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3 fathers footsteps are as long as 5 daughters footsteps. While the father makes 6 footsteps, the daughter makes 7. The daughter has already made 30 footsteps when the father went after her, after how many footsteps will the father reach the daughter?

I would be grateful for any kind of hint to solve this problem?

zermelovac
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  • Is the contest on-going? If so, we should wait until it is done before commenting. – lulu Mar 16 '17 at 15:41
  • It is over, this was on previous competitions. – zermelovac Mar 16 '17 at 15:47
  • Hint: say the daughter walks $1$ unit per step. Then start the clock with the daughter ahead by $30$ daughter steps. Suppose she has taken $n$ steps (so has gone $n$ units further). How many steps has the father taken? How many units has he walked? – lulu Mar 16 '17 at 15:54
  • I would say that he walked $\frac{7}{6}$ units per $\frac{5}{3}$ steps? – zermelovac Mar 16 '17 at 17:01
  • Well, the $\frac 53$ refers to units not number...Dad walks $\frac 53$ unit per step he takes. As to the number of steps dad takes, that's $\frac 67$ of the number of daughter steps. – lulu Mar 16 '17 at 17:28

2 Answers2

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Since the answer must be a number of steps the father takes, I think father-steps-distance (the distance the father travels in one step) and father-steps-time (the time that passes while the father takes one step) are the most convenient units of distance and time.

So the daughter's step has length $3/5 = 0.6$ and she takes $7/6$ steps per unit of time, which means her speed is $0.6 \times 7/6 = 0.7$ and she has a head start of $30 \times 0.6 = 18.$ The father's speed of course is $1,$ so their relative speed is $0.3$ toward each other. Now how long does it take to travel a distance $18$ at speed $0.3$? That is the number of steps the father takes in both time and distance.

Since the problem appears to want an integer answer, once I found that the number of steps is $t$ according to the relative speed and distance, I would then check that $t$ (number of steps the father takes) is an integer and also that the number of steps the daughter takes is an integer. In this problem, fortunately, the answer is already an integer so we do not have to be concerned with whether we need to round the steps up to an integer and with where exactly the father and daughter are when one of them is between the beginning and end of a step.

David K
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It is safe to assume that a father's footstep is $5$ units long while a daughter's footstep is $3$ units long. Now this means, according to the question, the daughter is $90$ units ahead of the father.

For every $30$ units ($6$ steps) the father moves, the daughter moves $21$ units ($7$ steps). In relative terms, the father moves $9$ units every $6$ of his steps. (In case you do not know what is relative, this means we assume that the daughter is stationary while the father moves $9$ units every $6$ of his steps). Since the father is behind by $90$ units, then he has to take $60$ of his steps to reach his daughter. I believe $60$ is the answer.

Icycarus
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