Question Number 88014 by ajfour last updated on 07/Apr/20
$${If}\:{there}\:{is}\:{no}\:{second}'{s}\:{hand}\:{on} \\ $$$${a}\:{clock}\:{and}\:{the}\:{minute}\:{and}\:{hour} \\ $$$${hand}\:{move}\:{in}\:{continuous}\:{fashion}, \\ $$$${then}\:{exactly}\:{at}\:{what}\:{time}\:{between} \\ $$$$\mathrm{02}:\mathrm{10}\:\:{and}\:\mathrm{02}:\mathrm{15}\:{does}\:{the}\:{position} \\ $$$${of}\:{the}\:{two}\:{hands}\:{exactly}\:{coincide}? \\ $$
Answered by mr W last updated on 08/Apr/20
$${say}\:{the}\:{time}\:{is}\:{x}\:{minutes}\:{after} \\ $$$$\mathrm{2}\:{o}'{clock}. \\ $$$${position}\:{of}\:{minute}\:{hand}:\:{x} \\ $$$${position}\:{of}\:{hour}\:{hand}:\:\mathrm{10}+\frac{\mathrm{5}{x}}{\mathrm{60}} \\ $$$${when}\:{both}\:{hands}\:{coincide}: \\ $$$${x}=\mathrm{10}+\frac{\mathrm{5}{x}}{\mathrm{60}} \\ $$$$\Rightarrow{x}=\frac{\mathrm{120}}{\mathrm{11}}=\mathrm{10}^{{m}} \mathrm{54}^{{s}} \\ $$
Commented by mr W last updated on 08/Apr/20
$${generally}:\:{y}\:{o}'{clock}\:{x}\:{minutes} \\ $$$${x}=\mathrm{5}{y}+\frac{\mathrm{5}{x}}{\mathrm{60}} \\ $$$$\Rightarrow{x}=\frac{\mathrm{60}{y}}{\mathrm{11}} \\ $$$${y}=\mathrm{0}^{{h}} :\:{x}=\mathrm{0} \\ $$$${y}=\mathrm{1}^{{h}} :\:{x}=\frac{\mathrm{60}}{\mathrm{11}}=\mathrm{5}^{{m}} \mathrm{27}^{{s}} \\ $$$${y}=\mathrm{2}^{{h}} :\:{x}=\frac{\mathrm{120}}{\mathrm{11}}=\mathrm{10}^{{m}} \mathrm{54}^{{s}} \\ $$$${y}=\mathrm{3}^{{h}} :\:{x}=\frac{\mathrm{180}}{\mathrm{11}}=\mathrm{16}^{{m}} \mathrm{21}^{{s}} \\ $$$${y}=\mathrm{4}^{{h}} :\:{x}=\frac{\mathrm{240}}{\mathrm{11}}=\mathrm{21}^{{m}} \mathrm{49}^{{s}} \\ $$$${y}=\mathrm{5}^{{h}} :\:{x}=\frac{\mathrm{300}}{\mathrm{11}}=\mathrm{27}^{{m}} \mathrm{16}^{{s}} \\ $$$${y}=\mathrm{6}^{{h}} :\:{x}=\frac{\mathrm{360}}{\mathrm{11}}=\mathrm{32}^{{m}} \mathrm{43}^{{s}} \\ $$$$...... \\ $$
Commented by ajfour last updated on 07/Apr/20
thank you for the general treatment mrW sir; i have presented roughly my approach too.
Answered by ajfour last updated on 07/Apr/20
$${let}\:{after}\:\mathrm{2}\:{o}'{clock}\:{hour}\:{hand} \\ $$$${moves}\:{by}\:{angle}\:\alpha. \\ $$$$\mathrm{12}\alpha=\mathrm{60}°+\alpha \\ $$$$\alpha=\frac{\mathrm{60}°}{\mathrm{11}} \\ $$$$\mathrm{6}°\:{of}\:{min}\:{hand}=\mathrm{1}{min} \\ $$$$\mathrm{60}°+\frac{\mathrm{60}°}{\mathrm{11}}=\left(\mathrm{10}+\frac{\mathrm{10}}{\mathrm{11}}\right){min} \\ $$$$=\frac{\mathrm{120}}{\mathrm{11}}{min} \\ $$$$\mathrm{2}{h}\mathrm{10}{min}+\mathrm{54}\frac{\mathrm{6}}{\mathrm{11}}{sec} \\ $$