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