Let d = the total distance the group walks.
First work on the fractions:
After Day 1: 3/4 of the distance remains
On day 2, they cover 1/4 of the remaining 3/4, which is 3/16, so they have 3/4 - 3/16 left or 9/16
On day 3, they cover 2/5 of the remaining 9/16, which is 9/40, so they have 9/16 - 9/40 left or 27/80
On day 4, they cover 1/2 of the remaining 27/80, which is 27/160, so they have 27/80 - 27/160 left or 27/160.
Whew! That fraction remaining is equal to the 14 miles remaining, so
27d/160 = 14
You can do the algebra here, I think, but it comes out to 2240/27 or just less than 83 miles (82.96).
Since the fractions used in the problem itself are easily convertible to decimals, you could do the problem that way, as well.