Two cars travel from city A to city B at speeds of 30 km/h and 44 km/h respectively. If one car takes 3.5 hours less than the other, what is the distance between city A and city B?

Introduction / Context:
This time and distance problem compares travel times of two cars between the same two cities. One car travels at 30 km/h and the other at 44 km/h. We are told that one car takes 3.5 hours less than the other, and we must find the distance between city A and city B. Such questions test your ability to set up equations using speed, distance and time relationships and to handle fractional time differences correctly.

Given Data / Assumptions:

• Speed of the slower car = 30 km/h.
• Speed of the faster car = 44 km/h.
• Both cars travel the same distance between city A and city B.
• Difference in their travel times = 3.5 hours.
• Motion is along the same route with constant speeds.

Concept / Approach:
Let the distance between the cities be D km. For each car, time taken is distance divided by speed. If the slower car takes more time, the difference in times is:
D / 30 - D / 44 = 3.5 We solve this equation for D. The main idea is to use a single variable for the distance and relate the two times through the given time difference.

Step-by-Step Solution:
Step 1: Write the time expressions. Time taken by car 1 (30 km/h) = D / 30 hours. Time taken by car 2 (44 km/h) = D / 44 hours. Step 2: Use the time difference condition. D / 30 - D / 44 = 3.5. Step 3: Simplify the left-hand side using a common denominator. Common denominator of 30 and 44 is 660. D / 30 = (22D) / 660 and D / 44 = (15D) / 660. So (22D / 660) - (15D / 660) = 3.5. This gives (7D / 660) = 3.5. Step 4: Solve for D. 7D = 660 * 3.5. 3.5 = 7 / 2, so 660 * 3.5 = 660 * (7 / 2) = (660 * 7) / 2 = 4620 / 2 = 2310. Therefore 7D = 2310, so D = 2310 / 7 = 330 km.
Verification / Alternative check:
Check the times for D = 330 km:
Time at 30 km/h = 330 / 30 = 11 hours. Time at 44 km/h = 330 / 44 = 7.5 hours. Difference in times = 11 - 7.5 = 3.5 hours, which matches the given condition. So D = 330 km is confirmed.

Why Other Options Are Wrong:
396 km: Time difference would be 396 / 30 - 396 / 44, which is greater than 3.5 hours.
264 km: Time difference for this distance is less than 3.5 hours.
495 km and 360 km similarly do not yield exactly 3.5 hours difference when substituted into the equation.
Only 330 km gives the correct time difference.

Common Pitfalls:
A frequent mistake is to write the time difference equation incorrectly, such as reversing which time is larger or adding times instead of subtracting them. Another issue is handling the fraction 3.5 incorrectly, for example treating it as 3.05. Some learners also approximate the fractions too early and end up with rounding errors. It is safer to work exactly with fractions, find a common denominator and only simplify at the end.

Final Answer:
The distance between city A and city B is 330 km.