Question

A train needs 4 hours to go from the first station to the second. If the third station is 120 miles away and the total travel time must be no more than 7 hours, write an inequality that would let you find the slowest average speed, v, the train could have.

Answer 1

A train needs 4 hours to go from the first station to the second. If the third station is 120 miles away and the total travel time must be no more than 7hours:120/v <= 3 
3v >= 120 
v >= 40 mph

Answer 2

A train needs 4 hours to go from the first station to the second and the total travel time must be no more than 7 hours, then the train needs at most 3 hours to go from the second station to the third station.

The average speed can be obtained by the formula:

$V_{\text{average}}=\dfrac{\text{total distance}}{\text{total time}}.$

Let V be the train speed between first and second stations, then the distance between the first and the second stations is 4V miles.

Therefore,

• the total distance is 4V+120 (V>0);
• the total time is T≤7 hours.

Now

$V_{\text{average}}=\dfrac{\text{4V+120}}{\text{T}}, \text{ where } T\le 7, V>0.$

The slowest average speed will be when nominator is the smallest and denominator is the greatest. The greatest denominator could be T=7, then

$V_{\text{average}}=\dfrac{\text{4V+120}}{7}=\dfrac{4V}{7}+17.14> 17.14$