Dudeney puzzle - catching the thief
"Now constable", said the defendant's council in cross-examination, "you say that the prisoner was exactly 27 steps ahead of you when you started to run after him?" "yes sir" "and you swear that he takes 8 steps to your 5?" "that is so" "then i ask you, constable, as an intelligent man to explain how you ever caught him if that is the case?" "well you see, i have got a longer stride. In fact, 2 of my steps are equalin length to 5 of the prisoner's. If you work it out you will find that the number of steps I required, would bring me exactly to the spot where I captured him".
Here the foreman of the jury asked for a few minutes to figure out the number of steps the constable must have taken.
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Let the distance where the defender and the prisoner met be D.
Then the distance the defender ran = D
Then, the distance the prisoner ran = D-27
The number of steps that the defender ran to D is x steps.
The number of steps that the prisoner ran to D is y steps.
2 steps of the defender = 5 steps of the prisoner.
==> 2x = 5 y
==> y= (2/5)x..............(1)
The defender run at a speed of 5 steps.
==> The defender's speed S1= 5x
The prisoner run at a speed of 8 steps.
The prisoner speed S2 = 8y
But y= (2/5)x
==> S2 = 8*(2/5)x = (16/5)x
Now we will determine the distance.
We will assume that the time is 1 unit.
==> D= S1 * T
==> D = 5x ..............(1)
Now we will determine the distance the prisoner ran,
==> D-27 = S2*T
==> D-27 = (16/5)x
==> D = (16/5) x + 27 ..............(2)
Now from (1) and (2).
==> 5x = (16/5)x + 27
==> 5x - (16/5)x = 27
==> (9/5)x = 27
==> x = 27*5/9 = 3*5 = 15
Then, the number of steps the defender ran is 15 steps.
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