Brain-Teaser 948: Journey north-east
From The Sunday Times, 21st September 1980 [link]
For the purpose of my problem I have to choose two towns 26 miles apart. I might have chosen Oxford and Newbury, but it would be more appropriate for me as a Scotsman to go much farther north and choose Kingussie and Grantown-on-Spey, where the roads are somewhat less busy.
Alf, Bert and Charles start off at the same time from Kingussie to make their way north-eastwards to Grantown-on-Spey, 26 miles distant.
Alf walks at a constant speed of four miles per hour. Bert and Charles drive together in a car. After a certain time, Bert leaves the car, and walks forward at the same rate as Alf, while Charles drives back to meet Alf.
Alf gets Into the car with Charles, and they continue to drive to Grantown-on-Spey, arriving there just as Bert does.
On each stretch Charles averages 40 miles per hour.
What is the time (in minutes) taken for them all to travel from Kingussie to Grantown-on-Spey?
This puzzle is included in the book The Sunday Times Book of Brainteasers (1994).
[teaser948]
Lise Andreasen and 
Hugo and 
S2T2 
Jim Randell 8:29 am on 17 March 2024 Permalink |
See: Teaser 3140, where we determined:
If there are k pedestrians to be transported a distance d, and each walks a distance x at velocity w and is transported a distance (d − x) at velocity v, and the total time taken is t, then we have:
We can plug the numbers for this puzzle in and calculate the result:
Run: [ @replit ]
from enigma import (fdiv, printf) # initial conditions k = 2 # number of walkers d = 26 # distance w = 4 # walking speed v = 40 # driving speed # calculate walking distance per person n = 2 * k - 1 x = fdiv(d * w * (n - 1), v + w * n) # calculate time taken t = fdiv(x, w) + fdiv(d - x, v) # output solution printf("t = {t:g} hours (= {m:g} min) [x={x:g} miles]", m=t * 60)Solution: The total time taken is 93 minutes.
Alf walks the first 4 miles (in 60 minutes), and is driven the remaining 22 miles (in 33 minutes).
Bert is driven 22 miles first, and walks the last 4 miles.
Charles drives 22 miles to drop off Bert, returns 18 miles to collect Alf, and then 22 miles to the destination, a total of 62 miles (in 93 minutes).
So each arrives at the destination after 93 minutes.
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Lise Andreasen 4:22 pm on 4 April 2024 Permalink |
Alf walks x miles. For symmetry reasons, Bert also walks x miles. In the middle we have y miles. 26 = x + y + x.
They all spend the same amount of time.
x/4 + (x+y)/40 = (2x+3y)/40.
Solve.
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