Question

Catalan drove 210 miles on her vacation. She drove an average of 1.4 times faster than the second 105 miles of her trip and she did on the first 105 miles of her trip. Which expression represents the time she spent driving? Let X equal her speed on the first half of the trip.

Answer 1

Logan, the way you have phrased this question makes it a bit hard to follow.  I'm going to take the liberty of paraphasing it:

"Catalan drives an average of 1.4 times faster during the first 105 miles of her trip than she does during the second 105 miles."

As we are told, let X represent her speed during the first 105 miles of her trip.  She drives more slowly during the second 105 miles.  Thus, her speed during the 2nd 105 miles is X/1.4.

Remember:  distance = (rate)(time), or time = (distance)(rate)

We need to determine an expression for the time she spends driving.  Let T1 be the time required to drive 105 miles at speed X mph and T2 be the time required to drive 105 miles at speed (X/1.4) mph.

What is the total time required to drive these 210 miles?

Total time = (time required to drive 105 miles at X mph) + (time required to drive 105 miles at X/1.4 mph).

This gives you TIME SPENT DRIVING as a function of X, her speed during the first 105 miles of driving.