 60) = 5.5.Step 6: Compute 1 / 45 − 1 / 60 = (60 − 45) / (45 * 60) = 15 / 2700.Step 7: Simplify 15 / 2700 to 1 / 180.Step 8: So D * (1 / 180) = 5.5.Step 9: Therefore D = 5.5 * 180.Step 10: 5.5 * 180 = 5 * 180 + 0.5 * 180 = 900 + 90 = 990 km.

Verification / Alternative check:
For D = 990 km, time for slower bus = 990 / 45 = 22 hours. Time for faster bus = 990 / 60 = 16.5 hours. The difference in time is 22 − 16.5 = 5.5 hours, which is exactly the difference stated in the problem. This confirms that the distance 990 km is correct.

Why Other Options Are Wrong:
900 km gives times of 900 / 45 and 900 / 60, whose difference is not 5.5 hours.
940 km and 780 km also do not yield the required 5.5 hour time difference when substituted into the time expressions.
1300 km is too large and results in a time difference much greater than 5.5 hours.

Common Pitfalls:
One typical error is to subtract speeds instead of times, which leads to a wrong relationship between the quantities. Another mistake is incorrect handling of the mixed decimal 5.5 hours, such as treating it as 5 hours and 50 minutes rather than 5 and a half hours. Carefully expressing the given condition and solving the linear equation step by step prevents such issues.

Final Answer:
The length of the journey is 990 km.