Enlarge image
Photo:
THE MIRROR
No puzzle collection without a train!
At this point there was already a puzzle by a man who is regularly picked up from the train station.
And also a puzzle with tickets.
The following task revolves around a double-track railway line that is used by trains of different speeds.
It connects the city A with the city B.
Two trains start at the same time in A and B and travel in the opposite direction to B or A. Both trains travel at different but constant speeds.
After the two trains have met, the fast ICE still needs exactly one hour to get to its destination.
The slower regional train, on the other hand, takes another four hours.
What is the ratio of the speeds of the two trains?
The ratio of ICE to regional train speed is 2: 1.
The solution does not require any special trick, you just have to calculate skillfully with the formula speed = distance / time - in short, v = s / t.
The ratio is the reciprocal of the square root of the ratio of travel times from the encounter to the destination station.
Why?
The distances traveled by both trains until they meet are proportional to the speed of the trains.
From the start to the meeting, both trains need time t1, because they leave at the same time.
Then the following applies:
vICE = sICE / t1 and vREGIO = sREGIO / t1
And thus
sICE = VICE * t1 and sREGIO = vREGIO * t1
After the encounter, each train must cover the distance that the other train has traveled before the encounter.
The time it takes a train to do this is calculated using the formula t = s / v.
The ICE has to cover the distance vREGIO * t1, for this it needs a time of tICE = vREGIO * t1 / vICE.
The regional train needs the time tREGIO = vICE * t1 / vREGIO for its remainder.
We don't know t1, but the ratio tICE / tREGIO = 1/4
In turn, we can calculate tICE / tREGIO from the equations above:
tICE / tREGIO = (vREGIO * t1 / vICE) / (vICE * t1 / vREGIO)
We can simplify the right side:
tICE / tREGIO = (vREGIO / vICE) 2
vREGIO / vICE is therefore the root of 1/4, i.e. 1/2.
The ICE is therefore twice as fast as the regional train.
Henry Dudeney, in whose book "Mathematical Amusements" I found this riddle, gives the solution in a more compact way: The distances from the encounter are inversely related to the speed - and so are the remaining travel times per kilometer. Draw 1 and get to 2/1.
If you missed a puzzle from the past few weeks, here are the most recent episodes:
How big is the yellow triangle?
Which stick is magnetic?
Who will get the last coin?
Difficult legacy
How big is the circular ring?
Buying a bicycle with counterfeit money
Six acute angles
How far is it to the neighboring village?
How old is sophie
Triangle searched on three circles
