Agroup of tourists went on three hikes with a distance of 12.5 km, 18 km, and 14 km. their speed on the first hike was 1km/hour slower than their speed on the second hike and 1 km/hour faster than their speed on the third hike. the third hike took 30 minutes longer than the second hike. how long did all three hikes take?

Let the speed on the second hike be r. Then the speed on the first hike is (r - 1), and the speed on the third hike is (r - 2). Let the time on the second hike be t, so the third hike's time is t + 0.5 (time in hours).
The second hike lasted 18 km, so 18 = rt. t = 18/r.
The third hike was 14 km, so 14 = (r - 2)(t + 0.5) = rt + 0.5r - 2t - 1
Substituting: 14 = (18) + 0.5r - 2(18/r) - 1
-3 = 0.5r - 36/r
-6 = r - 72/r
-6r = r^2 - 72
r^2 + 6r - 72 = 0
(r + 12)(r - 6) = 0
r = 6 (negative rate is not allowed for this)
Therefore, the speed of the first hike, r - 1 = 5 km/h. Since the distance was 12.5 km, the time was 12.5 / 5 = 2.5 h.
The second hike lasted 18 km / 6 km/h = 3 h.
The third hike lasted half an hour more than the second, which is 3.5 h.
Adding all three times gives 2.5 + 3 + 3.5 = 9 h.