If a bus took 4 hours to travel from town A to town B, what was its

Given:Total time taken by the bus = 4 hours

Statement 1

(1) In the first 2 hours, the bus covered a distance of 150 miles.

We are not given the information on the distance the bus covers in the subsequent two hours. Hence, the statement alone is not sufficient to find the average speed.

Eliminate A and D.

Statement 2

(2) The average speed of the bus for the first half of the distance was twice its average speed for the second half of the distance.

This statement is quite interesting!

As the average speed of the bus first half of the distance was twice the average speed for the second half of the distance, we can infer that the time taken for the bus to travel the first half of the distance was half the time taken to travel the second half of the distance.

Let's assume that the distance between A and B = 2d



Average speed = \(\frac{2d}{3t_1}\)

We also know that \(3t_1 = 4\), therefore the average speed = \(\frac{2d}{4} = \frac{d}{2}\)

While we find the values of the individual time, \(t_1 = \frac{4}{3}\) and \(t_2 = \frac{8}{3}\), we do not know the distance d, hence the statement is not sufficient on its own. We can eliminate B.

Combined

The statements combined tell us the following

1. From statement 2 - The bus took 4/3 hours to travel half the distance (i.e. d miles)
2. From statement 1 - The bus traveled 150 miles in 2 hours
3. From statement 2 - The average speed of the bus for the first half of the distance was twice its average speed for the second half of the distance.

We can visualize the information as shown below -



M ⇒ Is the midpoint between A and B
N ⇒ Is the distance the bus travels in 2 hours, i.e. AN = 150 miles

The bus travels a distance 'd', denoted by AM, in \(\frac{4}{3}\) hours.

We are given information on the average speed of the bus between points AM and MB, however, no information is given if the bus travels at a constant speed between these distances. Hence, we cannot find an exact value of 'd'. In other words, for different values of 'd' we can have different average speeds.

To understand this better let's take the following cases

Case 1



AB = 160 miles
AM = MB = 80 miles

Between A and M

Distance = 80 miles
Speed of the bus between points A and M = 60 mph
Time taken by the bus to cover AM = \(\frac{80}{60}\) = \(\frac{4}{3}\) hours
Average Speed between AM= 60 mph

Between M and N

Distance = 70 miles
Speed of the bus between points M and N = 105 mph
Time taken by the bus to cover AM = \(\frac{70}{105}\) = \(\frac{2}{3}\) hours

Between N and B

Distance = 10 miles (the distance is taken as 10 miles so as to keep MB = 80 miles)
Time taken by the bus to cover NB =2 hours (because total time is 4 hours)
Speed of the bus between points M and N = 5 mph
Average Speed between MN= \(\frac{80}{ 2+\frac{2}{3}\) = 30 mph

The average speed between AB = \(\frac{160}{ 4}\) = 40 mph

Case 2



AB = 240 miles
AM = MB = 120 miles

Between A and M

Distance = 120 miles
Speed of the bus between points A and M = 90 mph
Time taken by the bus to cover AM = \(\frac{120}{90}\) = \(\frac{4}{3}\) hours
Average Speed between AM= 90 mph

Between M and N

Distance = 30 miles
Speed of the bus between points M and N = 45 mph
Time taken by the bus to cover AM = \(\frac{30}{45}\) = \(\frac{2}{3}\) hours

Between N and B

Distance = 90 miles (the distance is taken as 90 miles so as to keep MB = 120 miles)
Time taken by the bus to cover NB =2 hours (because total time is 4 hours)
Speed of the bus between points M and N = 45 mph
Average Speed between MN= \(\frac{120}{ 2+\frac{2}{3}\) = 45 mph

The average speed between AB = \(\frac{240}{ 4}\) = 60 mph

As we are getting two different values for the average speed, the statements combined are not sufficient.

Option E