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A-and-B-are-running-along-the-wall-of-a-square-park-The-corners-of-the-park-are-facing-north-south-east-and-west-and-are-named-N-S-E-W-respectively-They-start-at-E-and-run-towards-S-If-the-sp

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Question Number 87034 by Zainal Arifin last updated on 02/Apr/20
A and B are running along the wall of a square park. The corners of the park are facing north, south, east and west and are named N, S, E, W respectively. They start at E and run towards S. If the speed of A is 6 tines that of B, where do they meet for the 27^(th) time?
$$\mathrm{A}\:\mathrm{and}\:\mathrm{B}\:\mathrm{are}\:\mathrm{running}\:\mathrm{along}\:\mathrm{the}\:\mathrm{wall}\:\mathrm{of} \\ $$$$\mathrm{a}\:\mathrm{square}\:\mathrm{park}.\:\mathrm{The}\:\mathrm{corners}\:\mathrm{of}\:\mathrm{the}\:\mathrm{park} \\ $$$$\mathrm{are}\:\mathrm{facing}\:\mathrm{north},\:\mathrm{south},\:\mathrm{east}\:\mathrm{and}\:\mathrm{west} \\ $$$$\mathrm{and}\:\mathrm{are}\:\mathrm{named}\:\mathrm{N},\:\mathrm{S},\:\mathrm{E},\:\mathrm{W}\:\:\mathrm{respectively}. \\ $$$$\mathrm{They}\:\mathrm{start}\:\mathrm{at}\:\mathrm{E}\:\mathrm{and}\:\mathrm{run}\:\mathrm{towards}\:\mathrm{S}.\:\mathrm{If} \\ $$$$\mathrm{the}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{A}\:\mathrm{is}\:\mathrm{6}\:\mathrm{tines}\:\mathrm{that}\:\mathrm{of}\:\mathrm{B},\:\mathrm{where} \\ $$$$\mathrm{do}\:\mathrm{they}\:\mathrm{meet}\:\mathrm{for}\:\mathrm{the}\:\mathrm{27}^{\mathrm{th}} \:\mathrm{time}? \\ $$
Answered by mr W last updated on 02/Apr/20
say speed of A is v_A and speed of B is v_B . say perimeter of square is l. say B has run a distance δ when they meet for the first time. (δ/v_B )=((l+δ)/v_A ) ((l+δ)/δ)=(v_A /v_B )=6 ⇒δ=(l/5) when they meet for the second time meeting point is at a distance ((2l)/5) from start point E. when they meet for the 5th time meeting point is at the start point. when they meet for the 25th time meeting point is at the start point. when they meet for the 27th time meeting point is at a distance ((2l)/5) from start point E. it is between S and W.
$${say}\:{speed}\:{of}\:{A}\:{is}\:{v}_{{A}} \:{and}\:{speed}\:{of}\:{B} \\ $$$${is}\:{v}_{{B}} .\:{say}\:{perimeter}\:{of}\:{square}\:{is}\:{l}. \\ $$$${say}\:{B}\:{has}\:{run}\:{a}\:{distance}\:\delta\:{when}\:{they} \\ $$$${meet}\:{for}\:{the}\:{first}\:{time}. \\ $$$$\frac{\delta}{{v}_{{B}} }=\frac{{l}+\delta}{{v}_{{A}} } \\ $$$$\frac{{l}+\delta}{\delta}=\frac{{v}_{{A}} }{{v}_{{B}} }=\mathrm{6} \\ $$$$\Rightarrow\delta=\frac{{l}}{\mathrm{5}} \\ $$$${when}\:{they}\:{meet}\:{for}\:{the}\:{second}\:{time} \\ $$$${meeting}\:{point}\:{is}\:{at}\:{a}\:{distance}\:\frac{\mathrm{2}{l}}{\mathrm{5}} \\ $$$${from}\:{start}\:{point}\:{E}. \\ $$$${when}\:{they}\:{meet}\:{for}\:{the}\:\mathrm{5}{th}\:{time} \\ $$$${meeting}\:{point}\:{is}\:{at}\:{the}\:{start}\:{point}. \\ $$$${when}\:{they}\:{meet}\:{for}\:{the}\:\mathrm{25}{th}\:{time} \\ $$$${meeting}\:{point}\:{is}\:{at}\:{the}\:{start}\:{point}. \\ $$$${when}\:{they}\:{meet}\:{for}\:{the}\:\mathrm{27}{th}\:{time} \\ $$$${meeting}\:{point}\:{is}\:{at}\:{a}\:{distance}\:\frac{\mathrm{2}{l}}{\mathrm{5}} \\ $$$${from}\:{start}\:{point}\:{E}.\:{it}\:{is}\:{between}\:{S} \\ $$$${and}\:{W}. \\ $$

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