Two cars travel from city A to city B along the same route. One car moves at 24 km/h and the other moves at 32 km/h. If the faster car takes exactly 2.5 hours less time than the slower one to complete the journey, what is the distance (in kilometres) between city A and city B?

Introduction / Context:
This question is very similar in structure to other time and distance comparison problems involving two vehicles travelling the same route at different speeds. The relationship between speed, time, and distance is used, along with the given difference in time, to determine the unknown distance between two cities.

Given Data / Assumptions:

Speed of the slower car = 24 km/h.
Speed of the faster car = 32 km/h.
Both cars cover the same distance from city A to city B.
The faster car takes 2.5 hours less than the slower car.
Speeds are constant and there are no breaks or changes in speed.
We need to find the distance in kilometres between the two cities.

Concept / Approach:
Let the common distance be D kilometres. The time taken by the slower car is D / 24 hours, and the time taken by the faster car is D / 32 hours. The difference between these times is given as 2.5 hours. Setting up the equation D / 24 − D / 32 = 2.5 allows us to solve for D. This type of equation uses basic algebra and fraction manipulation to find the unknown distance.

Step-by-Step Solution:
Step 1: Let the distance between city A and city B be D kilometres. Step 2: Time taken by the slower car = D / 24 hours. Step 3: Time taken by the faster car = D / 32 hours. Step 4: According to the question, D / 24 − D / 32 = 2.5. Step 5: Compute the difference in fractions: 1 / 24 − 1 / 32 = (32 − 24) / (24 * 32) = 8 / 768 = 1 / 96. Step 6: So D * (1 / 96) = 2.5, which implies D = 2.5 * 96. Step 7: Multiply 2.5 * 96 = (5 / 2) * 96 = 5 * 48 = 240. Step 8: Hence, D = 240 kilometres.

Verification / Alternative check:
Check the times for D = 240 km. Time for the slower car: 240 / 24 = 10 hours. Time for the faster car: 240 / 32 = 7.5 hours. The difference in times is 10 − 7.5 = 2.5 hours, which exactly matches the information in the question. Therefore, the distance of 240 km is correct.

Why Other Options Are Wrong:
For 288 km, times would be 288 / 24 = 12 hours and 288 / 32 = 9 hours, giving a 3 hour difference. For 360 km, the difference would be even larger. A distance of 192 km would give times of 8 hours and 6 hours, which differ by 2 hours. A distance of 320 km yields a difference that is not 2.5 hours. Only 240 km satisfies the exact 2.5 hour difference condition.

Common Pitfalls:
A common mistake is to attempt to use average speed or proportional reasoning without properly forming the time difference equation. Another error is miscalculating the fraction 1 / 24 − 1 / 32 or handling 2.5 incorrectly as a fraction. Writing 2.5 as 5 / 2 and proceeding step by step with fractional arithmetic helps avoid such mistakes.

Final Answer:
The distance between city A and city B is 240 kilometres.