A car travels from city A to city B. It covers the first 30% of the total…

Concept

Average speed equals total distance divided by total time. When a journey has segments at different speeds, first add the segment times using time = distance/speed; the segment speeds must not be averaged directly.

If a segment speed depends on the journey's average speed v, express both that segment's time and the total time in terms of v, and then solve the resulting equation.

Application

1. Let v km/h be the average speed for the whole journey. The three segment distances are 30% of 250 = 75 km, 40% of 250 = 100 km, and the remaining 75 km.

2. The times for the first two segments are 75/40 = 15/8 hours and 100/60 = 5/3 hours.

3. The last-segment speed is 1.2v = 6v/5 km/h, so its time is 75/(6v/5) = 125/(2v) hours.

4. Since the total distance is 250 km, the total journey time at average speed v is 250/v hours.

5. Therefore, 15/8 + 5/3 + 125/(2v) = 250/v. Combining the first two fractions gives 85/24, hence 85/24 = 375/(2v).

6. Cross-multiplication gives 85v = 4500, so v = 900/17 = 52.941... km/h. Rounded to the nearest integer, the average speed is 53 km/h.

Cross-check

1. With v = 900/17, the last-segment speed is 1.2v = 1080/17 km/h. The three times are 15/8, 5/3, and 85/72 hours; their sum is 85/18 hours.

2. Then total distance divided by total time is 250/(85/18) = 900/17 km/h, which reproduces the calculated unrounded average before rounding.