Two cars travel from city A to city B. One car moves at 30 km/h and the other at 36 km/h. The faster car takes 3 hours less than the slower car to cover the journey. What is the distance between city A and city B (in km)?

Introduction / Context:
This problem involves two vehicles travelling the same route at different speeds with a specified time difference between their journeys. It tests whether a candidate can relate speed, time, and distance through the simple formula distance = speed * time and construct an equation from a verbal description. By expressing both travel times in terms of the same distance, we can solve for that distance using basic algebra.

Given Data / Assumptions:

Speed of the slower car = 30 km/h.
Speed of the faster car = 36 km/h.
The faster car takes 3 hours less than the slower car for the same journey.
Let the distance between city A and city B be D km.
The cars travel at constant speeds along the same route.

Concept / Approach:
If the slower car takes time t hours, then the faster car takes t - 3 hours. Using time = distance / speed, we write one equation for each car in terms of D and t. Since both distances are the same and equal to D, we set 30 * t equal to 36 * (t - 3). Solving this equation yields t, and then D is found by multiplying 30 * t or 36 * (t - 3). This approach directly aligns with the relationship linking distance, speed and time.

Step-by-Step Solution:
Step 1: Let the time taken by the slower car be t hours. Then, time taken by the faster car = t - 3 hours. Step 2: Write expressions for the distance. For the slower car: D = 30 * t. For the faster car: D = 36 * (t - 3). Step 3: Equate the two expressions for D. 30 * t = 36 * (t - 3). Step 4: Solve for t. 30t = 36t - 108. Bring like terms together: 108 = 36t - 30t = 6t. t = 108 / 6 = 18 hours. Step 5: Find D. D = 30 * t = 30 * 18 = 540 km.

Verification / Alternative check:
Check the faster car's travel time: t - 3 = 18 - 3 = 15 hours. Distance covered at 36 km/h in 15 hours is 36 * 15 = 540 km, which matches the distance computed with the slower car. Thus, both cars travel the same distance, confirming D = 540 km as consistent with all given conditions.

Why Other Options Are Wrong:
If the distance were 648 km, then the slower car at 30 km/h would take 21.6 hours, and the faster car at 36 km/h would take 18 hours, giving a difference of 3.6 hours, not 3. Similarly, 810 km and 432 km do not produce a 3 hour difference at these speeds. Only 540 km yields exactly 3 hours difference between 18 and 15 hours.

Common Pitfalls:
Some students mistakenly set up the equation as 30 * (t - 3) = 36 * t, reversing the time difference. Others misinterpret the phrase "takes 3 hours lesser time" and subtract 3 from the slower car's time instead of the faster car's time. Carefully reading and translating the word description into algebra is crucial. Always verify which car is faster and which time must be smaller.

Final Answer:
The distance between city A and city B is 540 km.