A mathematical riddle

Question

I went through this mathematical riddle, but despite my attempts, I guess I'm missing the point. This is the riddle:

"Now, constable," said the defendant's counsel in cross-examination," you say that the prisoner was exactly twenty-seven steps ahead of you when you started to run after him?"

"Yes, sir."

"And you swear that he takes eight steps to your five?"

"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, two of my steps are equal in length to five 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. Can you also say how many steps the officer needed to catch the thief?

So I tried to make some calculations using the proportions between the real steps of the two persons, and then between the distance every step covers, but I believe something is wrong in my reasoning.

The correct answer shall be: $30$ steps, which is very different from the answer I obtain..

Any help? Thank you so much!

Answer 1

I think this comes down to keeping all the units straight, or rather it comes down to choosing good units to work with.

Let's fix a unit of time by saying that in each unit the prisoner travels $8$ steps and his pursuer travels $5$.

To calculate the amount of time involved, we work in "prisoner lengths". Since each pursuer step is $\frac 52$ of a prisoner step, in one unit of time the pursuer travels $5\times \frac 52$ lengths, and the prisoner travels $8$

Solve for the amount of time it takes to catch the prisoner. After $n$ units of time we must have $$n\times 5\times \frac 52=n\times 8 +27\implies n=6$$

But of course the pursuer then traveled $6\times 5=30$ "pursuer steps".

Answer 2

$@user46944$ made a good point regarding whose step is considered.

Alternative approach.

Let $m$ and $n$ be the step lengths of the constable and prisoner, respectively. Then: $$m=\frac{5}{2}n.$$

Let $c$ and $p$ be the numbers of steps of the constable and prisoner, respectively. Then: $$c=\frac58p.$$ Then the distance equation is (if constable's or prisoner's steps are considered, respectively): $$D_{c}=D_{p}+27m \ \ \ or \ \ \ D_{c}=D_{p}+27n \Rightarrow$$ $$mc=np+27m \ \ \ or \ \ \ mc=np+27n \Rightarrow $$ $$\frac52nc=\frac85nc+27\cdot \frac52n \ \ \ or \ \ \ \frac52nc=\frac85nc+27n \Rightarrow$$ $$c=75 \ \ (p=120) \ \ \ or\ \ \ c=30 \ \ (p=48).$$