Kate cycles the first 350 km of a 470 km journey at a certain average speed and the remaining distance at an average speed that is 15 km/h less than that for the first part of the journey. If the time taken for her to travel each part of her journey is the same, find the average speed for the second part of her journey.

Let x be her speed for the first part of the journey.

We know:

Time = distance / speed

For the first part:

time = 350/x

For the second part:

time = (470 - 350) / (x - 15) = 120 / (x - 15)

We are told the times are equal so:

350/x = 120 / (x - 15)

350 (x - 15) = 120x

230x = 5250

x = 22.8 km/hr

Speed for second part = 22.8 - 15 = 7.8 km/hr