Exercise 3 solution (Python)

def naismith(km, ascent_m):
    return km / 5 * 60 + ascent_m / 10

def clock(minutes):
    return f"{int(minutes // 60):02d}:{int(round(minutes % 60)):02d}"

start = 9 * 60 + 30
out = naismith(5, 400)    # 100 min
back = naismith(5, 0)     # 60 min
moving = out + back

for mult in (1.0, 1.3):
    total = moving * mult
    total += total / 60 * 5          # 5 minutes of breaks per hour
    print(mult, round(total), clock(start + total))

# latest turn-round time to be down by 15:30, using the slow multiplier
turn = 15 * 60 + 30 - (back * 1.3 * (1 + 5 / 60))
print("Turn round by", clock(turn))

Output:
1.0 173 12:23
1.3 225 13:15
Turn round by 14:06

(a) With the plain rule plus breaks the walk takes about 173 minutes and finishes at 12:23.
(b) With a 1.3 multiplier it takes about 225 minutes and finishes at 13:15. Either way there is a comfortable margin before 15:30.
(c) If the group is still going up at 14:06 (the latest time that still gets them down by 15:30 at the slow pace), they should turn round whatever the distance still to go. Fixing this time before you set out is the point: it takes the decision away from the tired, summit-hungry version of you.