Question Number 193565 by Mingma last updated on 16/Jun/23
Answered by mr W last updated on 17/Jun/23
![say at time x hours after 0 o′clock. assume the hands move continuously. position of hour hand: 30x ° position of minute hand: (x−[x])×60×6 ° (x−[x])×60×6 −30x=±45 22x−24[x]=±3 ⇒x=((24[x]±3)/(22)) with [x]=0, 1, 2, ..., 11 i.e. at following times: (totally 22 possibilities) x=(3/(22))⇔0h8m11s x=((21)/(22))⇔0h57m16s x=((27)/(22))⇔1h12m38s x=((45)/(22))⇔2h2m44s x=((51)/(22))⇔2h19m5s ...... x=((243)/(22))⇔11h2m44s x=((261)/(22))⇔11h51m49s](Q193601.png)
$${say}\:{at}\:{time}\:{x}\:{hours}\:{after}\:\mathrm{0}\:{o}'{clock}. \\ $$$${assume}\:{the}\:{hands}\:{move}\:{continuously}. \\ $$$${position}\:{of}\:{hour}\:{hand}:\:\mathrm{30}{x}\:° \\ $$$${position}\:{of}\:{minute}\:{hand}:\:\left({x}−\left[{x}\right]\right)×\mathrm{60}×\mathrm{6}\:° \\ $$$$\:\left({x}−\left[{x}\right]\right)×\mathrm{60}×\mathrm{6}\:−\mathrm{30}{x}=\pm\mathrm{45} \\ $$$$\:\mathrm{22}{x}−\mathrm{24}\left[{x}\right]=\pm\mathrm{3} \\ $$$$\Rightarrow{x}=\frac{\mathrm{24}\left[{x}\right]\pm\mathrm{3}}{\mathrm{22}}\:{with}\:\left[{x}\right]=\mathrm{0},\:\mathrm{1},\:\mathrm{2},\:...,\:\mathrm{11} \\ $$$${i}.{e}.\:{at}\:{following}\:{times}: \\ $$$$\left({totally}\:\mathrm{22}\:{possibilities}\right) \\ $$$${x}=\frac{\mathrm{3}}{\mathrm{22}}\Leftrightarrow\mathrm{0}{h}\mathrm{8}{m}\mathrm{11}{s} \\ $$$${x}=\frac{\mathrm{21}}{\mathrm{22}}\Leftrightarrow\mathrm{0}{h}\mathrm{57}{m}\mathrm{16}{s} \\ $$$${x}=\frac{\mathrm{27}}{\mathrm{22}}\Leftrightarrow\mathrm{1}{h}\mathrm{12}{m}\mathrm{38}{s} \\ $$$${x}=\frac{\mathrm{45}}{\mathrm{22}}\Leftrightarrow\mathrm{2}{h}\mathrm{2}{m}\mathrm{44}{s} \\ $$$${x}=\frac{\mathrm{51}}{\mathrm{22}}\Leftrightarrow\mathrm{2}{h}\mathrm{19}{m}\mathrm{5}{s} \\ $$$$...... \\ $$$${x}=\frac{\mathrm{243}}{\mathrm{22}}\Leftrightarrow\mathrm{11}{h}\mathrm{2}{m}\mathrm{44}{s} \\ $$$${x}=\frac{\mathrm{261}}{\mathrm{22}}\Leftrightarrow\mathrm{11}{h}\mathrm{51}{m}\mathrm{49}{s} \\ $$
Commented by otchereabdullai@gmail.com last updated on 17/Jun/23
$${I}\:{thank}\:{God}\:{my}\:{prof}\:{W}\:{is}\:{back} \\ $$
Commented by mr W last updated on 17/Jun/23
Commented by mr W last updated on 17/Jun/23
Commented by mr W last updated on 17/Jun/23
Commented by mr W last updated on 17/Jun/23
Commented by Mingma last updated on 17/Jun/23
Genius solutions! perfect ��
Commented by otchereabdullai@gmail.com last updated on 14/Aug/23
$${Thanks}\:{a}\:{lot}\:{prof}\:{W} \\ $$
Commented by mr W last updated on 14/Aug/23
$${you}\:{are}\:{welcome}! \\ $$