Two cars travel from city A to city B at speeds of 36 km/h and 54 km/h respectively. One car takes 3.5 hours less than the other to cover the journey. What is the distance (in km) between city A and city B?

Introduction / Context:
This time and distance problem involves two cars traveling the same route at different constant speeds. The time difference between their arrivals is known, and the speeds are known, but the distance is not. The question tests your ability to use the relationship between time, speed and distance and set up an equation using the difference in times.

Given Data / Assumptions:

• Speed of slower car = 36 km/h.
• Speed of faster car = 54 km/h.
• Both travel from city A to city B over the same distance.
• The faster car takes 3.5 hours less than the slower car.
• Time difference = 3.5 hours = 7 / 2 hours.

Concept / Approach:
Let the distance between the two cities be D km. Time taken by the slower car is D / 36 hours, and time taken by the faster car is D / 54 hours. The slower car takes 7 / 2 hours more than the faster car. This gives the equation D / 36 - D / 54 = 7 / 2. Solving this equation yields the value of D.

Step-by-Step Solution:
Let distance between A and B be D km. Time taken by slower car = D / 36 hours. Time taken by faster car = D / 54 hours. Given: D / 36 - D / 54 = 7 / 2. Take LCM of 36 and 54 which is 108. Rewrite: (3D - 2D) / 108 = 7 / 2. So D / 108 = 7 / 2. Therefore, D = 108 * 7 / 2 = 54 * 7 = 378 km.

Verification / Alternative check:
If distance is 378 km, time for slower car = 378 / 36 = 10.5 hours. Time for faster car = 378 / 54 = 7 hours. Time difference = 10.5 - 7 = 3.5 hours, which matches the given condition exactly. Thus the distance is consistent.

Why Other Options Are Wrong:

• 454 km, 567 km and 302 km produce time differences different from 3.5 hours when divided by 36 and 54 respectively.

Common Pitfalls:
Some students mistakenly add the times or set up the equation with reversed signs. Others mis-handle the fraction 3.5 hours or its conversion into 7 / 2. Accurate algebraic manipulation and careful use of LCM avoid these issues.

Final Answer:
The distance between city A and city B is 378 km.