M43-37

Official Solution:


Train A and Train B each departed simultaneously from Liverpool heading to London, which is 300 kilometers away. When Train A reached the destination, Train B still had 80 kilometers left to cover. What was Train A's average speed during this journey?

Essentially, we are given that Train A covered 300 kilometers in the same time Train B covered 220 kilometers and are asked to find the average speed of Train A during this 300-kilometer trip.

(1) During the first hour, Train B covered 40 kilometers less than Train A.

Don't fall into the trap of assuming the trains maintained the same constant speed throughout their journey. If that were true, then we could reason that in the first hour, Train B fell behind by 40 kilometers. Therefore, to fall behind by 80 kilometers, it would need 2 hours, which would mean that Train A covered 300 kilometers in 2 hours, making its average speed equal to \(\frac{300}{2} = 150\) kilometers per hour. However, this would not be correct because we cannot assume constant speed. It's possible, for example, that in the first hour, Train B covered 110 kilometers and Train A covered 150 kilometers. After that, they could have slowed down, and Train A could have covered the remaining 150 kilometers in 2 hours, spending a total of 3 hours on the journey and making its average speed equal to \(\frac{300}{3} = 100\) kilometers per hour. Not sufficient.

(2) Train B's average speed for the first 110 kilometers was 110 kilometers per hour.

Again, don't fall into the trap of assuming the trains maintained the same constant speed throughout their journey. If that were true, then we could reason that Train B covered 220 kilometers in 2 hours, which would mean that Train A covered 300 kilometers in 2 hours, making its average speed equal to \(\frac{300}{2} = 150\) kilometers per hour. However, this would not be correct because we cannot assume constant speed. It's possible, for example, that in the first hour, Train B covered 110 kilometers and Train A covered 150 kilometers. After that, they could have slowed down, and Train A could have covered the remaining 150 kilometers in 2 hours, spending a total of 3 hours on the journey and making its average speed equal to \(\frac{300}{3} = 100\) kilometers per hour. Therefore, Statement 2 is not sufficient.

(1) + (2) We can use the same example as above: the average speed of Train A can be 150 kilometers per hour, assuming constant speeds for both trains, or some other speed, for example, 100 kilometers per hour if in the first hour Train B covered 110 kilometers and Train A covered 150 kilometers, and after that they slowed down and Train A covered the remaining 150 kilometers in 2 hours, spending a total of 3 hours on the journey and making its average speed equal to \(\frac{300}{3} = 100\) kilometers per hour. Therefore, even together, the statements are not sufficient.


Answer: E